Masaryk Universityequadiff/1997/cd/equadiff/dalsi.pdfPreface The Conference on Di erential Equations...

CONFERENCE ON DIFFERENTIAL EQUATIONS ANDTHEIR APPLICATIONS, BRNO, AUGUST 25 – 29, 1997

Equadiff 9

Papers

edited by

Z. Dosla, J. Kuben, J. Vosmansky

EQUADIFF 9 • MASARYK UNIVERSITY BRNO 1998

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Preface

The Conference on Differential Equations and Their Applications (EQUA-DIFF 9) was held in Brno, August 25–29, 1997. It was organized by the MasarykUniversity, Brno in cooperation with Mathematical Institute of the Czech Acad-emy of Sciences, Technical University Brno, Union of Czech Mathematicians andPhysicists, Union of Slovak Mathematicians and Physicists and other Czechscientific institutions with support of the International Mathematical Union.EQUADIFF 9 was attended by 269 participants from 32 countries and morethan 50 accompanying persons and other guests.

The scientific program comprised 8 plenary lectures and 34 main lectures insections. In addition 208 papers were presented as communications, at the postersession and in the form of enlarged abstracts.

This volume contains 31 papers accepted for the presentation at the Equad-iff 9 Conference and submitted for this type of publication by the authors. Agreat part of them is in the final form and will not be published elsewhere, theothers are preliminary version or overview articles.

This volume is published in the electronic form, both on Equadiff 9 CDROM and on the Internet (http://www.math.muni.cz/Equadiff9CDROM/). Theauthors are provided with the hardcopies of their papers.

Our aim was to harness the possibilities of new computer technologies andfor this reason all Equadiff 9 publications were prepared in the hypertext PDFform. We hope that this form of publication will be accepted favourably.

Brno, May 1998 Editors

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Table of Contents

Stability Theorems for Nonlinear FDE’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Oleg Anashkin (Simferopol State University)

Analysis of Equations in the Phase-Field Model . . . . . . . . . . . . . . . . . . . . . . . . 17Michal Benes (Czech Technical University)

Summation of Polyparametrical Functional Series . . . . . . . . . . . . . . . . . . . . . . 37Andriy Blazhievskiy (Technological University of Podillia, Ukraine)

System of Differential Equations with Unstable Turning Point . . . . . . . . . . . 43V. N. Bobochko and I. I. Markush (Ukraine)

On the Symmetric Solutions to a Class of Nonlinear PDEs . . . . . . . . . . . . . . 53Gabriella Bognar (University of Miskolc)

The Abstract Cauchy Problem in Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Igor A. Brigadnov (North-West Polytechnical Institute St. Petersburg)

Thermoelastic Far-field Patterns for the Vector Thermoelastic Equation . . . 73Fioralba Cakoni (University of Tirana)

Transformations of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Jan Chrastina (Masaryk University Brno)

Abstract Differential Equations of Arbitrary (Fractional) Orders . . . . . . . . 93Ahmed M. A. El-Sayed (Alexandria University)

Parametric Representation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Peter L. Simon and Henrik Farkas (Technical University of Budapest)

Homogenization of Scalar Hysteresis Operators . . . . . . . . . . . . . . . . . . . . . . . . 111Jan Francu (Technical University Brno)

Global Qualitative Investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123Valery Gaiko (Belarus State University of Informatics and Radioelec-tronics)

The Existence of Global Solutions to the Elliptic-Hyperbolic DSI System . . 131Nakao Hayashi (Science University of Tokyo) and Hitoshi Hirata (Sop-hia University)

Scaling in Nonlinear Parabolic Equations : Locality versus Globality . . . . . . 137Grzegorz Karch (Uniwersytet Wroc lawski)

Almost Sharp Conditions for the Existence of Smooth Inertial Manifolds . . 139Norbert Koksch (Technical University Dresden)

The Property (A) for a Certain Class of the Third Order ODE . . . . . . . . . . 167Monika Kovacova (Slovak Technical University Bratislava)

On Factorization of Fefferman’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181Miroslav Krbec (Acad. Sci. Prague), Thomas Schott (FSU Jena)

A Time Periodic Solution of Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . 193Petr Kucera (Czech Technical University)

Numerical Solution of Compressible Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201Maria Lukacova-Medvid’ova (Technical University Brno)

Structure of Distribution Null-Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211Takeshi Mandai (Gifu University, Gifu)

A Functional Differential Equation in Banach Spaces . . . . . . . . . . . . . . . . . . . 223Nasr Mostafa (Suez Canal University)

On the Limit Cycle of the van der Pol Equation . . . . . . . . . . . . . . . . . . . . . . . 229Kenzi Odani (Aichi University of Education)

Lp Solutions of Non-linear Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 237Alejandro Omon and Manuel Pinto (Universidad de Chile)

Rothe’s Method for Degenerate Quasilinear Parabolic Equations . . . . . . . . . 247Volker Pluschke (University Halle)

A Posteriori Error Estimates for a Nonlinear Parabolic Equation . . . . . . . . . 255Karel Segeth (Academy of Sciences Praha)

The Solvability Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263G. M. Sklyar (Ukraine) and I. L. Velkovsky (Regensburg)

Elliptic Equations with Decreasing Nonlinearity I . . . . . . . . . . . . . . . . . . . . . . 269Tadie (Matematisk Institut Copenhagen)

Elliptic Equations with Decreasing Nonlinearity II . . . . . . . . . . . . . . . . . . . . . 275Tadie (Matematisk Institut Copenhagen)

Mathematical Models of Suspension Bridges . . . . . . . . . . . . . . . . . . . . . . . . . . . 281Gabriela Tajcova (University of West Bohemia, Pilsen)

Numerical Analysis of High-Temperature Strains . . . . . . . . . . . . . . . . . . . . . . . 307Jirı Vala (Brno)

Asymptotic Behavior of Solutions of Partial Difference Inequalities . . . . . . . 319Patricia J. Y. Wong (Nanyang Technological University)

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

iii

EQUADIFF 9 CD ROM, Brno 1997 PAPERS

Masaryk University pp. 1–15

Stability Theorems for Nonlinear Functional

Differential Equations

Oleg Anashkin

Department of Mathematics, Simferopol State University4, Yaltinskaya St., 333007 Simferopol, Ukraine

Email: [email protected]: http://www.ccssu.crimea.ua/~anashkin

Abstract. New approach in stability theory for a class of retarded non-linear functional differential equations is discussed. The problem of sta-bility of the zero solution is considered under assumption that the systemof interest has a trivial linearization, i.e. it is essentially nonlinear. Suffi-cient conditions for uniform asymptotic stability and instability are givenby auxiliary functionals of Lyapunov-Krasovskii type. The method is alsoapplicable to linear systems with a small parameter in standard form.Some examples are given.

AMS Subject Classification. 34K20

Keywords. asymptotic stability, instability, Lyapunov functionals, pa-rametric resonance

1 Introduction

The paper is devoted to the problem of stability of the zero solution of nonlinearsystem of functional differential equations (FDE) of retarded type

x = F (t, xt). (1.ana)

There is no need to elaborate on the role that Lyapunov-Krasovskii functionalsplay in the analysis of asymptotic properties of solutions of FDE. In particular,a suitable functional may ensure the uniform asymptotic stability of a trivialsolution of (1.ana). Due to classical results [1], [2]. suitable here may mean uni-formly positive definite on state space Ch = C([−h, 0]),Rn), which strictly anduniformly decreases along nontrivial solutions. There is the celebrated conversetheorem on Lyapunov-Krasovskii functional [2]. But to find actually such func-tionals in concrete examples is not an easy problem. In this paper we establishnew sufficient conditions on stability for a class of FDE in terms of functionalswhich satisfy less strong restrictions. We suggest a new approach in context ofgeneralized Lyapunov’s direct method [3] – [6]. Unlike the well-known theoremson stability for FDE [1], [2] suitable functionals satisfy main restrictions only in

This is the preliminary version of the paper.

http://www.ccssu.crimea.ua/~anashkin

2 Oleg Anashkin

some cone AhR ⊂ Ch not in the whole space Ch, moreover, they are nonmonotonealong nontrivial solutions of (1.ana). It gives a possibility to use a simple procedureto construct suitable functionals.

The paper is organized as follows. In the second section we give the statementof the problem, some definitions and mathematical facts. Theorems on uniformasymptotic stability and instability are stated and proved in sections 3 and 4,respectively. In the last section we consider two examples. The first one is anexample of nonlinear scalar equation with deviating argument which has unstablezero solution for all values of the constant delay h ∈ [a, b] but, from the otherhand, it is shown that there exist a time-varying delay h0(t) with the same rangeof values, h0(t) ∈ [a, b] for all t, such that this equation with delay h0(t) alreadyhave uniformly asymptotically stable trivial solution.

We also study the parametric resonance in linear equation with small param-eter of Mathieu type. It is shown that the delay being introduced may damp thedemultiplicative parametric resonances and make the equation either unstableor asymptotically stable.

2 Preliminaries

Consider a system of nonlinear functional differential equations with finite delaywritten as

x(t) = F (t, xt), (2.ana)

where F : GhH → Rn, GhH = R+×ΩhH , R+ = [0,∞), ΩhH = ϕ ∈ Ch : ‖ϕ‖ < His the open H – ball in the Banach space Ch = C([−h, 0],Rn) of continuousfunctions ϕ : [−h, 0] → Rn with the supremum norm ‖ϕ‖ = max|ϕ(s)|: −h ≤s ≤ 0, | · | is a norm in Rn. For a given function x(t) we denote by xt theelement in Ch defined by xt(s) = x(t + s), −h ≤ s ≤ 0. In the context of FDEthe element xt is called the state at time t.

Denote by UI(R+) a set of all functions L : R+ → R+ which are integrableon any finite segment [t0, t0 +∆] ⊂ R+ and for any ∆ > 0 there exists a constantL∆ > 0 such that

t0+∆∫t0

L(t) dt ≤ L∆ for any t0 ∈ R+. (3.ana)

We assume that there are exist functions L,M0 ∈ UI(R+) and a constantd0 > 1 such that for any t ∈ R+ and ϕ, ψ ∈ ΩhH

|F (t, ϕ)− F (t, ψ)| ≤ L(t)‖ϕ− ψ‖, (4.ana)

|F (t, ϕ)| ≤M0(t)‖ϕ‖d0 (5.ana)

for any t ∈ R+ and ϕ, ψ ∈ ΩhH .

Stability Theorems for Nonlinear FDE’s 3

A solution of (2.ana) through (t0, ϕ) ∈ R+ × ΩhH will be denoted by x(t0, ϕ) :R+ → Rn, t 7→ x(t; t0, ϕ), so that xt0(t0, ϕ) = ϕ. It is known that x(t0, ϕ)satisfies the integral equation

x(t; t0, ϕ) = ϕ(0) +

t∫t0

F (τ, xτ ) dτ, xt0 = ϕ, t ≥ t0. (6.ana)

Using this representation and Gronwall’s lemma it is easy to get the followingresults [7].

Lemma 1. Let t0 ∈ R+ and ϕ ∈ ΩhH be given and the functional F satisfiesLipschitz inequality (4.ana). Then until x(t0 +∆; t0, ϕ) ∈ Ωhh the following inequalityholds

‖xt0+∆(t0, ϕ)‖ ≤ ‖ϕ‖ exp(L∆), (7.ana)

where L∆ is a constant from the estimate (3.ana).

Note that the right-hand part of inequality (7.ana) does not depend on t0.

Lemma 2. Let t0 ∈ R+ and ϕ ∈ ΩhH be given and the functional F satisfiesinequalities (4.ana) and (5.ana) then

|x(t0 +∆; t0, ϕ)− ϕ(0)| ≤ ‖ϕ‖d0E∆, (8.ana)

where E∆ = M0∆ exp(d0L∆), M0

∆ and L∆ are the constants from the estimatesof the type (3.ana) for functions M0(t) and L(t) respectively.

Lemma 3. Let x : [t0− h,∞)→ Rn be a continuous function and there exist aconstant R > 1 such that |x(t)| ≤ ‖xt‖/R ≡ (1/R) max|x(t+ s)| : −h ≤ s ≤ 0for all t ≥ t0.

Then limt→∞

|x(t)| = 0, and

|x(t)| ≤ ‖xt0‖/RN+1 for t ≥ t0 +Nh,N = 0, 1, 2, . . . .

In this paper we use the known definition of stability.

Definition 4. The zero solution x = 0 of the system (2.ana) is said to bestable if for each σ ≥ 0, α > 0 there is β = β(α, σ) > 0 such that ϕ ∈ Ωhβ

implies that xt(σ, ϕ) ∈ Ωhα for any t ≥ σ;uniformly stable if it is stable and β is independent of σ;asymptotically stable, if it is stable and for each σ ≥ 0 there is β0 = β0(σ) > 0

such that ϕ ∈ Ωhβ0implies xt(t;σ, ϕ)→ 0 as t→∞;

uniformly asymptotically stable if it is uniformly stable and if there is a β0 > 0and for each η > 0 there exists t0(η) > 0 such that for any σ ≥ 0 xσ = ϕ ∈ Ωhβ0

implies xt(σ, ϕ) ∈ Ωhη for t ≥ σ + t0(η).

4 Oleg Anashkin

For given h0 > h and R ≥ 1 consider the set

Ah0R = ϕ ∈ Ch0 : ‖ϕ‖ ≤ R|ϕ(0)|.

It is easy to see thatAh0R 6= ∅ for R > 1,Ah0R1⊂ Ah0

R2for R1 < R2,

the boundary ∂Ah0R of the set Ah0

R is defined as

∂Ah0R = ϕ ∈ Ch0 : ‖ϕ‖ = R|ϕ(0)|,

Ah01 ≡ ∂Ah0

1 and Ah01 ⊂ Ah0

R for any R > 1,Ah0R is a nonconvex cone in Ch0 .

The cone Ah0R plays a crucial role in our approach. The fact is that the norm

‖xt(σ, ϕ)‖ may increase if and only if xt(σ, ϕ) ∈ Ah01 and |x(t;σ, ϕ)| tends to zero

when xt(σ, ϕ) 6∈ Ah0R for some R > 1. Therefore in the context of the stability

problem it is enough to investigate a behavior of the state xt(σ, ϕ) only in thecone Ah0

R not in the whole neighborhood Ωh0H .

3 Sufficient conditions on asymptotic stability

In this section we present sufficient conditions for uniform asymptotic stabilityof the zero solution of the system (2.ana) in terms of Lyapunov’s functionals v(t, ϕ)which can be nonmonotone along the solutions. It means that the derivativev|(2.ana) (σ, ϕ) of the functional v along the solution of (2.ana) can change the sign. Thisderivative is defined as

v|(2.ana) (σ, ϕ) = lim∆t→+0

v(σ +∆t, xσ+∆t(σ, ϕ)) − v(σ, ϕ)∆t

.

If v is differentiable v|(2.ana) (σ, ϕ) is obtained using the chain rule.We start with the following technical lemma.

Lemma 5. Let h0 ≥ h and R > 1 be given. Assume that for some τ0 ≥ 0 andψ0 ∈ Ah0

R ∩ Ωh0η a solution x(τ0, ψ0) of the system (2.ana) is defined for τ0 ≤ t ≤

τ0 + 2h0 and η < ηR,

ηR =[

R− 1(R+ 1)Rd0E0

]1/(d0−1)

, E0 = M02h0

exp(d0L2h0). (9.ana)

Then xt(τ0, ψ0) ∈ Ah0R for τ0 + h0 ≤ t ≤ τ0 + 2h0.

If, in addition, ‖xt(τ0, ψ0)‖ < ηR for all t ≥ τ0 + h0, then xt(τ0, ψ0) ∈ Ah0R

for all t ≥ τ0 + h0.

Proof. According to Lemma 2 the solution x(τ0, ψ0) satisfies the inequality

|x(t; τ0, ψ0)− x(τ0)| ≤ ‖xτ0‖d0E0 (10.ana)


for any t ∈ [τ0, τ0 +2h0], here xτ0 = ψ0, x(τ0) = ψ0(0). Using (10.ana), we obtain thefollowing upper and lower estimates for the norm |x(t; τ0, ψ0)| on the segment[τ0, τ0 + 2h0]:

|x(t; τ0, ψ0)| ≤ |x(t; τ0, ψ0)− x(τ0) + x(τ0)| ≤ |x(τ0)|+ ‖xτ0‖d0E0

≤ |x(τ0)|(1 +Rd0 |xτ0 |d0−1E0), (11.ana)

|x(t; τ0, ψ0)| ≥ |x(τ0)| − ‖xτ0‖d0E0 ≥ |x(τ0)|(1 −Rd0 |xτ0 |d0−1E0). (12.ana)

Hence for t ∈ [τ0 + h0, τ0 + 2h0] we have

‖xt‖|x(t)| ≤

max|x(t+ s), t− h0 ≤ s ≤ tmin|x(s)|, τ0 + h0 ≤ s ≤ τ0 + 2h0

≤ 1 +Rd0E0|x(τ0)|d0−1

1−Rd0E0|x(τ0)|d0−1

≤ 1 +Rd0E0ηd0−1

1−Rd0E0ηd0−1<

1 +Rd0E0ηd0−1R

1−Rd0E0ηd0−1R

= R. (13.ana)

It means, that ‖xt‖ < R|x(t)|, i.e. xt ∈ Ah0R , for t ∈ [τ0 + h0, τ0 + 2h0].

If ‖xt(τ0, ψ0)‖ < ηR for all t ≥ τ0 + h0, then |x(τ0 + kh0;σ, ψ0)| < ηR forany k = 1, 2, . . . . The constant E0 does not depend on xτ0 , consequently, theestimates (11.ana)–(13.ana) hold for τ0 + kh0 ≤ t ≤ τ0 + (k + 1)h0, k = 1, 2, . . . . Itmeans that xt ∈ Ah0

R for any t ≥ τ0. Lemma is proved.

Let K denotes the Hahn class, i.e. the set of all continuous strictly increasingfunctions a : R+ → R+ such that a(0) = 0.

Theorem 6. Suppose that for some h0 ≥ h and R > 1 the following assump-tions hold:

1) there exist functionals v, Φ : Gh0H → R and functions a, b ∈ K such that

a) v|(2.ana) (t, ϕ) ≤ Φ(t, ϕ),b) v(t, ϕ) ≤ b(‖ϕ‖) for (t, ϕ) ∈ Gh0

H ,c) v(t, ϕ) ≥ a(‖ϕ‖) for t ≥ 0 and ϕ ∈ Ah0

R ∩Ωh0H ;

2) there exist constants d > 1,m > 0 and a function M ∈ UI(R+) such that

|Φ(t, ϕ)| ≤M(t)‖ϕ‖d0 for (t, ϕ) ∈ Gh0H ,

|Φ(t, ϕ) − Φ(t, ψ)| ≤M(t)rd−1‖ϕ− ψ‖for ∀t ≥ 0 and ϕ, ψ ∈ Ωh0

r , 0 < r < H;3) there exist constants T > 0, β > 0 and δ > 0 such that for any t0 ≥ 0,

x0 ∈ Bβ and ∆t ≥ T

I(∆t, t0, x0) =

t0+∆t∫t0

Φ(t, x0) dt ≤ −2δ|x0|d∆t.

Then the zero solution of (1) is uniformly asymptotically stable.

6 Oleg Anashkin

Proof. First of all we will show that xt(t0, xt0) ∈ Ah0R for all t ≥ 0 if xt0 ∈ Ah0

R atsome moment t0 and ‖xt0‖ is small enough. Then we prove that the zero solutionof the system (2.ana) is uniformly stable and ‖xt‖ → 0 as t→∞. Uniformity of theasymptotic stability is guaranteed by the properties of the functional v in thecone Ah0

R .Let us fix an arbitrary small ε ∈ (0, ηR), where ηR is defined as in Lemma 5

by given h0 ≥ h and R > 1. Put ε1 = ε/2. Denote by

v < γτ = ϕ ∈ Ωh0H : v(τ, ϕ) < γ

a cut for t = τ of the region

v < γ = (t, ϕ) ∈ Gh0H : v(t, ϕ) < γ.

Due to the positive definiteness of the functional v in the region R+×(Ωh0H ∩A

h0R )

there exists a constant γ > 0 such that for all t ≥ 0

v < γt ∩ Ah0R ⊂ Ωh0

ε1 .

It is enough to take γ = a(ε1). Choose a value η0 > 0 such that

η0 < η0(1 + ηd0−10 Eh0) < b−1(γ), (14.ana)

where Eh0 = M0h0

exp[Lh0(d0 + 1)]. Note that (14.ana) implies that η0 < ηR.Let σ ≥ 0 and ψ ∈ Ωh0

η0be given. Inequality (14.ana) implies that v(σ, ψ) < γ. If

ψ /∈ Ah0R , then ‖xt(σ, ψ)‖ will decrease while xt /∈ Ah0

R . The rate of decreasingis given by Lemma 3. Suppose that xτ0 ∈ Ah0

R for some τ0 ≥ σ. According tothe choice of η0 (see the inequality (14.ana)) v(t, xt) < γ for t ∈ [τ0, τ0 + h0]. Due toLemma 5 xτ0+h0 ∈ Ah0

R , hence,

a(‖xτ0+h0‖) ≤ v(τ0 + h0, xτ0+h0) < γ = a(ε1),

therefore ‖xτ0+h0‖ < ε1 < ηR and xt ∈ Ah0R at least for t ∈ [τ0 + h0, τ0 + 3h0].

xt(σ, ψ) will be in Ah0R while ‖xt‖ < ηR. Remember that ‖xt‖ may increase only

if xt ∈ Ah01 ⊂ Ah0

R . According to the choice of γ we see that ‖xt‖ ≤ ηR/2 tillv(t, xt) ≤ γ. Suppose that v(t0, xt0) = γ for some t0 ≥ τ0 +h0 the point xt leavesthe domain v < γ. Note that xt0 ∈ Ah0

R , i.e. ‖xt0‖ ≤ R|x(t0)|. According tocondition 2a) of the theorem

v|(2.ana) (t, xt) ≤ Φ(t, xt).

Thus

v(t, xt(σ, ψ)) − v(t0, xt0(σ, ψ)) ≤∫ t

t0

Φ(τ, xτ ) dτ. (15.ana)

Define a function z : R → Rn such that z(t) = x(t;σ, ψ) for t0 − h0 ≤ t ≤ t0and z(t) = x(t0) for t ≥ t0 Adding and subtracting

∫ tt0Φ(τ, zτ ) dτ , we get

t∫t0

Φ(τ, xτ ) dτ =

t∫t0

Φ(τ, zτ ) dτ +

t∫t0

[Φ(τ, xτ )− Φ(τ, zτ )] dτ. (16.ana)


According to Lemma 2‖xτ − zτ‖ ≤M1‖xt0‖d0,

for τ ∈ [t0, t0 + T1], where the constant T1 ≥ T will be selected below, theconstant M1 depends only on T1. Using Lemma 1 and the Lipschitz conditionwe obtain

t0+T1∫t0

[Φ(τ, xτ )− Φ(τ, zτ )] dτ ≤ C0T1‖xt0‖d+d0−1. (17.ana)

The estimate (17.ana) is uniform with respect to t0 ≥ 0 and xt0 ∈ Ωh0H , i.e. the

constant C0 > 0 depends only on T1.To estimate first integral in the right-hand side of (16.ana) we represent it in the

form

t∫t0

Φ(τ, zτ ) dτ =

t0+h0∫t0

Φ(τ, zτ ) dτ +

t∫t0

Φ(τ, yτ ) dτ −t0+h0∫t0

Φ(τ, x(t0)) dτ, (18.ana)

Due to construction of the function z, condition 2) of the theorem and Lemma 1,we obtain ∣∣∣∣∣∣

t0+h0∫t0

Φ(τ, zτ ) dτ

∣∣∣∣∣∣ ≤ h0Mh0‖xt0‖d exp(dLh0), (19.ana)

∣∣∣∣∣∣t0+h0∫t0

Φ(τ, yτ ) dτ

∣∣∣∣∣∣ ≤ h0Mh0 |x(t0)|d exp(dLh0). (20.ana)

Choose T1 ≥ T such that

Mh0(Rd + 1)h0 exp(dLh0)/T1 ≤ δ/2, (21.ana)

where δ is a constant from condition 3) of the theorem. Using condition 3) andtaking into account that ‖xt0‖ < R|x(t0)|, from (18.ana)–(21.ana) we get

t0+T1∫t0

Φ(τ, zτ ) dτ =

t0+T1∫t0

Φ(τ, x(t0)) dτ −t0+h0∫t0

Φ(τ, x(t0)) dτ +

t0+h0∫t0

Φ(τ, zτ ) dτ

≤t0+T1∫t0

Φ(τ, x(t0)) dτ +

∣∣∣∣∣∣t0+h0∫t0

Φ(τ, x(t0)) dτ

∣∣∣∣∣∣ +

∣∣∣∣∣∣t0+h0∫t0

Φ(τ, zτ ) dτ

∣∣∣∣∣∣≤ −2δ|x(t0)|dT1 + h0Mh0 exp(dLh0)(|x(t0)|d + ‖xt0‖d)

≤ |x(t0)|dT1

(−2δ +

Mh0(Rd + 1)h0 exp(dLh0)T1

)≤ −3

2δ|x(t0)|dT1.

8 Oleg Anashkin

Thus

t0+T1∫t0

Φ(τ, zτ ) dτ ≤ −32δ|x(t0)|dT1. (22.ana)

Suppose that ε is small enough to be true the following inequality

C0Rd+d0−1(ε/2)d0−1 ≤ δ/2. (23.ana)

Then from (15.ana)–(17.ana) and (22.ana) we have

v(t0 + T1, xt0+T1) ≤ v(t0, xt0) +

t∫t0

Φ(τ, zτ ) dτ +

t∫t0

[Φ(τ, xτ )− Φ(τ, zτ )] dτ

≤ v(t0, xt0)− 32δ|x(t0)|dT1 + C0T1R

d+d0−1|x(t0)|d+d0−1

≤ v(t0, xt0) + |x(t0)|dT1(−32δ + C0T1R

d+d0−1|x(t0)|d0−1)

≤ v(t0, xt0)− δ|x(t0)|dT1 < v(t0, xt0).

It gives us the main inequality

v(t0 + T1, xt0+T1) ≤ v(t0, xt0)− δ|x(t0)|dT1. (24.ana)

We emphasize that the estimate (24.ana) is uniform with respect to t0 ≥ 0 andxt0 ∈ Ah0

R ∩ Ωh0ε1 . Inequality (24.ana) means that the state xt has returned into the

domain v < γt, moreover, we can choose ε > 0 so small that xt does not leavethe ball Ωh0

ε . Indeed, according to Lemma 2 for t ∈ [t0, t0 + T1]

|x(t; t0, xt0)− x(t0)| ≤ ‖xt0‖d0MT1 exp(d0LT1) ≤ |xt0 |d0Rd0MT1 exp(d0LT1),

therefore|x(t; t0, xt0)| ≤ |x(t0)|+ |x(t)− x(t0)| ≤ ε,

if ε is small enough to ensure

(ε/2)d0Rd0MT1 exp(d0LT1) ≤ ε/2.

According to Lemma 5 xt ∈ Ah0R for t ∈ [t0, t0 + T1]. It has been shown that

at the moment t1 = t0 + T1 xt1 belongs to the domain v < γt1 ⊂ Ωh0ε1 . If the

point xt will leave the domain v < γt again at some moment t′0 > t1 then itwill return back in finite time less than T1 because all estimates we employedabove to obtain the main inequality are uniform with respect to t ≥ t0 andxt0 ∈ Ah0

R ∩ Ωh0ε1 . Consequently, we have proved that xt ∈ Ah0

R ∩ Ωh0ε1 for all

t ≥ t0.Since ε is arbitrary small and η0 does not depend on initial moment σ, the

uniform stability of the zero solution of the system (2.ana) is proved.


To prove the asymptotic stability we note that in virtue of the uniformityof all estimates derived above with respect to t ≥ t0 and xt0 ∈ Ah0

R ∩ Ωh0ε1 (24.ana)

yields

0 < v(tk + T1, xtk+T1) ≤ v(t0, xt0)− δT1(|x(t0)|d + |x(t1)|d + . . .+ |x(tk)|d),(25.ana)

where tk = t0 + kT1, for any integer k ≥ 1. By condition 1c) of the theoremv(t0, xt0) ≥ a(‖xt0‖) > 0, and (25.ana) means that |x(tk)| → 0 as k → ∞, therefore‖xtk‖ → 0 as k →∞, because ‖xtk‖ < R|x(tk)|.

Uniformity of the asymptotic stability follows from Lemma 3 if xt /∈ Ah0R and

from conditions 1b) and 1c) if xt ∈ Ah0R . The theorem is proved.

4 Sufficient conditions for instability

Theorem 7. Suppose that for some h0 ≥ h, β > 0, σ > 0 and R > 1 there existfunctionals v, Φ : [σ,∞) ×Ωh0

β → R such that the following conditions satisfied:

1) v|(1) (t, ϕ) ≥ Φ(t, ϕ) for (t, ϕ) ∈ [σ,∞)×Ωh0β ;

2) for each t ≥ σ and η, 0 < η < β, there exists ϕ ∈ Ah0R ∩ Ωh0

η such thatv(t, ϕ) > 0;

3) there exists a function b ∈ K such that v(t, ϕ) ≤ b(‖ϕ‖) for each (t, ϕ) ∈[σ,∞)×Ah0

R ∩Ωh0β ;

4) the functional Φ satisfies condition 2) of Theorem 6 for (t, ϕ) ∈ [σ,∞)×Ωh0β

and 0 < r < β;5) there exist constants T > 0 and δ > 0 such that for any t0 ≥ σ, x0 ∈ Bβ

and ∆t ≥ TI(∆t, t0, x0) ≥ 2δ|x0|d∆t.

Then the zero solution of the system (1.ana) is unstable.

Proof. By way of contradiction, assume that the zero solution of (2.ana) is stable.By the conditions of the theorem choose a small enough value ε < minβ, ηRand a constant T1 ≥ h0 such that the inequalities (21.ana) and (23.ana) hold. Accordingto our assumption there exists η0 > 0 such that for any initial function ψ0 ∈ Ωh0

η0

xt(σ, ψ0) ∈ Ωh0ε for all t ≥ σ. Let us fix arbitrary small η ∈ (0, ε) and choose

ψ0 ∈ Ah0R ∩ Ωh0

η such that α = v(σ, ψ0) > 0. Lemma 5 implies that xt(σ, ψ0) ∈Ah0R for all t ≥ σ+h0. It means that xt ∈ Ah0

R ∩Ωh0ε for t ≥ σ+h0. By condition

3) of the theorem the functional v is bounded along the given solution x(σ, ψ0)of the system (2.ana).

Denote tk = σ + kT1, k = 0, 1, . . . . Since ψ0 ∈ Ah0R , then ‖ψ0‖ < R|ψ0(0)|.

According to condition 1)

v(t, xt(σ, ψ)) − v(t0, xt0(σ, ψ)) ≥∫ t

t0

Φ(τ, xτ ) dτ.

10 Oleg Anashkin

By the same way as in the proof of Theorem 6 we obtain the main inequality

v(t0 + T1, xt0+T1) ≥ v(t0, xt0)− δT1|x(t0)|d. (26.ana)

This inequality is valid for all (t0, xt0) ∈ [σ,∞)× (Ah0R ∩Ωh0

ε ).Denote tk = σ + kT1, k = 0, 1, . . . . Since ‖xt(σ, ψ0)‖ < ε for all t ≥ σ, we

obtain from (26.ana) that for any integer k = 0, 1, 2, . . .

v(tk + T1, xtk+T1(σ, ψ0)) ≥ v(σ, ψ0) + δT1(|x(t0)|d + |x(t1)|d + . . .+ |x(tk)|d).(27.ana)

Note that for any integer k |x(tk)| > (1/R)‖xtk‖ ≥ (1/R)b−1(v(tk, xtk)) >(1/R)b−1(α) > 0. Hence the right-hand side of (27.ana) tends to +∞ as k → +∞. Itcontradicts the boundedness of the functional v in the region [σ,∞)×(Ah0

R ∩Ωh0ε ).

The theorem is proved.

5 Remarks

Remark 8. Theorems 6 and 7 are valid also for the systems in the standard formof the type

x = µL(t, xt), (28.ana)

where µ is a positive small parameter, L is linear in xt and |L(t, ϕ)| ≤M(t)‖ϕ‖for any t ≥ 0 and ϕ ∈ ΩhH with some function M ∈ UI(R+).

Remark 9. Condition 3) of Theorem 6 (respectively, condition 5) of Theorem 7)will be fulfilled if there exists the average

Φ(t0, x0) = lim∆t→∞

1∆t

t0+∆t∫t0

Φ(t, x0) dt (29.ana)

and a constant δ0 > 0 such that for all t0 ≥ 0 Φ(t0, x0) ≤ 2δ0|x0|d (respectively,Φ(t0, x0) ≥ 2δ0|x0|d).

6 Examples

By simple examples we present an algorithm for construction of functionals whichsatisfy all conditions of new theorems on stability given in the previous sections.

Example 10. Consider a nonlinear equation with a time-varying delay

x = b(t)x3(ρ(t)), (30.ana)

where ρ is a differentiable function, t − h ≤ ρ(t) ≤ t for all t > 0 and somepositive constant h > 0. Suppose that the function b has a zero average b(t)


and a bounded antiderivative B,B′(t) = b(t), on t ∈ [0,∞). To construct anappropriate functional v we take a positive definite function v0(x) = x2/2. Then

v0|(30.ana) = b(t)x(t)x3(ρ(t)) = Φ0(t, x(t), x(ρ(t))).

Following to Remark 8, we have to evaluate the average Φ0(t0, x0) along theconstant solution x0 of the trivial system. The average Φ0(t, x0, x0) ≡ 0, sinceb(t) = 0. Consider a functional

v(t, xt) = v0(x(t)) + u(t, x(t), x(ρ(t))),

where the function u(t, p, q) is a bounded solution of the equation

∂u/∂t = −Φ0(t, p, q) = −b(t)pq3.

Putting u(t, p, q) = −B(t)pq3, we obtain a functional

v = v0 + u = x(t)2/2−B(t)x(t)x3(ρ(t)). (31.ana)

Calculating a full derivative of this functional in virtue of the system (30.ana), weget

v|(30.ana) = Φ1(t, x(t), x(ρ(t)), x(ρ(ρ(t))))

= −B(t)[b(t)x6(ρ(t)) + 3x(t)x2(ρ(t))x3(ρ(ρ(t)))b(ρ(t))ρ′(t)].

A sign of the average Φ1(t, x0, x0, x0) is defined by a sign of the average

δ0 − B(t)(b(t) + 3ρ′(t)b(ρ(t))). (32.ana)

According to Theorems 6 and 7 the zero solution of the system (30.ana) is uniformlyasymptotically stable if δ0 < 0 and it is unstable if δ0 > 0.

Let b(t) = cos t, ρ(t) = t−β+α sinωt, where α, β and ω are some constants.Then B(t) = sin t, ρ′(t) = 1 +αω cosωt. Substituting given functions to (30.ana), weobtain the equation

x = cos tx3(t− h(t)), (33.ana)

where h(t) = β − α sinωt. The index of stability (32.ana) now has the form

δ0 = −sin t(1 + αω cosωt) cos(t− β + α sinωt), (34.ana)

since the average sin t cos t = 0. If α = 0 the system (33.ana) takes the form

x = cos tx3(t− β) (35.ana)

with the constant delay β. In this case (34.ana) gives δ0 = − sinβ. Thus the zerosolution of the equation (35.ana) is unstable for any β ∈ (π, 2π), since sin β < 0 andall conditions of Theorem 7 are fulfilled.

12 Oleg Anashkin

It is interesting that it is possible to choose the values of the parameters α,β and ω such that the zero solution of the equation (33.ana) with time-varying delaybecomes already uniformly asymptotically stable although

h(t) = β − α sinωt ∈ (π, 2π),

i.e. for every t the value of the delay h(t) lies in the domain of instability of thetrivial solution of the same equation but with a constant delay (35.ana). Indeed, putβ = 1.5π, α = 1.55 < π/2 and ω = 2. Then (34.ana) gives δ0 = −0.04033 < 0 andall conditions of Theorem 6 on asymptotic stability are fulfilled, but sinh(t) < 0for all t ∈ (−∞,∞) because h(t) = 1.5π − 1.55 cos2t ∈ [3.16, 6.27] ⊂ (π, 2π)

This phenomenon of changing of the type of stability after replacing of theconstant parameter by the continuous function with the same range of values iswell-known for ordinary differential equations. It have been first demonstratedfor linear equation with deviating argument by A. D. Myshkis [8].

Example 11. [9] Consider the linear equation of Mathieu type with time delay

x+ ω2[x(t) − µ(2 cos νt)x(t − h)] = 0, (36.ana)

where µ is small parameter. This equation turns into the well-known Mathieuequation at h = 0:

x+ ω2[x(t)− µ(2 cos νt)x(t)] = 0. (37.ana)

There are infinite sequence of the so-called regions of dynamical instability forthe Mathieu equation (37.ana) at the critical values ν = 2ω/m,m = 1, 2, 3, . . . . Thisphenomenon is called parametric resonance. By Theorems 6 and 7 we show thatthe main resonance ν = 2ω also appears in the equation (36.ana) for any delay h. Atν 6= 2ω the type of stability depends greatly from h. The delay being introducedmay damp the demultiplicative parametric resonances and make the equationeither unstable or asymptotically stable.

Introducing complex conjugate variables ζ and ζ

ζ exp(iωt) = x− i xω, ζ exp(−iωt) = x+ i

x

ω, (38.ana)

and using more short notations for the variables with deviating argument:

ζh = ζ(t− h), ζ2h = ζ(t− 2h), . . . ,

we reduce (36.ana) to the linear system in standard form

ζ = µZ(t, ζh, ζh), ˙ζ = µZ(t, ζh, ζh), (39.ana)

where

Z(t, ζh, ζh) = −0, 5iω[ζhe−iωh(eiνt + e−iνt) + ζheiωh(e−i(2ω+ν)t + e−i(2ω−ν)t)].


To construct a suitable functional we start with the positive definite functionv0(ζ, ζ) = ζζ. Differentiating it in virtue of (39.ana), we obtain

v0|(39.ana) = µΦ0(t, ζh, ζh) = µ2Re(∂v0

∂ζZ

)= µRe[(−iω)(ζζhe−iωh(eiνt + e−iνt)

+ ζ ζheiωh(e−i(2ω+ν)t + e−i(2ω−ν)t)]. (40.ana)

It is obvious that average (29.ana) Φ0 = Φ0(t, ζ0, ζ0) = 0, if ν 6= 2ω.Let ν = 2ω, then

Φ0 = Re[(−iω)ζ20eiωh] = iω(ζ2

0e−iωh − ζ2

0eiωh)

Denotingw(ζ, ζ) = i(ζ2e−iωh − ζ2eiωh),

we have that

w|(39.ana) = µΨ(t, ζh, ζh) = 2µRe(∂w

∂ζZ

)= 2µRe[ω(ζζhe−i2ωh(ei2ωt + e−i2ωt) + ζζh(e−i4ωt + 1))].

The average Ψ(t, ζ0, ζ0) = 2ω|ζ0|2 is positive definite and for small enough |µ|all conditions of Theorem 7 on instability are fulfilled, moreover, Ψ does notdepend on h.

Suppose now that ν 6= 2ω, then Φ0 ≡ 0. Following to the theory of thegeneralized Lyapunov functions [5] we have to construct the so-called perturbedfunctional

V1 = v0 + µv1, (41.ana)

where a perturbation v1 of the functional (function) v0 is calculated by a boundedsolution of the equation

∂v1/∂t = −Φ0.

Since

−∫ (

∂v0

∂ζZ

)dt = (ω/2)

[ζζhe

−iωh(eiνt − e−iνt

ν

)− ζ ζheiωh

(e−i(2ω+ν)t

2ω + ν+e−i(2ω−ν)t

2ω − ν

)]+ const,

we can take the functional v1 in the form

v1(t, ζ, ζ, ζh, ζh) = ωRe[ζζhe

−iωh(eiνt − e−iνt

ν

)− ζ ζheiωh

(e−i(2ω+ν)t

2ω + ν+e−i(2ω−ν)t

2ω − ν

)].

14 Oleg Anashkin

Then

V1

∣∣∣(39.ana)

= v0|(39.ana) + µ∂v1

∂t+ µ22Re

(∂v1

∂ζZ +

∂v1

∂ζhZh

)= µ2Φ1.

Making necessary calculations, we obtain

∂v1

∂ζZ +

∂v1

∂ζhZh =

= (−iω2/4)[ζhζh

((e−i2νt − ei2νt)/ν − 4ω

4ω2 − ν2− ei2νt

2ω + ν− e−i2νt

2ω − ν

)+ ζ2

hei2ωh(e−i(2ω+2ν)t − e−i(2ω−2ν)t)/ν

− ζ2he−i2ωh

(ei(2ω+2ν)t + ei2ωt

2ω + ν+ei(2ω−2ν)t + ei2ωt

2ω − ν

)+ ζζ2he

−i2ωh(e−i2νte−iνh − e−iνh + eiνh − e−i2νteiνh)/ν

+ ζ ζ2h(1/ν)(e−i2ωtei(2ω+ν)h + e−i(2ω−2ν)tei(2ω−ν)h

− e−i(2ω+2ν)tei(2ω+ν)h − e−i2ωtei(2ω−ν)h)

− ζζ2he−i2ωh((e−iνhei(2ω+2ν)t + ei2ωteiνh)/(2ω + ν)

+ (ei(2ω−2ν)teiνh + e−iνhei2ωt)/(2ω − ν))

− ζζ2h((ei(2ω+ν)h + ei2νtei(2ω−ν)h)/(2ω + ν)

+ (ei(2ω+ν)hei2νt + ei(2ω−ν)h)/(2ω − ν))].

Here we use a notation: Zh = Z(t−h, ζ2h, ζ2h). Consequently, the average of thefunctional Φ1(t, ζh, ζh, ζ2h, ζ2h) for ν 6= 2ω, ν 6= ω and ζ = ζ0 has the form

Φ1 =ω3

2ν|ζ0|2

(sin(2ω + ν)h

2ω + ν− sin(2ω − ν)h

2ω − ν

).

It is not equal to zero for all values of h, in spite of the zeros of a function

σν(ωh) =sin(2 + ν/ω)ωh

2 + ν/ω− sin(2− ν/ω)ωh

2− ν/ω .

Since the functional (41.ana) is positively defined in the cone AR ⊂ Ch. R > 1,for small enough µ, the equation (36.ana) is uniformly asymptotically stable forσν(ωh) < 0 and it is unstable for σν(ωh) > 0.

References

[1] N. N. Krasovskii, Some Problems in the Stability Theory of Motions. Fizmatgiz,Moscow, 1959.

[2] J. Hale, Functional Differential Equations. Springer Verlag, New York-Heidelberg-Berlin, 1971.


[3] M. M. Khapaev, Theorem of Lyapunov Type, Soviet Math. Dokl., 8 (1967), 1329–1333.

[4] M. M. Khapaev, Instability for Constantly Operating Perturbations, Soviet Math.Dokl., 9 (1968), 43–47.

[5] M. M. Khapaev, Averaging in Stability Theory, Kluwer, Dordrecht-Boston-London1993.

[6] O. V. Anashkin, Asymptotic Stability in Nonlinear Systems, Differential Equations,14 (1978), 1061–1063.

[7] O. V. Anashkin, Averaging in the Theory of Stability for Functional DifferentialEquations, Differents. Uravneniya, 33 (1997), 448–457 (in Russian).

[8] A. D. Myshkis, Linear Differential Equations with Deviating Argument, Moscow,Nauka, 1972.

[9] O. V. Anashkin, Parametric Resonance in Linear System with Time Delay, Trans-actions of RANS, series MMMIC, 1 (1997), 3–18 (in Russian).



Analysis of Equations in the Phase-Field Model

Michal Benes

Department of Mathematics, Czech technical University, Trojanova 13,120 00 Prague, Czech Republic

Email: [email protected]: http://kmdec.fjfi.cvut.cz/~benes

Abstract. The article presents basic numerical analysis of equationsin the phase-field model which is performed using a FDM semi-discretescheme. The compactness technique allows to prove convergence of thescheme. Simultaneously, existence and uniqueness of weak solution tothe original system is shown. Additionally, the asymptotical behaviourof the solution with respect to the small parameter ξ is studied. Bothtemperature and phase fields converge in certain sense if ξ → 0. Thephase field gives rise to a step-wise function indicating the presence ofdifferent phases.

AMS Subject Classification. 80A22, 82C26, 35A40

Keywords. phase-field model, method of lines, compactness method

1 Introduction

The paper contains several remarks concerning basic analysis of the standardform of phase-field model. The system of equations in question reads as follows:

∂u

∂t= ∇2u+ L

∂p

∂t,

αξ2 ∂p

∂t= ξ2∇2p+ f0(p)− βuξ ,

(1.ben)

with initial conditions

u |t=0= u0 , p |t=0= p0 ,

and with boundary conditions of Dirichlet type

u |∂Ω= uΩ , p |∂Ω= pΩ ,

where L, α, β, ξ are positive constants, Ω is a bounded domain in Rn and f0

derivative of a quartic potential. For the sake of simplicity, we will consider rect-angular form of Ω in 2D, f0(p) = ap(1− p)(p− 1

2 ) with a > 0 and homogeneousboundary conditions.

This is the final form of the paper.

http://kmdec.fjfi.cvut.cz/~benes

18 Michal Benes

Such a system of equations has been studied by many authors throughoutlast decade (see, e.g. [5], [5], [1], [8], [16], [13], [10]). In the physical context,the system (1.ben) is treated as a regularization of the modified Stefan problemdescribing microstructure formation in solidification of a pure substance if ξ → 0,see [7], [2]:

∂u

∂t= ∇2u in Ωs and Ωl , (2.ben)

u |∂Ω= uΩ , (3.ben)u |t=0= u0 , (4.ben)

∂u

∂n|s −

∂u

∂n|l= −LvΓ , (5.ben)

6

√2aβu = −κ+ αvΓ , (6.ben)

Ωs(t) |t=0= Ωso , (7.ben)

where Ωs, Ωl are solid and liquid phases, respectively, L is latent heat per unitvolume, melting point is 0, u temperature field. Discontinuity of heat flux onΓ (t) is described by the Stefan condition (5.ben), the formula (6.ben) is the Gibbs-Thompson relation on Γ (t). The parameter α is the coefficient of attachmentkinetics. Following [2], the relation of (1.ben) and (2.ben)–(7.ben) is studied using asymp-totical analysis. The article presents the following results: convergence of thesemi-discrete scheme, existence and uniqueness of the original system of equa-tions, and convergence towards the sharp-interface state.

2 Interpolation theory for grid functions

The analysis of the system (1.ben) concerning the existence and uniqueness of theweak solution is performed using a semi-discrete scheme based on finite differ-ences. The following notations are introduced (see [15]):

h = (h1, h2) , h1 =L1

N1, h2 =

L2

N2, xij = [x1

ij , x2ij ], uij = u(xij), (8.ben)

ωh = [ih1, jh2] | i = 1, . . . , N1 − 1; j = 1, . . . , N2 − 1 , (9.ben)

ωh = [ih1, jh2] | i = 0, . . . , N1; j = 0, . . . , N2 , (10.ben)

γh = ωh − ωh , (11.ben)

ux1,ij =uij − ui−1,j

h1, ux1,ij =

ui+1,j − uijh1

, (12.ben)

ux2,ij =uij − ui,j−1

h2, ux2,ij =

ui,j+1 − uijh2

, (13.ben)

ux1x1,ij =1h2

1

(ui+1,j − 2uij + ui−1,j) , (14.ben)

Phase-Field equations 19

and

∇hu = [ux1 , ux2 ], ∇hu = [ux1 , ux2 ], ∆hu = ux1x1 + ux2x2 , (15.ben)

If Hh = f | f : ωh → R is a set of grid functions, the following notations willbe used (f, g ∈ Hh) :

‖f‖ph =

(N1−1,N2−1∑

i,j=1

h1h2|fij |p) 1p

for p > 1 , (16.ben)

(f, g)h =N1−1,N2−1∑

i,j=1

h1h2fijgij , ‖f‖2h = (f, f)h , (17.ben)

(f1, g1c =N1,N2−1∑i=1,j=1

h1h2f1ijg

1ij , ‖f1c|2 = (f1, f1c , (18.ben)

(f2, g2e =N1−1,N2∑i=1,j=1

h1h2f2ijg

2ij , ‖f2e|2 = (f2, f2e , (19.ben)

(f ,g] = (f1, g1c+ (f2, g2e , ‖f ]|2 = (f , f ] , (20.ben)

where f = [f1, f2] and g = [g1, g2].Referring to [2], we recall the following formulas

– Green formulas

(f, gx1x1)h = −(fx1 , gx1c+N2−1∑j=1

(fgx1 |N1,j −fgx1 |0,j)h2, (21.ben)

and

(f, gx2x2)h = −(fx2 , gx2e+N1−1∑i=1

(fgx2 |i,N2 −fgx2 |i,0)h1, (22.ben)

In a natural way, we define the space

lp(ωh) = Hh | ‖ · ‖ph . (23.ben)

– Poincare inequality. Let u ∈ l2(ωh) and u |γh= 0. Then

‖u‖2h ≤ C(Ω)[ ‖ux1c|2 + ‖ux2e|2 ] . (24.ben)

We continue by introducing an extension of grid functions, so that they aredefined almost everywhere on Ω. Such extensions are studied by the usual tech-nique of Lp and Hk spaces. The approach of [14] is adopted for the equationsin question. The limiting process requires a refinement of the FDM grid ωh, ifh → 0. For this purpose, a proper metric should be chosen. If we intent to usethe compactness technique, a mapping converting a grid function fh : ωh → Rinto a function f : Ω → R is needed. Then, the norm of Lp spaces will serve asa metric for convergence of the numerical scheme.

20 Michal Benes

Definition 1. Be ωh an uniform rectangular grid imposed on a domain Ω ⊂ R2.Let h = [h1, h2] is the mesh size. Then, the dual grid is a set

ω∗h =Σij ⊂ Ω | Σij =(

x1i −

h1

2, x1i +

h1

2

)×(x2j −

h2

2, x2j +

h2

2

)∩ Ω for [x1

i , x2j ] ∈ ωh

.

The dual simplicial grid is a set

ω∗sh = ω∗/h ∪ ω∗.h , (25.ben)

with

ω∗/h = Σ/ij ⊂ Ω | Σ/

ij = [xi,j , xi−1,j , xi,j−1]κ ∩ Ω for [x1i , x

2j ] ∈ ωh ,

ω∗.h = Σ.ij ⊂ Ω | Σ.

ij = [xi−1,j−1, xi−1,j , xi,j−1]κ ∩ Ω for [x1i , x

2j ] ∈ ωh ,

where [ ]κ denotes the convex hull.

Remark 2. Consequently,⋃Σ∈ω∗

hΣ = Ω - the system ω∗h covers the domain Ω.

Each (rectangular) set Σ ∈ ω∗h has the point [x1i , x

2j ] in its center. Similarly, the

system ω∗sh also covers Ω.

Definition 3. Let Hh be a set of grid functions on ωh. Define the followingmappings:

– Qh : Hh → C(Ω) such that for each u ∈ Hh

(Qhu)(x1, x2) = ui−1,j−1 +∇huh,i−1,j−1 · [x1 − x1i−1,j−1, x

2 − x2i−1,j−1] ,

if [x1, x2] ∈ Σ.ij , Σ

.ij ∈ ω∗.h ;

(Qhu)(x1, x2) = uij + ∇huij · [x1 − x1ij , x

2 − x2ij ] ,

if [x1, x2] ∈ Σ/ij , Σ

/ij ∈ ω∗/h .

– Sh : Hh → L1(Ω) such that for each u ∈ Hh

(Shu)(x1, x2) = uij ,

if [x1, x2] ∈ Σij , Σij ∈ ω∗h;– Ph : C(Ω)→ Hh such that for each u ∈ C(Ω)

(Phu)ij = u(xij) ,

if xij ∈ ωh.

Remark 4. The operator Ph is linear and continuous from C(Ω) to Hh, and canbe extended to H1(Ω) via density argument.Qhu is a continuous piecewise linearfunction, ∇(Qhu) exists a.e. in Ω. We proceed by determining basic propertiesof the above defined maps as proven in [2]:


1. If u, v |γh= 0 the scalar product coincides with the scalar product in l2(ωh)∫Ω

ShuShvdx = (u, v)h . (26.ben)

2. Let ωh is a grid on the domain Ω with the mesh h, let u, v ∈ Hh is such thatu, v |γh= 0. Then

(∇(Qhu),∇(Qhv)) = (∇hu, ∇hv] . (27.ben)

3. Let ωh is a grid on the domain Ω with the mesh h, let u ∈ Hh. Then

‖Qhu‖L2(Ω) ≤ ‖Shu‖L2(Ω) . (28.ben)

4. Let ωh is a grid on the domain Ω with the mesh h, let u ∈ Hh, u |γh= 0.Then ∫

Ω

|Qhu− Shu|2dx ≤|h|2

6‖∇hu]|2 , (29.ben)

if u |γh= 0.5. Let p ∈ C0,ν(Ω), ν ∈ (0, 1). Then,

Sh(Php)→ p in Ls(Ω), if h→ 0 , (30.ben)

for s > 1.6. Let u ∈ H1

0(Ω) ∩H2(Ω). Then

Qh(Phu)→ u (31.ben)

in H1(Ω), if h→ 0.7. Let p ∈ C2(Ω) and p |∂Ω= 0. Then

∇(Qh(Php))→ ∇p , (32.ben)

in L2(Ω), if h→ 0.

3 Main result

In this section, we give a proof of existence and uniqueness of the solution to(1.ben) regardless on values of coefficients. Compared to [5], we get a more generalresult. Similar procedure has been presented in [3].

Definition 5. Consider a bounded domain Ω ⊂ R2, T > 0. The classical solu-tion of the system of phase-field equations is a couple of functions

[u, p] : 〈0, T 〉 × Ω → R2 ,

22 Michal Benes

satisfying the equations

∂u

∂t= ∇2u+ L

∂p

∂tin (0, T )×Ω

u |∂Ω = 0 , t ∈ (0, T ) ,

u |t=0 = u0 in Ω ,

αξ2 ∂p

∂t= ξ2∇2p+ f0(p)− βξu in (0, T )×Ω ,

p |∂Ω = 0 , t ∈ (0, T ) ,

p |t=0 = p0 in Ω .

(33.ben)

Remark 6. The form of the phase-field equations is referred to [5]. For the sakeof simplicity, we consider a 2-D rectangular domain and homogeneous boundarycondition. Obviously, the extension to higher dimensions, and to other boundaryconditions is possible. Let [u, p] is a classical solution such that u, p ∈ C2(〈0, T 〉×Ω) and let v, q ∈ D(Ω). Multiplying the first one of equations (1.ben) by v and thesecond one by q (scalar product in L2(Ω)), and using the Green formula, we get

d

dt(u, v) + (∇u,∇v) = L

d

dt(p, v) a.e. in (0, T ) ,

u(0) = u0 ,

αξ2 d

dt(p, q) + ξ2(∇p,∇q) = (f0(p), q)− βξ(u, q) a.e. in (0, T ) ,

p(0) = p0 .

(34.ben)

This leads to the next definition:

Definition 7. Weak solution of the boundary-value problem for the phase-fieldequations is a couple of functions [u, p] from (0, T ) to [H1

0(Ω)]2 such that itsatisfies (34.ben) for each q, v ∈ H1

0(Ω).

The term f0(p) requires that p ∈ L4(Ω) for almost all t ∈ (0, T ). As Ω ⊂ R2,it suffices to take p ∈ H1

0(Ω) for almost all t ∈ (0, T ) due to the continuousimbedding into Lq(Ω) for each q ∈ (1,+∞). If

[u, p] ∈ [L∞(0, T ; H10(Ω))]2 ,

[u, p] is continuous mapping from 〈0, T 〉 to H−1(Ω), as shown in [11]).Next statement gives an information about the existence and uniqueness

of the solution to (34.ben); the proof by its virtue contains the investigation ofconvergence of a semi-discrete scheme based on method of lines.

Theorem 8. Consider the problem (34.ben) in a rectangular domain Ω = (0, L1)×(0, L2), where

u0, p0 ∈ H2(Ω) ∩H10(Ω) .


Then, there is a unique solution of the problem (34.ben) satisfying

u, p ∈ L∞(0, T ; H10(Ω) ∩ H2(Ω)) ,

∂tu, ∂tp ∈ L2(0, T ; L2(Ω)) .

Proof. The proof is constructive. Cover Ω by an uniform grid with the meshh = [h1, h2], use the previously introduced notations. Consider the semi-discretescheme

uh = ∆huh + Lph on (0, T )× ωh ,

uh |γh = 0 ,

uh |t=0 = Phu0 on ωh ,

αξ2ph = ξ2∆hph + f0(ph)− βξuh in (0, T )× ωh ,

ph |γh = 0 ,

ph |t=0 = Php0 on ωh .

(35.ben)

where dot denotes the time derivative. In the proof, the major role is played bythe a priori estimate for both equations in question. Multiply the first one ofequations (35.ben) by uh, and the second one by ph; sum over ωh.

‖uh‖2h +12d

dt‖∇huh]|2 = L(ph, uh)h ,

αξ2‖ph‖2h + ξ2 12d

dt‖∇hph]|2 = (f0(ph), ph)h − βξ(uh, ph)h .

(36.ben)

Using Schwarz and Young inequalities, we get12‖uh‖2h +

12d

dt‖∇huh]|2 ≤ 1

2L2‖ph‖2h ,

12αξ2‖ph‖2h + ξ2 1

2d

dt‖∇hph]|2 ≤ − d

dt(w0(ph), 1)h +

β2

2α‖uh‖2h .

(37.ben)

Combining these estimates, we have

14αξ2‖ph‖2h +

αξ2

4L2‖uh‖2h +

αξ2

4L2

d

dt‖∇huh]|2 + ξ2 1

2d

dt‖∇hph]|2 +

+d

dt(w0(ph), 1)h ≤

β2

2α‖uh‖2h . (38.ben)

Using the discrete Poincare inequality (24.ben)

‖uh‖2h ≤ C(Ω)‖∇huh]|2 ,

and adding non-negative terms on the right-hand side,

14αξ2‖ph‖2h +

αξ2

4L2‖uh‖2h +

αξ2

4L2

d

dt‖∇huh]|2 +

+ ξ2 12d

dt‖∇hph]|2 +

d

dt(w0(ph), 1)h ≤

≤ 2β2L2

α2ξ2C(Ω)

αξ2

4L2‖∇huh]|2 + ξ2 1

2‖∇hph]|2 + (w0(ph), 1)h

. (39.ben)

24 Michal Benes

Integrating over (0, t), we haveαξ2

4L2‖∇huh]|2 + ξ2 1

2‖∇hph]|2 + (w0(ph), 1)h

(t) ≤αξ2

4L‖∇huh]|2 + ξ2 1

2‖∇hph]|2 + (w0(ph), 1)h

(0) exp

2β2L2

α2ξ2C(Ω)t

,

(40.ben)

which implies∇huh, ∇hph ∈ L∞(0, T ; l2(ωh)) ,

ph ∈ L∞(0, T ; l4(ωh)) .

Integrating the preceding result over (0, T ) again, we get∫ T

0

αξ2

4L2‖∇huh]|2 + ξ2 1

2‖∇hph]|2 + (w0(ph), 1)h

(t)dt ≤

≤αξ2

4L‖∇huh]|2 + ξ2 1

2‖∇hph]|2 +

+ (w0(ph), 1)h

(0)1

2β2L2

α2ξ2 C(Ω)

exp(2β2L2

α2ξ2C(Ω)T

)− 1

, (41.ben)

which implies∇huh, ∇hph ∈ L2(0, T ; l2(ωh)) ,

ph ∈ L2(0, T ; l4(ωh)) .

Extending these results into the continuum of Ω, we see that ∇Qh(Php0) and∇Qh(Phu0) are bounded in L2(Ω) (by (31.ben)), and Sh(Php0) is bounded in L4(Ω)(by (30.ben)). Therefore

∇Qhuh,∇Qhph ∈ L∞(0, T ; L2(Ω)) ,

Shph ∈ L∞(0, T ; L4(Ω)) ,

from which,∇Qhuh,∇Qhph ∈ L2(0, T ; L2(Ω)) ,

Shph ∈ L2(0, T ; L4(Ω)) ,

are bounded independently on h. Moreover, we obtain that

Shuh,Shph ∈ L2(0, T ; L2(Ω)) ,

are bounded independently on h as follows from (39.ben). We conclude that

Qhuh,Qhph ∈ L∞(0, T ; H10(Ω)) ,

Qhuh,Qhph ∈ L2(0, T ; H10(Ω)) ,

are bounded independently on h. According to (28.ben),

Qhuh,Qhph ∈ L2(0, T ; L2(Ω)) .

Passing to a subsequence, we have


– Qhnuhn ,Qhnphn ∗ u, p in L∞(0, T ; H10(Ω));

– Qhnuhn ,Qhnphn u, p in L2(0, T ; H10(Ω));

– Shn phn ,Qhn phn ∂tu, ∂tp in L2(0, T ; H−1(Ω));– Shn uhn ,Qhn uhn ∂tu, ∂tp in L2(0, T ; H−1(Ω));– Shnuhn ,Shnphn u, p in L2(0, T ; L2(Ω)).

The non-linear terms in the equation (1.ben) require stronger convergence result.Using the lemma on the compact imbedding, we conclude that Qhnphn convergesstrongly in L2(0, T ; L2(Ω)). Relation (29.ben) implies the same result for Shnphn .Denote their common limit as p and the weak limit of Shn phn in L2(0, T ; L2(Ω))as q1. The estimate

‖f0(Shph)‖L4/3(Ω) ≤

≤ a[1

2‖Shph‖L4/3(Ω) +

32‖(Shph)2‖L4/3(Ω) + ‖(Shph)3‖L4/3(Ω)

]=

= a[1

2‖Shph‖L4/3(Ω) +

32‖Shph‖2L8/3(Ω) + ‖Shph‖3L4(Ω)

], (42.ben)

justifies the existence of weak limit of f0(Shnphn) in L2(0, T ; L4/3(Ω)) denotedby q2 (dual space).

These limits exist as a consequence of the a priori estimate and of (29.ben), (28.ben).We prove that q1 = ∂tp, q2 = f0(p). First relation is implied by the uniquenessof the limit in D′(0, T ), as∫ T

0

(Shn phn −Qhn phn , q)ψ(t)dt = −∫ T

0

(Shnphn −Qhnphn , q)ψ(t)dt ,

where q ∈ D(Ω), ψ ∈ D(0, T ). The remaining equality is proven in the followinglemma.

Lemma 9. If p denotes the weak limit of Shnphn in L2(0, T ; L2(Ω)), then

f0(Shnphn)→ f0(p) weakly in L 43(0, T ; L 4

3(Ω)) .

Proof. According to the compact imbedding, we have that Shnphn convergesstrongly in L2(0, T ; L2(Ω)) and it can be considered to converge a.e. in this space(see [9]). Furthermore, we observe that as Shnphn was bounded in L∞(0,T ;L4(Ω))(see (42.ben), f0(Shnphn) is bounded in L∞(0, T ; L 4

3(Ω)). These two facts together

with the Aubin lemma [2] give the final result. ut

Before proceeding in the proof, we show more about regularity of p.

Lemma 10. Under the assumptions of the theorem, the function p belongs toH1

0(Ω) ∩H2(Ω).

Proof. Multiply the equation of phase by a function Phnq, where q ∈ D(Ω).

αξ2(phn ,Phnq)h + ξ2(∇hnphn , ∇hnPhnq] =

= (f0(phn),Phnq)h − βξ(uhn ,Phnq)h . (43.ben)

26 Michal Benes

In terms of L2(Ω), this means that

αξ2(Shn phn ,Shn(Phnq)) + ξ2(∇(Qhnphn),∇Qhn(Phnq)) =

= (f0(Shnphn),Shn(Phnq))− βξ(Shnuhn ,ShnPhnq) . (44.ben)

According to (32.ben), we realize that Qhn(Phnq)n→∞→ q in H1

0(Ω), and similarlyShn(Phnq)

n→∞→ q in L2(Ω) (see (30.ben)). We can pass to the limit in the sense ofD′(0, T ) obtaining

αξ2(∂tp, q) + ξ2(∇p,∇q) = (q2, q)− βξ(u, q) . (45.ben)

Consequently, the function p is continuous from 〈0, T 〉 into L2(Ω). We rewritethe previous equality in the sense of D′(Ω),

αξ2∂tp = ξ2∆p + q2 − βξu . (46.ben)

Note that q2 = f0(p) and p ∈ L∞(0, T,Ls(Ω)) for any s > 1. Consequently,q2 ∈ L2(0, T,L2(Ω)). As ∂tp, q2 belong to L2(Ω), this means that ∆p ∈ L2(Ω)for each t ∈ (0, T ). Consequently, we find that p must be in the domain of ∆ —see [11], [2]:

p(t) ∈ D(∆) = H2(Ω) ∩H10(Ω) for t ∈ (0, T ) .

ut

Next statement investigates the convergence of gradient.

Lemma 11. The sequence ∇Qhnphn converges strongly to ∇p in L2((0, T )×Ω).

Proof. Following the technique of [12], the statement of the lemma is shown.Multiply the equation of phase in (35.ben) by phn − Phnp and sum over ωh.

αξ2(phn , phn − Phnp)h + ξ2(∇hnphn , ∇hn(phn − Phnp)] =

= (f0(phn), phn − Phnp)h − βξ(uhn , phn − Phnp)h . (47.ben)

Rewrite this equality in terms of L2(Ω), and integrate over (0, T ).

αξ2

∫ T

0

(Shn phn ,Shn(phn − Phnp))dt+

+ ξ2

∫ T

0

(∇(Qhnphn),∇Qhn(phn − Phnp))dt =

=∫ T

0

(f0(Shnphn),Shn(phn − Phnp))dt−

− βξ∫ T

0

(uhn ,Shn(phn − Phnp))dt. (48.ben)


As we have shown that p ∈ L2(0, T ; H2(Ω)) satisfies (45.ben), it means that p(t) ∈C0,1(Ω), t ∈ (0, T ), and consequently, Shn(Phnp) → p, and ∇Qhn(Phnp) → ∇pin L2(0, T ; L2(Ω)) (see (30.ben), (31.ben)). We add and subtract a term

ξ2

∫ T

0

(∇(Qhn(Phnp)),∇Qhn(phn − Phnp))dt

to the equality (48.ben) knowing that it tends to 0 as

∇Qhn(phn − Phnp)→ 0 ,

weakly in L2(0, T ; L2(Ω)), if n→∞. Then, we have

ξ2

∫ T

0

(∇(Qhnphn − Phnp),∇Qhn(phn − Phnp))dt =

=− αξ2

∫ T

0

(Shn phn ,Shn(phn − Phnp))dt+

+∫ T

0

(f0(Shnphn),Shn(phn − Phnp))dt+

− βξ∫ T

0

(uhn ,Shn(phn − Phnp))dt+

+ ξ2

∫ T

0

(∇(Qhn(Phnp)),∇Qhn(phn − Phnp))dt . (49.ben)

As all terms in the right hand side tend to 0 if n→∞, we see that ∇(Qhn(phn−Phnp))→ 0 in L2(0, T ; L2(Ω)), which together with (32.ben) gives the desired result.

ut

Passage to the limit. Take the system of (35.ben) into the consideration, mul-tiply by test functions Phnw, Phnq where w, q ∈ D(Ω). Integrate it over ωh.Then, we have, in terms of L2(Ω),

(Shn uhn ,ShnPhnw) + (∇Qhnuhn ,∇QhnPhnw) = L(Shn phn ,ShnPhnw) ,

αξ2(Shn phn ,ShnPhnq) + ξ2(∇Qhnphn ,∇QhnPhnq) =

= (f0(Shnphn),ShnPhnq)− βξ(Shnuhn ,ShnPhnq) . (50.ben)

Knowing that

1. Shn phn , Shn uhn converge weakly in L2(0, T ; L2(Ω)) to ∂tp, ∂tu;2. ∇Qhnphn , ∇Qhnuhn converge strongly in L2(0, T ; L2(Ω)) to ∇p, ∇u;3. ShnPhnp0, ShnPhnu0 converges strongly to p0, u0 in H1

0(Ω),

multiply (50.ben) by a scalar function ψ(t) ∈ C1〈0, T 〉, for which ψ(T ) = 0. Weintegrate by parts. Taking into account all previous results, the fact that

Shnphn(0) = ShnPhnp0, Shnuhn(0) = ShnPhnu0

28 Michal Benes

and the Lebesgue theorem, we are able to pass to the limit.

(u0 − Lp0, w)ψ(0) −∫ T

0

(u− Lp,w)ψdt+∫ T

0

ψ[(∇u,∇w) = 0 ,

αξ2(p0, q)ψ(0)−∫ T

0

αξ2(p, q)ψdt+∫ T

0

ψ[ξ2(∇p,∇q)−

− (f0(p), q) + βξ(u, q)]dt = 0 . (51.ben)

If ψ ∈ D(0, T ), we have

d

dt(u − Lp,w) + (∇u,∇w) = 0 ,

αξ2 d

dt(p, q) + ξ2(∇p,∇q) = (f0(p), q)− βξ(u, q) .

(52.ben)

It remains to show that the weak solution satisfies the initial condition. Mul-tiplying (51.ben) by a scalar function ψ(t) ∈ C1〈0, T 〉, for which ψ(T ) = 0, andintegrating by parts, we obtain

(u(0)− Lp(0), w)ψ(0)−∫ T

0

(u− Lp,w)ψdt+∫ T

0

ψ[(∇u,∇w) = 0 ,

αξ2(p(0), q)ψ(0)−∫ T

0

αξ2(p, q)ψdt+

+∫ T

0

ψ[ξ2(∇p,∇q)− (f0(p), q) + βξ(u, q)]dt = 0 . (53.ben)

Subtracting this equation from (51.ben), we get

(u0 − Lp0 − u(0) + Lp(0), w)ψ(0) = 0, (p0 − p(0), q)ψ(0) = 0 .

From this we see that u(0) = u0, p(0) = p0 in L2(Ω). To prove uniqueness, con-sider two solutions of the problem (34.ben), denoted as [u, p] and [v, q]. Subtractingtwo systems of equations and denoting [w, r] = [u − v, p − q], multiplying thefirst equation by w and the second equation by r via the semi-discrete scheme,we have

12d

dt‖w‖2 + (∇w,∇w) = (r, w) in (0, T ) , (54.ben)

w(0) = 0 ,

αξ2‖r‖2 + ξ2 12d

dt(∇r,∇r) = (f0(p)− f0(q), r)− βξ(w, r) in (0, T ) ,

r(0) = 0 . (55.ben)

Denote Ψ(p, q) = − 12a+ 3

2a(p+q)−a(p2+pq+q2). The existence proof guaranteesthat there is a constant C such that

‖Ψ(p, q)‖L4(Ω) ≤ C ,


(as implied by the continuous imbedding H10(Ω) ⊂> Lq(Ω) for q ∈< 0,+∞)).

Therefore, we have

|(Ψ(p, q)r, r)| ≤ ‖Ψ(p, q)‖L4(Ω)‖r‖L4(Ω)‖r‖L2(Ω) ≤ C‖r‖L4(Ω)‖r‖L2(Ω) .

Using the Poincare and Schwarz inequalities, we get

12d

dt‖w‖2 ≤ C(Ω)

4‖r‖2

αξ2‖r‖2 + ξ2 12d

dt‖∇r‖2 ≤ C‖r‖L4(Ω)‖r‖L2(Ω)+

+12αξ2‖r‖2 +

β

2αξ‖w‖2 , (56.ben)

or, considering the fact, that there is a constant C4 > 0 such that

‖r‖L4(Ω) ≤ C4‖∇r‖ ,

we obtain

12d

dt‖w‖2 ≤ C(Ω)

4‖r‖2

αξ2‖r‖2 + ξ2 12d

dt‖∇r‖2 ≤ 1

4αξ2‖r‖2 +

C2

αξ2C2

4‖∇r‖2+

+12αξ2‖r‖2 +

β

2αξ‖w‖2 . (57.ben)

Combining these inequalities, we have

1C(Ω)

αξ2 12d

dt‖w‖2 + ξ2 1

2d

dt‖∇r‖2 ≤ C2

αξ2C2

4‖∇r‖2 +β

2αξ‖w‖2 .

Such inequality implies, together with the initial conditions, that

r(t) = w(t) = 0 ∀t ∈ (0, T ) in L2(Ω) .

ut

4 Convergence towards the sharp interface model

This paragraph deals with the relation of the phase-field model to a sharp-interface formulation of the Stefan problem. It uses estimates derived above toshow certain compactness statements leading to the existence of a step functiondefining the position of solid domain in time. Consider the weak formulation ofthe standard phase-field model.

d

dt(u, v) + (∇u,∇v) = L

d

dt(p, v) in (0, T ) , (58.ben)

u(0) = u0 ,

αξ2 d

dt(p, q) + ξ2(∇p,∇q) = (f0(p), q)− βξ(u, q) in (0, T ) ,

p(0) = p0 . (59.ben)

30 Michal Benes

Main purpose of next investigation will be the dependence on ξ. Consider thesolution of the semidiscrete scheme (35.ben). We multiply the first equation by uh

and the second one by ph.

12d

dt‖uh‖2h + ‖∇huh]|2 = L(ph, uh)h , (60.ben)

αξ2‖ph‖2h + ξ2 12d

dt‖∇hph]|2 = − d

dt(w0(ph), 1)h − βξ(uh, ph)h . (61.ben)

Combining previous equalities, we get

αξ2‖ph‖2h + ξ2 12d

dt‖∇hph]|2 =

= − d

dt(w0(ph), 1)h −

βξ

L

12d

dt‖uh‖2h + ‖∇huh]|2

, (62.ben)

or

αξ2‖ph‖2h + ξ2 12d

dt‖∇hph]|2 +

d

dt(w0(ph), 1)h +

βξ

L

12d

dt‖uh‖2h = 0.

We integrate over (0, t), which gives

ξ2 1

2‖∇hph]|2 + (w0(ph), 1)h +

βξ

L

12‖uh‖2h

(t) ≤

≤ξ2 1

2‖∇hph]|2 + (w0(ph), 1)h +

βξ

L

12‖uh‖2h

(0) . (63.ben)

Passing to the limit, if h→ 0, which is justified by the proof of the Theorem 8,we get

12β

L‖uξ(t)‖2 + Eξ[pξ](t) ≤

12β

L‖uξ(0)‖2 + Eξ[pξ](0) t ∈ (0, T ) , (64.ben)

where we denoted (ph h→0−−−→ pξ),

Eξ[pξ](t) =∫Ω

[ξ12|∇pξ|2E +

1ξw0(pξ)]dx .

Additionally, there is an estimate for the time derivative, if we integrate (62.ben)over (0, T ) and pass to the limit for h→ 0.

αξ

∫ T

0

‖∂tpξ‖2dt+ Eξ[pξ](T )− Eξ[pξ](0) +β

2L(‖u(T )‖2 − ‖u(0)‖2) = 0 (65.ben)

Consequently, there is a constant C1 such that

12αξ

∫ T

0

‖∂tp‖2hdt+ Eξ[pξ](T ) ≤ Eξ[pξ](0) + C1 . (66.ben)


These estimates allow to use the method proposed by [4] and used in [2]. Definethe following monotone function

G(s) =∫ s

0

|1− (1− 2r)2|dr . (67.ben)

We prove next lemma

Lemma 12. Be pξ the solution of (34.ben) where Eξ[pξ](0) ≤M0 independently onξ. Then there are constants M > 0 and M1 > 0 such that

sup∫Ω

|∇G(pξ)|dx | t ∈ 〈0, T 〉 ≤M (68.ben)

and, for 0 ≤ t1 < t2,∫ t2

t1

∫Ω

|∂tG(pξ)|dxdt ≤M1(t2 − t1)0.5 . (69.ben)

Proof. We have shown that

Eξ[pξ](t) ≤M0 + C1 ,

on 〈0, T 〉. We write

Eξ[p](t) =∫Ω

[ξ12|∇p|2E +

1ξw0(p)]dx ≥

≥∫Ω

√2|∇pξ|

√w0(pξ)dx =

√2∫Ω

|∇G(pξ)|Edx , (70.ben)

which shows (68.ben) by setting M = 1√2(M0 + C1). Furthermore, if

∫ t2

t1

dt

∫Ω

dx|∂tG(pξ)| =∫ t2

t1

dt

∫Ω

dx|pξ||G′(pξ)| ≤

≤(∫ t2

t1

dt

∫Ω

dx|pξ|2) 1

2(∫ t2

t1

dt

∫Ω

dx|G′(pξ)|2) 1

2

≤

≤ (2α

(C1 +M0)2)12 (t2 − t1)

12 , (71.ben)

then (69.ben) is shown, if setting M1 =√

2α (C1 +M0). ut

The previous statement leads to the existence of a step function as expected.

Theorem 13. Let uξ, pξ is the solution of the problem (34.ben) with the initial datasatisfying Eξ[pξ](0) < M0 and uξ, pξ ∈ H2(Ω) ∩ H1(Ω), and let∫

Ω

|pξ(0,x)− v0(x)|dx→ 0, ‖uξ(0)‖L2(Ω) ≤ C2 ,

32 Michal Benes

as ξ → 0, for a function v0 ∈ L1(Ω). Then for any sequence tending to 0 thereis a subsequence ξn′ such that

limξn′→0

pξn′ (t,x) = v(t,x), uξ′n(t,x) u(t,x) in L2((0, T )×Ω),

and u, v are defined a.e. in (0, T )× Ω. The function v reaches values 0 and 1,and satisfies ∫

Ω

|v(t1,x)− v(t2,x)|dx ≤ C|t2 − t1|12 ,

where C > 0 is a constant, and

supt∈〈0,T 〉

∫Ω

|∇v|Edx ≤ C1 ,

in the sense of BV (Ω), where C1 > 0 is a constant. The initial condition is

limt→0

v(t,x) = v0(x) ,

a.e.

Proof. The proof follows steps presented in [4]. We find that

G(s) = 2s2 − 43s3 for s ∈ 〈0, 1〉 ,

G(s) =43s3 − 2s2 +

43

for s ∈ (1,+∞) .

Consequently, a direct computation justifies that

|G(s)| ≤ 43

+ [1− (1− 2s)2]2 .

Then, we are able to obtain the upper bounds for the function G and its spa-tiotemporal gradient. According to (64.ben), we have∫ T

0

∫Ω

w0(p)dxdt ≤M2ξ . (72.ben)

Putting (72.ben), (68.ben) and (69.ben) together, we conclude, thatG(pξ) is in BV ((0, T )×Ω)regardless the value ξ > 0. Following [6],

BV ((0, T )×Ω) ⊂>⊂> L1((0, T )×Ω) .

Consequently, there is a sequence G(pξn) converging to an element G∗ in thespace L1((0, T ) × Ω). According to [9], there is a further subsequence G(pξn′ )converging to G∗ almost everywhere in (0, T )×Ω.

The function G :< 0,+∞)→< 0,+∞) is monotone, which implies existenceof the unique function v such that

G∗ = G(v) ,


andpξn′ → v a.e. in (0, T )×Ω

According to (72.ben) and by the Fatou lemma, we obtain∫ T

0

∫Ω

w0(v)dxdt = 0 , (73.ben)

from which follows that the function p takes only the values 0, 1.Now, we prove that G is Holder-continuous in the time variable. The function

pξn′ satisfies

|G(pξn′ (t1,x))−G(pξn′ (t2,x))| ≤∫ t2

t1

|∂tG(pξn′ (t,x))dt ,

for 0 ≤ t1 ≤ t2 ≤ T . Integrating over Ω,∫Ω

|G(pξn′ (t1,x))−G(pξn′ (t2,x))|dx ≤M1|t1 − t2|0.5 , (74.ben)

according to the Lemma 12. Passing to the limit for n′ →∞, we find∫Ω

|G∗(t1,x)−G∗(t2,x)|dx ≤M1|t1 − t2|0.5 ,

for almost all t1, t2 ∈ (0, T ). The statement of theorem is obtained by the factthat

|G∗(t1,x)−G∗(t2,x)| = (G(1) −G(0))|v(t1,x)− v(t2,x)| .

The a.e. argument makes from the function v a continuous map from 〈0, T 〉 toL1(Ω). Taking t1 = 0 in (74.ben) and according to the assumption∫

Ω

|pξ(0,x)− v0(x)|dx→ 0 ,

as ξ → 0, (similarly for G-valued function) we have∫Ω

|G(v0(x)) −G(v(t2,x))|dx ≤M1t0.52 ,

from which ∫Ω

|v0(x) − v(t2,x)|dx ≤ M1

G(1)−G(0)t0.52 .

This concludes the proof. It remains to show the boundedness of the total varia-tion of v ([6]). The lower semicontinuity of the total variation in L1-space togetherwith the Lemma 12 yields

ess sup0<t<T

∫Ω×t

|∇G∗|dx ≤M0 ,

34 Michal Benes

and by the continuity of v in time, we have

sup0<t<T

∫Ω×t

|∇v|dx ≤ M0

G(1)−G(0).

It remains to show the convergence of uξ. The relation (64.ben) implies that uξ isbounded in L2((0, T )×Ω). Then, using the subsequence argument, uξn′ convergesweakly to an element u ∈ L2((0, T )×Ω). This completes the proof. ut

5 Conclusion

The purpose of the paper was to show the convergence property of the semi-discrete scheme based on the method of lines. The compactness technique al-lowed to prove existence and uniqueness of the original weak solution. The phasefunction depending on the small parameter ξ is bounded in the BV sense. Con-sequently, it converges to a step-wise function indicating different phases. Thetechnique of the recovery of the sharp-interface relation can also be applied tothe presented problem. The presented approach is applicable even in case ofdifferent modifications of the model.

Acknowledgement.The author was partially supported by the grant No. 3097486 of the CzechTechnical University.

References

[1] N. D. Alikakos and P. W. Bates. On the singular limit in a phase field model ofphase transitions. Ann. Inst. Henri Poincare, 5:141–178, 1988.

[2] M. Benes. Phase-Field Model of Microstructure Growth in Solidification of PureSubstances. PhD thesis, Faculty of Nuclear Sciences and Physical Engineering,Czech Technical University, 1997.

[3] D. Brochet, X. Chen, and D. Hilhorst. Finite dimensional exponential attractorfor the phase-field model. J. Appl. Anal., 49:197–212, 1993.

[4] L. Bronsard and R. Kohn. Motion by mean curvature as the singular limit ofGinzburg-Landau dynamics. J. Differential Equations, 90:211–237, 1991.

[5] G. Caginalp. An analysis of a phase field model of a free boundary. Arch. RationalMech. Anal., 92:205–245, 1986.

[6] E. Giusti. Minimal Surfaces and Functions of Bounded Variation. Birkhauser,Basel, 1984.

[7] M. Gurtin. On the two-phase Stefan problem with interfacial energy and entropy.Arch. Rational Mech. Anal., 96:200–240, 1986.

[8] R. Kobayashi. Modeling and numerical simulations of dendritic crystal growth.Physica D, 63:410–423, 1993.

[9] A. Kufner, O. John, and S. Fucık. Function Spaces. Academia, Prague, 1977.[10] P. Laurencot. Solutions to a Penrose-Fife model of phase-field type. J. Math.

Anal. Appl., 185:262–274, 1994.


[11] J. L. Lions. Quelques Methodes aux Resolution des Problemes Nonlineaires. DunodGauthiers-Villars, Paris, 1969.

[12] K. Mikula and J. Kacur. Evolution of convex plane curves describing anisotropicmotions of phase interfaces. SIAM J. Scientific Computing, 17:1302–1327, 1996.

[13] O. Penrose and P. C. Fife. Thermodynamically consistent models of phase-fieldtype for the kinetics of phase transitions. Physica D, 43:44–62, 1990.

[14] P. A. Raviart. Sur la resolution de certaines equations paraboliques non lineaires.J. Func. Anal., 5:299–328, 1970.

[15] A. A. Samarskii. Theory of Difference Schemes. Nauka, Moscow, 1977.[16] A. A. Wheeler, B. T. Murray, and R. J. Schaefer. Computation of dendrites using

a phase field model. Physica D, 66:243–262, 1993.



Summation of Polyparametrical Functional

Series by the Method of Finite Hybrid IntegralTransforms (Fourier, Bessel)

Andriy Blazhievskiy

Department of High Mathematics, Technological University of Podillia,11 Institutskaya st., Khmelnitsky, Ukraine

Email: [email protected],[email protected]

WWW: http://www.hirup.khmelnitskiy.ua

Abstract. The design of physical and mechanical characteristics of thinisotropic nonhomogeneous (partly-homogeneous) plates of limited sizeaccording to the degree low lead to the construction of solution for theseparate system of differential equations on the given initial conditions,boundary conditions and the conditions of thermomechanical contact inthe connecting plains. Let one end of the plates is under the actions ofthe spasmodic heat regime, or the heat sources act on the plane partaccording to the spasmodic law. The stationary state of system is de-scribed by the functions depending on functional series consisting of thecombination of the trigonometric and Bessel functions. Since we dealeasier with functions than with series we encounter with the problem offunction series summation. The article is devoted to the summation ofjust such series by the method of finite hybrid integral transforms Hankel2-Hankel 2-Fourier, Hankel 1-Hankel 2-Fourier, Fourier-Hankel 2-Hankel2, . . . [1].

By the Cauchy’s method for the separate system of the ordinary differ-ential equations we have constructed the solution of the correspondingboundary problem in the case of general assumption on the differen-tial and connected operators. The condition of the nonlimited solvingand the structure of the general solution for the boundary problem havebeen written in the explicit form. On the other hand solution of thisproblem has been constructed by the method of the finite hybrid inte-gral transforms. Since this problem has one and only one solution wemay compare the first solution with the second and, as a result, get thesums of functional series.

AMS Subject Classification. 40G99, 35K55

Keywords. hybrid integral integral transforms, polyparametrical func-tional series


http://www.hirup.khmelnitskiy.ua

38 Andriy Blazhievskiy

1 Introduction

There are many engineering problems, which occur in design and calculate ofstability for machine constructive elements, in the designing engineer structureand in the research of kinetic for physical and chemical processes. Since theconstructive elements is under action of instantaneous heat-stroke and after itare work in stationary state, one would like to know the value of stationary heatstrength. This problem is very impotent with respect to composite materials. Thestationary state of system is described by the functions depending on functionalseries consisting of the combination trigonometric and Bessel functions. Sincewe deal easier with functions than with series we encounter with the problem offunction series summation.

Let us describe our reasoning on the next heat problems.Problem 1. Let heat characteristics of thin isotropic plate

Π = r : r ∈ (0, R), R < +∞

are designed according to the continuous law. If one end r = 0 of the plate Πis under action of the spasmodic heat regime, other end r = R is under zerotemperature. The structure problem of non-stationary heat field in this platelead to mathematical construction in region

D = (t, r) : t > 0, r ∈ Π

of finite solution for heat equation

∂T

∂t+ χ2T − ∂2T

∂r2= 0, t > 0

on boundary conditions

T |r=0 = S+(t), T |r=R = 0

and zero initial condition. Note S+(t) =

1, t > 0,0, t ≤ 0.

Solution of this problem is

T (t, r) =2RS+(t)

∞∑n=1

λnλ2n + χ2

(1− 1−(λ2

n + χ2)t)

sinλnr.

The stationary state is described by function

TST. = limt→∞

T (t, r) =2R

∞∑n=1

λnλ2n + χ2

sinλnr ≡shχ(R − r)

shχR= S1(r, χ).

Notice that S1(r, χ) is finite on [0, R] solution of boundary problem(d2

dr2− χ2

)S1 = 0, S1|r=0 = 1, S1|r=R = 0.

Summation of Polyparametrical Functional Series 39

Problem 2. Let now heat characteristics of thin isotropic plate are designedaccording to the linear law. The structure problem of non-stationary heat field inthis plate lead to mathematical construction of finite solution for heat equation

∂T

∂t+ χ2T −

(∂2T

∂r2+

1r

∂

∂r

)T = 0, χ > 0, t > 0


∂T

∂r

∣∣∣∣r=0

, T |r=R = S+(t)

and zero initial condition. Solution of this problem is

T (t, r) =2RS+(t)

∞∑n=1

λnλ2n + χ2

(1− 1(λ2

n + χ2)t)J1(λnr)

[J20 (λnR) + J2

1 (λnR)].

The stationary state is described by function

TCT. =2R

∞∑n=1

λnλ2n + χ2

J1(λnR)J0(λnr)J2

0 (λnR) + J21 (λnR)

=I0(χr)I0(χR)

≡ S2(r, χ),

where Iν(x) – modify Bessel function of the first kind of order ν.Notice that S2(r, χ) is finite on [0, R] solution of the boundary problem(

d2

dr2+

1r

d

dr− χ2

)S2 = 0,

dS2

dr

∣∣∣∣r=0

, S2|r=R = 1.

Problem 3. Let us consider the finite thin plate

Π1 = r : r ∈ (0, R1) ∪ (R1, R2), R2 <∞

with continuous heat characteristics on the first part and linear heat character-istics on the second part. The structure problem of non-stationary heat field inthis plate lead to mathematical construction in region

D1 = (t, r) : t > 0, r ∈ Π1

of finite solution for separate system of equations

1a2

1

∂T1

∂t+ χ2

1T1 −∂2T1

∂r2= 0, t > 0, r ∈ (0, R1),

1a2

2

∂T2

∂t+ χ2

2T2 −(∂2

∂r2+

1r

∂

∂r

)T2 = 0, χ2 > 0, t > 0, r ∈ (R1, R2)

on zero initial condition, boundary conditions

T1|r=0 = S+(t),∂T2

∂r

∣∣∣∣r=R2

= 0

40 Andriy Blazhievskiy

and conditions of non-ideal heat contact[(R0

∂

∂r+ 1)T1 − T2

] ∣∣∣∣r=R1

= 0,(∂T1

∂r− γ1

∂T2

∂r

) ∣∣∣∣r=R1

= 0.

Solution of this problem is

T1(t, r) = S+(t)∞∑n=1

β1nβ2n2ω2

1(λn) sinβ1nr

(λ2n + a2

2χ22)‖V (r, λn)‖2

(1− 1−(λ2

n + a22χ

22)t),

T2(t, r) = S+(t)γ1

R1

∞∑n=1

β1nβ2n2ω1(λn)

(λ2n + a2

2χ22)‖V (r, λn)‖2

(1− 1−(λ2

n + a22χ

22)t)×

× [ω3(λn)N0(β2nr) − ω2(λn)J0(β2nr)].

Notice that β1n =√λ2n + a2

2χ22 − a2

1χ21, β2n = λn, βjn = a−1

j βjn, j = 1, 2,

ω1(λn) = J1(β2nR2)N0(β2nR1)−N1(β2nR2)J0(β2nR1),

ω2(λn) = (R0β1n cosβ1nR1 + sinβ1nR1)N1(β2nR2),

ω3(λn) = (R0β1nR1 + sinβ1nR1)J1(β2nR2).

If (a21χ

21 − a2

2χ22) ≥ 0 then β1n = λn, β2n =

√λ2n + a2

1χ21 − a2

2χ22. In this case

T1,CT.(t, r) =∞∑n=1

β1nβ2n2ω2

1(λn) sinβ1nr

(λ2n + a2

2χ22)‖V (r, λn)‖2 = chχ1r +

∆1(χ∆(χ)

shχ1r

χ1= S3(r, χ),

T2,CT.(t, r) =γ1

R1

∞∑n=1

β1nβ2n2ω1(λn)[ω3(λn)N0(β2nr) − ω2(λn)J0(β2nr)]

(λ2n + a2

2χ22)|V (r, λn)‖2 =

= − χ2

∆(χ)[K1(χ2R2)I0(χ2r) + I1(χ2R2)K0(χ2r)]≡S4(r, χ),

χ=χ1, χ2, χj=ajχj ,

∆(χ)=γ1χ22

(R0chχ1R1+

shχ1R1

χ1

)(I1(χ2R1)K1(χ2R2)−I1(χ2R2)K1(χ2R1)−

− χ2chχ1R1 (I0(χ2R1)K1(χ2R2) + I1(χ2R2)K0(χ2R1)) ,

∆1(χ) = −ν1χ22(R0χ1shχ1R1 + chχ1R1)(I1(χ2R1)K1(χ2R2)− I1(χ2R2)×

×K1(χ1R1)) + χ1χ2chχ1R1(I0(χ2R1)K1(χ2R2)− I1(χ2R2)K0(χ2R1)),

‖V (r, λn)‖2 =ω2

1(λn)2

(R21 −

sin 2β1nR1

β1n

+2

(πβ1n)2− R2

1

2ω4(λn),

ω4(λn) = ω22(λn)(J0(β2nR1) + J1(β2nR1)) − 2ω2(λn)ω3(λn)[J0(β2nR2)×

×N0(β2nR1) + J1(β2nR1)N1(β2nR1)] + ω23(λn)[N2

0 (β2nR1) +N21 (β2nR1)].

Summation of Polyparametrical Functional Series 41

Notice that S3(r, χ) and S4(r, χ) is finite on Π1 solution of the boundaryproblem (

d2

dr2− χ1

2

)S3(r, χ) = 0, r ∈ (0, R1),(

d2

dr2+

1r

d

dr− χ2

2

)S4(r, χ) = 0, χ2 > 0, r ∈ (R1, R2)


S3|r=0 = 1,dS4

dr

∣∣∣∣r=R2

= 0

and contact conditions[(R0

d

dr+ 1)S3 − S4

]∣∣∣∣r=R1

= 0,(dS3

dr− γ1

dS4

dr

)∣∣∣∣r=R1

= 0.

My research is devoted to the summation of just such series by the methodof finite hybrid integral transforms.

By the Cauchy’s method for the separate systems of the ordinary differen-tial equations we have constructed the solution of the corresponding boundaryproblem in the case of general assumption on the differential and connected op-erators. The condition of the non-limited solving and the structure of the generalsolution for the boundary problem have been written in the explicit form. Onthe other hand solution of this problems has been constructed by the method ofthe finite hybrid integral transforms. Since this problems has one and only onesolution we may compare the first solution with the second and, as a result, getthe sums of functional series.

References

[1] A. Blazhievskiy, Summation of polyparametrical functional series by the methodof finite hybrid integral transforms, Thesis for a degree of Doctor of Philosophy(Ph.D.) in Physics and Mathematics, Chernivtsy, Ukraine 1996.



System of Differential Equations with Unstable

Turning Point and Multiple Elementof Spectrum of Degenerate Operator

V. N. Bobochko1 and I. I. Markush2

1 St. Gerojev Stalingrada, 12-1-101, 316 031 Kirovograd, UkraineEmail: [email protected]

2 Square Theatralnaya 17/7, 294000 Uzhgorod, UkraineEmail: [email protected]

Abstract. A uniform asymptotics of a solution is constructed for a sys-tem of singularly perturbed differential equations with a turning point.The paper investigates the case when the spectrum of a phase operatorconsists of multiple elements.

AMS Subject Classification. 34B05, 34E05, 34E20

Keywords. Operator, spectrum, turning point, differential equation,parameter, system, singular solution, manifolds, asymptotic

1 Introduction

The systems of singular perturbation differential equations (SSPDE) play thegreat role in many mathematical models of biological problems and in medicine.This can be seen from monograph [1] in which kinetic rule of Michaelis-Menten isdescribed by Tikhonov SSPDE (see equations (1.20), (1.21)). One of the knownproblems for investigation such problems is as follows: Is there a biological pro-cess stable or not? If the spectrum of the degenerate operator is stable, thensuch a system is known pretty well. Remembering classical results (Vasilieva’smethod, Lomov’s method and others) which give the answer of the discussedproblems concerning bounded solutions of the adequate SSPDE. If the systemcontains turning points, i.e. some elements of the spectrum of the degenerateoperator are unstable, then the general theory of investigating such problemsis not constructed yet, however some special problems have the solution (see[2,3,4]).

In the presented article we consider the problem:

LεW (x, ε) ≡ ε2W ′′(x, ε)−AW (x, ε) = h(x) ,

E1W (m, ε) = E1(µ−2αm. + Wm) ,

E2dW (m, ε)

dx= E2(µ−4αm. + µ−3Wm) ,

(1.mar)


44 V. N. Bobochko and I. I. Markush

whereε→ +0, x ∈ I = [0, 1], m = 0, 1, µ = 3

√ε .

Here A denotes a linear operator on Rn, αm and Wm are given vectors, Ek(k = 1, 2) — diagonal matrices of the nth order of the formE1 = diag1, 0, . . . , 0,E2 = diag0, 1, . . . , 1), h(x) — given vector-function, W (x, ε) — a sought vector-function.

We shall consider the problem (1.mar) under the following conditions:

1 A(x), h(x) ∈ C∞[I]2 Spectrum of the degenerate operator A is real and fulfills the following con-

dition

0 ≤ λ1(x) ≡ xλ1(x) < λ2(x) < · · · < λp.(x) ≡ · · · ≡ λn(x), (2.mar)

where λ1(x) > 0 for all x ∈ I.

One can see from (2.mar) that the point x = 0 is a turning point for equation (1.mar),and moreover of λ1(x) ≥ 0, then it is unstable turning point. Thus it is necessaryto construct a solution of (1.mar) in the case when the simplified equation

−Aω(x) = h(x) (3.mar)

has, in general case, a point of discontinuity at 0. System (1.mar) with several con-ditions for the spectrum of degenerate operator A was consider in [2,3,4,5] andin other articles of authors.

In the present article we shall prove, that ignoring the unstability of theturning point some partial solutions of the vector equation (1.mar) can be boundedin the domain.

2 Extension of the perturbation problem

One of the main problem of asymptotics of solution of singular perturbed prob-lem (SPP) (1.mar) is: choice, description and conserve them as a whole all essentiallysingular manifolds (ESM) contained in the solutions SPP (1.mar).For this purpose, together with the independent variable x, we shall considervector-variable

t = tik , i = 1, p , k = 1, 2 according to

t1k ≡ t1 = µ−2

(32

∫ x

0

√λ1(x)dx

) 23

≡ µ−2ϕ1(x) ≡ φ1(x, ε),

tjk = µ−3(−1)k∫ x

(k−1)l

√λj(x)dx ≡ µ−3ϕjk(x) ≡ φjk(x, ε),

k = 1, 2, j = 2, p

(4.mar)

System of Differential Equations 45

Then instead of vector-function W (x, ε) we shall consider a new “extended” vec-tor-function W (x, t, ε), where in view of the regularization method (see [5]), theextension is taken in such a way that

W (x, t, ε)∣∣∣t=φ(x,ε)

≡W (x, ε), (5.mar)

where

φ(x, ε) = φ1(x, ε), φjk(x, e), k = 1, 2; j = 2, p .

Differentiating twice the identity (5.mar) and substituting the derivative of thesecond order of W (x, t, ε) to equation (1.mar), we obtain, for the extension W (x, t, ε)the following “extended” problem:

LεW (x, t, ε) = h(x) , Mm = (m, t(m)) ,

E1W (Mm, ε) = E1(µ−2αm + Wm) ,

E2dW (Mm, ε)

dx= E2(µ−4αm + µ−3Wm) .

(6.mar)

Here

Lε ≡ µ2ϕ′21 (x)∂2

∂t21+ µ4d1

∂

∂t1+ µ6 ∂

2

∂x2−

−A+2∑

k=1

p∑j=2

[ϕ′2jk(x)∂

∂tjk+ µ3djk]

∂

∂tjk+ Y 1

ε , (7.mar)

where

djk ≡ 2ϕ′jk(x)∂

∂x+ ϕ′′jk(x) . (8.mar)

Y 1ε — operator that plays the role of annihilator. Then there is no idea to

give its full form.

3 Spaces of nonresonance solutions

We shall describe the sets (subspaces) of functions in which we shall solve theextended problem (6.mar). We have

Yrilk = bi(x)[Vrik(x)Uk(t1) +Qrik(x)U ′k(t1)] ,Yrijk = bi(x)αrijk(x) exp(tjk), j = 2, p ,Vri = bi(x)[fri(x)ν(t1) + gri(x)ν′(t1)] ,Xri = bi(x)ωri(x) , i = 1, n; k = 1, 2,

(9.mar)


where the coefficients in ESM and the functions ωri(x) are arbitrary sufficientlysmooth functions for x in I, and bi(x) (i = 1, n) — a complete system of propervectors for proper values λi(x) (i = 1, p). Since the operator A is operator ofa simple structure, then for a multiple proper value λp(x) ≡ · · · ≡ λn(x) thereexists a system of linearly independent vectors (fundamental solutions) bi(x)(i = p, n).

By b∗i (x) (i = 1, n) we shall denote the complete system of proper vectorsof the conjugate operator A∗, where those vectors are chosen in such a waythat together the vectors bi(x) they form a biorthogonal system of vectors. Theexistence of such a system is proved since A is an operator of a simple structure(see [6, p. 218]).

Later on U1(t1) ≡ Ai(t1), U2(t1) ≡ Bi(t1) — are the Eiry-functions, whichproperties are described in monograph [7, chapter 1.1].

Essentially singular manifolds ν(t1) ≡ −Gi(t1) — are Skorera functions (see[7, p. 412]).

From the subspaces (9.mar) we shall construct the space of the form:

Yr =n⊕i=1

Yri ≡n⊕i=1

[ 2⊕k=1

p⊕j=1

Yrijk ⊕ Vri ⊕Xri

], (10.mar)

which in view of the known terminology [5] will be called the space of nonreso-nance solutions (SNS).

The element Wr(x, t) ∈ Yr has the form

Wr(x, t) =n∑i=1

bi(x)Wri(x, t) ≡n∑i=1

Wri(x, t) , (11.mar)

where

Wri(x, t) ≡2∑k=1

[Vrik(x)Uk(t1) +Qrik(x)U ′k(t1) +

+p∑j=2

αrijk(x) exp(ttj)]

+ fri(x)ν(t1) + gri(x)ν′(t1) + ωri(x) .

4 Regularization of the singular perturbed problem

We have to find the properties of the extended operator Lε on elements fromSNS (10.mar). Method of obtaining such procedure is described in articles [2,3,4,5].That is why we shall write only the final result of that property. We have:

LεWr(x, t) ≡ [R0 + µ2R2 + µ3R3 + µ4R4 + µ6R6]Wr(x, t) . (12.mar)


Operators Rs can be written in the form

R0Wr(x, t) ≡n∑i=1

bi(x)

(λ1 − λi)[ 2∑k=1

[Vrik(x)Uk(t1) +Qrik(x)U ′k(t1)] +

+ fri(x)ν(t1) + gri(x)ν′(t1)]

+

+p∑j=2

(λj − λi)2∑

k=1

αrijk(x) exp(tjk)− λi(x)ωri(x), (13.mar)


bi(x)[ n∑

s=1

ϕ1 · Tsi1 + Di1

]×

×[ 2∑k=1

Qrik(x)Uk(t1) + gri(x)ν(t1)]− π−1ϕ′21 (x)fri(x)

, (14.mar)


bi(x) p∑j=2

2∑k=1

[Dijkαrijk(x) +

n∑s6=is=1

Tsijkαrsjk(x)]

exp(tjk),

(15.mar)


bi(x)[ n∑

s6=is=1

Tsi1 +Di1

]·[ 2∑k=1

Vrik(x)U ′k(t1) +

+ fri(x)ν′(t1) + π−1gri(x)]

, (16.mar)

R6Wr(x, t) ≡∂2Wr(x, t)

∂x2. (17.mar)

Here

Dijk ≡ 2ϕ′jk(x)[∂

∂x+ (b′i(x), b∗i (x))

],

Di1 ≡ ϕ1(x)Di1 + ϕ′21 (x) ≡ ϕ1(x) · 2ϕ′1(x)[∂

∂x+ (b′i(x), b∗i (x))

]+ ϕ′21 (x) ,

(18.mar)

Tsi1 ≡ 2ϕ′1(x)(b′s(x), b∗i (x)), Tsijk ≡ 2ϕ′jk(x)(b′s(x), b∗i (x)) .

Analyzing the obtained identities we can make the following implications.

1. Spaces of nonresonance solutions Yr are invariant with respect to operatorsR0, R2, R3, R6 and consequently with respect to the extended operator Lεwhich is represented in the form (12.mar).

2. Operator R0 is a main operator of the extended operator Lε in SNS (10.mar).3. Extended problem (6.mar) depends regularly on a small parameter µ > 0 in SNS

(10.mar).


5 Formalism of construction of solution of extendedproblem

Since the extended problem (6.mar) is regularly dependent on a small parameterµ > 0 in SNS (10.mar), then the asymptotic solution of that problem can be foundin the form of a series

W (x, t, ε) =+∞∑r=−2

µrWr(x, t) , (19.mar)

whereWr(x, t) ∈ Yr .

Let us substitute (19.mar) to the extended problem (6.mar) and compare the coefficientsby the parameter µ > 0. Then for defining of the coefficients of the series (19.mar)we shall get the following recurrence system of problems:

R0W−2(x, t) = 0, E1W−2(Mm) = E1αm ,

GmW−2(x, t) ≡ E2

2∑k=1

p∑j=2

ϕ′jk(m)∂W−2(Mm)

∂tjk= 0 , (20.mar)

R0W−1(x, t) = 0, E1W−1(Mm) = 0 ,

GmW−1(x, t) = E2

[αm − ϕ′1(m)

∂W−2(Mm)∂t1

], (21.mar)

R0W0(x, t) = h(x)−R2W−2 , E1W0(Mm) = E1Wm ,

GmW0(x, t) = E2

[Wm − ϕ′1(m)

∂W−1(Mm)∂t1

], (22.mar)

In this way we obtain a series of recurrence problems with point boundaryconditions which is in the general case insufficient for uniqueness of solutionsof each of those separate problems. However, further considerations show thatin SNS (10.mar) the series of problems (20.mar)–(22.mar) is asymptotically correct, i.e. eachof the problems (20.mar)–(22.mar) has the unique solution in SNS if we consider thoseproblems step by step. Further on we should ask a question of the existence ofa solution in SNS (10.mar) of the iteration equation

R0Wr(x, t) = Hr(x, t) . (23.mar)

From identity (13.mar) one can describe the structure of the kernel of the operatorR0. We have

KerR0 =bj(x)αrjjk(x) exp(tjk), j = 2, p− 1,

bj(x)αrjjk(x) exp(tpk), j = p, n ,

b1(x)[Vr1k(x)Uk(t1) +Qr1k(x)U ′k(t1)],

b1(x)[fr1(x)ν(t1) + gr1(x)ν′(t1)], k = 1, 2

(24.mar)


We introduce the following notations: bi(x)Sri(x), i = 1, n projection of thevector-function Hr(x, t) to the subspace Xri.

Then we have the following

Theorem 1. For equation (1.mar) let

a) the conditions 1 and 2 hold,b) right side of the equation (23.mar) belongs to SNS (10.mar) and contains no element

of the kernel of the operator R0,c) Sr1(0) = 0.

Then there exists a solution of (23.mar) in the space Yr and it can be represented inthe form

Wr(x, t) = Zr(x, t) + yr(x, t) . (25.mar)

Here

Zr(x, t) = b1(x)[ 2∑k=1

[Vr1k(x)Uk(t1) +Qr1k(x)U ′k(t1)] +

+ fr1(x)ν(t1) + gr1(x)ν′(t1)]

+2∑

k=1

n∑j=2

bj(x)αrjjk(x) exp(tjk) , (26.mar)

where the coefficients of ESM are arbitrary of the sufficiently smooth functionswith x ∈ I and yr(x, t) uniquely defined and sufficiently smooth functions for allx ∈ I, in particular for x = 0.

Remark 2. Since the point x = 0 is unstable, then ESM U2(t1) ≡ Bi(t1) and itsderivative is unboundedly increasing when t1 → +∞. However in spite of thatin ESM the ν(t) and ν′(t) contain unboundedly increasing functions Bi(t1) andB′i(t1), they all are still bounded functions for t ≥ 0 i.e. for t ∈ [0, µ−2ϕ1(1)].

Because of the shortness of the article we are not able to construct the fullsolution of at least three equations (20.mar)–(22.mar). We are bound to give the onlyremark.

Remark 3. Coefficients αrjjk(x), j = 2, p− 1 derived from the simple elementsλj(x) can be defined by scalar linear differential equations of the first order.Multiplicity of the element λp(x) ≡ · · · ≡ λn(x) introduces in the constructionof asymptotic solutions of the extended problem (6.mar) the following changes. Co-efficients αrjjk(x) (j = p, n) derived from multiple elements of the spectrum canbe determined by the system of (n− p+ 1) differential equations.


6 Asymptotic correctness of iterated problems

Applying Theorem 1 consequently we can obtain solutions of the iterated equa-tions (20.mar)–(22.mar) and so on. Each of those solutions contains 2n arbitrary constantswhich were obtained by integrations of differential equations and systems of dif-ferential equations with respect to unknown functions Vrlk(x), αrjjk(x), j = 2, n;k = 1, 2.

Substituting the obtained solutions Wr(x, t) to the adequate boundary con-ditions one can obtain the system of 2n algebraic equations from which one canfind unknown constants:

∆(ε)Cr = Γr , (27.mar)

where

Cr = (V 00111, V

00112, α

00221, . . . , α

00nn1, α

00222, . . . , α

00nn2)

is an unknown vector and Γr a given vector.One can show that the asymptotic equality holds:

∆(ε) = KBi(t1(l)

)[1 +O

(Bi−1(t1(l))

)], (28.mar)

where

K = 2−23Γ−1(2/3)

2∏k=1

b11((k − 1)l)×

×p∏j=2

ϕ′jk((k − 1)l) ·∣∣B1((k − 1)l)

∣∣ · ∣∣B2((k − 1)l)∣∣ , (29.mar)

B1(x) = (bij(x))p−1i,j=2 , B2(x) = (bij(x))ni,j=p,

where bij(x) is ith coordinate of the vector bj(x).

Lemma 4. Let ∣∣Bs((k − 1)l)∣∣ 6= 0, k, s = 1, 2 (30.mar)

Then for sufficiently small values of the parameter ε > 0 the determinant of thematrix ∆(ε) is distinct from zero.

Thus if the conditions (30.mar) holds, then each of the systems (19.mar), wherer ≥ −2, has a unique solution, i.e. each of the functions Wr(x, t) is uniquelydetermined as a solution of the adequate iterated problem.


7 Estimation of the last part of the asymptotic ofa solution

Let us write the formal solution of the extended problem (6.mar) in the form:

W (x, t, µ) ≡Wεq(x, t, µ) + µq+1ξq+1(x, t, ε) , (31.mar)

where

Wεq(x, t, µ) ≡q∑

r=−2

µrWr(x, t) — q-partial sum of the series (7.mar).

If in the equality (31.mar) we realized the narrowing by t = φ(x, ε) then we obtainthe equality

W (x, ε) ≡W (x, φ(x, ε), ε) ≡Wεq(x, φ, ε) + εq+1

3 ξq+1(x, φ, ε) . (32.mar)

Lemma 5. If the conditions 1 and 2 hold then for sufficiently small values ofthe parameter ε > 0 :

a) the series (19.mar) is an asymptotic series for the solution of the extended prob-lem (6.mar)

b) the narrowing of the series (19.mar) by t = φ(x, ε), i.e. the series (32.mar), is asymp-totic series for a solution SSPDE (1.mar).

Applying Lemma 5 one can prove that the following asymptotic equality holds:

ξq+1(x, φ, ε) ∼= O(µ−12 expµ−3(2/3)ϕ

32 (x)) . (33.mar)

The results obtained in the article can be formulated in the form of thefollowing theorem:

Theorem 6. If the conditions 1 and 2 hold then for sufficiently small valuesof the parameter ε > 0:

a) one can construct (applying the above described methods) a unique asymp-totic series (19.mar) as a solution of the extended problem (6.mar) in SNS;

b) the narrowing of the series (19.mar) for t = φ(x, ε) (32.mar) is asymptotic seriesfor a solution SSPDE (1.mar);

c) the last part of the asymptotic series of the solution (1.mar) has the estimation(33.mar).

Remark 7. Let (h(0), b∗1(0)) = 0. Then

1) a solution of the degenerate equation (3.mar) is a sufficiently smooth function foreach x ∈ [0, 1];

2) a solution SSPDE (1.mar) contains no negative degrees of a small parameter µ > 0;3) αm = 0 (m = 0, 1) in the boundary conditions for SSPDE (1.mar), i.e. they

represent the form like in the problems with stable spectrum of degenerateoperator.


References

[1] Marry J., Nonlinear differential equations in biology (lectures on models), MoscowMir, 1983, 400 p.

[2] Bobochko V. N., Asymptotic integration of systems of differential equations with aturning point, Diff. Equat. 1991, t. 27, No 9, 1505–1518.

[3] Bobochko V. N., Turning point in a system of differential equations with analyticoperator, Ukrainian Math. J. 1996, t. 48, No 2, 147–160.

[4] Bobochko V. N., Asymptotics solutions of a system differential equations with amultiple turning point, Diff. Equat. 1996, t. 32, No 9, 1153–1155.

[5] Lomov S. A., Introduction to the general theory of singular perturbations, Moscow,Nauka, 1981, 400 p.

[6] Gantmacher F. R., Theory of matrices, Moscow, 1953, 492 p.[7] Olver F. W. J., Asymptotic and special functions, Moscow, Nauka, 1990, 528 p.



On the Symmetric Solutions to a Class of

Nonlinear PDEs

Gabriella Bognar

Mathematical Institute, University of Miskolc,3515 Miskolc-Egyetemvaros, HungaryEmail: [email protected]

Abstract. The existence and uniqueness of symmetric solutions to theboundary value problem of nonlinear partial differential equations areestablished. The Dirichlet boundary condition is given on the ball inRN .

AMS Subject Classification. 35A05, 34B15

Keywords. Boundary value problem, symmetric solutions

1 Introduction

We consider the following boundary value problem

N∑i=1

∂

∂xi

(∂u

∂xi

)∗p+ f(u,

∣∣gradp u∣∣p) = 0 in Bp, (1.bog)

u = 0 on ∂Bp, (2.bog)

where 0 < p <∞ and the function u∗p is defined as follows:

u∗p = |u|p−1 u,

and the domain Bp ∈ RN is an open unit “ball” centered at the origin and ∂Bpmeans the boundary of the domain Bp. In (1.bog) gradp u denotes the expression

gradp u = (u∗px1, u∗px2, .., u

∗pxN ), u = u(x1, x2, . . . , xN )

and |(x1, x2, . . . , xN )|p =(N∑i=1

|xi|1p+1

) pp+1

.

If p = 1, the operatorN∑i=1

∂∂xi

(∂u∂xi

)∗pin the equation (1.bog) is reduced to ∆u.


54 Gabriella Bognar

For the problem (1.bog)–(2.bog) we shall define the distance ρ between the point andthe origin in RN as follows:

ρ1p+1 =

N∑i=1

|xi|1p+1 . (3.bog)

In the case ρ = 1 the equation (3.bog) gives the equation of the unit “ball” Bpin RN . We mention that the curve ρ = 1 in R2 is a central symmetric convexcurve which plays the same role in the case of nonlinear differential equation (1.bog)as the unit circle in the case of linear (p = 1) partial differential equation.

For the “ball” Bp we introduce now instead of rectangular coordinates x1, x2,x3, . . . , xN a new type of polar coordinates ρ, ϕ1, . . . , ϕN−1 as follows

x1 = ρ

N−1∏i=1

[S′(ϕi)] ,

xk = ρ [S(ϕk−1)]N−1∏i=k

[S′(ϕi)] if 1 < k ≤ N,(4.bog)

where S = S(ϕi), 1 < i ≤ N − 1 is the generalized sine function given byA. Elbert [6]. The Pythagorean relation for this generalized sine function hasthe form

|S|1p+1 + |S′|

1p+1 = 1, where S′ = dS(ϕ)

dϕ .

The unit “ball” Bp in RN is defined by

Bp =

(x1, x2, . . . , xN ) :N∑i=1

|xi|1p+1 ≤ 1

, 0 < p <∞.

When we study the radially symmetric solution u(x) = ν(ρ) of the nonlinearboundary value problem (1.bog)–(2.bog) the nonlinear partial differential equation (1.bog) isreduced to the following nonlinear ordinary differential equation (5.bog)

∂

∂ρ

(∂ν

∂ρ

)∗p+N − 1ρ

(∂ν

∂ρ

)∗p+ f(ν, |ν′|) = 0, ρ ∈ (0, 1) , (5.bog)

where f(u,∣∣gradp u

∣∣p) = f(ν, |ν′|) since

∣∣gradp u∣∣p

= |ν′|p2

p+1 . We note that theequation (5.bog) can be written also in the form

(ρN−1ν′∗p(ρ))′ + ρN−1f(ν, |ν′|) = 0, ρ ∈ (0, 1) .

Instead of the boundary condition (2.bog) we shall consider the conditions

ν(1) = 0, (6.bog)

On the Symmetric Solutions 55

ν′(0) = 0. (7.bog)

Now let us take the boundary value problem of another nonlinear partialdifferential equation instead of (1.bog)–(2.bog)

N∑i=1

∂

∂xi

[|∇u|p−1∇u

]+ f(u, |grad u|) = 0 in B, (8.bog)

u = 0 on ∂B, (9.bog)

where the unit ball B in RN is defined by

B =

(x1, x2, . . . , xN ) :N∑i=1

x2i ≤ 1

,

as it is usual in the Euclidean metric and |grad u| =(

N∑i=1

u2xi

) 12

(p = 1). The

expression

N∑i=1

∂

∂xi

[|∇u|p−1∇u

]in (8.bog) is used to call p-Laplacian.This operator appears in many contexts inphysics: non-Newtonian fluids, reaction-diffusion problems, non-linear elasticity,and glaceology, just to mention a few applications( see [3], [9], [10], [11], [12]). Ifp = 1 the equation (8.bog) is also reduced to the semilinear problem

∆u+ f(u, |grad u|) = 0,

the existence of these problems are investigated in [1], [2], [7].The radially sym-metric solutions of the Dirichlet problem of

∆u+ f(u) = 0

were examined by Gidas, Wei-Ming Ni, and Nirenberg for the ball B [8]. Ifp > 0 then, applying the usual spherical transformation, the radially symmetricsolutions of equation (8.bog) has to satisfy formally the same equation as (5.bog). So, ifwe examine the solutions of (5.bog) we get results on the radially symmetric solutionsboth for the nonlinear partial differential equation (1.bog) in the “ball” Bp and alsofor the nonlinear partial differential equation (8.bog) in the Euclidean ball B. Hereour aim is to show existence and uniqueness results of symmetric solutions forthe problem

∂

∂ρ

(∂ν

∂ρ

)∗p+N − 1ρ

(∂ν

∂ρ

)∗p+ eλν+κ|ν′| = 0, ρ ∈ (0, 1) ,

ν(1) = 0, ν′(0) = 0,

56 Gabriella Bognar

where λ, κ are negative real numbers. In the case p = 1 the existence anduniqueness results of the problem

∂

∂ρ

(∂ν

∂ρ

)+N − 1ρ

(∂ν

∂ρ

)+ eλν+κ|ν′| = 0, ρ ∈ (0, 1) ,

ν(1) = 0, ν′(0) = 0,

are established in [4].

2 Results

Let us consider the following boundary value problem

∂

∂ρ

(∂ν

∂ρ

)∗p+N − 1ρ

(∂ν

∂ρ

)∗p+ eλν+κ|ν′| = 0, ρ ∈ (0, 1)

ν(1) = a, a ∈ R+, ν′(0) = 0.

(10.bog)

We shall say that the function is the positive solution of problem (10.bog) if

i) ν (ρ) is continuous on [0, 1] and ν (ρ) > 0 in the interval (0, 1];ii) ν′ (ρ) exists and is continuous, moreover ν′ (ρ) ≤ 0 in the interval [0, 1];

iii) ν (ρ) satisfies the boundary conditions: ν (1) = a, for a ≥ 0, ν′ (0) = 0;iv) ν′′ (ρ) exists almost everywhere and locally Lebesgue integrable in the inter-

val [0, 1];v) ν (ρ) satisfies the differential equation

∂

∂ρ

(∂ν

∂ρ

)∗p+N − 1ρ

(∂ν

∂ρ

)∗p+ eλν+κ|ν′| = 0, ρ ∈ (0, 1) .

Theorem 1. If a ≥ 0 then the boundary value problem (10.bog) has at most onepositive radial solution.

Proof. Let us denote by ν1 (ρ) and ν2 (ρ) two different positive solutions to theboundary value problem (10.bog). Without loss of generality we may suppose, thatthere exists a point ρ = γ, γ ∈ [0, 1) such that ν1 (ρ) ≥ ν2 (ρ). If ν1 (ρ) −ν2 (ρ) < 0 in the interval [0, 1) then we change the notations of ν1 (ρ) andν2 (ρ) for the opposite. Let us denote by δ ∈ (γ, 1], the first zero of the functionν1 (ρ)− ν2 (ρ) which lays to the right from γ. By the Lagrange’s theorem thereexists β ∈ (γ, δ) for which ν′1 (β)− ν′2 (β) < 0 and ν1 (β)− ν2 (β) > 0. We shalldenote by α ∈ [0, β) the zero of the function ν′1 (ρ) − ν′2 (ρ). If there are morezeroes α1, α2, . . . , αk of ν′1 (ρ) − ν′2 (ρ) in the interval [0, β) then let us take thenotation α = max(α1, α2, . . . , αk).

In this case we can summarize that ν1 (ρ)−ν2 (ρ) > 0 and ν′1 (ρ)−ν′2 (ρ) < 0,ρ ∈ (α, β], ν′1 (α)− ν′2 (α) = 0.


Since the functions ν1 (ρ) and ν2 (ρ) satisfy the nonlinear differential equationin (10.bog) therefore substituting them into the differential equation and subtractingthe two equations we get the equation

[ρN−1(ν′∗p

1 − ν′∗p

2 )]′ + ρN−1[eλν1+κ|ν′1| − eλν2+κ|ν′2|] = 0. (11.bog)

We introduce the following notations

J(ρ) = ν∗p1 (ρ)− ν

∗p2 (ρ) ,

K(ρ) = ν′∗p

1 (ρ)− ν′∗p

2 (ρ) ,

moreover J(ρ) and K(ρ) have the properties

J(1) = 0,J(γ) > 0,

J(ρ) > 0, ρ ∈ (α, β],

K(0) = 0,K(β) < 0,K(α) = 0,

K(ρ) < 0, ρ ∈ (α, β].

(12.bog)

Rearranging the differential equation (11.bog) we obtain[ρN−1K(ρ)

]′+ ρN−1K(ρ)A(ρ) − ρN−1J(ρ)B(ρ) = 0 ,

where the expressions A(ρ) and B(ρ) have the forms

A(ρ) =eλν1+κ|ν′1| − eλν1+κ|ν′2|

ν′∗p

1 (ρ)− ν′∗p

2 (ρ),

B(ρ) =eλν2+κ|ν′2| − eλν1+κ|ν′2|

ν∗p1 (ρ)− ν

∗p2 (ρ)

.

Using the properties of the function eλν+κ|ν′| we get that A(ρ) ≥ 0 andB(ρ) ≥ 0 when ρ ∈ (α, β]. Thus from the equation (11.bog) we obtain the inequality

[ρN−1K(ρ)

]′+ ρN−1K(ρ)

eλν1+κ|ν′1| − eλν1+κ|ν′2|

ν′∗p

1 (ρ)− ν′∗p

2 (ρ)≥ 0 , ρ ∈ (α, β]. (13.bog)

If we multiply the inequality in (13.bog) by the expression

exp

−∫ b

ρ

etaeλν1+κ|ν′1| − eλν1+κ|ν′2|

ν′∗p

1 (τ)− ν′∗p

2 (τ)dτ

,

and take the integral on the interval [δ, β] where δ ∈ (α, β) we get the inequality

βN−1K(β)− δN−1K(δ) exp

−∫ γ

δ

eλν1+κ|ν′1| − eλν1+κ|ν′2|

ν′∗p

1 (τ)− ν′∗p

2 (τ)dτ

≥ 0.

58 Gabriella Bognar

If we take δ → α then we get that

K(β) ≥ 0,

since K(α) = 0. This is contradiction with (12.bog).

In the next theorem we establish the existence result:

Theorem 2. If a ≥ 0 then the boundary value problem (10.bog) has a unique posi-tive solution.

In the following we need some subsidiary statements.

Lemma 3. If a ≥ 0 then there is a positive solution to problem (10.bog).

Proof. Let us define the mappings

(Φµ) (t) = a+∫ 1

t

µ(τ)dτ,

(Ψµ) (t) =[∫ t

0

(τt

)N−1

eλ(Φµ)(τ)+κµ(τ,a)dτ

] 1p

,

H =µ (τ, a) ∈ C[0, 1), 0 ≤ µ (τ, a) ≤

(eλa

N

) 1p

, t ∈ (0, 1) , µ (0, a) = 0.

The functions which belong to the set ΦH are uniformly bounded and equicon-tinuous functions therefore H is compact. Since every Cauchy sequence beingcontained in the set H converges in H then H is closed. Thus the set H isbounded, convex, closed and compact in the Banach space C[0, 1).

The mapping Ψ is a continuous mapping fromH to H . Applying the Schauderfixed point theorem the mapping Ψ has a fixed point.

Using notation µ (ρ, a) = −ν′ (ρ, a) the positive solution to problem (10.bog) hasthe form

ν (t, a) = a+∫ 1

t

µ(τ)dτ = a+∫ 1

t

[∫ τ

0

(ρτ

)N−1

eλν(ρ,a)+κµ(ρ,a)dρ

] 1p

dτ. (14.bog)

Lemma 4. Let ν (t, a) be the unique positive solution to the problem (10.bog). If0 ≤ a2 < a1, then ν (t, a1) ≥ ν (t, a2) and ν′ (t, a1) ≥ ν′ (t, a2) for all t ∈ [0, 1).

Proof. Let ν (t, a1) and ν (t, a2) be the unique positive solution to the problem(10.bog) for a1and a2, respectively. Let us take the notation

j(t) = ν∗p (t, a1)− ν

∗p (t, a2) and k(t) = ν′

∗p (t, a1)− ν′

∗p (t, a2) .

Clearly j(1) = a∗p1 − a

∗p2 > 0 and k(0) = 0. Hence there exists at least one point

t = α, α ∈ [0, 1) such that k(α) = 0 and j(t) > 0 in the interval (α, 1]. If


there are more values of α (α1, α2, . . . , αk) for which k(α) = 0 in the interval[0, 1) then let us take the notation α = max(α1, α2, . . . , αk). We may assume thatthere exists a point β ∈ [α, 1) where ν (t, a1) > ν (t, a2) and ν′ (t, a1) < ν′ (t, a2),that is j(t) > 0, and k(t) < 0 in the interval (α, β]. In an analogous way as inthe proof of Theorem 1 one can obtain that k(β) ≥ 0. It is contradiction sincewe supposed that k(β) < 0.

The inequality ν (t, a1) ≥ ν (t, a2) we get in a similar way as in the proof ofTheorem 1.

Proof of Theorem 2. When a → 0 we get that ν (t, a) and ν′ (t, a) convergesuniformly to ν (t, 0) and ν′ (t, 0) in the interval [0, 1], respectively. Taking a→ 0in the expression (14.bog) we shall get the positive solution to the problem

∂

∂ρ

(∂ν

∂ρ

)∗p+N − 1ρ

(∂ν

∂ρ

)∗p+ eλν+κ|ν′| = 0, ρ ∈ (0, 1) ,

ν(1) = 0, ν′(0) = 0,

in the following form

ν(t, 0) =∫ 1

t

[∫ τ

0

(ρτ

)N−1

eλν(ρ,0)−κν′(ρ,0)dρ

] 1p

dτ.

Supported by the Grant No. OTKA 019095 (Hungary).

References

[1] F. V. Atkinson, L. A. Peletier, Ground states of ∆u = f(u) and the Emden-Fowlerequation, Archs. Ration. Mech. Analysis, 93 (1986), 103–107.

[2] J. V. Baxley, Some singular nonlinear boundary value problems, SIAM J. Math.Analysis, 22 (1991), 463–479.

[3] T. Bhattacharya, Radial symmetry of the first eigenfunction for the p-Laplacianin the ball, Proc. of the Amer. Math. Soc., 104 (1988), 169–174.

[4] G. Bognar, On the radially symmetric solutions to a nonlinear PDE, Publ. Univ.of Miskolc, Series D. Natural Sciences. 36 No.2. Mathematics (1996), 13–20.

[5] G. Bognar, On the radial symmetric solutions of a nonlinear partial differentialequation, Publ. Univ. of Miskolc, Series D. Natural Sciences. 36 No.1. Mathe-matics (1995), 13–22.

[6] A. Elbert, A half-linear second order differential equation, Coll. Math. Soc. JanosBolyai, 30. Qualitative theory of differential equations, Szeged, (1979), 153–179.

[7] A. M. Fink, J. A. Gattica, G. E. Hernandez, P. Waltman, Approximation of solu-tions of singular second order boundary value problems, SIAM J. Math. Analysis,22 (1991), 440–462.

[8] B. Gidas, Wei-Ming Ni, L. Nirenberg, Symmetry and related properties via Max-imum Principle, Commun. Math. Phys., 68 (1979), 209–243.

[9] B. Kawohl, On a family of torsional creep problems, J. Reine Angew. Math. Mech.,to appear.

60 Gabriella Bognar

[10] P. Lindqvist, Note on a nonlinear eigenvalue problem, Rocky Mountain J. of Math.,23 (1993), 281–288.

[11] M. Otani, A remark on certain nonlinear elliptic equations, Proc. of Faculty ofScience, Tokai Univ., 19 (1984), 23–28.

[12] F. de Thelin, Quelques resultats d’existence et de non-existence pour une E.D.P.elliptique non lineaire, C. R. Acad. Sci. Par., 299 Serie I. Math. (1986), 911–914.



The Abstract Cauchy Problem in Plasticity

Igor A. Brigadnov

North-West Polytechnical InstituteMillionnaya Str. 5, St. Petersburg, 191186, Russia

Email: [email protected]

Abstract. The boundary-value problem of plasticity is formulated asthe evolution variational problem (EVP) over the parameter of externalloading for the displacement in the framework of the small deformationstheory. The questions of the mathematical correctness of the plasticityEVP are discussed. The general existence and uniqueness theorem is for-mulated. The main necessary and sufficient condition has the simplestalgebraic form and does not coincide with the classic Drucker’s hypothe-sis and similar thermodynamical postulates. By means of finite elementapproximation the plasticity EVP transforms into the Cauchy problemfor a non-linear system of ordinary differential equations unsolved re-garding derivative. Moreover, this system can be stiff. Therefore, for thenumerical solution the implicit Euler scheme with the decompositionmethod of adaptive block relaxation (ABR) is used. The numerical re-sults show that, for finding the displacement and the time of calculation,the ABR method has advantages over the standard method.

AMS Subject Classification. 73E05, 73V20, 35J55

Keywords. Plasticity BVP, evolution variational equation, mathemat-ical correctness, stiff system, adaptive finite element method

1 Introduction

The solution of plasticity boundary-value problems (BVPs) is of particular inter-est in both theory and practice. At present there are many models of plasticityin the framework of the small deformations theory [1,2,3]. Adequacy and thefield of application of every model must be found only by correlation betweenexperimental data and solutions of appropriate BVPs. Therefore, the analysisof mathematical correctness and the treatment of numerical methods for theseproblems is very important [4,5,6,7].

In this paper the plasticity BVP is formulated as the evolution variationalproblem (EVP) (i.e. as the abstract Cauchy problem in the weak form) for thedisplacement in the Hilbert space [8]. For this reason the parameter of externalloading in the interval [0, 1] is used. The general existence and uniqueness theo-rem for the plasticity EVP is formulated. The proof of this theorem is based on


62 Igor A. Brigadnov

the monotonous operators theory and the theory of the abstract Cauchy problemin the Hilbert space [7]. The main necessary and sufficient condition has the sim-plest algebraic form and does not coincide with the classic Drucker’s hypothesisand similar thermodynamical postulates [2,3,9,10]. This condition is the generalcriterion of mathematical correctness for plasticity models. Its independence isillustrated for the plasticity model of linear isotropic-kinematic hardening withideal Bauschinger’s effect, dilatation and internal friction [11,12].

For the numerical solution of the plasticity EVP the standard spatial piece-wise linear finite element approximation is used [13]. For some models the appro-priate finite dimensional Cauchy problem can be stiff [14,15]. The main cause ofthis phenomenon consists of the following: the global shear stiffness matrix haslines with significantly different factors (it is badly determined). Moreover, forreal plasticity models both initial continuum and discrete Cauchy problems areprincipally unsolved regarding derivative [1,2,12]. Therefore, for the numericalsolution the implicit Euler scheme with the decomposition method of adaptiveblock relaxation (ABR) is used [4,5,6,7]. The main idea of this method consistsof iterative improvement of zones with ”proportional” deformation by specialdecomposition of domain (variables), and separate calculation in these zones(on these variables). The global convergence theorem for the ABR method isformulated. The proof of this theorem is based on the monotonous operatorstheory [4,5].

The numerical results show that, for finding the displacement and the timeof calculation, the ABR method has advantages over the standard method.

2 Evolution formulation of the plasticity BVP

Let a homogeneous rigid body in the undeformed reference configuration occupya domain Ω ⊂ R3 with boundary Γ . In the deformed configuration each pointx ∈ Ω moves into a position x+u(x) ∈ R3, where u : Ω → R3 is the displacement.In the framework of the small deformations theory the strain Cauchy tensorε = ε(u) = 1

2

(∂iuj + ∂jui

): Ω → S3 is used as the measure of deformation,

where ∂i = ∂/∂xi; i, j = 1, 2, 3. The symbol S3 denotes the subspace of realsymmetrical 3× 3 matrices.

In the mathematical theory of plasticity the isotropic material is describedby the constitutive relation for speeds [1,2,3,7,10,12]

σij = Sij (ε, ε) = Cijkm

(εkm − Pkm(ε, ε)

),

Cijkm = 2µ δikδjm +(k0 − 2

3µ)δijδkm,

(1.bri)

where σ : Ω → S3 is the Cauchy stress tensor, P : Ω → S3 is the plastic partof the Cauchy strain tensor, Cijkm are the components of elasticity acoustictensor [2,3], µ > 0 and k0 > 0 are the shear and bulk moduli, respectively; δijis the Kronecker symbol, the above point is d/dt and t ∈ [0, 1] is the parameterof external loading. Here and in what follows we use the rule of summing over

The Abstract Cauchy Problem in Plasticity 63

repeated indices and the designation |A| = |Akm| = (AijAij)1/2 for modulus of

matrix A.We consider the following boundary-value problem. The quasi-static influ-

ences acting on the body are: a mass force with density f in Ω, a surface forcewith density F on a portion Γ 2 of the boundary, and a surface displacement uγon a portion Γ 1 of the boundary is also given. Here Γ 1 ∪ Γ 2 = Γ , Γ 1 ∩ Γ 2 = ∅and area(Γ 1) > 0.

According to the evolution description [8] the external influences, internaldisplacement and stress tensor are taken as continuous and piecewise smoothabstract functions acting from interval [0, 1] to appropriate Banach spaces, sup-posing that Γ 1 = const(t) and f = 0, F = 0, uγ = 0 for t = 0.

The plasticity BVP is formulated as the evolution variational problem (EVP)(i.e. as the abstract Cauchy problem in the weak form): the sought displacementcorresponds to the abstract function u∗(t) = u0(t) + u(t), where the piecewisesmooth abstract function u0(t) with u0(0) = 0 corresponds to the surface dis-placement uγ , and unknown abstract function u : [0, 1] → V 0 must satisfy theinitial condition u(0) = 0 and the differential equation for every v ∈ V 0 andalmost every t ∈ (0, 1)∫

Ω

Sij(ε(u0 + u), ε(u0 + u)

)∂jvi dx = L(t, v),

L(t, v) =∫Ω

fi(t)vi dx+∫Γ 2

Fi(t)vi dγ.(2.bri)

Here V 0 = v : Ω → R3; v(x) = 0, x ∈ Γ 1 — is the set of kinematicallyadmissible variations of the displacement. For real plasticity models this equationis principally unsolved regarding u [2,7,12].

Concerning the constitutive relation S, the domain Ω and the functions f ,F , uγ we make the following hypotheses:

(H1) Matrix function S(A,B) is the continuous and strongly monotonous in B,i.e. there exists a constant m0 > 0 such that for every A,B1, B2 ∈ S3 thefollowing estimate is true(

Sij(A,B1)− Sij(A,B2)) (B1ij −B2

ij

)≥ m0

∣∣B1 −B2∣∣2 .

(H2) Matrix function S(A,B) is the Lipschitz continuous in A, i.e. there existsa scalar function M0 : S3 → (0,+∞) such that for every A1, A2, B ∈ S3

the following estimate is true∣∣S(A1, B)− S(A2, B)∣∣ ≤M0(B)

∣∣A1 −A2∣∣ .

(H3) Matrix function S(A,B) has the growth in A and B no above linear, i.e.there exists a constant M1 > 0 such that for every A,B ∈ S3 the followingestimate is true

|S(A,B)| ≤M1 (|A| + |B|).


(H4) Ω is a connected bounded domain in R3 with a Lipschitz boundary Γ .(H5) f ∈ C0,1

([0, 1], L6/5(Ω,R3)

).

(H6) F ∈ C0,1([0, 1], L4/3(Γ 2, R3)

).

(H7) uγ ∈ C0,1([0, 1], L2(Γ 1, R3)

).

We define the set of kinematically admissible variations of the displacementin the following way:

V 0 =v ∈ H1 : v(x) = 0, x ∈ Γ 1

,

where H1 := W 1,2(Ω,R3) is the Hilbert space.

Theorem 1 (was proved in [7]). In the framework of the hypotheses (H1)–(H7) the following statements are true:

(i) The unique strict solution of the EVP (2.bri) exists, i.e. the absolutely con-tinuous function u ∈ C0,1([0, 1], V 0), u(0) = 0 with the strong derivative u,satisfying the equation (2.bri) for a.e. t ∈ (0, 1).

(ii) The map (f, F, uγ) 7→ u is continuous.

Remark 2. For the constitutive relation S the main condition (H1) is necessaryand sufficient. It is the general criterion of mathematical correctness for plasticitymodels. This question is in detail discussed in [7]. Therefore, we rewrite thiscondition for the matrix function P (ε, ε), usually used in the modern theory ofplasticity [2,7,12].

(H1) Matrix function P (A,B) is continuous in B and satisfies the followingestimate for every A,B1, B2 ∈ S3

Cijkm

(Pkm(A,B1)− Pkm(A,B2)

) (B1ij −B2

ij

)< 2µ

∣∣B1 −B2∣∣2 . (3.bri)

This condition does not coincide with the Lipschitz condition of the matrixfunction P (A,B) over second matrix argument. It is easily to get convinced thatthe Lipschitz condition is stronger than the condition (3.bri). In the following sectionwe show that this condition is independent and does not coincide with the classicDrucker’s hypothesis based on the thermodynamical postulates [2,3,9,10].

3 Example of analysis of plasticity models

The independence of the main necessary and sufficient condition (3.bri) of mathe-matical correctness of plasticity EVP (2.bri) we illustrate for the generalized modelof plasticity with linear isotropic-kinematic hardening, ideal Bauschinger’s effect,


dilatation and internal friction [12]

Pkm = (1 + h0 + 3λΛ)−1H (ρe − ε∗ + λ tr(ε− P ))××H

(cos γ + λε−1

e tr(ε))ρ−2e (ρkm + Λρeδkm)(ρpq + λρeδpq)εpq,

ρkm = εDkm − (1 + h0)PDkm, cos γ = (ρeεe)−1ρij εDij , (4.bri)

ρe = |ρij | , εe =∣∣εDij ∣∣ ,

H(q) = 0 for q < 0 and H(q) = 1 for q ≥ 0,

where h0 is the parameter of plastic hardening, λ ≥ 0 and Λ ≥ 0 are theparameters of dilatation and internal friction, respectively; ε∗ ≥ 0 is the limit ofelastic strain, ADij = Aij − 1

3 tr(A)δij are the components of deviatoric part andtr(A) = δijAij is the trace (first invariant) of matrix A.

For λ = Λ = 0 model (4.bri) equals the classic model of plasticity with linearisotropic-kinematic hardening and ideal Bauschinger’s effect [1,2,3]. In this casetr(P ) = 0 and the constitutive relation (4.bri) is associated with the Mises yieldsurface ρe − ε∗ = 0 [2,3].

For λ = Λ 6= 0 the constitutive relation (4.bri) is associated with the yield sur-face for strain ρe − ε∗ + λ tr(ε− P ) = 0. This surface for h0 = 0 corresponds tothe Mises-Schleiher yield surface for stress

∣∣σD∣∣+ c−1λ tr(σ)− 2µ ε∗ = 0, wherec = 3k0/(2µ) [11]. For λ 6= Λ the constitutive relation (4.bri) is non-associated withsome yield surface. In both cases tr(P ) 6= 0 what is a well known experimentalphenomenon of dilatation [11,12].

Let matrices A,B1, B2 ∈ S3 be arbitrary. Then from condition (3.bri) for model(4.bri) we have

Cijkm(Pkm(A,B1)− Pkm(A,B2)

) (B1ij −B2

ij

)≤

≤ (1 + h0 + 3λΛ)−1[ ∣∣B1 −B2

∣∣+ λ tr(B1 −B2

) ]×

×[

2µ∣∣B1 −B2

∣∣+ 3k0Λ tr(B1 −B2

) ]≤

≤ 2µΨ(λ, Λ, h0)∣∣B1 −B2

∣∣2 ,where

Ψ =(1 +

√3 λ)(1 +

√3 cΛ)

1 + h0 + 3λΛ.

The constant c = (1 + ν)/(1 − 2ν) ≥ 1, because for real materials the Poissonratio 0 ≤ ν < 1/2 [1,2,3]. Therefore, for parameters λ, Λ ≥ 0 the condition (3.bri) istrue (Ψ < 1) only for the positive parameter of plastic hardening, satisfying thefollowing estimate

h0 > 3(c− 1)λΛ +√

3 (λ+ cΛ) ≥ 0. (5.bri)

If this condition is disturbed then the effects of bifurcation and internalinstability exist in the plasticity EVP (2.bri) [9,12].

The classic Drucker’s hypothesis σij Pij ≥ 0 is the only necessary conditionfor the uniqueness of solution of EVP (2.bri). For the model (4.bri) it has the following


form

h0 ≥ 3 (cΛ− λ)Λ. (6.bri)

For example, if Λ = 0, λ > 0 then the condition (5.bri) is carried out onlyfor h0 >

√3λ > 0, but the condition (6.bri) is fulfilled for h0 ≥ 0. This simple

example proves that the classic Drucker’s hypothesis, based on thermodynamicalpostulates [2,3,9,10], does not provide the existence of solution of the plasticityEVP (2.bri).

4 Computational method

For the numerical solution of the plasticity EVP (2.bri) the standard spatial piece-wise linear finite element approximation is here used: Ωh = ∪Th, Γh = ∂Ωh andvol(Ω\Ωh)→ 0, area(Γ\Γh)→ 0 for h→ 0 regularity, where Th is the simplestsimplex and h is the step of approximation [13].

For the displacement the following piecewise linear approximation is used

uh(t, x) = Uγ(t)Φγ(x) (γ = 1, 2, . . . ,m),

where Uγ ∈ R3 is the displacement in the node xγ , Φγ : Ωh → R is the standardcontinuous piecewise linear function such that Φγ(xα) = δαγ (α, γ = 1, 2, . . . ,m),m is the number of nodes. In this case the subspace V 0 ⊂ H1 is approximatedby the subspace V 0

h ⊂ R3m

V 0h =

U ∈ R3m : Uα = 0, xα ∈ Γ 1

h

.

The plasticity EVP (2.bri) is approximated by the Cauchy problem for nonlinearsystem of ordinary differential equations: vector function U : [0, 1] → V 0

h mustsatisfy the initial condition U(0) = 0 and the following differential equation foralmost every t ∈ (0, 1)

Apq(U, U)Uq = Bp, (7.bri)

where U is the global vector of free nodal displacements, A is the global shearstiffness matrix and in the end B is the global vector of nodal speeds of in-fluences; p, q = 1, 2, . . . , 3m. Here Up = Uγi with index p = 3(γ − 1) + i. Due tothe properties of the real plasticity models this equation is principally unsolvedregarding U in the explicit form.

For some plasticity models the differential system (7.bri) can be stiff. The maincause of this phenomenon consists of the following: matrix A has lines withsignificantly different factors (it is badly determined) for the small parameter ofplastic hardening h0 1 [4,5,6,7].

Example 3. Let the bounded rigid body Ω ⊂ R3 with the regular boundary Γconsist of incompressible material describing by the model (4.bri) with parametersλ = Λ = 0. The body is fastened on a portion Γ 1 of the boundary (i.e. uγ ≡ 0)


and deformed by the external forces. In this case the set of kinematically admis-sible displacements is

V 0div =

u ∈ V 0 : div(u(x)) = 0, x ∈ Ω

.

We use the following approximation for unknown displacement

uN (t, x) = Yr(t)wr(x) (r = 1, . . . , N),

where wrNr=1 ⊂ V 0div are the basic functions.

In this case the plasticity EVP (2.bri) is approximated by the Cauchy problem fornonlinear system of ordinary differential equations: vector function Y : [0, 1]→RN must satisfy the initial condition Y (0) = 0 and the following differentialequation for almost every t ∈ (0, 1)

Aqr(Y, Y )Yr = Bq (q, r = 1, 2, . . . , N), (8.bri)

where

Aqr(Y, Y ) =∫Ω

Ψqr(Y, Y )|ε(wq)| |ε(wr)| dx,

Ψqr = cos γqr − (1− ψ)H(ρe − ε∗)H(cos γ) cosγq cos γr,

Bq = (2µ)−1L(t, wq).

Here and in what follows the summing over indices q, r, s does not used, ρe =ρe(uN ), γ = γ(uN , uN ) from (4.bri), the parameter ψ = h0/(1 + h0) and

cos γs = (ρe |ε(ws)|)−1ρijεij(ws) (s = q, r),

cos γqr = (|ε(wq)| |ε(wr)|)−1εij(wq)εij(wr).

Due to the properties of the finite element approximation the matrix A issymmetrical and has the largest elements on the main diagonal

Aqq(Y, Y ) =∫Ω

Ψqq(Y, Y )|ε(wq)|2 dx,

Ψqq =[1− (1 − ψ)H(ρe − ε∗)H(cos γ) cos2 γq

].

If the solution of problem (8.bri) has the zone of active deformation with a smallcurvature trajectory (γ ∼ 0) then for basic functions wq with cos γq ≈ 1 thefactors Ψqq ≈ ψ. In the zone of passive deformation, or for a large curvaturetrajectory (γ ∼ π/2), or for basic functions wr with cos γr ≈ 0 the factorsΨrr ≈ 1.

It is easily seen that for the small parameter of plastic hardening (h0 1)the global shear stiffness matrix A has lines with significantly different factors(it is badly determined). As a result, the following estimate was proved in [4,6]

cond(A) :=νmax

νmin≥ C N2h−1

0 1,

where cond(A) is the condition number of matrix A; νmax and νmin are the largestand smallest eigenvalues of the matrix A, respectively, and C = const(N, h0).


According to standard technique [14,15] for the solution of the unsolved re-garding derivative and stiff problem (7.bri) the implicit Euler scheme is used

Apq(Uk + τV, V )Vq = Bk+1p , V ∈ V 0

h ,

Uk+1 = Uk + τV, U0 = 0,(9.bri)

where index k corresponds to the parameter tk = kτ , k = 0, 1, . . . , K − 1;τ = 1/K and K 1. Here and in what follows the summing over index k doesnot used.

For the numerical solution of algebraic system (9.bri) for every k = 0, 1, . . . , K−1the decomposition method of adaptive block relaxation (ABR) is used. Thismethod disregards the condition number of the matrix A and has the followingdescription [4,5,6,7].

Step 1. As the initial approach the explicit solution is used (here O is thezero vector)

Y (0)q = A−1

pq (Uk, O)Bk+1p .

Step 2. Due to the properties of the finite element approximation the ma-trix A has the largest elements on the main diagonal. Therefore, by the cur-rent approach Y (m) variables are separated on quick and slow ones by theproximity criterion of appropriate diagonal elements of the matrix A(m) =A(Uk + τY (m), Y (m)

)I(m)s =

p = 1, 2, . . . , N : ∆(s−1)/L ≤ A(m)

pp /d(m) < ∆s/L,

I(m)L = 1, 2, . . . , N\

L−1⋃s=1

I(m)s ,

where s = 1, 2, . . . , L − 1; ∆ = D(m)/d(m); D(m) and d(m) are the largest andsmallest diagonal elements of the matrix A(m), respectively, L = int(ω lg∆) + 1is the number of blocks (1 ≤ L ≤ N), ω ≥ 0 is the decomposition parameter.

Step 3. The block version of the Seidel method is used [16]. In practice onestep of this method is enough (here the summing over index s does not used)

Y (m+1) =w1 ⊕ w2 ⊕ · · · ⊕ wL

T,

wsi = [Λss]−1ij

(Ξsj −

s−1∑t=1

Λstjrwtr −

L∑t=s+1

Λstjrvtr

),

Λst =A(m)pq : p ∈ I(m)

s , q ∈ I(m)t

,

Ξs =Bk+1p : p ∈ I(m)

s

, vt =

Y (m)q : q ∈ I(m)

t

.

It is easily seen that the ABR method practically disregards the conditionnumber of the matrix A(m) because

cond (Λss) ∼ cond1/L(A(m)

) cond

(A(m)

)


for every s = 1, 2, . . . , L even if L = 2.By the new approach Y (m+1), variables are separated on quick and slow ones

too, etc.Step 4. For termination of the iteration process the following condition is

used ∣∣∣A(m)pq Y (m)

q −Bk+1p

∣∣∣ < ξ, (10.bri)

where ξ is the prescribed precision.

Theorem 4. In the framework of the hypotheses (H1)–(H7) the following state-ments are true:

(i) The solutions of systems (7.bri) and (9.bri) exist.(ii) The ABR method converges: lim

m→∞Y (m) = V .

Proof. According to the properties of the finite element approximation for theconstitutive relation satisfying the conditions (H1)–(H3) the vector functionApq(U + τY, Y )Yq : R3m → R3m is strongly monotonous in Y for everyU ∈ R3m and τ ∈ [0, 1] [8,13]. Therefore, according to the classic results ofthe theory of ordinary differential equations [15] and algebra [16] the statements(i) and (ii) are true.

Remark 5. In the computational mathematics the Schwarz decomposition meth-ods are well known [17]. But they are used only for linear BVPs without the mainidea of adaptiveness (see References in [17]).

5 Numerical results

The numerical analysis was realized on series of BVPs with model (4.bri) for the ax-isymmetrical kinematic deformation of long round tube fastened on the internalradius ρ = a. The complicated plane deformation was given by different regimesof the displacement on the external radius ρ = b [4,6,7]: (here the summing overindices ϕ and ρ does not used)

u0ϕ(t) = CϕZϕ(t), u0

ρ(t) = CρZρ(t)

where t ∈ [0, 1], Cϕ = ε∗b(1−a2/b2) and Cρ =√

32 Cϕ are the maximum external

displacements for which the clearly elastic deformation is realized in the frame-work of the classic model of plasticity (i.e. for the model (4.bri) with parametersλ = Λ = 0) [6].

In the computer experiments the following data were used: a = 10, b = 20(mm), k0 = 105, µ = 7.5 · 104 (MPa), ε∗ = 5 · 10−3, h0 = 0.001 and λ = Λ = 0 inthe model (4.bri). The radius [a, b] was discretized by 50 segments and the standardpiecewise linear approximation was used for unknown functions uϕ(t, ρ) anduρ(t, ρ) such that uϕ ≡ 0, uρ ≡ 0 for ρ = a and uϕ ≡ u0

ϕ(t), uρ ≡ u0ρ(t) for ρ = b.


Fig. 1. The tangential displacement in the end of the simplest radial regime

The ABR method with the decomposition parameter ω = 0.5 was comparedwith the standard method of simple iterations which equals the ABR methodwith parameter ω = 0.

For the simplest radial regime of clear twisting Zϕ(t) = 10t, Zρ(t) ≡ 0 inthe implicit Euler scheme (9.bri) K = 100 steps over the parameter of loading wereused. In figure 1 the following solutions in the end of process are shown: curves1 and 2 correspond to the standard method with the single ξ = 10−3 and doubleξ = 10−5 precision in the criterion (10.bri), respectively; curve 3 corresponds to theABR method with the single precision. The last numerical solution (curve 3)practically equals the analytical solution which was built in [4,5].

For the complicated cyclic regime Zϕ(t) = 10 sin(4πt), Zρ(t) = 10 sin(2πt) inthe scheme (9.bri) K = 800 steps over parameter t ∈ [0, 1] were used. In figure 2 thefollowing solutions in the end of process are shown: curves 1 and 2 correspond tothe standard method with the single and double precision, respectively; curve 3corresponds to the ABR method with the single precision.

In all experiments the time of calculation with single precision was approx-imately equal for both methods; whereas with double precision, the time ofcalculation was longer for the standard method than for the ABR method.

It is easily seen that, for finding the displacement and the time of calculation,the ABR method has advantages over the standard method.


Fig. 2. The tangential displacement in the end of the cyclic regime

6 Conclusion

The questions of mathematical correctness and effective numerical solution forthe plasticity BVP have been discussed. By using the evolution variationalmethod: 1) the general algebraic criterion of mathematical correctness for plas-ticity models has been constructed; 2) the effective qualitative FE analysis hasbeen realized. As a result, an original implicit adaptive strategy has been pre-sented for the numerical simulation of practically important plastic and similareffects in the Mechanics of Solids.

Acknowledgement

I would like to thank Professor Yu. I. Kadashevich for consultations and theOrganizing Committee and sponsors of the conference on Differential Equationsand their Applications (EQUADIFF 9) for their support of my visit to Brno,Czech Republic.


References

[1] Novozhilov, V. V., Kadashevich, Yu. I.: Microstresses in Technical Materials.Mashinostrojenije, Leningrad (1990)

[2] Hill, R.: Classical plasticity: a retrospective view and a new proposal. J. Mech.and Phys. Solids. 42(11) (1994) 1803–1816

[3] Kljushnikov, V. D.: Mathematical Theory of Plasticity. Moscow State Univ. press.,Moscow (1979)

[4] Brigadnov, I. A.: Methods of solution of elastoplastic boundary value problemsfor small hardening materials. Ph.D. Thesis. Leningrad Polytech. Inst., Leningrad(1990)

[5] Brigadnov, I. A., Repin, S. I.: Numerical solution of plasticity problems for ma-terials with small strain hardening. Mekh. Tverd. Tela 4 (1990) 73–79; Englishtransl. in Mech. of Solids 4 (1990)

[6] Brigadnov, I. A.: On the numerical solution of boundary value problems for elasto-plastic flow. Mekh. Tverd. Tela 3 (1992) 157–162; English transl. in Mech. of Solids3 (1992)

[7] Brigadnov, I. A.: Mathematical correctness and numerical methods for solution ofthe plasticity initial-boundary value problems. Mekh. Tverd. Tela 4 (1996) 62–74;English transl. in Mech. of Solids 4 (1996)

[8] Gajewski, H., Groger, K., Zacharias, K.: Nichtlineare Operatorgleichungen undOperatordifferentialgleichungen. Akademia-Verlag, Berlin (1974)

[9] Noll, W.: Lectures on the foundations of continuum mechanics and thermodynam-ics. Arch. Rat. Mech. Anal. 52 (1973) 62–92

[10] Pal’mov, V. A.: Reological models in the non-linear mechanics of solids. Advancesin Mech. 3(3) (1980) 75–115

[11] Novozhilov, V. V.: On the plastic loosening. Prikl. Mat. Mekh. 29(4) (1965) 681–689; English transl. in J. Appl. Math. Mech. 29(4) (1965)

[12] Garagash, I. A., Nikolajewskiy, V. N.: Non-associated laws of flow and localizationof plastic strains. Advances in Mech. 12(1) (1989) 131–183

[13] Ciarlet, Ph. G.: The Finite Element Method for Elliptic Problems. North-Hollandpubl. co., Amsterdam etc. (1980)

[14] Rakitskiy, Yu. V., Ustinov, S. M., Chernorutskiy, I. G.: Numerical Methods forSolution of Stiff Systems. Nauka, Moscow (1979)

[15] Hairer, E., Wanner, G.: Solving Ordinary Differential Equations. Pt.2. Stiff Dif-ferential Algebraic Problems. Springer, Berlin etc. (1991)

[16] Collatz, L.: Funktionalanalysis und Numerische Mathematik. Springer, Berlin(1964)

[17] Dryja, M., Widlund, O. B.: Schwarz methods of Neumann-Neumann type forthree-dimensional elliptic finite element problems. Communic. Pure & Appl. Math.48 (1995) 121–155



Dense Sets and Far-field Patterns for the VectorThermoelastic Equation

Fioralba Cakoni

Department of Mathematics, Faculty of Natural Sciences,University of Tirana

Tirana, AlbaniaEmail: [email protected]

[email protected]

Abstract. We study the set of far-field patterns which are generated byentire incident thermoelastic fields scattered by a boundary nopenetra-ble obstacle. Necessary and sufficient conditions are given for the set tobe dense in the set of all square integrable vector fields defined on theunit sphere. The method of Herglotz thermoelastic function is utilizedto prove the dense properties of the asymptotic fields.

AMS Subject Classification. 35L, 73D, 47A

Keywords. Scattering theory, thermoelastic equation, far-field pattern

1 Introduction

A basic problem in inverse scattering theory is the classification of far field pat-terns corresponding to the scattering of a time harmonic thermoelastic incidentwave by a bounded, connected obstacle. Indeed, if T denotes the operator map-ping the incident field and scattering obstacle onto the far field patterns, thenthe inverse scattering problem is to construct T −1 defined on the range of T ,and the determination of this range is nothing more than the description of theclass of far field patterns. It is easily verifiable that the class of functions thatcan be far field patterns is a subset of the class of the entire functions for eachpositive fixed value of the wave numbers. The crucial point is the question if thefar field patterns for a fixed obstacle and all incident plane wave are completein a production of L2(Ω). Colton, Kress and Kirsch [2,3] gave an answer of thisquestion for some acoustic and electromagnetic scattering problems. Dassios [4,5]has investigated the case of elastic rigid scattering problems where the situationbecomes much harder since in elasticity, besides the vectorial (displacement, sur-face traction) as well as the tensorial (stress, strain) characteristics of the fields,there are two separate wave solutions propagating at different phase velocities.The purpose of this work is to extend the mentioned results to coupled ther-moelasticity. In this case, there are five types of waves present, two of which


74 Fioralba Cakoni

are longitudinal elastic, one is transverse elastic, and two are thermal waves[7]. Consequently, there are four complex dimensionless parameters by means ofwhich the different wave numbers are connected. The situation is much morecomplicated.In particular we shall show that the set of thermoelastic far field patterns cor-responding to the scattering of the entire incident fields by a bounded ther-moelastic rigid at zero temperature obstacle is dense in the production spaceof square integrable function on ∂Ω if and only if does not exist a eigenfunc-tion of an eigenvalue problem which is a thermoelastic Herglotz function. Thisresult will be established by first constructing an appropriate complete set ofthe function defined on the boundary of the scattering obstacle and then estab-lishing an integral representation for the displacement part of the thermoelasticHerglotz function. In order to avoid the difficulties come from the existence ofpolarization of the transverse displacement wave, we have to raise the rank ofthe tensorial character of the fields involved by one. This idea is used by Twerskyin electromagnetic scattering and after by Dassios in elastic case.

2 Scattering Problems

The direct scattering problem asks: given an open domain V ⊂ R3 with connectedC2 boundary S and V e = R3 \ V , given a plane incident wave of time harmonicdependence e−iωt

Ui(r, k) : V e → R4 (1.cak)

(k the direction of propagation), determine in V e a solution

U(r, k) = Ui(r, k) + Us(r, k) (2.cak)

of the equation

L(∂r)U(r, k) =(

(µ∆+ ρω2)I3 + (λ+ µ)∇∇· −γ∇qκη∇· ∆+ q

)(u(r, k)Θ(r, k)

)= 0 (3.cak)

such that

Bk(∂r, n)U(r, k) = 0 (4.cak)

on S, where the boundary differential operator Bk(∂r, n) is expressed via thethermoelastic surface traction operator.More over U(r, k) has to satisfy the asymptotic Kupradze condition as r → ∞[7].In the theory of direct problems in thermoelasticity [1,4], it is shown how thesolution of a boundary value problem and related far-field corresponding to anincident field and to a given obstacle can be calculated. We call the set of vectors

Pj0 : Ω → C3; j = 1, 2, s

=

=P 1r0(r, k)r, P 2

r0(r, k)r, P sθ0(r, k)θ + P sφ0(r, k)φ

(5.cak)

Thermoelastic Far-field Patterns 75

defined on the unite sphere Ω, as set of thermoelastic far-field patterns corre-sponding to the thermoelastic radiation solution U(r), propagating in the di-rection k. The thermal component is cancelled in this definition because of lin-ear dependence between the module of asymptotic displacement fields P 1

r0(r, k),P 2r0(r, k) and the asymptotic thermal fields t10(r, k), t20(r, k) [1,4] with known co-

efficient. Let us do the following symbolic notation U∞ =Pj

0, j = 1, 2, s∈

[L2(Ω)]3. In terms of the mapping

F : U→ U∞ (6.cak)

we want to solve the equation FU = U∞We have proved in [1] by means of Atkinson expansion theorem the followingresult known as the correspondence theorem.There exists one to one correspondence between:elastothermal far-field pattern P 1

r0 and elastothermal 4-dimensional part of radi-ation solution U1 ;thermoelastic far-field pattern P 2

r0 and thermoelastic 4-dimensional part of radi-ation solution U2 ;transverse far-field patterns P sθ0, P

sφ0 and elastothermal 3-dimensional part of ra-

diation solution Us = (us, 0) .In other words, the mapping (6.cak) is an one to one in its range.

Theorem 1. The regular solution of thermoelastic equation (3.cak) allows the fol-lowing representation of the displacement part and the temperature part

u(r) = ∇[−λ+ 2µ

ρω2Φ1 +

γ

ρω2Θ1 −

λ+ 2µρω2

Φ2 +γ

ρω2Θ2

]+∇×

[rΨ(r) +

1ks∇× (rχ(r))

], (7.cak)

Θ(r) = Θ1(r) +Θ2(r) , (8.cak)

where the potentials Φ1, Φ2, Ψ, χ,Θ1, Θ2 solve the scalar Helmholtz equation

(∆+ k21)Φ1 = (∆+ k2

1)Θ1 = 0, (9.cak)

(∆+ k22)Φ2 = (∆+ k2

2)Θ2 = 0, (10.cak)

(∆+ k2s)Ψ = (∆+ k2

s)χ = 0, (11.cak)

where k1, k2, k3 are wave numbers.

The prove is a consequence of Kupradze decomposition [7] by straightforwardcalculation. We write again the above relations as U = U1 + U2 + Us

U1 =(∇[−λ+2µ

ρω2 Φ1 + γρω2Θ1]

Θ1

), (12.cak)

U2 =(∇[−λ+2µ

ρω2 Φ2 + γρω2Θ2]

Θ2

), (13.cak)

Us =(∇× [rΨ(r) + 1

ks∇× (rχ(r))]

0

), (14.cak)

76 Fioralba Cakoni

where the irrotational and solenoidal part of the displacement fields are distin-guished.

3 Herglotz thermoelastic function

We restrict so far to the rigid at zero temperature scattering thermoelastic prob-lem.

Definition 2. An thermoelastic Herglotz function is defined to be a classicalsolution of the thermoelastic equation (3.cak) U(r) in all of R3, which satisfies thegrowth condition

limr→∞

sup1r

∫B(o,r)

‖U(r′)‖2dv(r′) <∞ . (15.cak)

Using orthogonality of the vector spherical harmonics we can easily verify

Proposition 3.

[L2(Ω)]3 = [L2r(Ω)]3 ⊕ [L2

t (Ω)]3 , (16.cak)

where [L2r(Ω)]3 is the subspace spanned by the set Pm

n and [L2t (Ω)]3 is the

subspace spanned by the set Bmn ∪ Cm

n .

This implies that for every f ∈ [L2r(Ω)]3 we have the unique expansion in L2

sense

f(r) = fr(r) + ft(r) =∞∑n=0

n∑m=−n

[αmn Pmn (r)] + [βmn Bm

n (r) + γmn Cmn (r)] . (17.cak)

Moreover U(r) = fn(kr)Yn(r) where fn(r) denotes any spherical Bessel func-tion, is a solution of the Helmholtz equation (∆+ k2)U(r) = 0. Obviously, fromthese considerations we get

Proposition 4. Every solution of thermoelastic equation in a regular domainsatisfies the following unique decomposition.

u(r) = u1(r) + u2(r) + us(r) =∞∑n=0

n∑m=−n

[αm1nLm1n(r) +

+ αm2nLm2n(r) + βmn Mmn (r) + γmn Nm

n (r)] , (18.cak)

Θ(r) = Θ1(r) +Θ2(r) =

=∞∑n=0

n∑m=−n

δm1nfn(k1r)Yn(r) + δm2nfn(k2r)Yn(r) . (19.cak)

The convergence is considered to be in the L2 sense.


The Navier elastostatic eigenvectors are given by

Lm(1,2)n(r) =1

k(1,2)∇[fn(k(1,2)r)Yn(r)

]=

=[ d

d(k(1,2)r)fn(k(1,2)r)

]Pmn (r) +

+√n(n+ 1)

fn(k(1,2)r)k(1,2)r

Bmn (r) , (20.cak)

Mmn (r) = ∇×

[rfn(ksr)Yn(r)

]=√n(n+ 1)fn(ksr)Cm

n (r) , (21.cak)

Nmn (r) =

1ks∇×∇×

[rfn(ksr)Yn(r)

]=

= n(n+ 1)fn(ksr)ksr

Pmn (r) +

+√n(n+ 1)

1ksr

[ d

d(ksr)(ksrfn(ksr))

]Bmn (r) (22.cak)

and satisfy the vector Helmholtz equation

(∆+ k2(1,2))L

m(1,2)n = (∆+ k2

s)Mmn = (∆+ k2

s)Nmn = 0 . (23.cak)

We recall the result from [5] that there is a one to one correspondence betweenvector spherical harmonics and elastostatic eigenvectors. In the terminology thatwill be introduced latter, spherical harmonics Pm

n ,Cmn ,B

mn are the vector Her-

glotz kernels of the elastostatic eigenvectors Lmn ,Mmn ,N

mn with fn = jn respec-

tively. In other words, if the entire elastostatic eigenvectors are to be decomposedin plane waves propagating in all directions, then the corresponding vector spher-ical harmonics provide the distribution of the amplitudes over directions. Thesame between the solid harmonics and spherical harmonics.The behavior of Herglotz solution of thermoelastic equation (3.cak) and in particu-lar the connection of the its displacement part with the far-field patterns theygenerate are the main subject of the following.Let U : R3 → C4 an Herglotz function that satisfies the thermoelastic equationin the classical sense. Then, by the completeness of the elastostatic eigenvectorsand spherical harmonics and the general theory of eigenfunction expansions weobtain for the displacement and temperature part the expansions (18.cak), (19.cak) re-spectively. The asymptotic analysis of the above expansions as r →∞ gives forthe far-field patterns generated by the Herglotz solution the following formulas

L1(r) =ik1

2

∞∑n=0

n∑m=−n

i−nαm1nPmn (r) , (24.cak)

L2(r) =ik2

2

∞∑n=0

n∑m=−n

i−nαm2nPmn (r) , (25.cak)

78 Fioralba Cakoni

T(r) =12

∞∑n=0

n∑m=−n

in√n(n+ 1)[βmn Cm

n (r) + iγmn Bmn (r)] (26.cak)

and for the temperature part

l1(r) =ik1

2

∞∑n=0

n∑m=−n

i−nδm1nYmn (r), l2(r) =

ik2

2

∞∑n=0

n∑m=−n

i−nδm2nYmn (r) . (27.cak)

By virtue of the Riesz-Fisher theorem and the relations involving the Fouriercoefficients of (18.cak), (19.cak) we claim that the far field patterns L(1,2), T are welldefined in the L2-sense. The coefficient of the linear dependence between thedisplacement far fields and temperature far fields is qκηik(1,2)

k2(1,2)−q

see [1].

The most important result in the theory of Herglotz functions is given by thefollowing representation theorem.

Theorem 5 (Representation). If U is an Herglotz solution of the thermoe-lastic equation (3.cak), then there are functions L1,L2,T : Ω → C4 which belongsto L2(Ω) (i.e. the corresponding far field patterns), such that

U(r) =1

2π

∫Ω

L1(k)ek1ik·rds(k) +1

2π

∫Ω

L2(k)ek2ik·rds(k) +

+1

2π

∫Ω

T(k)eksik·rds(k) . (28.cak)

Conversely, if U is given by L1,L2,T in L2(Ω), then it is a thermoelastic Her-glotz function.

The L2 functions L1,L2,T are known as the Herglotz kernels. The proof argu-ment of the first part is the interpretation of the series (24.cak), (25.cak), (26.cak), (27.cak) as thecorresponding Herglotz kernels. Using orthogonality arguments of the consideredeigenfunctions and the uniform convergence we obtain the growth condition ofthe Herglotz function provided the L2 Herglotz kernels exist. Theorem of therepresentation furnishes a proof that U is uniquely determined by L1,L2,T.This result is also obtainable from the unique determination of the Fourier co-efficients of the expansions for U and L1,L2,T in the appropriate eigenvectorsfor both components, displacement and temperature.

4 Dense properties of the far field patterns

Let us turn back to the functional equation (6.cak). The correspondence theoremprovides the uniqueness of its solution. As we proved the thermoelastic far fieldpatterns (which are defined as the displacement amplitudes) must satisfy theexpantion (24.cak), (25.cak), (26.cak). By using the technique of Colton, Kress [2] (theorem2.15) and the fact that the elastostatic eigenvectors are expressed via sphericalwave functions we easily verify that the existence of a solution requires a kind ofgrowth condition of the Fourier coefficients (it is analytically complicated that is


why we do not present it here)to be satisfied, for a given function U∞ ∈ [L2(Ω)]3.So, the solution of equation (6.cak) will, in general, not exist. The argument of[2, p. 36] is valid here to claim more over that, if a solution U does exist itwill not depend continuously on U∞ in any reasonable norm. That is that theequation (3.cak) is ill-posed. The image of the linear operator F is not equal to[L2(Ω)]3.But, is the far field patterns for a fixed rigid obstacle at zero temperature and allincident thermoelastic plane waves complete in [L2(Ω)]3?In order to have an answer of this question we need the following dense result.To avoid long repetitions of requirements upon fields, we introduce the followingspaces:

– the space of incident Herglotz-type field

H(R3) =Φ : R3 → R4

∣∣ Φ(r) =

=∫Ω

L1(k)eik1k·r + L2(k)eik2k·r + T(k)eiksk·r; L1,L2,T ∈ L2(Ω)

(29.cak)

– the space of scattered fields

S(V e) =U : V e → R4

∣∣U ∈ C2(V e) ∩ C(Ve) s.t. U

satisfies the equation (3.cak) and the Kupradze condition at ∞ (30.cak)

– the space of rigid at zero temperature solutions

P(V e) = Ψ = U + Φ : V e → R4 | U ∈ S, Φ ∈ H, r ∈ ∂V Ψ(r) = 0 (31.cak)

– the space of traction traces

RP(∂V ) =RΨ : ∂V → R4;Ψ ∈ P

(32.cak)

Theorem 6. The space RP(∂V ) is dense in L2(∂V )

Proof. From [6, Theorem 2] we have that if g ∈ L2(∂V ) and

(T Ψ,g) =∫∂V

g · T(∂r, n)Ψ(r)ds(r) = 0

for every T Ψ ∈ RP(∂V ) · I3, then g = 0 almost everywhere on ∂V . The samefor the operator ∂n. Now, let us consider a 4-dimensional vector G ∈ L2(∂V ).

The shape of operator R(∂r, n) =(

R(∂r, n) −γn0 ∂n

)implies that G = 0 if for

every RΨ ∈ RP(∂V ),

(RΨ,G) =∫∂V

G · R(∂r, n)Ψ(r)ds(r) = 0

which ends the proof of the theorem.

80 Fioralba Cakoni

It is well known that there are three types of plane displacement fields andtow types of plane temperature fields that can propagate in a thermoelasticmedium. The displacement waves depend on two orthogonal vectors which arek and p ⊥ k respectively for two types of longitudinal waves and one typeof transversal wave. These reflects the same vectorial nature of the far fieldpatterns (5.cak). The existence of two perpendicular vector complicates the study ofthe far field patterns. To avoid this difficulty we raise the rank of the tensorialcharacter of the fields involved by one. Then the incident displacement fieldsdepend only on one vector, the direction of propagation, while the transversepolarization vector is now replaced by 2-dimensional complement of the directionof propagation.Our tensorial thermoelastic model is as following.The scatterer is exited by a tensorial 4× 3 thermoelastic time harmonic wave

Ui(r; k) = A1(k⊗ k, β1k)eik1k·r +A2(β2k⊗ k, k)eik2k·r +

+As(I3 − k⊗ k, 0)eiksk·r . (33.cak)

The tensorial incident field can be interpreted as a tensor superposition of threevector fields which appear as the first vectors of tensors, while the second vectorsare provided by the incident orthogonal base k, θk, ϕk. This tensorial characterof the incident field is inherited in the scattered field Us = (us, Θs) it generates

us(r; k) = us1(r; k, k)⊗ k + us2(r; k, k)⊗ k +

+ uss(r; k, θk)⊗ θk + uss(r; k, ϕk)⊗ ϕk , (34.cak)

Θs(r; k) = Θs1(r; k)k +Θs2(r; k)k .

Then the total tensorial field

U(r; k) = Ui(r; k) + Us(r; k) (35.cak)

solve the tensorial thermoelastic coupled system

µ∆u(r; k) + (λ+ µ)∇⊗∇ · u(r; k) = γ∇⊗Θ(r; k) , (36.cak)

∆Θ(r; k) +iω

κΘ(r; k) = −iω∇ · u(r; k) (37.cak)

and the same Kupradze asymptotic conditions as r →∞. More over U(r; k) = 0on S.The asymptotic analysis uniform over Ω leads to the tensorial shape of far fieldpatterns

U∞ =Pj

0, j = 1, 2, s∈ [L2(Ω)]9 . (38.cak)

Note that in accord with known results the radial patterns P10, P

20 are the lon-

gitudinal wave of displacement part propagating along r and Ps0 is transversal

spherical wave propagating along r and polarized orthogonally to r. Obviously,


the definition and the results of thermoelastic Herglotz function may be trans-lated to the terms of tensors. Everything remains true provided the vectors arereplaced by tensors.An answer of the question whether is the set of far field patterns complete in[L2(Ω)]3 is given by the following theorem

Theorem 7. Let (kn) be a sequence of unit vectors that is dense on Ω.

[L2(Ω)]9 = spanU∞(r, kn) (39.cak)

if and only if there does not exist a Herglotz thermoelastic function

U =(

uΘ

)=(

u1 + u2 + us

Θ1 +Θ2

)(40.cak)

such that

u = u1 ⊗ k + u2 ⊗ k + us ⊗ θ + us ⊗ ϕ (41.cak)

is an eigenfunction of the interior eigenvalue problem(∆∗ + ω2)u = 0∇ · u = 0 in V

u = 0 on S

(42.cak)

where ∆∗ is elastostatic operator.

Proof. Recall that W is complete in the Hilbert space X if and only if (w,ϕ) = 0for all w ∈ W implies that ϕ = 0.Let us write the dual relation in the space [L2(Ω)]9 for every l1, l2, t ∈ [L2(Ω)]9∫

Ω

[P1r0(r; kn) + P2

r0(r; kn) + Pst0(r; kn)

]:

:[l1(r; kn) + l2(r; kn) + t(r; kn)

]ds(r) = 0 , (43.cak)

which in a simpler symbolic way should be written∫Ω

U∞(r; kn)K(r; kn)ds(r) = 0 , (44.cak)

where r ∈ Ω and n = 1, 2, . . . The relation (43.cak) implies that exists a nontrivialthermoelastic Herglotz function, which we take as incident wave Ui, (with ker-nels for displacement part l1, l2, t ∈ [L2(Ω)]9) for which the far field patterns ofthe corresponding scattered wave Us is U∞ = 0. By the one to one correspon-dence between the radial solution and corresponding far field patterns we havethat the vanishing far field U∞ = 0 on Ω is equivalent to Us = 0 in V e. By theboundary condition Us+Ui = 0 on S and the uniqueness Dirichlet thermoelasticeigenvector problem [7], this is equivalent to Ui = 0 on S. Again by Kupradze

82 Fioralba Cakoni

[7], the first thermoelastic eigenvalue problem is equivalent with the eigenvalueproblem (42.cak). The above result gives the basic tool to construct a Colton-Monktype algorithm for the thermoelastic inverse scattering problem correspondingto a rigid scatterer at zero temperature of the reconstruction the structures.This inverse frame means: given the thermoelastic incident field Ui ∈ H andby the knowledge of the far field patterns U∞ correspond to the scattered fieldUs ∈ S(V e), Dirichlet boundary condition Ψ = 0, Ψ ∈ P(V e) and the equa-tions (3.cak) which govern the phenomena, one determines the geometrical shape ofthe boundary ∂V . Apriori conditions are given about the boundary, for exampleto be star shape.

References

[1] F. Cakoni, G.Dassios, The Atkinson-Wilcox expansion theorem for thermoelasticwaves, (accepted to be published in Quart. Appl. Math.).

[2] D. Colton, R. Kress, Inverse acoustic and electromagnetic scattering theory,Springer-Verlag, (1992).

[3] D. Colton, A. Kirsch, Dense sets and far-field patterns in acoustic wave propagation,SIAM J. Appl. Math., 15 (1984), 996–1006.

[4] G. Dassios, V. Kostopoulos, The scattering amplitudes and cross sections in thetheory of thermoelasticity, SIAM J. Appl. Math., 48, 1 (1988), 1283–1284.

[5] G. Dassios, Z.Rigou, Elastic Herglotz function, SIAM J. Appl. Math., (in print).[6] G. Dassios, Z.Rigou, On the density of traction traces in scattering of elastic waves,

SIAM J. Appl. Math., 53/1 (1993), 141–153.[7] V. Kupradze, Tree-dimensional problems of the mathematical theory of elasticity

and thermoelasticity, North-Holland, New York. (1979).



Transformations of Differential Equations

Jan Chrastina

Department of Mathematics, Faculty of Science, Masaryk University,Janackovo nam. 2a, 662 95 Brno, Czech Republic

Abstract. The article concerns the second order differential equationswith one unknown function and the aim is twofold: to compare some re-sults for the well-known linear with the more complicated nonlinear case,and to point out some distinctions between ordinary and partial differen-tial equations. We shall mention automorphisms permuting the conjugatepoints, moving frames for particular fiber-preserving mappings, the Dar-boux transformations of ordinary differential equation, and the Laplaceseries for the hyperbolic case of two independent variables.

AMS Subject Classification. 34K05, 35A30, 35L10

Keywords. Dispersions, contact transformation, moving coframe, diffi-ety, Laplace series

Our reasoning will be developed in the real smooth category. As a rule, weshall not specify the definition domains and our primary aim is to outline somenew ideas and methods rather than to derive certain definite theorems.

For the convenience of a possible reader, let us outline the contents. We be-gin with the family of all equations (2.chr). It is preserved if transformations (1.chr)are performed, and certain self-transformations of this kind (so called centraldispersions) of a given equation (2.chr) are determined by the location of roots ofsolutions: they permute the roots. This result can be easily adapted for the non-homogeneous family (5.chr) subjected to a broader class (6.chr) of transformations, thenthe intersections of solutions undertake the previous role of roots. These well-known results can be verified by a manner which can be carried over the class ofall nonlinear equations (7.chr) subjected to contact transformations. In particular,certain automorphisms of a given equation (7.chr) exist which permute the intersec-tion points of infinitesimally near couples of solutions. They may be regardedfor nonlinear generalization of dispersions.

Our next aim is to determine some subfamilies of the family of all equations(7.chr) which are preserved if all transformations of the kind (6.chr) are applied. We usethe moving frames.

On the other hand, a given equation (2.chr) can be transformed into the familyof all equations (2.chr) by many rather peculiar transformations, even if the inde-pendent variable x is kept fixed. They can be explicitly found and the famous


84 Jan Chrastina

Darboux transformation of eigenvalue problems appears as a very particular sub-case.

As yet the transformations did not much change the order of derivatives.However passing to partial differential equations, already the classical Laplaceseries has quite other properties: invertible transformations of not too specialequations (16.chr) exist where the new unknown function U(x, y) may depend onderivatives of arbitrarily high order of the primary unknown u(x, y). This is awell-known result but we again pass to a nonlinear generalization: the Laplacecoframes permit to determine all invertible mappings of a given equation (18.chr)into the class of all equations (18.chr), at least in principle since the calculationsare rather complicated. The cases when the independent variables need not bepreserved are involved. We can state only a modest illustrative example of theequation ∂2u/∂x∂y = g(∂u/∂x) + u here with new independent variables X =x − g′(∂u/∂x), Y = y and new unknown function (23.chr). The method can begeneralized and applied to higher order equations, as well.

1 The dispersion theory [1], [7]

We find ourselves in the plane x, y, where new variables X = X(x, y), Y =Y (x, y) can be introduced. In particular transformations of the kind

X = X(x), Y = c|X ′(x)|1/2 y (c = const. 6= 0, X′(x) 6= 0) (1.chr)

are the most general ones which preserve the family of all equations

d2y/dx2 = q(x)y, (2.chr)

i.e., which turn every equation (2.chr) into certain d2Y/dX2 = Q(X)Y . It may beproved that under transformations (1.chr), equations (2.chr) are locally like each other.Roots of solutions y are obviously transformed into roots of solutions Y andthis trivial remark can be developed to give the global theory. In particular,in the oscillatory subcase, there exist automorphisms (1.chr) of equation (2.chr) per-muting the roots of solutions y, the so called central dispersions of (1.chr). Sincethe transformations (1.chr) between two mentioned equations can be determined assolutions of a certain nonlinear third order differential equation (depending onq,Q) for the function X, it follows in the particular case of automorphisms thatthe distribution of roots of solutions y is governed by a third order differentialequation.

2 A note to proofs [3]

The shortest way to the mentioned results consists in introduction of functionζ = y/y, where y, y are two independent solutions of (1.chr). The value ∞ at theroots of y with obvious rules of calculations should be admitted. Then

ζ ′ = c/y2, y = (c/ζ′)1/2, q = y′′/y = |ζ′|1/2

(|ζ′|−1/2

)′′, (3.chr)

Transformations of DE 85

where c = const. 6= 0 and the last expression is the familiar Schwarz deriva-tive independent of the choice of y, y. Conversely, every such a function ζ(x)with ζ ′(x) 6= 0 can be arbitrarily chosen in advance. Then the equation (2.chr) isdetermined by (3.chr3), automorphisms (1.chr) of (2.chr) are (obviously) given by formula

ζ (X(x)) =αζ(x) + β

γζ(x) + δ(α, β, γ, δ constants with αδ 6= βγ)

which provides a third order equation for the function X(x) by applying theSchwarz derivative, and the central dispersions appear as a particular subcaseζ (X(x)) = ζ(x). Continuing in this way, analogous function Z = Y /Y and theformula

Z(X) =αζ(x) + β

γζ(x) + δ(α, β, γ, δ constants with αδ 6= βγ) (4.chr)

(obviously) provides all transformation into the equation d2Y/dX2 = Q(X)Y .

3 Nonlinear dispersions

The above results can be carried over the broader family of all equations

d2y/dx2 = q(x)y + r(x) (5.chr)

subjected to the transformations of the kind

X = X(x), Y = c|X ′(x)|1/2 y + Z(x) (c = const. 6= 0, X ′(x) 6= 0). (6.chr)

The previous role of the roots of solutions is undertaken by the points of inter-section of pairs of solutions in this non-homogeneous case. We shall be howeverinterested in still broader family of all nonlinear equations

d2y/dx2 = f(x, y, dy/dx). (7.chr)

It may be easily seen that contact transformations

X = X(x, y, y′), Y = Y (x, y, y′), Y ′ = Y ′(x, y, y′) (8.chr)

are the most general ones which preserve the family (7.chr), that is, which turnevery equation (6.chr) into certain d2Y/dX2 = F (X,Y, dY/dX). (Indeed, owingto (8.chr), differential form dY − Y ′dX should be a linear combination of formsdy − y′dx and dy′ − fdx with arbitrary f , hence a multiple of dy − y′dx.) Itmay proved that under contact transformations, equations (7.chr) are locally likeeach other. Instead of common methods, we shall derive this well-known resultby a geometrical reasoning which will be subsequently related to (nonlinear)dispersions.

86 Jan Chrastina

Let y = y(x, a, b) be a complete solution of (7.chr). Keeping a, b fixed for a mo-ment, choose a near solution with a common point x, y . In other terms, wesuppose

y = y(x, a, b) = y(x, a+ ε, b+ δ), y′ = yx(x, a, b). (9.chr)

Analogously let Y = Y (X,A,B) be a complete solution of the equation d2Y/dX2

= F . Choose fixed A = A(a, b), B = B(a, b) such that there exists a commonpoint X, Y with the corresponding near solution, that means, we may write

Y = Y (X, A,B) = Y (X, A(a+ ε, b+ δ), B(a+ ε, b+ δ)),Y ′ = Y X(X, A,B).

(10.chr)

Keeping ε, δ fixed but a, b (hence x, y) variable, the invertible transformation(x, y, y′) −→ (X, Y , Y ′) appears. If ε, δ = δ(ε) −→ 0, we obtain even a con-tact transformation (as follows by simple geometrical arguments or by directverification) implicitly given by formulae

y = y(x, a, b), y′ = yx(x, a, b), Y = Y (X, A,B), Y ′ = Y X(X, A,B),

ya(x, a, b) + λyb(x, a, b) = 0 = YA(X, A,B)(Aa + λAb)+ YB(X, A,B)(Ba + λBb),

where A = A(a, b), B = B(a, b), λ = δ′(0) and the parameters a, b, λ should beeliminated. Since every curve y = y(x, a, b) is (obviously) transformed into thecurve Y = Y (X,A,B), the equation (7.chr) turns into d2Y/dX2 = F .

We shall mention two particular kinds of this construction.Assuming f(x, y, y′) = F (x, y, y′), we deal with automorphisms of equation

(7.chr). Since the functions A = A(a, b), B = B(a, b) can be (in principle) quitearbitrarily chosen, there is a huge family of them. In the case of oscillatoryequation, the simple choice A = a and B = b gives (besides the identity) theautomorphisms permuting the conjugated points: the common point x, y of twoinfinitesimally near solutions (cf. (9.chr1) with ε, δ near to zero) can be transformedinto the next intersection point X, Y of the same pair of solutions (cf. (10.chr1) withY = y,A = a,B = b). So we have a nonlinear generalization of dispersions.

Assuming f(x, y, y′) = q(x)y, F (X,Y, Y ′) = Q(X)Y , we deal with the equa-tion (2.chr) and the above construction gives (besides the contact transformations)the point transformations (1.chr) for a particular choice of functions A = A(a, b),B = B(a, b). In more detail, let

y = ay(x) + by(x), Y = AY (X) +BY (X)

be complete solutions in our linear case of equations. For our point transforma-tion (x, y, y′) −→ (X, Y , Y ′), the couple (X, Y ) should depend only on (x, y)and not on y′. Recall formulae (9.chr1, 10.chr1) in our particular case:

y = ay(x+ by(x), Y = AY (X) +BY (X). (11.chr)


Choosing arbitrary A = αa − β,B = γa − δb, where αδ 6= βγ, and assumingy = Y = 0 for a moment, it follows ζ(x) = −a/b hence

Z(X) = −AB

= −αa− βbγa− δb =

αζ(x) + β

γζ(x) + δ

by using notation of Section 2. However this is just formula (4.chr) and we alreadyknow that such functions X = X(x) completed by Y = c|X ′(x)|1/2y providetransformations into d2Y/dX2 = Q(X)Y .

4 The moving frames method [2]

Our aim is to determine some kinds of the second order differential equations(7.chr) which are preserved under the family of all transformations (6.chr).

For better clarity, we shall deal with the pseudogroup of all transformations(6.chr), where X ′(x) > 0. Then

dX = u2dx, dY = vdx+ cudy (u2 = X ′, v = cX ′′y/2u+ Z ′)

and it follows (from group composition properties) that two families of forms

ω1 = u2dx, ω = vdx+ cudy (u, v are parameters) (12.chr)

are preserved by mappings (6.chr). One can verify that the converse is also true:transformations (8.chr) preserving families (12.chr) are just of the kind (6.chr). On theother hand, the system

dy − y′dx = dy′ − fdx = 0 turns into dY − Y ′dX = dY ′ − FdX = 0

and it follows that two families of forms

ω = λ(dy − y′dx), ¯ω = µ(dy′ − fdx) + ν(dy − y′dx),

where λ, µ, ν are new variables make the intrinsical sense: they are transformedinto the relevant “capital families”. Comparing ω with ω (hence cu = λ, v =−λy′), we obtain better intrinsical family of forms

ω2 = cu(dy − y′dx), ω3 = µ(dy′ − fdx) + νω2,

where u, µ, ν = ν/cu are new variables (and c is an unknown constant). So weoccur ourselves in the space x, y, y′, u, µ, ν, equipped with intrinsical families offorms ω1, ω2, ω3.

Exterior derivatives are intrinsical, too. However dω1 = 2ω4 ∧ ω1 with themost general factor

ω4 =du

u− ξω1 (ξ a new parameter)

88 Jan Chrastina

which provide still an intrinsical family. Analogously

dω2 =du

u∧ ω2 − c

dy′

u∧ ω1 = ω4 ∧ ω1 + (ξ − cν

uµ)ω1 ∧ ω2 +

c

uµω1 ∧ ω3

and we may introduce intrinsical restrictions c/uµ = 1, ξ = cν/uµ hence uµ =c, ξ = ν (then dω2 = ω1 ∧ (ω3 − ω4)). In the same manner

dω3 = ω5 ∧ ω2 +∂f/∂y′

u2ω1 ∧ ω3 − ω4 ∧ ω3, (13.chr)

where

ω5 = dν + βω1 + γω2 + 2νω4

(β = ν2 +

µ

cu3

∂f

∂y− ν

u2

∂f

∂y′

)is intrinsical family with a new variable γ. However

dω4 = −(dξ + 2ξω4) ∧ ω1 = (γω2 − ω5) ∧ ω1

owing to ξ = ν, and we may suppose γ = 0. Returning to (12.chr), we have todistinguish two subcases A : ∂f/∂y′ = 0, B = ∂f/∂y′ 6= 0.

It follows that the family of all equations d2y/dx2 = f(x, y) is preserved bytransformations (6.chr), and one can directly verify that other transformations donot have such property. Assuming A, then

dω5 = dβ ∧ ω1 + 2βω4 ∧ ω1 + 2(dν ∧ ω4 − νω5 ∧ ω1) = 2ω5 ∧ ω4 + ζ ∧ ω1,

where ζ ∼= dβ+4βω4−2νω5 (mod ω1) is intrinsical form. However β = ν2+fy/u4

(use uµ = c) therefore ζ ∼= fyyω2/cu5 after short calculation and we have to

distinguish two subcases C : fyy = 0,D : fyy 6= 0 of our case A.Subcase B is the classical one f = q(x)y + r(x) mentioned above. Surveying

the results, we have structural formulae

dω1 = 2ω4 ∧ ω1, dω2 − ω1 ∧ (ω3 − ω4)

dω3 = ω5 ∧ ω2 − ω4 ∧ ω3, dω4 = ω1 ∧ ω5, dω5 = 2ω5 ∧ ω4

of a Lie group of automorphisms of an equation (5.chr) and, since invariants are lack-ing, all equation (5.chr) are (locally) like each other with respect to transformations(6.chr) which is the already mentioned result.

In subcase D, we may introduce the requirement cu5 = ∂2f/∂y2 which im-plies 5cu4du = fyyxdx+ fyyydy or, in terms of intrinsical forms,

5cu5(ω4 + νω1) =∂3f

∂x∂y2

ω1

u2+∂3f

∂y3

(ω2

cu+ y′

ω1

u2

).

It follows 5ω4 = Mω1 +Nω2 with intrinsical coefficients. In particular

N =∂3f

∂y3

/cu∂2f

∂y2=∂3f

∂y3

/c4/5

(∂2f

∂y2

)6/5


does not change after transformations (6.chr). The constant c is not fixed here.Consequently if N is a conical (containing constant multiples)set of functionsg(x, y), then the family of all equations d2y/dx2 = f(x, y) such that N ∈ N

(equivalently: (fyyy)5/(fyy)

6 ∈ N) is preserved when transformations (6.chr) areapplied. Possibly some narrower families could be obtained by using coefficientM but we shall not continue further.

Let us conclude with the remaining subcase B. Owing to (13.chr), we may assumeu2 = ∂f/∂y′ and, analogously as above, one can obtain an identity of the kindω4 = Mω1 +Nω2 + Pω3 with intrinsical coefficients. We shall mention only thesimplest one

P =∂2f

∂y′2

/2µu2 =

∂2f

∂y′2

/2c(∂f

∂y′

)1/2

,

which yields the following result: if P is a conical set of functions g(x, y, y′) thenthe family of all equations d2y/dx2 = f(x, y, y′) such that P ∈ P (equivalently:f2y′y′/fy′ ∈ P) is preserved when transformations (6.chr) are applied.

5 On the Darboux transformation [4]

There exist many mappings (8.chr) of the space x, y, y′ which transform a given(single) equation (2.chr) into an equation d2Y/dX2 = Q(X)Y . For the sake ofbrevity, we shall mention only the particular case q(x) = 0 and the x-preservingmappings (hence X = x) with Y = Y (x, y, y′) arbitrary. One can then findY ′ = ∂Y/∂x+ y′∂Y/∂y and the requirement

Q(x)Y = ∂2Y/∂x2 + 2y′∂2Y/∂x∂y + y′2∂2Y/∂y2 (14.chr)

for the function Y = Y (x, y, y′). Denoting by Y = φ(x), Y = ψ(x) two linearlyindependent solutions of equation d2Y/dX2 = Q(X)Y , then (14.chr) is satisfied if

Y (x, y, y′) = α(y − y′x, y′)φ(x) + β(y − y′x, y′)ψ(x),

where α, β are arbitrary functions.On the other hand, let us suppose the conjecture Y = A(x)y+B(x)y′ which

leads to useful particular results. Then (14.chr) gives Q(Ay +By′) = A′′y +B′′y′ +2y′A′ and if follows QA = A′′, QB = B′′ + 2A′ whence

A′′B = A(B′′ + 2A′) (15.chr)

by elimination of function Q. Requirement (15.chr) can be explicitly resolved. In par-ticular, for the choice A = 1, one obtains the famous transformation of equationy′′ = 0 into the nontrivial equations with Q(x) = cos−2 x or Q(x) = cosh−2 x.

The general case of the equation(2.chr) can be investigated by the same manner,of course. Then the above conjecture leads to slight generalizations of the familiarDarboux transformation.

90 Jan Chrastina

6 The Laplace series [5], [6]

Turning to partial differential equations, we begin with the general linear hyper-bolical equation

∂2u

∂x∂y= a(x, y)

∂u

∂x+ b(x, y)

∂u

∂y+ c(x, y)u (16.chr)

to transparently illustrate the most important distinctive feature, the possibili-ty of higher order invertible transformations. One can verify that the functionU = uy − au satisfies a certain equation Uxy = AUx + BUy + CU of the samekind as (16.chr), and the iteration provides an infinite series of higher order transfor-mations in the family of equations of the kind (16.chr). Moreover, if ax+ab 6= c thenanalogous substitution with variables x, y exchanged yields the inversion. (Theexceptional case ax + ab = c is much easier and may be omitted: then (16.chr) canbe replaced by certain first order linear equations.) So we obtain an infinite inboth direction series of invertible substitutions in the general case, the Laplaceseries. (The equation (16.chr) moreover admits a change X = X(x), Y = Y (y)of independent variables and a linear change of function u; these are howeverwell-known adaptations. In general, together with the Laplace series, invertibletransformations do not exist, see below.)

The existence of higher order substitutions is possible thanks to the fact thatequation (16.chr) is considered in the infinite-dimensional space with coordinates

x, y, u, ur ≡ ∂ru/∂xr, us ≡ ∂su/∂ys (r, s > 0). (17.chr)

Other derivatives usr ≡ ∂r+su/∂xr∂ys (r, s > 0) can be expressed in terms ofthem by virtue of the equation (16.chr) and its derivatives.

7 The Laplace coframe

Passing to the nonlinear case, we shall mention a hyperbolical equation

∂2u

∂x∂y= f

(∂2u

∂x2,∂2u

∂y2,∂u

∂x,∂u

∂y, u, x, y

)(18.chr)

again in the space of variables (17.chr). Since then the contact transformations canbe applied, it follows that the previous prominent role of variables x, y, u lostthe sense: it is better to employ the contact form ω = du − u1dx − u1dy andin general the higher order contact forms should replace the functions usr. Quiteanalogously the characteristic vector fields

Z+ = a+∂ + b+δ, Z− = a−∂ + b−δ,

where a+, b+anda−, b− are (real and distinct) roots of the equation a2∂f/∂u2 +ab+ b2∂f/∂u2 = 0, and

∂ = ∂/∂x+∞∑ur+1 ∂/∂ur +

∞∑Y s−1f∂/∂us,

δ = ∂/∂y +∑

Xr−1 f∂/∂ur +∞∑us+1∂/∂us (19.chr)


are so called formal derivatives will replace the previous derivative operators∂/∂x, ∂/∂y in equation (16.chr).

In full detail, (18.chr) expressed in terms of contact forms means that

ω11 =

∂f

∂u2ω2 +

∂f

∂u2ω2 +

∂f

∂u1ω1 +

∂f

∂u1ω1 +

∂f

∂uω (20.chr)

(we abbreviate ω0r ≡ ωr, ω

s0 ≡ ωs) and using the Lie derivatives L satisfying

L∂ωsr ≡ ωsr+1, Lδω

sr ≡ ωs+1

r , one can express the last identity in the manner

LZ− LZ+ ω = aLZ− ω + bLZ+ ω + cω (21.chr)

quite analogous to (16.chr). (We shall not state rather clumsy formulae for coeffi-cients a, b, c in terms of the function f .) Then the procedure of Section 6 can besimulated in terms of contact forms: the form Ω = LZ+ω−aω satisfies a certainidentity

LZ− LZ+Ω = ALZ−Ω +BLZ+Ω + CΩ (22.chr)

and the procedure can be iterated. Moreover, if Z−a + ab 6= c, analogous sub-stitution with Z+, Z− exchanged yields the inversion. In general, one obtains aninfinite in both direction series of certain differential forms which constitute abasis in the module of all contact forms, the Laplace coframe.

8 Applications

If the equation Ω = 0 can be expressed by five functions, that means, the Pfaff-Darboux shape is dU − PdX − QdY = 0 for appropriate U,X, Y, P,Q, thenthe functions X,Y may be regarded for independent variables and U for newunknown satisfying a certain second order equation (as follows from (22.chr)). Thesame conclusion can be made for any other term of the Laplace coframe andit may be proved that all possible invertible transformations into some secondorder equation arise only in this manner. Other applications as the Darbouxmethod, representation of solution of an equation (18.chr) by means solutions ofanother such equation, Backlund correspondences, are also possible.

9 Example

One can easily investigate the Laplace coframes for the linear equations (16.chr)and verify that they give the well-known Laplace series of transformations. Ingeneral, Laplace coframes are rather complicated. Therefore we shall mentiononly few results concerning the particular equation ∂2u/∂x∂y = g(∂u/∂x) + ufor a transparent example. Identity (20.chr) means that ω1

1 = g′ω1 + ω and (sinceZ− = ∂, Z+ = δ are formal derivatives in our case and therefore a = g′, b =0, c = 1 in identity (21.chr) we have to introduce the form Ω = ω1 − g′ω. One

92 Jan Chrastina

can find the Pfaff-Darboux shape Ω = dU − PdX − QdY of this form, whereX = x− g′(u1), Y − y may be taken for new independent variables and

U = u1 − ug′(u1)−G(u1) (G(u1) =∫

(g − u1g′)g′′du1) (23.chr)

for new unknown function. Moreover coefficients

P = ∂U/dX = g(u1) + u− u1g′(u1), Q = ∂U/∂Y = u2 − u1g′(u1) (24.chr)

may be identified with new partial derivatives.Then, looking at the differentialdP , one can find (surprisingly simple) formulae

∂2U/dX2 = u1, ∂2U/∂X∂Y = u1 (25.chr)

which means that functions (23.chr, 24.chr, 25.chr) are related by the equation

∂2U

∂X∂Y= U +

(∂U

∂X− g

(∂2U

∂X2

)+∂2U

∂X2g′(∂2U

∂X2

))g′(∂2U

∂X2

)+G

(∂2U

dX2

).

Analogous transformation with the role of x, y exchanged leads to the formΩ = ω1, the independent variables are preserved, and the new unknown functionU = u1 (obviously) satisfies ∂2U/∂x∂y = g′(U)∂U/∂x + U . Modulo contacttransformations, these are the only possible first order invertible transformationsof the equation under consideration.

Acknowledgment

This work was supported by the grant 201/96/0410 of Grant Agency of theCzech Republic.

References

[1] O. Boruvka, Lineare Differentialgleichungen 2. Ordnung, VEB Verlag 1967.[2] E. Cartan, Les Problemes d’Equivalence, Seminaire de Math. expose D, 1937.[3] J. Chrastina, On Dispersions of the 1st and 2nd Kind of Differential Equation

y′′ = q(x)y, Spisy prırod. fak. UJEP 580, 1969.[4] V. Chrastinova, On the Darboux Transformation II, Arch. Math. 30, 1995.[5] G. Darboux, Lecons sur la Theorie Generale des Surfaces II, Gauthier Villars

1894.[6] E. Goursat, Lecons sur l’Integration des Equations aux Derivees Partielles du

Second Order II, Hermann 1898 (second printing 1924).[7] F. Neuman, Global Properties of Linear Ordinary Diff. Equations, Kluwer Acad.

Publ. 1991



Abstract Differential Equations of Arbitrary

(Fractional) Orders

Ahmed M. A. El-Sayed

Department of Mathematics, Faculty of Science,Alexandria University,

Alexandria, EgyptEmail: [email protected]

Abstract. The arbitrary (fractional) order integral operator is a sin-gular integral operator, and the arbitrary (fractional) order differentialoperator is a singular integro-differential operator. And they generalize(interpolate) the integral and differential operators of integer orders. Thetopic of fractional calculus ( derivative and integral of arbitrary orders)is enjoying growing interest not only among Mathematicians, but alsoamong physicists and engineers (see [1]–[18]).Let α be a positive real number. LetX be a Banach space and A be alinear operator defined on X with domain D(A).In this lecture we are concerned with the different approaches of the def-initions of the fractional differential operator Dα and then (see [5,6,7])study the existence, uniqueness, and continuation (with respect to α)of the solution of the initial value problem of the abstract differentialequation

Dαu(t) = Au(t) + f(t), D =d

dt, t > 0, (1.els)

where A is either bounded or closed with domain dense in X.Fractional-order differential-difference equations, fractional-order diffu-sion-wave equation and fractional-order functional differential equationswill be given as applications.

AMS Subject Classification. 34C10, 39A10

Keywords. Fractional calculus, abstract differential equations, differ-ential-difference equations, nonlinear functional equations.

1 Introduction

Let X be a Banach space. Let L1(I,X) be the class of (Lebesgue) integrablefunctions on the interval I = [a, b], 0 < a < b <∞,


94 Ahmed M. A. El-Sayed

Definition 1. Let f(x) ∈ L1(I,X), β ∈ R+. The fractional (arbitrary order)integral of the function f(x) of order β is (see [1]–[11]) defined by

Iβa f(x) =∫ x

a

(x − s)β−1

Γ (β)f(s)ds. (2.els)

When a = 0 and X = R we can write Iβ0 f(x) = f(x)∗φβ(x), where φβ(x) = xβ−1

Γ (β)

for x > 0, φβ(x) = 0 for x ≤ 0 and φβ → δ(x) (the delta function) as β → 0 (see[11]).

Now the following lemma can be easily proved

Lemma 2. Let β and γ ∈ R+. Then we have

(i) Iβa : L1(I,X) → L1(I,X), and if f(x) ∈ L1(I,X), then Iγa Iβa f(x) =

Iγ+βa f(x).

(ii) limβ→n Iβa f(x) = Ina f(x), uniformly on L1(I,X), n = 1, 2, 3, . . . , where

I1af(x) =

∫ xaf(s)ds.

For the fractional order derivative we have (see [1]–[10] and [15]) mainly thefollowing two definitions.

Definition 3. The (Riemann-Liouville) fractional derivative of order α ∈ (0, 1)of f(x) is given by

dαf(x)dxα

=d

dxI1−αa f(x), (3.els)

Definition 4. The fractional derivative Dα of order α ∈ (0, 1] of the functionf(x) is given by

Dαa f(x) = I1−α

a Df(x), D =d

dx. (4.els)

This definition is more convenient in many applications in physics, engineeringand applied sciences (see [15]). Moreover, it generalizes (interpolates) the defi-nition of integer order derivative. The following lemma can be directly proved.

Lemma 5. Let α ∈ (0, 1). If f(x) is absolutely continuous on [a, b], then

(i) Dαa f(x) = dαf(x)

dxα + (x−a)−α

Γ (1−α) f(a)

(ii) limα→1Dαa f(x) = Df(x) 6= limα→1

dαf(x)dxα .

(iii) If f(x) = k, k is a constant, then Dαa k = 0, but dαk

dxα 6= 0.

Definition 6. The finite Weyl fractional integral of order β ∈ R+ of f(t) is

W−βb f(t) =1

Γ (β)

∫ b

t

(s− t)β−1f(s) ds , t ∈ (0, b), (5.els)

and the finite Weyl fractional derivative of order α ∈ (n− 1, n) of f(t) is

Wαb f(t) = W

−(n−α)b (−1)nDnf(t), Dnf(t) ∈ L1(I,X). (6.els)

Differential Equations of Fractional Orders 95

The author [6] stated this definition and proved that if f(t) ∈ C(I,X), then

limβ→p

W−βb f(t) = W−pb f(t), p = 1, 2, . . . , W−1b f(t) =

∫ b

t

f(s)ds, (7.els)

and if g(t) ∈ Cn(I,X) with g(j)(b) = 0, j = 0, 1, . . . , (n− 1), then

limα→q

Wαb g(t) = (−1)qDqg(t), q = 0, 1, . . . , (n− 1), W 0

b g(t) = g(t). (8.els)

2 Ordinary Differential Equations

Let A be a bounded operator defined on X , consider the initial value problemDαau(t) = Au(t) + f(t), t ∈ (a, b], α ∈ (0, 1],u(a) = uo.

(9.els)

Definition 7. By a solution of (9.els) we mean a function u(t) ∈ C(I,X) thatsatisfies (9.els).

Theorem 8. Let uo ∈ X and f(t) ∈ C1(I,X). If ||A|| ≤ Γ (1+α)bα , then (9.els) has

the unique solution

uα(t) = Tαa (t)uo + Iαa Tαa (t)f(t) ∈ C1((a, b], X), (10.els)

where

Tαa g(t) =∞∑k=o

Ikαa Akg(t), g(t) ∈ L1(I,X). (11.els)

And

(1) Tαa (a)uo = uo,(2) Dα

aTαa (t)uo = ATαa (t)uo,

(3) limα→1 Tαa (t)uo = e(t−a)Auo.

Moreover

limα→1

uα(t) = e(t−a)Auo +∫ t

a

e(t−s)Af(s) ds . (12.els)

Proof. See [8].

As an application let 0 < β ≤ α ≤ 1 and consider the two (forward andbackward) initial value problems of the fractional-order differential-differenceequation

(P )

Dαau(t) + CDβ

au(t− r) = Au(t) +Bu(t− r), t > a,

u(t) = g(t), t ∈ [a− r, a], r > 0,(13.els)


(Q)

Wαb u(t) + CW β

b u(t+ r) = Au(t) +Bu(t+ r), t < b, α ≥ β,u(t) = g(t), t ∈ [b, b+ r], r > 0,

(14.els)

where A , B and C are bounded operators defined on X .

Theorem 9. Let g(t) ∈ C1([a−r, a], X). If ||A|| ≤ Γ (1+α)bα , then the problem (P)

has a unique solution u(t) ∈ C((a, b], X), Du(t) ∈ C(Inr , X) and Dαa+nru(t) ∈

C(Inr , X), where Inr = (a, a+ nr].Moreover if C = 0 then u(t) ∈ C1(I,X) and Dα

au(t) ∈ C(I,X).

Proof. See [8].

Theorem 10. Let u(t) be the solution of (P). If the assumptions of Theorem 9are satisfied, then there exist two positive constants k1 and k2 such that

||u(t)|| ≤ k1e(t−a)k2 , (15.els)

i.e., the solution of (P) is exponentially bounded.

Proof. See [8].

The same results can be proved for the problem (Q) (see [8]).

3 Fractional-Order Functional Differential Equation

Consider the two initial value problems

Dαax(t) = f(t, x(m(t))), x(a) = xo, α ∈ (0, 1], (16.els)

Wαb y(t) = f(t, y(m(t))), y(b) = yo, α ∈ (0, 1], (17.els)

with the following assumptions

(i) f : (a, b) × R+ → R+ = [0,∞), satisfies Caratheodory conditions andthere exists a function c ∈ L1 and a constant k ≥ 0 such that f(t, x(t)) ≤c(t) + k|x|, for all t ∈ (a, b) and x ∈ R+. Moreover, f(t, x(t)) is assumed tobe nonincreasing (nondecreasing) on the set (a, b) × R+ with respect to tand nondecreasing with respect to x,

(ii) m : (a, b) → (a, b) is increasing, absolutely continuous and there exists aconstant M > 0 such that m′ ≥M for almost all t ∈ (a, b),

(iii) k/M < 1.

Theorem 11. Let the assumptions (i)–(iii) be satisfied. If xo and yo are positiveconstants, then the problem (16.els) has at least one solution x(t) ∈ L1 which is a.e.nondecreasing (and so Dx(t) ∈ L1) and the problem (17.els) has at least one solutiony(t) ∈ L1 which is a.e. nonincreasing (and so Dy(t) ∈ L1).

Proof. See [9].


4 Fractional-Order Evolution Equations

Let A be a closed linear operator defined on X with domain D(A) dense in Xand consider the two initial value problems

Dγu(t) = Au(t), t ∈ (0, b], γ ∈ (0, 1],u(0) = uo,

(18.els)Dβu(t) = Au(t), t ∈ (0, b], β ∈ (1, 2],u(0) = uo, ut(0) = u1.

(19.els)

Remark 12. Some special cases of these two equations have been studied by someauthors (see [12] and [16] e.g.).

Definition 13. By a solution of the initial value problem (18.els) we mean a func-tion uγ(t) ∈ L1(I,D(A)) for γ ∈ (0, 1] which satisfies the problem (18.els). Thesolution uβ(t) of the problem (19.els) is defined in a similar way.

Consider now the following assumption

(1) Let A generates an analytic semi-group T (t), t > 0 on X . In particularΛ = λ ∈ C : |argλ| < π/2 + δ1, 0 < δ1 < π/2 is contained in the resolventset of A and ||(λI − A)−1|| ≤ M/|λ|, Reλ > 0 on Λ1, for some constantM > 0, where C is the set of complex numbers.

Theorem 14. Let u1, uo ∈ D(A2). If A satisfies assumption (1), then thereexists a unique solution uγ(t) ∈ L1(I,D(A)) of (18.els) given by

uγ(t) = uo −∫ t

0

rγ(s)esuods, Duγ(t) ∈ D(A), (20.els)

and a unique solution uβ(t) ∈ L1(I,D(A)) of (19.els) given by

uβ(t) = uo + tu1 −∫ t

0

rβ(s)es(uo + (t− s)u1)ds, D2uβ(t) ∈ D(A). (21.els)

Here rγ(t) and rβ(t) are the resolvent operators of the the two integral equations

uγ(t) = uo +∫ t

0

φγ(t− s)Auγ(s)ds, (22.els)

uβ(t) = uo + tu1 +∫ t

0

φβ(t− s)Auβ(s)ds, (23.els)

respectively.

Proof. See [6].


Now one of the main results in this paper is the following continuation the-orem. To the best of my knowledge, this has not been studied before.

Theorem 15. Let the assumptions of Theorem 14 be satisfied with u1 = 0, then

limγ→1−

uγ(t) = limβ→1+

uβ(t) = T (t)uo, (24.els)

limγ→1−

Dγuγ(t) = limβ→1+

Dβuβ(t) = AT (t)uo = Du(t), (25.els)

where T (t), t ≥ 0 is the semigroup generated by the operator A and so u(t) =T (t)uo is the solution of the problem

du(t)dt

= Au(t), t > 0

u(0) = uo.

(26.els)

Proof. See [6].

5 Fractional-Order Diffusion-Wave Equation

Let X = Rn and u(t, x) : Rn × I → Rn, I = (0, T ].

Definition 16. The fractional D-W (diffusion-wave) equation is the equation(see [7])

∂αu(x, t)∂tα

= Au(x, t), t > 0, (27.els)

and the fractional diffusion-wave problem is the Cauchy problem

(D-W)

∂αu(x, t)∂tα

= Au(x, t), t > 0, x ∈ Rn, 0 < α ≤ 2,

u(x, 0) = uo(x), ut(x, 0) = 0, x ∈ Rn.(28.els)

From the properties of the fractional calculus we can prove (see [7])

Theorem 17 (Continuation of the problem). If the solution of the (D-W)problem exists, then as α → 1 the (D-W) problem reduces to the diffusionproblem

∂u(x, t)∂t

= Au(x, t), t > 0, x ∈ Rn,

u(x, 0) = uo(x), x ∈ Rn,(29.els)

and as α→ 2 the (D-W) problem reduces to the wave problem∂2u(x, t)∂t2

= Au(x, t), t > 0, x ∈ Rn,

u(x, 0) = uo(x), ut(x, 0) = 0, x ∈ Rn.(30.els)


Proof. See [7].

Theorem 18. Let uo ∈ D(A2). If A satisfies the condition (1) with X = Rn,then the (D-W) problem has a unique solution uα(x, t) ∈ L1(I,D(A)) and thissolution is continuous with respect to α ∈ (0, 2]. Moreover

limα→1

uα(x, t) = u1(x, t) and limα→2−

uα(x, t) = u2(x, t), (31.els)

where u1(x, t) and u2(x, t) are the solutions of (29.els) and (30.els), respectively.

Proof. See [7].

References

[1] Ahmed M. A. El-Sayed, Fractional differential equations. Kyungpook Math. J. 28(2) (1988), 18–22.

[2] Ahmed M. A. El-Sayed, On the fractional differential equations. Appl. Math. andComput. 49 (2–3) (1992).

[3] Ahmed M. A. El-Sayed, Linear differential equations of fractional order. Appl.Math. and Comput. 55 (1993), 1–12.

[4] Ahmed M. A. El-Sayed and A. G. Ibrahim, Multivalued fractional differential equa-tions. Apll. Math. and Compute. 68 (1) (1995), 15–25.

[5] Ahmed M. A. El-Sayed, Fractional order evolution equations. J. of Frac. Calcu-lus 7 (1995), 89–100.

[6] Ahmed M. A. El-Sayed, Fractional-order diffusion-wave equation. Int. J. Theoret-ical Physics 35 (2) (1996), 311–322.

[7] Ahmed, M. A. El-Sayed, Finite Weyl fractional calculus and abstract fractionaldifferential equations. J. F. C. 9 (May 1996).

[8] Ahmed M. A. El-Sayed, Fractional Differential-Difference equations, J. of Frac.Calculus 10 (Nov. 1996).

[9] Ahmed M. A. El-Sayed, W. G. El-Sayed and O. L. Moustafa, On some fractionalfunctional equations. PU. M. A. 6 (4) (1995), 321–332.

[10] Ahmed M. A. El-Sayed, Nonlinear functional differential equations of arbitraryorders. Nonlinear Analysis: Theory, Method and Applications (to appear).

[11] I. M. Gelfand and G. E. Shilov, Generalized functions, Vol. 1, Moscow 1958.[12] F. Mainardy, Fractional diffusive waves in viscoelastic solids. Wenger J. L. and

Norwood F. R. (Editors), IUTAM Symposium-Nonlinear Waves in Solids, FairfieldNJ: ASME/AMR (1995).

[13] F. Mainardy, Fractional relaxation in anelastic solids. J. Alloys and Compounds211/212 (1994), 534–538.

[14] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and FractionalDifferential Equations. John Wiley & Sons. Inc., New York (1993).

[15] Igor Podlubny and Ahmed M. A. El-Sayed, On Two Definitions of Fractional Cal-culus. Slovak Academy Of Sciences, Institute of Experimental Physics, UEF-03-96ISBN 80-7099-252-2 (1996).

[16] W. R. Schneider and W. Wyss, Fractional diffusion and wave equations. J. Math.Phys. 30 (134) (1989).

[17] S. Westerlund, Dead matter has memory. Phisica Scripta 43 (1991), 174–179.[18] S. Westerlund and L. Ekstam, Capacitor theory. IEEE Trans. on Dielectrics and

Electrical Insulation 1 (5) (Oct. 1994), 826–839.



Some Examples for the Extended Use

of the Parametric Representation Method

Peter L. Simon1 and Henrik Farkas2

1 Department of Applied Analysis, Eotvos Lorand University, BudapestEmail: [email protected]

2 Institute of Physics, Department of Chemical Physics,Technical University of Budapest, Budapest H-1521, Hungary


Abstract. The Parametric Representation Method had been appliedsuccessfully to construct bifurcation diagrams relating to equilibria ofdynamical systems whenever the equilibria are determined from a singleequation containing two control parameters linearly. The Discriminant-curve (that is the saddle-node bifurcation curve parametrized by thestate variable remained after the elimination) is the base of this method,as it had been shown. The number and even the value of the stationarystate variables can be derived from that.Here we show some possible extensions of the method via two examples.1. Nonlinear parameter dependence2. Reaction-diffusion equations,Similarly to the above simple case, the PRM provides us with informationabout the stationary solutions. Although some features do not remainvalid for these extensions.

AMS Subject Classification. 58F14, 34C23, 35B32,

Keywords. Bifurcation diagrams, multistationarity

1 Introduction

The parametric representation method is a geometric tool for the study of sta-tionary solutions of differential equations depending on two parameters. Thereare well-known methods [12,20,22] giving the bifurcation parameter values (whe-re the number or stability of stationary solutions changes), and serving withinformation on the stationary solutions if the parameters are in a small neigh-bourhood of the bifurcation values.

Our aim is to divide the whole parameter space according to the numberand the type of the stationary points. We shall call this separation the globalbifurcation diagram;‘global’ refers here to the parameter space, while our inves-tigation is local in the phase space. The first theoretical result in this directionwas achieved by Rabinowitz [16]. He followed the changes of one stationary state


102 P. L. Simon and H. Farkas

varying a single parameter value. Details and other references can be found in [5].In [3,4,9] there are methods for constructing global bifurcation diagrams whichgive the number of roots of polynomials. In many practical applications theconstruction of such bifurcation diagrams was carried out by ad hoc methods[11,13,15].

The Parametric Representation Method (PRM) [9] is a systematic approach,which is especially useful if the parameter dependence of the system is simplerthan the dependence on the state variables. As an example, in chemical dynam-ical systems the parameter dependence is usually linear, therefore the PRM iseasy to apply [2,14,17]. Some general features of the method together with apictorial algorithm for determination of the exact number of stationary pointscan be found in [6,7]. PRM was also applied to study the root structure of poly-nomials and extended to study their complex roots [8]. This method is also auseful tool to reveal some relations between the saddle-node and Hopf bifurcationdiagrams [17,19].

We summarize the main results concerning the case of linear parameter de-pendence in Section 2, the detailed study can be found in [18]. In Section 3 itis shown on an example how can be used the PRM, when the equation (de-termining the stationary states) contains parameters non-linearly. In Section 4we illustrate that the PRM may be useful for the determination of stationarysolutions of a scalar reaction-diffusion equation.

2 Linear parameter dependence

We want to give the number of the stationary points of the following ODE:

x(t) = F (x(t), u),

where F : Rn × Rk → Rn is a differentiable function, x(t) ∈ Rn is the vectorof state variables and u ∈ Rk is the vector of parameters. The first step beforeexecuting the global bifurcation analysis is the reduction of the dimension ofthe system. There is no general method for that, the optimal one depends onthe structure of the concrete system. The Liapunov-Schmidt reduction or — forpolynomials — the Euclidean algorithm are often useful tools. In this section weassume that— the system of algebraic equations F (x, u) = 0 giving the stationary points isalready reduced to a single equation and— we have two control parameters, u1 and u2, which are involved in the righthand side of the reduced equation linearly. These control parameters may also befunctions of the original parameters of the system. (Two control parameters arechosen regularly in practical applications, primarily because of the visualization.)

With these assumptions the above general problem reduces to the followingone:Problem. Let us divide the parameter plane (u1, u2) with respect to the numberof the solutions of equation

f(x, u1, u2) := f0(x) + f1(x)u1 + f2(x)u2 = 0, (1.far)

Parametric Representation Method 103

where fi ∈ C2 and f21 (x)+f2

2 (x) 6= 0 for all x ∈ R; and give a geometric methoddetermining the number and values of the solutions at a given parameter pair(u1, u2).

According to the implicit function theorem the number of solutions maychange when the parameter values cross the singularity set:

S = (u1, u2) ∈ R2 : ∃x ∈ R , f(x, u1, u2) = f ′(x, u1, u2) = 0,

where prime denotes differentiation with respect to x. The detailed study ofsingularities can be found in [1,10]. The PRM has the following advantages:1. the singularity set can be easily constructed as a curve parametrized by x,called D-curve; 2. the solutions belonging to a given parameter pair can bedetermined by a simple geometric algorithm based on the tangential property;3. the global bifurcation diagram, that divides the parameter plane accordingto the number of solutions can be geometrically constructed with the aid of theD-curve.

Now let us see how to apply the PRM for (1.far). Concerning the singularity setthe determinant

∆(x) := f1(x)f ′2(x)− f ′1(x)f2(x)

plays a crucial role. For simplicity we assume that ∆(x) 6= 0 for all x ∈ R (thegeneral case, when ∆ may have zeros is considered in [18]). Then the system

f0(x) + f1(x)u1 + f2(x)u2 = 0, (2.far)f ′0(x) + f ′1(x)u1 + f ′2(x)u2 = 0, (3.far)

has one and only one solution for (u1, u2). These equations determine the D-curvefor this case:

Definition. The solution of the system (2.far), (3.far) for u1 and u2 is called D-curve(or discriminant curve). The point belonging to x will be denoted by D(x) =(D1(x), D2(x)), i.e.

u1 =f2(x)f ′0(x)− f ′2(x)f0(x)

∆(x)=: D1(x),

u2 =f1(x)f ′0(x)− f ′1(x)f0(x)

∆(x)=: D2(x).

Thus we produced the singularity set as a curve parametrized by x.Let us introduce the straight lines:

M(x) := (u1, u2) ∈ R2 : f(x, u1, u2) = 0,

i.e. the set of parameter pairs for which a given number x is a solution of (1.far).The main point of the PRM is the fact that the D-curve (the singularity set) isthe envelope of these lines. This fact is involved in the following theorem, whichwill be referred to as tangential property.


-2 u1

1

u2

3

1

Fig. 1.

Theorem 1. The line M(x) is tangential to the D-curve at the point D(x).

For the proof see [18] Theorem 4.

Corollary 1. The number of solutions of (1.far) belonging to a given parameterpair (u1, u2) is equal to the number of tangents, which can be drawn to theD-curve from the point (u1, u2).

Thus as a solution of our problem we got the following:Geometric algorithm. Draw the D-curve belonging to our equation. Givena parameter pair (u1, u2) any tangent from this point to the D-curve gives asolution x of the equation; the value of x can be read on the D-curve at thetangential point.

As an illustration let us consider the equation

x3 + u1x+ u2 = 0.

The D-curve is determined by the system

x3 + u1x+ u2 = 0,

3x2 + u1 = 0.

From this system we get

D1(x) = −3x2, D2(x) = 2x3.

The D-curve is depicted in Fig. 1. If (u1, u2) is on the left side of the D-curve,then the equation has three solutions, because we can draw three tangents from(u1, u2) to the D-curve. If (u1, u2) is on the right side of the D-curve, then thereis one solution, because we can draw one tangent from (u1, u2). The value of xon the D-curve is increasing with increasing u2 and it is zero at the origin.

The determination of the number of the tangents is facilitated by the so-calledconvexity property: the D-curve consists of convex arcs that join together incusp points. To be more formal we cite Theorem 5 from [18]:


Theorem 2. Suppose that the roots of the function

B(x) := f ′′0 (x) + f ′′1 (x)D1(x) + f ′′2 (x)D2(x)

are isolated.

(i) If B changes its sign at x0 then the D-curve has a cusp point at x0.(ii) If B does not change its sign at x0 then the D-curve is locally on the left

(right) side of its tangent belonging to x0 if ∆(x0) is positive (negative).

The D-curve also gives the global bifurcation diagram (GBD) i.e. the curve(or system of curves) which divides the parameter plane into regions withinwhich the number of solutions of (1.far) is constant. The construction of the GBDis based on the fact that the number of roots of a function may change in twoways:

1. it has a multiple root (the derivative vanishes at a root),2. a root goes to (or comes from) the infinity.The GBD consists of the D-curve and its tangents or asymptotes (if they

exist) at the points belonging to x→∞ and x→ −∞. For the exact formulationsee Theorem 6 in [18].

3 Nonlinear parameter dependence

In this section we apply the PRM for the special equation

x2 + u21x+ u2 = 0. (4.far)

Our aim is to divide the parameter plane (u1, u2) according to the numberof solutions (x ∈ R) of (4.far). The singularity set is determined by (4.far) and

2x+ u21 = 0. (5.far)

The solution of the system (4.far)–(5.far) is

u21 = −2x, u2 = x2. (6.far)

Thus the singularity set can not be parametrized by x, but we can define theD-curve with two branches D+ and D− as follows (see Fig. 2.):

Definition. The two solutions of (4.far)–(5.far) for (u1, u2) are called the two branchesof the D-curve for x ≤ 0, i.e.

D+1 (x) =

√−2x, D+

2 (x) = x2,

D−1 (x) = −√−2x, D−2 (x) = x2.


Similarly as in Section 2 let us introduce

M(x) := (u1, u2) ∈ R2 : x2 + u21x+ u2 = 0,

which is in this case a parabola for a given x ∈ R. The singularity set is theenvelope of the parabolas belonging to x < 0, therefore the tangential propertyholds in the following form:

Theorem 3. For a fixed x < 0 the parabola M(x) is tangential to the D+ andD− curves at the points D+(x) and D−(x).

The tangential property does not refer to the values x > 0, however, it isobvious from (6.far) that there is no singularity for these values. Therefore theparabolas M(x) belonging to x > 0 do not intersect each other, they form aone-fold cover of the lower half plane. Thus the number of solutions can be givenby the following:Geometric algorithm. Given a parameter pair (u1, u2) any tangential parabolaof the form (4.far) from this point to the D-curve gives a solution x of the equa-tion (4.far); the value of x can be read on the D-curve at the tangential point (thevalue of x is the same on the D+ and on the D− branches). If the parameter pairis in the upper half plane, then the number of the tangential parabolas is equalto the number of solutions. If the parameter pair is in the lower half plane, thenthe number of solutions is more by one than the number of tangential parabolas.

Using this algorithm we get that the number of solutions of (4.far) is 0 if (u1, u2)is above the D-curve, and it is 2 if (u1, u2) is below the D-curve, see Fig. 2.

2 u1

10

u2

0

2

x=1

x=-2

D + D -

Fig. 2.


4 Application of the PRM to a reaction-diffusionequation

Let us consider the reaction-diffusion equation

∂tu(t, x) = ∂xxu(t, x) + f(u(t, x), a, b)

with the boundary condition

u(t, 0) = u(t, 1) = 0,

where f : R3 → R is a differentiable function. The stationary solutions aredetermined by the following boundary value problem:

v′′(x) + f(v(x), a, b) = 0, (7.far)v(0) = v(1) = 0. (8.far)

We will study the following:Problem. Divide the parameter plane (a, b) according to the number of thesolutions of (7.far)–(8.far).

The boundary value problem (7.far)–(8.far) is usually [20,21] reduced to the phaseplane analysis of the system

v′ = w, (9.far)w′ = −f(v, a, b). (10.far)

If we have a p ∈ R, such that the trajectory t → (v(t), w(t)) of (9.far)–(10.far)starting from (0, p) reaches the vertical axis (v = 0) at time 1 (i.e. v(1) = 0),then v is a solution of (7.far)–(8.far). Therefore the time map T is defined that measuresthe time an orbit takes to get from the point (0, p) to the vertical axis. This timeis the double of that one the orbit takes to get from the point (0, p) to thehorizontal axis (say, at point (0, q)), because the flow of (9.far)–(10.far) is symmetricto the horizontal axis. System (9.far)–(10.far) has the first integral

H(v, w) =w2

2+ F (v),

where F (v) =∫ v

0 f(s) ds. This first integral enables us to calculate the time mapexplicitly:

T (p) = 2∫ q

0

1√2(F (q)− F (v))

dv.

The relation between p and q is given by the first integral: F (q) = p2

2 . Thus thetime map can be regarded as a function of q, a and b:

S(q, a, b) = 2∫ q

0

1√2(F (q)− F (v))

dv.


A solution q of

S(q, a, b) = 1 (11.far)

gives a solution of (7.far)–(8.far). However, several solutions of (7.far)–(8.far) may give thesame q as a solution of (11.far). Therefore our problem partially reduces to thefollowing one:Problem. Divide the parameter plane according to the number of solutionsof (11.far).

This problem is similar to that one dealt with in Section 3 (the parameter de-pendence may be more complicated). Using the PRM we can define the D-curve(singularity set) belonging to equation (11.far). It is determined by the equations:

S(q,D1(q), D2(q)) = 1,∂qS(q,D1(q), D2(q)) = 0.

Solving these equations numerically one can get a curve on the parameterplane (a, b), which divides it into regions according to the number of solutionsof (11.far).

References

[1] Arnol’d, V. I., The Theory of Singularities and its Applications, Cambridge Uni-versity Press, Cambridge, 1991.

[2] Balakotaiah, V. Luss, D., Global analysis of the multiplicity features of multi-reaction lumped-parameter systems, Chem. Engng. Sci. 39., 865–881, 1984.

[3] Callahan, J., Singularities and plane maps, Am. Math. Monthly 81, 211–240, 1974.[4] Callahan, J., Singularities and plane maps II: Sketching catastrophes, Am. Math.

Monthly 84, 765–803, 1977.[5] Chow, S. N., Hale, J. K., Methods of Bifurcation Theory, Springer-Verlag, New

York, 1982.[6] Farkas, H., Gyoker, S., Wittmann, M., Investigation of global equilibrium bifurca-

tions by the method of parametric representation, Alk. Mat. Lapok, 14, 335–364,in Hungarian. 1989.

[7] Farkas, H., Gyoker, S., Wittmann, M., Use of the parametric representationmethod in bifurcation problems, In Nonlinear Vibration Problems, Vol. 25, p. 93,1993.

[8] Farkas, H., Simon, P. L., Use of the parametric representation method in revealingthe root structure and Hopf bifurcation, J. Math. Chem. 9, 323–339, 1992.

[9] Gilmore, R., Catastrophe Theory for Scientists and Engineers Wiley, New York,1981.

[10] Golubitsky, M., Schaeffer, D. G., Singularities and Groups in Bifurcation TheoryVol. I. Springer, New York, 1985.

[11] Gray, P., Scott, S. K., Chemical Oscillations and Instabilities: Non-linear ChemicalKinetics, Clarendon Press, Oxford, 1994.

[12] Guckenheimer, J. Holmes, P., Nonlinear Oscillations, Dynamical Systems and Bi-furcations of Vector Fields, Springer-Verlag, New York, 1983.


[13] De Kepper, P., Boissonade, J., From bistability to sustained oscillations in homo-geneous chemical systems in a flow reactor mode, in Oscillations and TravelingWaves in Chemical Systems, eds.: R.J. Field and M. Burger, (Wiley, New York),p. 223, 1985.

[14] Kertesz, V., Farkas, H., Local investigation of bistability problems in physico-chemical systems, Acta Chim. Hung. 126, 775, 1989.

[15] Murray, J. D., Mathematical Biology, Springer-Verlag, New York, 1989.[16] Rabinowitz, P., Some global results for nonlinear eigenvalue problems, J. Func.

Anal. 7, 487–513, 1971.[17] Simon, P. L., Nguyen Bich Thuy, Farkas, H., Noszticzius, Z., Application of the

parametric representation method to construct bifurcation diagrams for highlynon-linear chemical dynamical systems, J. Chem.Soc., Faraday Trans. 92 (16),2865–2871, 1996.

[18] Simon, P. L., Farkas, H., Wittmann, M., Constructing global bifurcation diagramsby the parametric representation method, submitted for publication.

[19] Simon, P. L., Farkas, H., Relation between the saddle-node and Hopf bifurcation,work in progress.

[20] Smoller, J. A, Shock Waves and Reaction Diffusion Equations Springer, 1983.[21] Smoller, J. A., Wasserman, A., Global bifurcation of steady-state solutions, J.Diff.

Equ. 39, 269–290, 1981.[22] Wiggins, S., Introduction to Applied Nonlinear Dynamical Systems and Chaos,

Springer-Verlag, New York, 1990.



Homogenization of Scalar Hysteresis Operators

Jan Francu?

Department of Mathematics. Technical University BrnoTechnicka 2,

616 96 Brno, Czech RepublicEmail: [email protected]

Abstract. Scalar hysteresis operators of Ishlinskii stop type are studied.Homogenization problem for hyperbolic equation with hysteresis opera-tor is formulated. Formula for the homogenized operator is derived.

AMS Subject Classification. 35B27, 73B27, 73E05

Keywords. Scalar hysteresis operators, Ishlinskii operator, homogen-ization

Introduction

The title of the contribution contains words homogenization and hysteresis. Ho-mogenization is a mathematical method used in modelling composite materialswith periodic structure. It consists in replacing the heterogeneous material mod-elled by equations with periodic coefficients with an equivalent homogeneousmaterial modelled by constant coefficients, see e. g. [1], [2], [3], [4] and manyothers.

Hysteresis is one of nonlinear phenomena that appears in evolutionary non-linear problems of continuum mechanics, see e. g. [5], [6], [7], [8], [9], [10]. Thebasic feature of hysteresis behavior is a memory effect and irreversibility of theprocess, the response of the material by loading differs from the response byunloading. The behavior of the material is well characterized by the hysteresisloop. In mechanics hysteresis operators model plastic deformation.

We shall deal with homogenization of a one-dimensional boundary valueproblem, that can be interpreted as a vibration of a plastic rod. We assumethat all quantities depend on length variable x ∈ R only. The plasticity of therod is modelled by Ishlinskii hysteresis operator based on the stop operators.Existence of the solutions of these problems was proved e. g. in [10]. The proofneeds properties of stop and play operators and Ishlinskii operators of stop andplay type studied in Section 2 and 3.

? This research has been supported by grant No. 201/97/0153 of Grant Agency ofCzech Republic.


112 Jan Francu

The plan of the contribution is the following. In Section 1 we start with basicelements used in modelling of materials. Section 2 deals with the stop Sh and theplay Ph operators and their properties. Combination in parallel of the stop or theplay operators leads to Ishlinskii operator studied in Section 3. One-dimensionalhomogenization problem is introduced in Section 4. The last Section 5 deals withform of the homogenized operator. The complete proof and the convergence ofthe solutions is subject of future research.

1 Basic elements

In this section we briefly recall basic one-dimensional models of deformation ofsolid materials. We consider a one-dimensional homogeneous solid (a homoge-neous rod of unit length and of uniform small cross-section). We assume a uni-form (independent of place) strain e caused by loading. The strain correspondsto a displacement u(x) satisfying e = ux(≡ du/dx) or equivalently u(x) = e · x.Thus fixing one end u(0) = 0 the displacement in the second end x = 1 cor-responds to the strain: e = u(1). The response of the material is the stress σ.It is also supposed to be uniform in the rod. The material is described by theconstitutive relation between σ(t) and e(t) represented by the stress-strain σ-e(or strain-stress e-σ) diagram. The behavior is often modelled by a mechanicaldevice (string, pipe, friction element etc.). Piston-in-cylinder element representsa geometric model.

Elastic element E

The elastic element means linear stress-strain relation

σ = A · e , (1.fra)

where A is the elasticity constant (Young modulus of elasticity). In this modelthe stress depends on instant value of strain independently on preceding courseof deformation. The elastic element is modelled by a string and its graph in thestress-strain diagram forms a line crossing the origin with slope A.

Rigid-plastic element R

The element is characterized by a parameter r (r > 0). Three cases occur: eitherthe stress σ is inside the interval (−r, r), then the strain e does not change, orthe stress σ reach r, then e can grow, or σ = −r then e may decrease. Withcyclic loading and deloading we obtain a rectangular hysteresis loop.

The element can be modelled by a mechanical system with friction, the pa-rameter r of the model represents the friction coefficient of the Coulomb frictionlaw. The behavior can be also described by a variational inequality

σ(t) ∈ [−h, h] , such thatde(t)

dt(σ(t) − σ) ≥ 0 ∀ σ ∈ [−h, h] . (2.fra)

Homogenization of Hysteresis Operators 113

Other basic elements

In material modelling also other elements are used, e. g. viscous and brittle el-ement, but they do not correspond to our framework of behaviour of plasticmaterials.

Combination of elements

The elements can be combined either in series or in parallel or in more compli-cated systems. Let us consider n elements and denote their strains by ei, theirstresses by σi, i = 1, 2, . . . , n, the total strain by e and the total stress of thesystem by σ. Then in case of combination in parallel the deformation is commonand the stresses are added:

e = e1 = e2 = · · · = en , σ = σ1 + σ2 + · · ·+ σn (3.fra)

while in case of combination in series the stress is common and the deformationsare added:

e = e1 + e2 + · · ·+ en , σ = σ1 = σ2 = · · · = σn . (4.fra)

Many other combinations are used, we shall deal with the following serial elasto-plastic combination.

Elasto-plastic element.

Combining elastic and rigid-plastic elements in series E −R we obtain an elasto-plastic element. Its hysteresis loop is similar to that of plastic element, only itsvertical segments are slanted, the slope reflects the elasticity constant.

Geometric model: Piston-in-cylinder

Piston-in-cylinder model represents a geometric equivalent of elasto-plastic ele-ment. Let us consider a cylinder of length 2h with a piston moving inside thecylinder. Denoting the input — the absolute (with respect to the coordinate sys-tem) position of the piston by u, the relative position of the piston with respectto the cylinder by Sh and the absolute position of the cylinder by Ph. Clearly,u = Ph + Sh. The relative position of the piston is proportional to the stressσ = η · Sh.

Let us consider an increasing deformation e(t). The piston is moving in thecylinder, Sh(t) which is proportional to the stress σ(t) is increasing (elastic defor-mation). If the piston reaches the end of the cylinder and e(t) is still increasing,then Sh(t) and the stress σ(t) remain constant but the cylinder Ph(t) starts tomove (plastic deformation). Now, if the e(t) starts decreasing, the piston startsmoving backwards. In the beginning the piston is moving in the cylinder, Sh(t)(and the stress σ(t)) is decreasing (elastic deformation) down to the opposite

114 Jan Francu

end of the cylinder when again Sh(t) stops and Ph(t) starts to decrease (plas-tic deformation). The position of the cylinder Ph(t) describes an internal stateyielding memory effects.

Let us remark the degenerated cases. For h =∞ the stop operator convertsto identity: S∞ = I and for h = 0 the play operator becomes identity: P0 = I;in both cases the operators degenerate to the elastic element with A = 1.

2 Stop and Play operator

The introduced piston-in-cylinder model yields two operators Sh called the stopand Ph the play. They form the base for more complicated models. The operatorsare functional, they map a function to a function. They have a parameter h —it corresponds to the half length of the cylinder. Besides the input function u(t)we need to set the initial states sh, ph — the value of the operators in the initialtime. The operators are complementary, i. e. they are connected by the relation

(Shu)(t) + (Phu)(t) = u(t) ∀u, ∀ t (5.fra)

and satisfy

|(Shu)(t)| ≤ h and |(Phu)(t)− u(t)| ≤ h. (6.fra)

The initial states sh, ph ∈ R are supposed to satisfy the compatibility condi-tions in the initial time t = a analogous to (5.fra), (6.fra):

sh + ph = u(a) and |sh| ≤ h , |ph − u(a)| ≤ h , . (7.fra)

If no initial values are stated we can assigns to the input u(t) the natural initialvalues sh for the stop operator Sh

sh =

h if u(a) ≥ hu(a) if |u(a)| < h−h if u(a) ≤ −h

(8.fra)

and similarly the natural initial value ph for the play operator Ph

ph =

u(a)− h if u(a) ≥ h0 if |u(a)| < hu(a) + h if u(a) ≤ −h

. (9.fra)

The operators are introduced by preceding geometric model, nevertheless letus give their exact definition.

Max-min definition

The definition, see e. g. [5], [8] starts with defining the operator for piecewisemonotone functions, then it is extended to all continuous functions:


Let I = [a, b] be a time interval and u a continuous piecewise monotonefunction on I. Let us denote by ti the turning points of u i. e. a = t0 < t1 <· · · < tk = b and the continuous input u is monotone (non-decreasing or non-increasing) on each subinterval I1, . . . , In (Ii = (ti−1, ti]). Moreover we mayassume that in each ti (0 < i < n) the function u changes from non-decreasingto non-increasing or the other way round.

Then the value of the stop operator Sh is function Shu : I → R defined by

(Shu) (t) =

sh for t = a ,min(Shu) (ti−1) + u(t)− u(ti−1), h for t ∈ Ii if u is non-

decreasing on Ii ,max(Shu) (ti−1) + u(t)− u(ti−1),−h for t ∈ Ii if u is non-

increasing on Ii .

(10.fra)

Since the play operator is complementary it can be defined either by a similarmax-min definition or simply by

(Phu) (t) = u(t)− (Shu) (t) . (11.fra)

Since both operators are Lipschitz continuous, see Lemma 2, the definitioncan be extended to continuous functions by continuity.

Definition by variational inequality

The stop and play operator can be also introduced by a variational inequality,similar to (4.fra), see [10], [9].

Let u ∈ W 1,1(I) be an input function on the interval I = [a, b] and sh theinitial state satisfying |sh| < h. Let x(t) be a solution of the following problem:

Find x(t) ∈W 1,1(I) such that: x(t) ∈ [−h, h] , x(a) = sh(a) ,and for almost all t ∈ I the following inequality holds(

dudt

(t)− dxdt

(t))

(x(t) − x) ≥ 0 ∀ x ∈ [−h, h] . (12.fra)

Since the problem admits unique solution x(t) we use it to definition of thestop operator Sh and play operator Ph:

(Shu) (t) = x(t) , (Phu) (t) = u(t)− x(t) . (13.fra)

It can be proved that both ways of introducing the play and the stop opera-tors define the same operators.

Inverse operators

Both operators S and P are not invertible, since they are not injective. Adding amultiple of identity denoted by I the operators become injective and invertible:inverse of the stop type operator is the play type operator and vice versa.

116 Jan Francu

Lemma 1. Let a, b, c, d, h, k be positive constants. Then we have the followingrelations

(a I + bSh)−1 = c I + dPk , (14.fra)

where

c =1

a+ b, d =

1a− 1a+ b

, k = (a+ b)h ,

and conversely

(c I + dPk)−1 = a I + bSh , (15.fra)

where

a =1

c+ d, b =

1c− 1c+ d

, h = ck .

Scaling, continuity and monotony properties

Due to definition we have the following dependence on the parameter h:

Shu = hS1

(uh

), Phu = hP1

(uh

). (16.fra)

The stop and the play operators are Lipschitz continuous, see e. g. [5], [8]:

Lemma 2. Let u1(t), u2(t) be two inputs on I = [a, b] with the same initialvalues sh, ph. Then for all t ∈ (a, b] we have

‖(Shu1) (t)− (Shu2) (t)| ≤ 2 maxs≤t|u1(s)− u2(s)| (17.fra)

‖(Phu1) (t)− (Phu2) (t)| ≤ maxs≤t|u1(s)− u2(s)| (18.fra)

The operators conserve monotony properties, non-decreasing inputs yieldsnon-decreasing output and non-increasing input yields non-increasing output.The properties are formulated in [5], [8], [9].

3 Ishlinskii operators

Simple stop Sh and play Ph operators yield hysteresis loops consisting of seg-ments with two slopes. By weighted parallel finite or infinite or continuum com-bination of stop (or play) operators (including a multiple of identity I) we obtainmore general hysteresis loops.


Stop type Ishlinskii operators

Let us consider a parallel weighted combination of n different stop operators,in physical setting elasto-plastic elements. Each element has two parameters: hihalf-length of the cylinder and ηi “stiffness” of the element, it describes stresscontribution of the element σi = ηi ·fh. Thus the combination is determined by afamily of n pairs (hi, ηi) (i = 1, 2, . . . , n), ordered 0 < h1 < h2 < · · · < hn ≤ ∞,and ηi ≥ 0. Admitting the case hn =∞ we include a multiple of identity ηn I —simple elastic element combined in parallel. Let us consider an input e(t). Thenthe output stress is

σ(t) = (F(e)) (t) ≡n∑i=1

ηi (Shie) (t) . (19.fra)

The operator F is called discrete Ishlinskii operator. It has to be completed bythe initial values of stop operators shi . They can be given or set by (8.fra) in caseof intact initial state.

The stress-strain diagram to the increasing loading e(t) from the intact stateof this material forms a concave non-decreasing function Φ — another charac-terization of the hysteresis behavior called virgin curve.

Using (19.fra) we can compute the function Φ. Put the deformation e(r) = r forr ∈ [0,∞) with shi = 0. Then we have

Φ(r) =∑i

ηiShi(r) =∑hi≤r

ηiShi(r) +∑hi>r

ηiShi(r) .

Since in course of this loading Sh(r) = h for h ≤ r and Sh(r) = r for h > r weobtain

Φ(r) =∑hi≤r

ηihi + r∑hi>r

ηi . (20.fra)

The function Φ(r) is clearly concave on [0,∞). Considering the negative loadingfrom the intact state the function Φ(r) can be extended to negative values to anodd function by Φ(r) = −Φ(−r).

The Ishlinskii operator represents a generalization of parallel combinationof elastic-plastic elements to the case of infinite elements with continuously in-creasing parameter h. The family of pairs (hi, ηi) is replaced by a non-negative“stiffness” function η(h). Further we replace the sum in (21.fra) by an integral(possible η∞ remains) and we obtain the Ishlinskii operator F

σ(t) = (F(e)) (t) ≡∫ ∞

0

η(h) (She) (t)dh+ η∞e(t) . (21.fra)

The operator F(e) should be completed by the initial values sh that can be givenor defined by (8.fra).

In the stress-strain diagram the cyclic loading has concave increasing archesand convex decreasing arches.

118 Jan Francu

Introducing a function µ(h) by

µ(h) =∫ h

0

η(r)dr or µ′(h) = η(h) , µ(0) = 0 (22.fra)

we can replace the Riemann integral by Stieltjes integral

σ(t) = (F(e)) (t) ≡∫ ∞

0

(She) (t)dµ(h) + η∞e(t) . (23.fra)

The advantage of the formula is the fact that it includes even the discrete caseof the Ishlinskii operator (19.fra) taking piecewise constant function

µ(r) =∑hi<r

ηi . (24.fra)

Characteristics of the Ishlinskii operator

Besides the function η(h) or µ(h) the Ishlinskii operator can be characterizedby the function Φ of the stress-strain diagram of monotone loading from intactstate. Using Shr = h for h ≤ r and Sh(r) = r for h > r from (21.fra) or (23.fra) we cancompute the function Φ by means of η∞ and η(h) or µ(h):

Φ(r) =∫ r

0

hη(h)dh+ r

∫ ∞r

η(h)dh+ η∞r , (25.fra)

Φ(r) =∫ r

0

hdµ(h) + r [µ(∞)− µ(r) + η∞] . (26.fra)

On the other hand since Φ′(r) = µ(∞)−µ(r)+η∞ we can express µ(r) by meansof function Φ(r):

η∞ = Φ′(∞) , µ(r) = Φ′(0)− Φ′(r) . (27.fra)

The function Φ(r) is continuous and concave on [0,∞) with Φ(0) = 0. Ifη∞ > 0 or Φ(∞) =∞, then the function Φ(r) is increasing and it has an inversefunction Ψ(s) on [0,∞) defined by

Ψ(s) = r iff Φ(r) = s .

If Φ(∞) ≡ Φ∞ < ∞, then its inverse function Ψ is defined on [0, Φ∞] only.Moreover, it may be multivalued for s = Φ∞. In both cases Ψ is a convexfunction.

Play type Ishlinskii operators

Replacing the stop operator Sh by the play operator Ph we obtain Ishlinskiioperators of play type. Let us state the formula with input u(t), weight functionζ(h) and ζ0 multiple of identity:

(G(u)) (t) =∫ ∞

0

ζ(h) (Phu) (t)dh+ ζ0u (28.fra)


or with function ν(r) =∫ r

0ζ(s)ds and Stieltjes integral

(G(u)) (t) =∫ ∞

0

(Phu) (t)dν(h) + ζ0u . (29.fra)

The operator G can be characterized by a function Ψ(s) describing the responseof the operator to monotone increasing load from the intact state. Using Ph(s) =s− h for h < s and Ph(s) = 0 for h ≥ s from (28.fra) or (29.fra) we obtain formula forthe function Ψ(s):

Ψ(s) =∫ s

0

(s− h)ζ(h)dh + ζ0s = sν(s)−∫ s

0

hdν(h) + ζ0s . (30.fra)

On the other hand since Ψ ′(s) = ν(s) + ζ0 the function Ψ(s) yields ν(s) and ζ0:

ζ0 = Ψ ′(0) , ν(s) = Ψ ′(s)− Ψ ′(0) . (31.fra)

The hysteresis loops have convex increasing arches and concave decreasingarches. Again if the function Ψ is defined on [0,∞) it is invertible, its inverseis the Ishlinskii operator of stop type and vice versa. Thus the operators of theplay type are used in modelling the inverse hysteresis constitutive relations i. e.dependence of strain on stress.

Ishlinskii operators satisfy similar continuity and monotony properties likethe stop and play operators.

4 Homogenization problem

Let us consider a two component periodically ordered layered material. We as-sume that both materials denoted as material A and B are characterized byIshlinskii operators FA and FB with corresponding characteristics ηA, µA, ΦAand ηB, µB, ΦB .

Material with a periodic layered structure

The homogenization approach, see [1], [2], [3], [4] consists in considering a se-quence of periodically ordered materials with a diminishing period ε.

Let dA : dB (dA + dB = 1) be the proportion of the thicknesses of thecomponents A and B. We define a space dependent Ishlinskii operator periodicin space

F(y)(e)(t) =∫ ∞

0

(She) (t) dµ(y, h) + η∞(y)e(t) , (32.fra)

where µ(y, h) and η∞(y) are functions periodic in y with period one defined by

µ(y, h) =µA(h) for y ∈ [k, k + dA) k ∈ Z ,µB(h) for y ∈ [k − dB, k) k ∈ Z . (33.fra)

The function η∞(y) is defined similarly.Now we define a sequence of operators Fε with diminishing period ε→ 0

Fε(x)(e) = F(xε

)(e) . (34.fra)

120 Jan Francu

Boundary value problems

We shall consider a problem modelling longitudinal vibration of the rod. Let usconsider a thin rod with space variable x ∈ [0, l] made of a periodically layeredmaterial. Let us denote the longitudinal displacement by u(x, t) and the stressby σ(x, t). We combine the equation of motion σx + f = ρ utt, where ρ is thedensity and f(x, t) the applied force, with the stress-strain relation σ = F(e),where F(x)(e) is the space dependent Ishlinskii operator and e(x, t) = ux(x, t).Since the material has periodic structure, the Ishlinskii operator F(x) is periodicwith the period ε. We denote it by a superscript ε, see (34.fra). We also denote thecorresponding solution by uε.

Thus for any ε > 0 we obtain the following equation

[Fε(x)(uεx)]x + f = ρ uεtt for x ∈ (0, l) , t > 0 . (35.fra)

We complete the equation (35.fra) with suitable boundary conditions, e. g. fixedends of the rod

uε(0, t) = 0 , uε(l, t) = 0 for t > 0 (36.fra)

and initial conditions with a given initial displacement u0 and a given initialvelocity u1

uε(x, 0) = u0(x) , uεt (x, 0) = u1(x) for x ∈ (0, l) . (37.fra)

Finally we have to add the initial state of the stop operators Sh in the Ishlinskiioperator

sh(x) for x ∈ (0, l) h > 0 , (38.fra)

satisfying |sh(x)| ≤ h. In case of the intact initial state we use (8.fra).

Homogenization problem

Taking a sequence εi converging to zero (we shall write ε→ 0 only) we obtain asequence of boundary value problems (35.fra) with a sequence of Ishlinskii operators(34.fra) completed by boundary and initial conditions (36.fra), (37.fra) and the Ishlinskiioperator initial state condition (38.fra). Assuming the solvability of the problems,see [10], we have a sequence of the corresponding solutions uε.

The following steps of homogenization are to be carried out. We have toshow that the sequence uε converge to a function u0 which is a solution to theso called homogenized problem of the similar form. The homogenized problemconsists of the equation[

F0u0x

]x

+ f = ρ u0tt for x ∈ (0, l) t > 0 . (39.fra)

with the homogenized operator F0 independent of the space variable x. Theproblem is completed by analogous boundary and initial conditions

u0(0, t) = 0 , u0(l, t) = 0 for t > 0 , (40.fra)


u0(x, 0) = u0(x) , u0t (x, 0) = u1(x) for x ∈ (0, l) . (41.fra)

Assuming that F0 is also an Ishlinskii operator we add the same condition givingthe initial state of its stop operators sh(x).

5 Homogenized operator

We try to derive the form of the homogenized operator F0. Let us consider theuniformly loaded layered rod. Both components described by Ishlinskii operatorsFA,FB can be considered as combination in series of the elements, namely thelayers satisfy the rule (4.fra).

Denoting the deformation and stress in both materials A,B by eA, eB andσA, σB, the serial combination rule yields

σ = σA = σB . (42.fra)

for the total stress. Taking into account the ratio dA : dB of the thickness of thelayers A,B (dA + dB = 1) we can write

e = dA eA + dB eB . (43.fra)

Inserting inverses eA = GA(σA), eB = GB(σB) of the constitutive relationsσA = FA(eA), σB = FB(eB) into (43.fra) we obtain

e = dA GA(σ) + dB GB(σ) = (dA GA + dB GB) (σ) ,

which implies the limit constitutive relation

σ = F0(e) ≡ (dA GA + dB GB)−1 (e) . (44.fra)

The last equality gives little information on the homogenized operator F0. Ifwe consider an increasing uniform loading from the intact state, we can replacethe Ishlinskii operator by the corresponding functions ΦA, ΦB and rewrite therelation (44.fra) to

Φ0(r) =(dA Φ

−1A + dB Φ

−1B

)−1(r) = (dA ΨA + dB ΨB)−1 (r) . (45.fra)

We have arrived to the following result:

Theorem 3. Under assumptions that the homogenized operator F0 of the ho-mogenized problem (39.fra)–(41.fra) is the Ishlinskii operator of stop type, it is definedby

F 0(e)(t) =∫ ∞

0

(She) (t)dµ0(h) . (46.fra)

The weight function µ0(r) is given by

µ0(r) =(Φ0)′

(0)−(Φ0)′

(r) , (47.fra)

where Φ0(r) is given by (45.fra).

Justification of the assumption and proof of the convergence uε → u0 issubject of the future research.

122 Jan Francu

References

[1] I. Babuska: Homogenization approach in engineering. In Computing methods inapplied sciences and engineering, Lecture notes in Econ. and Math. Systems 134,Springer, Berlin 1976, 137–153.

[2] A. Bensoussan, J. L. Lions, G. Papanicolaou: Asymptotic analysis for periodicstructure, North Holland, 1978.

[3] E. Sanchez Palencia: Non-homogeneous media and vibration theory, Lecture Notesin Physics 127, Springer 1980.

[4] N. S. Bakhvalov, G. P. Panasenko: Averaging of processes in periodic media (Rus-sian), Nauka, Moscow 1984.

[5] P. Krejcı: Methods of solving equations with hysteresis, Proceedings of 12 Seminaron P.D.E. (Czech), Lipovce 1987, 7–60, Union of Czechoslovak Mathematiciansand Physicists Prague, Technical University Plzen, 1987.

[6] M. A. Krasnosel’skii, A. V. Pokrovskii: Systems with hysteresis, Springer, Berlin1989. Russian edition: Nauka, Moscow 1983.

[7] P. Krejcı: Modelling of singularities in elastoplastic materials with fatigue, Appl.Math. 39 (1994), 137–160.

[8] M. Brokate: Hysteresis Operators. In: Phase Transitions and Hysteresis, LectureNotes of C.I.M.E., Montecatini Terme July 13–21, 1993, 1–38, Springer Berlin,1994.

[9] A. Visintin: Differential models of hysteresis, Springer, Berlin Heidelberg 1994.[10] P. Krejcı: Hysteresis, convexity and dissipation in hyperbolic equations, Gakuto

Int. Series Math. Sci. & Appl., Vol. 8, Gakkotosho, Tokyo 1996.



Global Qualitative Investigation,

Limit Cycle Bifurcations and Applicationsof Polynomial Dynamical Systems

Valery Gaiko

Department of MathematicsUniversity of Informatics and Radioelectronics

Koltsov Str. 49-305, Minsk 220090, BelarusEmail: [email protected]

Abstract. Two-dimensional polynomial dynamical systems are mainlyconsidered. By means of Erugin’s two-isocline method we carry out theglobal qualitative investigation of such systems, construct canonical sys-tems with field-rotation parameters and study limit cycle bifurcations.It is known, for example, that generic quadratic systems with limitcycles have three field-rotation parameters and bifurcation surfaces ofmultiplicity-two and three limit cycles are familiar saddle-node and cuspbifurcation surfaces respectively. We use the canonical systems, cyclicityresults and apply Perko’s termination principle, stating that the bound-ary of a global limit cycle bifurcation surface typically consists of Hopfbifurcation surfaces and homoclinic (or heteroclinic) loop bifurcation sur-faces, to prove the non-existence of swallow-tail bifurcation surface ofmultiplicity-four limit cycles for quadratic systems.We discuss also possibilities of application of obtained results to thestudy of higher-dimensional dynamical systems and systems with morecomplicated dynamics.

AMS Subject Classification. 34C05, 34C23, 58F14, 58F21

Keywords. Hilbert’s 16th Problem, Erugin’s two-isocline method, Win-tner’s principle of natural termination, Perko’s planar termination prin-ciple, field-rotation parameter, bifurcation, limit cycle, separatrix cycle

1 Introduction

This paper is connected with the development of a global bifurcation theory ofdynamical systems and discussing possibilities of its application to more compli-cated systems. First of all, it is directed to the solution of Hilbert’s 16th Problemon the maximum number and relative position of limit cycles of two-dimensionaldynamical systems given by the equations

dx

dt= Pn(x, y),

dy

dt= Qn(x, y), (1.gai)


124 Valery Gaiko

where Pn(x, y) and Qn(x, y) are polynomials of real variables x, y with realcoefficients and not greater than n degree.

This is the most difficult problem in the qualitative theory of systems (1.gai).There are a lot of methods and results for the study of limit cycles [1], [2]. How-ever, the Problem has not been solved completely even for the case of simplest(quadratic) systems. It is known only that a quadratic system has at least fourlimit cycles in (3:1) distribution [3]. Besides, we can finally state that a generalpolynomial system has at most a finite number of limit cycles [4]–[6].

A new impulse to the study of limit cycles was given by the introductionof ideas coming from Bifurcation Theory [7]–[10]. We know three principal bi-furcations of limit cycles: 1) Andronov-Hopf bifurcation (from a singular pointof centre or focus type); 2) separatrix cycle bifurcation (from a homoclinic orheteroclinic orbit); 3) multiple limit cycle bifurcation. The first bifurcation wasstudied completely only for quadratic systems: the number of limit cycles bi-furcating from a singular point (its cyclicity) is equal to three [11]. For cubicsystems the cyclicity of a singular point is not less than eleven [12]. The secondbifurcation was studying in [13]–[15]. Now we have the classification of separatrixcycles and know the cyclicity of the most of them (of elementary graphics). Thelast bifurcation is the most complicated. Multiple limit cycles were considering,for example, in [16] and [17]–[19]. All mentioned bifurcations can be generalizedfor higher-dimensional dynamical systems [20]–[22] and can be used for the studyof systems with more complicated dynamics [23]–[25].

But how to find a way to the solution of Hilbert’s 16th Problem? Unfortu-nately, all known limit cycle bifurcations are local. We consider only a neigh-borhood of either the point or the separatrix cycle, or the multiple limit cycle.We consider also local unfoldings in the parameter space. It needs a qualitativeinvestigation on the whole (both on the whole phase plane and on the wholeparameter space), i.e., it needs a global bifurcation theory. This is the first ideaintroduced for the first time by N. P. Erugin in [26]. Then we should connect alllimit cycle bifurcations. This idea came from the theory of higher-dimensionaldynamical systems. It was contained in Wintner’s principle of natural termina-tion [27] and was used by L. M. Perko for the study of multiple limit cycles intwo-dimensional case [17]–[19]. At last, we must understand how to control thelimit cycle bifurcations. The best way to do it is to use field-rotation parametersconsidered for the first time by G. F. Duff in [28]. If we are able to realize theseideas we will answer many questions: 1) How to prove that the maximum num-ber of limit cycles in a quadratic system is equal to four and their only possibledistribution is (3:1)? 2) How to construct a cubic system with more than elevenlimit cycles? 3) What is a strategy of the qualitative investigation on the wholefor cubic and general polynomial systems? 4) How to generalize obtained resultsfor the study of higher-dimensional dynamical systems and to use them for morecomplicated systems?

Global Qualitative Investigation 125

2 Methods and technical difficulties

Methods and approaches used for the study of limit cycles are very different:analytic, algebraic, geometric, etc. There are various combinations. In [4], forexample, classical analytic methods are applied to the analysis of normal formsin special cases of polycycles. The techniques of [5] and [6] are much more sophis-ticated. However, by means of these methods we can prove only the finiteness ofthe number of limit cycles. In the particular case of quadratic (cubic) systemswe do not need such powerful methods, since the number of possible situations israther limited. It is enough to show that limit cycles cannot accumulate on anyseparatrix cycle. Other techniques are used in [13]–[15] where families of planarquadratic vector fields are considered and the cyclicity of unfoldings for variouslimit periodic sets is estimated. This is a new combination: of analytic and bi-furcation methods. But it does not work for non-elementary (non-monodromic)limit periodic sets. It needs a global blow-up of some unfoldings. Even after sucha desingularization we will have only an upper bound of the number of limitcycles. We must find the least upper bound of the number and estimate therelative position of limit cycles!

Purely algebraic methods, for instance, are not able to solve even simplerproblems: to distinguish centre and focus or to give the number of small am-plitude limit cycles bifurcating from a singular point at least for cubic systems.These problems are algorithmically solvable. Nevertheless, it is still complicatedto calculate all the Liapunov focus quantities and to estimate their independence.There are some types of integrable cubic systems: reversible, Hamiltonian, Dar-boux integrable. For the study of limit cycles we perturbate such systems, con-sider linear and higher order Abelian integrals (monodromy variations). But onlyeleven small amplitude limit cycles can be obtained in this way at present [12].

We will develop a geometric aspect of Bifurcation Theory. It will give a globalapproach to the qualitative investigation and will help to combine all other ap-proaches, their methods and results. We will use the two-isocline method, whichwas developed by N. P. Erugin for two-dimensional systems [26] and then wasgeneralized by his pupil V. A. Pliss for three-dimensional case [29]. An isoclineportrait is the most natural construction in the corresponding polynomial equa-tion. It is enough to have only two isoclines (of zero and infinity) to obtaina principal information on the original system, because these two isoclines areright-hand sides of the system. We know geometric properties of isoclines (con-ics, cubics, etc.) and can easily get all isoclines portraits. By means of themwe can obtain all topologically different qualitative pictures of integral curvesto within a number of limit cycles and distinguishing centre and focus. So, wewill be able to carry out a rough topological classification of the phase portraitsfor the polynomial systems. It is the first application of Erugin’s two-isoclinemethod.

Studying contact and rotation properties of isoclines we can also construct thesimplest (canonical) systems containing limit cycles. Two groups of parameterscan be distinguished in such systems: static and dynamic. Static parametersdetermine a behavior of the phase trajectories in principle, since they control

126 Valery Gaiko

the number, position and type of singular points in finite part of the plane(finite singularities). Parameters from the first group determine also a possiblebehavior of separatrices and singular points at infinity (infinite singularities)under the variation of parameters from the second group. Dynamic parametersare rotation parameters. They do not change the number, position and indexof finite singularities and involve a directional rotation in the vector field. Therotation parameters allow to control infinite singularities, a behavior of limitcycles and separatrices. The cyclicity of singular points and separatrix cycles,the behavior of semistable and other multiple limit cycles are controled by theseparameters as well. Thus, with the help of rotation parameters, we can controlall limit cycle bifurcations, i.e., we can solve the most fine qualitative problemsand carry out the global qualitative investigation of the polynomial systems.

Of course, some technical difficulties may arise in such investigation. Wehave a good tool: rotation parameters. But we have no enough experience to usethem. To control all limit cycle bifurcations (especially, of multiple limit cycles),we should know the properties and combine the effects of all rotation parameters.These difficulties can be overcome by means of the development of new methodsbased on Perko’s planar termination principle stating that the maximal one-parameter family of multiple limit cycles terminates either at a singular point,which is typically of the same multiplicity, or on a separatrix cycle, which is alsotypically of the same multiplicity [19]. This principle is a consequence of Wint-ner’s principle of natural termination, which was stated for higher-dimensionaldynamical systems in [27] where A. Wintner studied one-parameter families ofperiodic orbits of the restricted three-body problem and used Puiseux series toshow that in the analytic case any one-parameter family of periodic orbits canbe uniquely continued through any bifurcation except a period-doubling bifurca-tion. Such a bifurcation can happen, for example, in a three-dimensional Lorenzsystem. Besides, the periods in a one-parameter family of a higher-dimensionalsystem can become unbounded in strange ways: for example, the periodic orbitsmay belong to a strange invariant set (strange attractor) generated at a bifur-cation value for which there is a homoclinic tangency of the stable and unstablemanifolds of the Poincare map [17]. This cannot happen for planar systems. Itwould be interesting (in the case of success) to generalize results on multiplelimit cycle bifurcations to more complicated systems.

3 Aims and preliminary results

Global bifurcation theory of quadratic systems. It is known that thegeneric quadratic system with limit cycles has three rotation parameters and bi-furcation surfaces of multiplicity-two and three limit cycles are familiar saddle-node and cusp bifurcation surfaces respectively. We will apply Perko’s termi-nation principle to prove the non-existence of swallow-tail bifurcation surfaceof multiplicity-four limit cycles, i.e., using the data on the cyclicity of singularpoints and separatrix cycles we will prove by contradiction that a quadratic sys-tem cannot have more than four limit cycles, that the distributions (4:0), (2:2)


are impossible and the multiplicity of a limit cycle is not higher than three.Thus we intend to prove that for quadratic systems the maximum number oflimit cycles is equal to four and the only possible their distribution is (3:1).

Cubic and general polynomial systems. First of all, a general strategyof the qualitative investigation on the whole will be developed. The main resultsby analogy with quadratic systems will be obtained and systematized. We willapply Erugin’s two-isocline method to get all isocline portraits of cubic systems,to carry out the rough topological classification of their phase portraits and toconstruct the canonical systems with field-rotation parameters, which will beused for various aims: study of limit cycle bifurcations, classification of separa-trix cycles, obtaining bifurcation diagrams. We will use Poincare topographicalsystems and small parameter method, Abelian integrals and variational methodsto construct a cubic system with more than eleven limit cycles. All these resultswill be generalized to develop a global bifurcation theory of planar polynomialsystems.

Higher-dimensional dynamical systems and applications. We will ap-ply the theory of planar dynamical systems to the qualitative investigation ofhigher-dimensional systems. Various bifurcations in reversible, Hamiltonian andconservative systems will be considered: Hopf bifurcation, bifurcations of homo-clinic and heteroclinic orbits (including degenerate cases). Multiple limit cycle bi-furcations with the application of Wintner’s principle of natural termination willbe studied as well. Since theory of dynamical systems and bifurcation methodscan be used for the mathematical modelling natural systems with complicateddynamics, we will consider possibilities of the application of global bifurcationtheory, for instance, to the study of Josephson junctions in forsed non-lineardynamical networks, non-linear evolution systems in Belousov-Zhabotinskii re-action, etc.

Results. A particular preliminary work in this direction has already beencarried out in [30]–[41]. By means of Erugin’s two-isocline method we carried outthe global qualitative investigation of quadratic systems, constructed the canon-ical systems with field-rotation parameters and applied them for the study oflimit cycle bifurcations: Andronov-Hopf bifurcation, bifurcations of homoclinicand heteroclinic orbits (separatrix cycles), multiple limit cycle bifurcations. Westudied the bifurcations of various codimensions and introduced so-called a func-tion of limit cycles: a cross-section of Andronov-Hopf surface formed by limit cy-cles and the corresponding values of rotation parameters. Using numerical andanalytic methods, we constructed concrete examples of systems with differentnumber and relative position of limit cycles. In particular, the following theoremshave been proved:

Theorem 1. A quadratic system has at least four limit cycles in (3:1) distribu-tion.

Theorem 2. . Any quadratic system with limit (separatrix) cycles can be re-duced to one of the systems:

128 Valery Gaiko

dx

dt= −(x+ 1)y + αQ(x, y),

dy

dt= Q(x, y) (2.gai)

or

dx

dt= −y + αy2,

dy

dt= Q(x, y), (3.gai)

whereQ(x, y) = x+ λy + ax2 + b(x+ 1)y + cy2.

We developed a new approach to the classification of separatrix cycles. Itis based on the application of canonical systems (2.gai) and (3.gai). The classificationwas carried out according to the number and type of finite singularities of theoriginal reversible systems and with the help of the successive variation of ro-tation parameters. That approach allowed not only to define all possible typesof separatrix cycles, but also to determine their cyclicity and relative position,to obtain both the corresponding phase portraits and the division of parameterspace. By means of the field-rotation parameters and function of limit cycles weshowed how to control semistable limit cycles: we were changing the rotationparameters so that to push the semistable cycles either to a singular point offocus (centre) type or to some separatrix cycle and to obtain the contradictionwith their cyclicity. On the basis of reversible systems we constructed Poincaretopographical systems and with the help of small parameter method studiedvarious periodic orbits: limit cycles, centre curves. We developed also a controltheory of quadratic systems and considered possibilities of the application of ourresults to general polynomial systems.

References

[1] A. A. Andronov, E. A. Leontovich, I. I. Gordon, A. G. Maier, Theory of Bi-furcations of Dynamical Systems on a Plane, Israel Progr. Scient. Transl.,Jerusalem 1971.

[2] A. A. Andronov, E. A. Leontovich, I. I. Gordon, A. G. Maier, Qualitative Theoryof Second Order Dynamical Systems, Wiley, New York 1973.

[3] Y. Ye et al., Theory of Limit Cycles, AMS Transl. Math. Monogr. 66, Providence,RI 1986.

[4] H. Dulac, Sur les cycles limites, Bull. Soc. Math. France, 51 (1923), 45–188.[5] Yu. Iliashenko, Finiteness theorems for limit cycles, Russian Math. Surv.,

40 (1990), 143–200.

[6] J. Ecalle, Finitude des cycles limites et accelero-sommation de l’application deretour, Lect. Not. Math., 1455 (1990), 74–159.

[7] S.-N. Chow, J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag,New York 1982.

[8] J. Guckenheimer, P. Holms, Nonlinear Oscillations, Dynamical Systems, and Bi-furcations of Vector Fields, Springer-Verlag, New York 1983.


[9] M. Golubitsky, D. G. Shaeffer, Singularities and Groups in Bifurcation Theory,Springer-Verlag, New York 1985.

[10] S.-N. Chow, C. Li, D. Wang, Normal Forms and Bifurcations of Planar VectorFields, Cambridge Univ. Press, Cambridge 1994.

[11] N. N. Bautin, On the number of limit cycles which appear with the variation ofthe coefficients from an equilibrium point of focus or center type, Matem. Sbor.,30 (1952), 181–196. (in Russian)

[12] H. Zo ladek, Eleven small limit cycles in a cubic vector field, Nonlinearity, 8 (1995),843–860.

[13] F. Dumortier, R. Roussarie, C. Rousseau, Hilbert’s 16th problem for quadraticvector fields, J. Diff. Equations, 110 (1994), 86–133.

[14] F. Dumortier, R. Roussarie, C. Rousseau, Elementary graphics of cyclicity 1 and2, Nonlinearity, 7 (1994), 1001–1043.

[15] F. Dumortier, M. El Morsalani, C. Rousseau, Hilbert’s 16th problem for quadraticsystems and cyclicity of elementary graphics, Nonlinearity, 9 (1996), 1209–1261.

[16] J.-P. Francoise, C. C. Pough, Keeping track of limit cycles, J. Diff. Equations, 65(1986), 139–157.

[17] L. M. Perko, Global families of limit cycles of planar analytic systems, Trans-act. Amer. Math. Soc., 322 (1990), 627–656.

[18] L. M. Perko, Homoclinic loop and multiple limit cycle bifurcation surfaces, Trans-act. Amer. Math. Soc., 344 (1994), 101–130.

[19] L. M. Perko, Multiple limit cycle bifurcation surfaces and global families of mul-tiple limit cycles, J. Diff. Equations, 122 (1995), 89–113.

[20] S.-N. Chow, B. Deng, B. Fiedler, Homoclinic bifurcation at resonant eigenvalues,J. Dynam. Diff. Equations, 2 (1990), 177–244.

[21] R. Roussarie, C. Rousseau, Almost planar homoclinic loops in R3, J. Diff. Equa-tions, 126 (1996), 1–47.

[22] J. Knobloch, A. Vanderbauwhede, A general reduction method for periodic solu-tions in conservative and reversible systems, J. Dynam. Diff. Equations, 8 (1996),71–102.

[23] U. Schalk, J. Knobloch, Homoclinic points near degenerate homoclinics, Nonlin-earity, 8 (1995), 1133–1141.

[24] T. Bartsch, M. Kern, Bifurcation of steady states in a modified Belousov-Zhabotinskii reaction, Topol. Meth. Nonlin. Analysis, 2 (1993), 105–124.

[25] T. Bartsch, Bifurcation of stationary and heteroclinic orbits for parametrizedgradient-like flows, Topol. Nonlin. Analysis, 35 (1996), 9–27.

[26] N. P. Erugin, Some questions of motion stability and qualitative theory of differ-ential equations on the whole, Prikl. Mat. Mekh., 14 (1952), 459–512. (in Russian)

[27] A. Wintner, Beweis des E. Stromgrenschen dynamischen Abschlusprinzips derperiodischen Bahngruppen im restringierten Dreikorperproblem, Math. Z., 34(1931), 321–349.

[28] G. F. D. Duff, Limit cycles and rotated vector fields, Ann. Math, 67 (1953), 15–31.[29] V. A. Pliss, Nonlocal Problems of Oscillation Theory, Nauka, Moscow 1964.

(in Russian)[30] L. A. Cherkas, V. A. Gaiko, Bifurcations of limit cycles of a quadratic system

with two field-rotation parameters, Dokl. Akad. Nauk BSSR, 29 (1985), 694–696.(in Russian)

[31] L. A. Cherkas, V. A. Gaiko, Bifurcations of limit cycles of a quadratic systemwith two critical points and two field-rotation parameters, Diff. Urav., 23 (1987),1544–1553 (in Russian); Diff. Equations, 23 (1987), 1062–1069.

130 Valery Gaiko

[32] V. A. Gaiko, L. A. Cherkas, Bifurcations of limit cycles of the vector fields onC2 and R2, in Problems of Qualitative Theory of Differential Equations, Nauka,Novosibirsk 1988, 17–22. (in Russian)

[33] V. A. Gaiko, Separatrix cycles of quadratic systems, Dokl. Akad. Nauk Belarusi,37 (1993), 18–21. (in Russian)

[34] V. A. Gaiko, The limit cycle bifurcations of quadratic autonomous systems, inNonlinear Phenomena in Complex Systems, Inst. Phys. Press, St.Petersburg 1993,60–65.

[35] V. A. Gaiko, On application of two-isocline method to investigation of two-dimensional dynamical systems, Adv. Synerg., 2 (1994), 104–109.

[36] V. A. Gaiko, Bifurcations of limit cycles and classification of separatrix cyclesof two-dimensional polynomial dynamical systems, Vest. Bel. Gos. Univ. Ser. 1(1995), 69–70. (in Russian)

[37] V. A. Gaiko, Geometric approaches to qualitative investigation of polynomial sys-tems, Adv. Synerg., 6 (1996), 176–180.

[38] V. A. Gaiko, On development of new approaches to investigation of limit cyclebifurcations, Adv. Synerg., 8 (1997), 162–164.

[39] V. A. Gaiko, Application of topological methods to qualitative investigationof two-dimensional polynomial dynamical systems, to appear in Univ. Iagel-lon. Acta Math., 36 (1997).

[40] V. A. Gaiko, Qualitative theory of two-dimensional polynomial dynamical sys-tems: problems, approaches, conjectures, Nonlin. Analysis, Theory, Meth. Appl.,30 (1997), 1385–1394.

[41] V. A. Gaiko, Control of multiple limit cycles, to appear in Adv. Synerg.



The Existence of Global Solutions to the

Elliptic-Hyperbolic Davey-Stewartson System

with Small Initial Data

Nakao Hayashi1 and Hitoshi Hirata2

1 Department of Applied Mathematics, Science University of Tokyo1-3, Kagurazaka, Shinjuku-ku, Tokyo 162, Japan

Email: [email protected] Department of Mathematics, Sophia University

7-1, Kioicho, Chiyoda-ku, Tokyo 102, JapanEmail: [email protected]

Abstract. We study the initial value problem for the elliptic-hyperbolicDavey-Stewartson system. Our purpose in this paper is to prove globalexistence of small solutions of this system in the usual weighted Sobolevspace H3,0 ∩ H0,3. Furthermore we prove L∞ time decay estimates inL∞ of solutions u such that ‖u(t)‖L∞ ≤ C(1 + |t|)−1.

AMS Subject Classification. 35D05, 35E15, 35Q55, 76B15

Keywords. Davey-Stewartson system, commutator calculus

We study the initial value problem for the Davey-Stewartson systemsi∂tu+ c0∂

2x1u+ ∂2

x2u = c1|u|2u+ c2u∂x1ϕ, (x, t) ∈ R3,

∂2x1ϕ+ c3∂

2x2ϕ = ∂x1 |u|2,

u(x, 0) = φ(x),(1.hir)

where c0, c3 ∈ R, c1, c2 ∈ C, u is a complex valued function and ϕ is areal valued function. The systems (1.hir) for c3 > 0 were derived by Davey andStewartson [4] and model the evolution equation of two-dimensional long wavesover finite depth liquid. Djordjevic-Redekopp [5] showed that the parameter c3can become negative when capillary effects are significant. When (c0, c1, c2, c3)= (1,−1, 2,−1), (−1,−2, 1, 1) or (−1, 2,−1, 1) the system (1.hir) is referred as theDSI, DSII defocusing and DSII focusing respectively in the inverse scatteringliterature. In [7], Ghidaglia and Saut classified (1.hir) as elliptic-elliptic, elliptic- hy-perbolic, hyperbolic-elliptic and hyperbolic-hyperbolic according to the respec-tive sign of (c0, c3) : (+,+), (+,−), (−,+) and (−,−). For the elliptic-ellipticand hyperbolic-elliptic cases, local and global properties of solutions were stud-ied in [7] in the usual Sobolev spaces L2, H1 and H2. In this paper we consider

The final version of this paper is published in Nonlinearity 9, which is cited in [10].

132 N. Hayashi and H. Hirata

the elliptic-hyperbolic case. In this case after a rotation in the x1x2-plane andrescaling, the system (1.hir) can be written as

i∂tu+∆u = d1|u|2u+ d2u∂x1ϕ+ d3u∂x2ϕ,

∂x1∂x2ϕ = d4∂x1 |u|2 + d5∂x2 |u|2,(2.hir)

where ∆ = ∂2x1

+ ∂2x2

, d1, · · · , d5 are arbitrary constants. In order to solve thesystem of equations, one has to assume that ϕ(·) satisfies the radiation condition,namely, we assume that for given functions ϕ1 and ϕ2

limx2→∞

ϕ(x1, x2, t) = ϕ1(x1, t) and limx1→∞

ϕ(x1, x2, t) = ϕ2(x2, t). (3.hir)

Under the radiation condition (3.hir), the system (2.hir) can be written as

i∂tu+∆u = d1|u|2u+ d2u

∫ ∞x2

∂x1 |u|2(x1, x2′, t)dx2

′

+ d3u

∫ ∞x1

∂x2 |u|2(x1′, x2, t)dx1

′ + d4u∂x1ϕ1 + d5u∂x2ϕ2 (4.hir)

with the initial condition u(x, 0) = φ(x). In what follows we consider the equa-tion (4.hir).

In this paper we use the following notations.

Notations. We define the weighted Sobolev space as follows

Hm,l = f ∈ L2; ‖(1− ∂2x1− ∂2

x2)m/2(1 + x1

2 + x22)l/2f‖ <∞,

Hm,l(Rxj ) = f ∈ L2(Rxj ); ‖(1− ∂2xj )

m/2(1 + xj2)l/2f‖L2(Rj) <∞,

where ‖·‖ denotes the usual L2 norm. We let ∂ = (∂x1 , ∂x2), J = (Jx1 , Jx2), Jxj =xj+2it∂xj . For simplicity we write Lpxj = Lp(Rxj ), L

px1Lqx2

= Lp(Rx1 ;Lq(Rx2)),Hm,lxj = Hm,l(Rxj ), ‖ · ‖Xm,l(t) =

∑|α|≤m ‖∂α · ‖ +

∑|α|≤l ‖Jα · ‖, where α =

(α1, α2), |α| = α1 + α2, α1, α2 ∈ N ∪ 0.

Local existence of small solutions to (4.hir) was shown when the initial functionis in Hm,l in [13] for H12,0 ∩H0,6, [8] for Hm,0 ∩H0,l, (m, l > 1), [1] for Hm,0,(m is sufficiently large integer) and [9] for Hm,0, (m ≥ 5/2). Furthermore in [11]without smallness condition on the data local existence of solutions was provedin the analytic function space which consists of real analytic functions. Globalexistence of small solutions to (4.hir) was also given in [11] when the data are realanalytic and satisfy the exponential decay condition.

Recently, H. Chihara [1] established the global existence of small solutionsto (4.hir) in higher order Sobolev spaces. Our purpose in this paper is to prove theglobal existence of small solutions to (4.hir) in the usual weighted Sobolev spacesH3,0 ∩H0,3, which is considered as lower order Sobolev class compared to oneused in [1], by the calculus of commutator of operators. We shall prove

The Existence of Global Solutions to DSI System 133

Theorem 1. Let φ ∈ H3,0 ∩H0,3, ∂j+1x1

ϕ1 ∈ C(R;L∞x1), ∂j+1

x2ϕ2 ∈ C(R;L∞x2

),(0 ≤ j ≤ 3), ε3 and δ3 be sufficiently small, where

εm = supt∈R

∑0≤j≤m

(1 + t)1+a(‖(t∂x1)j∂x1ϕ1(t)‖L∞x1

+ ‖∂j+1x1

ϕ1(t)‖L∞x1

+ ‖(t∂x2)j∂x2ϕ2(t)‖L∞x2+ ‖∂j+1

x2ϕ2(t)‖L∞x2

), a > 0,

δm ≥( ∑|α|+|β|≤m

‖∂α1x1∂α2x2x1β1x2

β2φ‖2)1/2

.

Then there exists a unique global solution u of (4.hir) such that

u ∈ L∞local(R;H3,0 ∩H0,3) ∩ C(R;H2,0 ∩H0,2), (5.hir)

supt∈R

( ∑|α|+|β|≤2

‖∂αJβu(t)‖+∑

|α|+|β|≤3

(1 + t)−Cδ3‖∂αJβu(t)‖)≤ 4δ3. (6.hir)

Corollary 2. Let u be the solution constructed in Theorem 1. Then we have

‖u(t)‖L∞ ≤ C(1 + |t|)−1(‖φ‖H3,0 + ‖φ‖H0,3).

Moreover, for any φ ∈ H3,0 ∩H0,3 there exist u± such that

‖u(t)− U(t)u±‖H2,0 → 0 as t→ ±∞,

where U(t) = eit(∂2x1

+∂2x2

).

The rate of decay obtained in Corollary 2 is the same as that of solutions tolinear Schrodinger equations. Time decay of solutions for the Davey-Stewartsonsystems (1.hir) was obtained in [3,7] when (c0, c3) = (+,+) and (c0, c3) = (−,+)and in [11] when (c0, c3) = (+,−) and (c0, c3) = (−,−) under exponential decayconditions on the data.

We try to explain our strategy of the proof of Theorem 1. For simplicity weconsider following equation

i∂tu+∆u = u

∫ ∞x2

∂x1 |u|2dx2′,

which have only main nonlinear term. We use following operator Kx1 and Kx2 ,where

Kx1 = Kx1(v) =∞∑m=0

Am

m!

(∫ x1

−∞‖v(t, x1

′)‖2L2x2dx1′ Dx1

〈Dx1〉

)mand

Kx2 = Kx2(v) =∞∑m=0

Am

m!

(∫ x2

−∞‖v(t, x2

′)‖2L2x1dx2′ Dx2

〈Dx2〉

)m.

So, if we take A2 = 1/δ3 (for the definition of δ3, see Theorem 1), by virtue ofcommutator estimates and following lemma,


Lemma 3.

‖[〈Dx1〉1/2, f ]g‖L2x1

+ ‖[〈Dx1〉, f ]g‖L2x1≤ C‖〈Dx1〉f‖L∞x1

‖g‖L2x1,

which follows from Coifman and Meyer’s result (see [2, p. 154]), we have

12d

dt

∑|α|+|β|≤3

(‖Kx1∂αJβu(t)‖2 + ‖Kx2∂

αJβu(t)‖2)

+1

4δ1/23

∑|α|+|β|≤3

(∥∥‖u(t)‖L2x2‖〈Dx1〉

1/2Kx1∂

αJβu(t)‖L2x2

∥∥2

L2x1

+∥∥‖u(t)‖L2

x1‖〈Dx2〉

1/2Kx2∂

αJβu(t)‖L2x1

∥∥2

L2x2

)≤ C(1 +A)2(1 + t)−1‖u(t)‖2X2,2(t)(1 + ‖u(t)‖2X2,2(t))‖u(t)‖2X3,3(t)

+∑

|α|+|β|≤3

(∣∣∣∣Im(Kx1∂αJβu

∫ ∞x2

∂x1 |u|2dx2′, Kx1∂

αJβu)∣∣∣∣

+∣∣∣∣Im(Kx2∂

αJβu

∫ ∞x2

∂x1 |u|2dx2′, Kx2∂

αJβu)∣∣∣∣). (7.hir)

The second term of the left hand side of (7.hir) means smoothing properties ofsolutions to the equation. So we have to estimate the term

∑|α|+|β|≤3

(∣∣∣∣Im(Kx1∂αJβu

∫ ∞x2

∂x1 |u|2dx2′, Kx1∂

αJβu)∣∣∣∣

+∣∣∣∣Im(Kx2∂

αJβu

∫ ∞x2

∂x1 |u|2dx2′, Kx2∂

αJβu)∣∣∣∣).

For this purpose, we pay attention to the special structure of the nonlinear term

u

∫ ∞x2

∂x1 |u|2dx2′ = u

12it

∫ ∞x2

uJx1u− uJx1udx2′. (8.hir)

This deformation shows this nonlinear term has own time decay in some sence.Using this structure, we can estimate as following,

12d

dt

∑|α|+|β|≤3

(‖Kx1∂αJβu(t)‖2 + ‖Kx2∂

αJβu(t)‖2)

+(

1

4δ1/23

− CeCδ3) ∑|α|+|β|≤3

(∥∥‖u(t)‖L2x2‖〈Dx1〉

1/2Kx1∂

αJβu(t)‖L2x2

∥∥2

L2x1

+∥∥‖u(t)‖L2

x1‖〈Dx2〉

1/2Kx2∂

αJβu(t)‖L2x1

∥∥2

L2x2

)≤ C(1 + t)−1δ3‖u(t)‖2X3,3(t)

(9.hir)

The Existence of Global Solutions to DSI System 135

provided that δ3 is sufficiently small and

sup−T≤t≤T

‖u(t)‖2X2,2(t) ≤ 4δ23, (10.hir)

sup−T≤t≤T

(1 + |t|)−Cδ3‖u(t)‖2X3,3(t) ≤ 4δ23 (11.hir)

for some time T > 0. We choose δ3 satisfying

1

4δ1/23

− CeCδ3 ≥ 0.

Then we have

‖u(t)‖2X3,3(t) ≤ eCδ3δ23 + Cδ3

∫ t

0

(1 + s)−1‖u(s)‖2X3,3(t)ds. (12.hir)

Thus (9.hir) shows that the nonliear term is controlled by the second term of theleft hand side of (7.hir) and the right hand side of (9.hir). Global existence theorem isobtained by showing that (10.hir) and (11.hir) hold for any T . In order to prove (10.hir)and (11.hir) for any T > 0 we need (12.hir) and the following inequality

‖u(t)‖2X2,2(t) ≤ eCδ3δ23 + Cδ3

∫ t

0

(1 + s)−1−2Cδ3‖u(s)‖2X3,3(t)ds. (13.hir)

The inequality (13.hir) is obtained by the structure of nonlinear term (8.hir) again.Theorem 1 is obtained by applying the Gronwall inequality to (12.hir) and (13.hir).

It seems to be difficult to get the inequality (12.hir) through the methods used in[8,9], because nonlinear terms are not taken into account to derive smoothingproperties of solutions in [8,9]. On the other hand the operators Kx1 and Kx2

are made based on the nonlocal nonlinear terms (the second and the third termson the right hand side of (4.hir)). The similar operators as those of Kx1 and Kx2

have been used in [1] to obtain Theorem 0.1 and the local existence theorem ofsmall solutions to (4.hir) in the usual order Sobolev space.

Remark 4. We cannot apply above method to hyperbolic-hyerbolic Davey-Ste-wartson system. In fact, if we estimate similarly as above, we have

12d

dt

∑|α|+|β|≤3

(‖Kx1∂αJβu(t)‖2 + ‖Kx2∂

αJβu(t)‖2)

+1

4δ1/23

∑|α|+|β|≤3

(∥∥‖u(t)‖L2

x2‖〈Dx1〉

1/2Kx1∂

αJβu(t)‖L2x2

∥∥2

L2x1

+∥∥‖u(t)‖L2

x1‖〈Dx2〉

1/2Kx2∂αJβu(t)‖L2

x1

∥∥2

L2x2

)

≤ C(1 +A)2(1 + t)−1‖u(t)‖2X2,2(t)(1 + ‖u(t)‖2X2,2(t))‖u(t)‖2X3,3(t)

+ CeCδ3∑

|α|+|β|≤3

∥∥‖u(t)‖L2x2‖〈Dx1〉1/2Kx1∂

αJβu(t)‖L2x2

∥∥2

L2x1

(14.hir)


under the condition (10.hir) and (11.hir), where

Kx1 =∞∑m=0

Am

m!

(∫ x2

−∞‖v(t, x2

′)‖2L2x1dx2′ Dx1

〈Dx1〉

)m= e

A∫ x2−∞ ‖v(t,x2

′)‖2L2x1dx2′ Dx1〈Dx1 〉

and

Kx2 =∞∑m=0

Am

m!

(∫ x1

−∞‖v(t, x′1)‖2L2

x2dx1′ Dx2

〈Dx2〉

)m= e

A∫ x1−∞ ‖v(t,x′1)‖2

L2x2dx1′ Dx2〈Dx2 〉 ,

but we can easy to see that the last term of the right-hand side of (14.hir) cannotcontrolled by the second term of the left-hand side of (14.hir).

Remark 5. For Davey-Stewartson systems, we can define formally the energysimilar as the usual nonlinear Schrodinger equation if c1, c2 ∈ R. But un-fortunately, this energy is not conserved in elliptic-hyperbolic and hyperbolic-hyperbolic cases. So, we cannot use the usual H1 a-priori estimate by energy.This is one reason that the global existence theorem of this system is difficult.

References

[1] H. Chihara, The initial value problem for the elliptic-hyperbolic Davey-Stewartsonequation, preprint (1995).

[2] R. R. Coifman and Y. Meyer, Au dela des operateurs pseudodifferentieles,Asterique 57, Societe Mathematique de France, 1978.

[3] P. Constantin, Decay estimates of Schrodinger equations, Comm. Math. Phys. 127(1990), 101–108.

[4] A. Davey and K. Stewartson, On three-dimensional packets of surface waves, Proc.R. Soc. A. 338 (1974), 101–110.

[5] V. D. Djordjevic and L. G. Redekopp, On two-dimensional packets of capillary-gravity waves, J. Fluid Mech. 79 (1977), 703–714.

[6] A. S. Fokas and L. Y. Sung, On the solvability of the N-wave, Davey-Stewartsonand Kadomtsev-Petviashvili equations, Inverse Problems 8 (1992), 673–708.

[7] J. M. Ghidaglia and J. C. Saut, On the initial value problem for the Davey-Stewartson systems, Nonlinearity 3 (1990), 475–506.

[8] N. Hayashi, Local existence in time of small solutions to the Davey-Stewartsonsystems, Ann. Inst. Henri Poincare, Physique theorique 65-4 (1996), 313–366.

[9] N. Hayashi and H. Hirata, Local existence in time of small solutions to the elliptic-hyperbolic Davey-Stewartson system in the usual Sobolev spaces, Proc. Royal Soc.of Edinburgh Sec. A (1997) (to appear).

[10] N. Hayashi and H. Hirata, Global existence and asymptotic behaviour in time ofsmall solutions to the elliptic-hyperbolic Davey-Stewartson system, Nonlinearity 9(1996), 1387–1409.

[11] N. Hayashi and J. C. Saut, Global existence of small solutions to the Davey-Stewartson and the Ishimori systems, Diff. Integral Eqs. 8 (1995), 1657–1675.

[12] H. Kumano-go, Pseudo-Differential Operators, The MIT press, 1974.[13] F. Linares and G. Ponce, On the Davey-Stewartson systems, Ann. Inst. Henri

Poincare, Anal. non lineaire 10 (1993), 523–548.



Scaling in Nonlinear Parabolic Equations :

Locality versus Globality

Grzegorz Karch

Instytut Matematyczny, Uniwersytet Wroc lawskipl. Grunwaldzki 2/4, 50-384 Wroc law, Poland

Email: [email protected]: http://www.math.uni.wroc.pl/~karch

Abstract. The Cauchy problem for parabolic equations with quadraticnonlinearity is studied. We investigate the existence of global-in-timesolutions and their large-time behavior assuming some scaling propertyof the equation as well as of the norm of the Banach space in which thesolutions are constructed.

AMS Subject Classification. 35K55, 35K15

Keywords. the Cauchy problem, self-similar solutions

We study the Cauchy problem for the parabolic equation

ut = ∆u+ B(u, u)

supplemented by the initial condition

u(x, 0) = u0(x).

Here u = u(x, t), x ∈ Rn, and t ∈ [0, T ) for some T ∈ (0,∞]. We assume that thenonlinear term B(·, ·) is defined by a bilinear form acting on u(x, t) with respectto x only. This nonlinearity will also be assumed to satisfy a scaling property. Toset it up, first given f : Rn → R we define the rescaled function fλ(x) = f(λx)for each λ > 0. We extend this definition for all f ∈ §′ in the standard way.

Definition 1. The bilinear form B(·, ·) is said to have the scaling order equalto b ∈ R if

B(fλ, gλ) = λb(B(f, g)

)λ

for any λ > 0 and all f, g ∈ §′(Rn), for which the both sides make sense.

Our main requirement is that the bilinear form B(·, ·) has the scaling orderequal to b < 2.

Now suppose we are able to construct local-in-time solutions in C([0, T );E),where the Banach space E consists of tempered distributions. Assume, more-over, that the equation is invariant under some scaling transformations of the

The paper is the extended abstract of the paper [1].

http://www.math.uni.wroc.pl/~karch

138 Grzegorz Karch

independent and dependent variables. We show that these two assumptions com-bined with a scaling property of ‖ ·‖E allow us to obtain global-in-time solutionsfor suitably small initial data. To get such results we introduce a new Banachspace of distributions which, roughly speaking, is a homogeneous Besov typespace modeled on E. This approach allows us to get solutions for initial dataless regular than those from E. In this abstract setting, we also study large-timebehavior of constructed solutions. We find a simple condition (in terms of decayproperties of the heat semigroup) which guarantees that solutions have the sameasymptotic behavior as t→∞.

References

[1] Karch, G., Scaling in nonlinear parabolic equations: locality versus globality, Reportof the Mathematical Institute, University of Wroclaw 92 (1997) 1–29, submitted.



Almost Sharp Conditions for the Existence of

Smooth Inertial Manifolds

Norbert Koksch

Departement of Mathematics, Technical University Dresden,01062 Dresden, Germany


Abstract. We consider the nonlinear evolution equation u+Au = f(u)in a separable, real Hilbert space H assuming that A is a linear, self-adjoint, positive operator on H with compact resolvent. The nonlinearityf is assumed to belong to Ckb (D(Aα),D(Aβ)) with k ∈ N>0 ∪ 1− andnonnegative α, β satisfying 0 ≤ α − β ≤ 1

2. Let PN be the orthogonal

projection of H onto the subspace generated by the eigenvectors corre-sponding to the first N eigenvalues λi of A. We state an existence theo-rem for an inertial Ck manifold graph(ϕ) with ϕ ∈ Ckb (PND(Aα), (I −PN)D(Aα)) using an almost sharp spectral gap condition

λN+1 − kλN >√

2 Lip (f)(λα−βN+1 + kλα−βN

).

Assuming the existence of an absorbing ball BD(Aα)(r) in dom(Aα), and

assuming only f |BD(Aα)(√

2r) ∈ Ckb (BD(Aα)(√

2r),D(Aβ)), we state theexistence of a globally attracting, locally positively invariant Ck manifoldgraph(ϕ) ∩BD(Aα)(r) using the spectral gap condition

λN+1 − kλN >√

2 Lip(f |BD(Aα)(

√2r))(

λα−βN+1 + kλα−βN

)where r > r. For it a special preparation of f is used.The proofs of the theorems base on comparison theorems for special two-point boundary value problems and for inequalities in ordered Banachspaces.

AMS Subject Classification. 34C30, 35K22, 34G20, 47H20

Keywords. smooth inertial manifolds, spectral gap condition, graphtransformation, boundary value problems, comparison theorems

1 Introduction

Let H be a separable, real Hilbert space with inner product 〈·|·〉 and norm | · |.We consider the nonlinear evolution equation

u+Au = f(u) (1.kok)

for u ∈ H where A satisfies


140 Norbert Koksch

Assumption 1. A is a linear, self-adjoint, positive operator on H with compactresolvent.

Thus, −A is the infinitesimal generator of an analytic semigroup on H.Let λ1 ≤ λ2 ≤ λ3 ≤ · · · denote the eigenvalues of A repeated with their

multiplicities, and let ei denote corresponding orthonormal eigenvectors of A.By the properties of A the eigenvectors ei form an orthonormal basis in H.

We can define the fractional powers Aα for α ∈ R, see [Hen81]. The do-mains Uα := D(Aα) of Aα are Hilbert spaces with respect to the scalar prod-uct 〈u|v〉α := 〈Aαu|Aαv〉, and the corresponding norm | · |α is equivalent tothe graph norm. With PN we denotes the orthogonal projection of H onto

spane1, . . . , eN. Since Uα = u ∈ H :∞∑j=1

〈u, ej〉2 λ2αj < ∞, we have PNH ∩

Uα = PNUα and (I − PN )H ∩ Uα = (I − PN )Uα. Further PN commutes withAγ for γ ≥ 0.

The nonlinear term f is assumed to satisfy at least

Assumption 2. There are k ∈ N>0 ∪ 1−, κ ∈ [0, 1[ with κ = 0 iff k = 1−and nonnegative constants α, β satisfying 0 ≤ α− β ≤ 1

2 such that f |Ω belongsto Ck+κ

bu (Ω,Uβ) for any bounded set Ω ⊂ Uα.

Here f |Ω denotes the restriction of f onto Ω. C1−bu (E,F) denotes the Banach

space of the bounded continuous functions from E into F being uniformly Lip-schitz. For k ≥ 1, Ck+κ

bu (E,F) denotes the Banach space of the k-times κ-Holdercontinuously differentiable functions from E into F with bounded derivatives upto the order k. In the following, we calculate with 1− as with 1. We denote byLip (g) the smallest Lipschitz constant of g on its domain dom(g).

For a subspace U of Uα endowed with the induced topology let

BU(r) := u ∈ U : |u|α < r

be the open ball in U centered at 0 with radius r ≤ ∞.Applying the results of [Hen81], equation (1.kok) generates a (local) semigroup

S in Uα, such that the (classical) solution at time t in the existence intervalthrough an initial point u0 ∈ Uα is given by u(t) = S(t)u0. For t > 0, u(t) ismore regular than the initial point, with u(t) ∈ U1+β ⊆ D(A) and u(t) ∈ Uβ .These regularity results make it possible to work with the equation itself andtake inner products rather than have to use the variation of constant formula.In particular expressions such as

12d

dt|u(t)|2α =

⟨Aβ u(t)|A2α−βu(t)

⟩make sense for t > 0 since U1+β ⊆ U2α−β because of α − β ≤ 1

2 and sinceu(t) ∈ Uβ and u(t) ∈ U1+β for t > 0.

Recall that an inertial Ck manifold M is a subset of H with the followingproperties (see [MPS88,FST88,Tem88] for k = 1−):

Smooth Inertial Manifolds 141

1. M is a finite dimensional Ck manifold in Uα ⊆ H.2. M is positively invariant; i.e., if u0 ∈M then S(t)u0 ∈M for all t ∈ [0,∞[.3. M is exponentially attracting; i.e., there is a γ > 0 such that for every η ∈ Uα

there is a C such that

dist(S(t)η,M) ≤ Ce−γt (t ≥ 0).

In some papers the exponential attracting property is supplemented by theexponential tracking property ([FST89]) or asymptotical completeness property([CFNT89,Rob96,Tem97]):

There is γ > 0 such that for every η ∈ Uα there are η ∈ M andC ≥ 0 with |S(t)η − S(t)η|α ≤ Ce−γt dist(η,M) for all t > 0.

Usually we are looking for an inertial Ck manifold M which is constructedas the graph graph(ϕ) := ξ + ϕ(ξ) : ξ ∈ PNUα of a Ck function ϕ : PNUα →(I − PN )Uα.

Because of the attraction property of M, the asymptotical behavior of thesolutions of (1.kok) is governed by the asymptotical behavior of the solutions on thefinite-dimensional manifold M. The dynamic on M is determined by the ordinarydifferential equation (inertial form)

x+Ax = PNf(x+ ϕ(x))

in the N -dimensional Banach space PNUα.Instead of Assumptions 2 usually one assumes

f ∈ Ckb (Uα,Uβ) (2.kok)

with suitable α ≥ β: For k = 1− we have for example α = 1, β = 12 in [FST88],

β = α − 12 in [Tem88], α = β = 0 in [MPS88], 0 = β ≤ α < 1 in [Rom94],

0 ≤ α − β ≤ 12 in [Rob93], 0 ≤ α − β < 1 in [CLS92]. Thus our assumption

0 ≤ α− β ≤ 12 assumed for technical reason is not the weakest possible one.

A spectral gap condition mostly of the form

λN+1 − λN > C1 Lip (f) (λα−βN+1 + λα−βN ) (3.kok)

plays an important role where C1 is a number depending on α, β, and Lip (f).Romanov [Rom94] found (3.kok) with C1 = 1 ensuring the existence of a Lipschitz

inertial manifold for (1.kok) with 0 = β ≤ α < 1. He gave counter-examples satisfyinga spectral gap condition (3.kok) with C1 < 1 but not having an inertial manifold.That means, the spectral gap condition (3.kok) with C1 = 1 is a sharp condition forLipschitz inertial manifolds. As corollary of our Theorem 8 we have a spectralgap condition (3.kok) with C1 =

√2, i.e. our spectral gap condition is a little stronger

than Romanov’s one.The weakest known spectral gap condition in the form (3.kok) for inertial C1

manifolds was found by Ninomiya [Nin92] with C1 = 2 for 0 ≤ α−β < 1/2. Our

142 Norbert Koksch

Theorem 8 will allow C1 =√

2, i.e. our spectral gap condition is a little weakerthan Ninomiya’s one.

For k > 1 and (2.kok), Chow et al. [CLS92] have a spectral gap condition of theform

λN+1 − kλN > C1

(λα−βN+1 + λα−βN

), λN > C0

but with unknown C0, C1 depending on α, β, k and Lip (f). Additionally theyget the a priori estimate Lip (ϕ) ≤ 1. Inserting (10.kok) with Q = Uα in (14.kok) weobtain the spectral gap condition

λN+1 − kλN >√

2 Lip (f)(λα−βN+1 + kλα−βN

)for the existence of an inertial Ck manifold of (1.kok) even for k ≥ 1. Moreover, weget the better a priori estimate Lip (ϕ) ≤ χ1 where the number χ1 < 1 is definedin Lemma 4.

Let Q be an open set in Uα. In order to include also manifolds which aresubsets of Q, we introduce the following notion: A set M is called inertial Ck

manifold in Q if

1. M is a finite dimensional Ck manifold in Uα.2. M ∩ Q is locally positively invariant; i.e., if u0 ∈ M ∩ Q then there is ε > 0

such that S(t)u0 ∈M ∩ Q for all t ∈ [0, ε[.3. M is exponentially attracting for all orbits in Q; i.e., there is a γ > 0 such

that for any u0 with S(t)u0 ∈ Q for t > 0 there is a constant C such that

dist(S(t)η,M) ≤ Ce−γt (t ≥ 0).

If M is an inertial manifold in Q then the asymptotical behavior of the orbits of(1.kok) in Q is determined by the orbits of (1.kok) in M ∩ Q.

If Q = Uα and dom(ϕ) = PNUα then an inertial manifold M = graph(ϕ) inQ is an inertial manifold in the usual sense.

If f does not satisfy (2.kok), it is usually modified by a trunctation method toa new function f so that the asymptotic behavior of the solutions of (1.kok) is notchanged but f satisfies (2.kok): If B(r) is an absorbing set of (1.kok) then f is modifiedoutside of B(r) in such a way that f(u) = 0 outside of B(2r), and such thatLip (f) of the new function is not greater than Lip (f |B(2r)) of the old function.Then an inertial manifold M of the prepared equation is an inertial manifold inB(r) of the original equation (1.kok).

Let Q = BPNUα(r) + B(I−PN )Uα(r) and Q = BPNUα(r) + B(I−PN )Uα(r) withr > r > r arbitrary close to r. Theorem 8 allows to use Lip

(f |Q)

instead ofLip (f |B(2r)) even for k ≥ 1 such that we get an additional weakening in thespectral gap condition since Lip

(f |Q)≤ Lip

(f |B(

√2r))≤ Lip (f |B(2r)) for

r <√

2r.A crucial role in the proof of Theorem 8 plays comparison system (8.kok). System

(8.kok) has a linear inertial manifold in R2≥0 if and only if the spectral gap condition

(5.kok) is satisfied.


2 Main Results

2.1 Existence of Smooth Inertial Manifolds in a Set Q

Let N ∈ N be suitable chosen. In order to simplify notation we shall use

Λ1 := λN , Λ2 := λN+1,

π1 := PN , π2 := I − PN ,Uα1 := π1U

α, Uα2 := π2Uα

such that Uα = Uα1 ⊕Uα2 . To avoid repetition, we agree that i always ranges overthe integers 1 and 2.

We assume that the set Q has the special form

Q := BUα1(r1) + BUα2

(r2),

where ri ∈ R≥0 or r1 = r2 =∞.In order to ensure the existence of an inertial manifold in Q, we introduce

Assumption 3. There are numbers γi > 0 and ri with ri < ri < ∞ or ri =ri =∞ such that the one-sided Lipschitz inequalities⟨

A2α−βπ1u∆|Aβπ1[f(u1)− f(u2)]⟩≥ −γ1Λ

β−α1 |π1u∆|2α−β |u∆|α,⟨

A2α−βπ2u∆|Aβπ2[f(u1)− f(u2)]⟩≤ γ2Λ

β−α2 |π2u∆|2α−β |u∆|α

(4.kok)

hold for any ui ∈ Q ∩ U1+β where u∆ = u1 − u2 and


(r2).

The following technical lemma gives a connection between the spectral gapcondition (5.kok) and a comparison problem (8.kok) in the plane:

Lemma 4. Let the spectral gap condition

Λ2 − Λ1 >(γ

2/31 + γ

2/32

)3/2

(5.kok)

be satisfied. Then we have:

1. There are χ2 >3√γ2/γ1 > χ1 > 0 and %2 < %1 which are uniquely deter-

mined by

%i = −Λ1 − γ1

√1 + χ2

i = −Λ2 + γ2

√1 + χ−2

i . (6.kok)

Moreover,

%2 < 0. (7.kok)

144 Norbert Koksch

2. The sets Ψi := w ∈ R2≥0 : w2 = χiw

1 are integral manifolds of thecomparison system

w1 = −Λ1w1 − γ1|w|, w2 = −Λ2w

2 + γ2|w|, (8.kok)

where |w| =√

(w1)2 + (w2)2. The function ψi : R≥0 → R2≥0 defined by

ψi(t) := e%it(1, χi) (t ≥ 0)

is the solution of (8.kok) through (1, χi) ∈ Ψi at t = 0.

Proof. First we note that Ψ = w ∈ R2≥0 : w2 = χw1 is an integral manifold

of (8.kok) if χ ≥ 0 is a zero of the function p : R>0 → R defined by

p(χ) = Λ1 − Λ2 + γ1

√1 + χ2 + γ2

√1 + χ−2.

The function p is strongly convex with limχ→0 p(χ) = limχ→∞ p(χ) = +∞.Hence p has at most two positive zeroes. p is minimized at χ0 := 3

√γ2/γ1 and

p(χ0) < 0 because of (5.kok). Therefore, the existence of positive zeroes χ1, χ2 of pwith χ1 < χ0 < χ2 follow. By definition of p, these numbers χ1, χ2 satisfy (6.kok).Thus Ψi are integral manifolds of the comparison system (8.kok) and the functionsψi are solutions on Ψi with the stated properties.

Since Λ1 > 0 we have %2 < %1 < 0 and hence (7.kok). ut

Remark 5. Ψ1 is an inertial manifold of (8.kok) in R2≥0.

Remark 6. Requiring p(1) < 0 one gets the little stronger gap condition

Λ2 − Λ1 >√

2(γ2 + γ1). (9.kok)

Assuming (9.kok) we have χ1 < 1 < χ2 and

%1 > −Λ1 −√

2γ1, %2 < −Λ2 +√

2γ2.

Remark 7. If ri < ∞ and Lip(f |Q)> 0 the existence of numbers γi satisfying

Assumption 3 follows from Assumption 2: We can choose

γi = Lip(f |Q)Λα−βi . (10.kok)

The spectral gap conditions (5.kok), (9.kok) read now

Λ2 − Λ1 > Lip(f |Q)(Λ

2(α−β)/32 + Λ

2(α−β)/31

)3/2

and

Λ2 − Λ1 >√

2 Lip(f |Q) (Λα−β2 + Λα−β1

)in the well-known form.


Theorem 8 (Inertial manifold in Q). Let the Assumption 1, 2, 3 be satis-fied. If (5.kok) then there is a ϕ ∈ C1−

b (BUα1(r1),Uα2 ) with Lip (ϕ) ≤ χ1 and being

uniquely defined if ri =∞ such that M := graph(ϕ) is an inertial C1− manifoldin Q with

dist(S(t)u0,M) ≤ χ2 + χ1

χ2 − χ1|π2u0 − ϕ(π1u0)|αe%2t (t ≥ 0) (11.kok)

for any u0 with S(t)u0 ∈ Q for t ≥ 0.Moreover, for any Q ⊆ Q with positive distance to ∂Q if ri <∞ and any u0

with S(t)u0 ∈ Q for t ≥ 0 there are u0 ∈M ∩ Q and T ≥ 0 with S(t)u0 ∈M ∩ Q

for t ≥ 0 and

|πi[S(t+ T )u0 − S(t)u0]|α ≤|π2u0 − ϕ(π1u0)|α

χ2 − χ1ψi2(t+ T ) (t ≥ 0) (12.kok)

where T = 0 if r2 =∞. If in addition k ≥ 1 and

%2 > k%1 (13.kok)

then ϕ ∈ Ckb (BUα1(r1),BUα2

(r2)).

Theorem 8 will be proved by means of Theorem 11 concerning the existenceof special overflowing invariant manifolds.

Remark 9. Since χ1 <3√γ2/γ1 < χ2 we have

%1 > −Λ1 − γ2/31

√γ

2/31 + γ

2/32 , %2 < −Λ2 + γ

2/32

√γ

2/31 + γ

2/32

such that (13.kok) can be replaced by

Λ2 − kΛ1 > (γ2/32 + kγ

2/31 )

√γ

2/31 + γ

2/32 .

Assuming (9.kok), this inequality can be replaced by the stronger condition

Λ2 − kΛ1 >√

2(γ2 + kγ1). (14.kok)

2.2 Overflowing Invariant Manifolds

Theorem 8 will be reduced to the following Theorem 11 concerning the existenceof an overflowing invariant manifold for the prepared evolution equation

u+Au = f(u). (15.kok)

A set M∗

= graph(ϕ∗) with ϕ∗ : cl W0 → Uα2 and W0 ⊆ Uα1 is calledoverflowing invariant with respect to (15.kok) (compare [Wig94]) if:

– M∗ := graph(ϕ∗|W0) is locally positively invariant with respect to (15.kok).

146 Norbert Koksch

– The vector field of (15.kok) is pointing strictly outward on the boundary ∂M∗ =M∗ \M∗.

– The vector field of (15.kok) is nonzero on ∂M∗.

Besides Assumption 1 we need

Assumption 10. There are ri with 0 < r1 < r1 <∞ or 0 < r1 ≤ r1 ≤ ∞ and0 < r2 < r2 ≤ ∞ such that f |Q belongs to Ckb (Q,Uβ), and such that f satisfies⟨

A2α−βπ1u∆|Aβπ1[f(u1)− f(u2)]⟩≥ −γ1Λ

β−α1 |π1u∆|2α−β |u∆|α,⟨

A2α−βπ2u∆|Aβπ2[f(u1)− f(u2)]⟩≤ γ2Λ

β−α2 |π2u∆|2α−β |u∆|α

(16.kok)

for ui ∈ Q ∩U1+β where u∆ = u1 − u2, and⟨A2α−βπ1u| −A1+βπ1u+Aβπ1f(u)

⟩> 0 if |π1u|α = r1,⟨

A2α−βπ2u| −A1+βπ2u+Aβπ2f(u)⟩< 0 if |π2u|α = r2

(17.kok)

for u ∈ Q ∩ U1+β .

The inequalities (17.kok) ensure some inflowing and outflowing properties of thevector field on the boundary of


(r2).

Let S denote the local semiflow of (15.kok) in Q.

Theorem 11 (Overflowing invariant manifold). Let Assumption 1 and 10as well as (5.kok) be satisfied and let W0 := BUα1

(r1). Then there is a unique ϕ∗ ∈C1−

b (cl W0,Uα2 ) with Lip (ϕ) ≤ χ1 and |ϕ(ξ)|α ≤ r2 for ξ ∈ cl W0 such that

M∗

:= graph(ϕ∗) is overflowing invariant with respect to the prepared evolutionequation (15.kok). Moreover, for any u0 ∈ Q with S(t)u0 ∈ Q for t ≥ 0 there isu0 ∈M∗ with S(t)u0 ∈M∗ for t ≥ 0 and

|πi[S(t)u0 − S(t)u0]|α ≤|π2u0 − ϕ∗(π1u0)|α

χ2 − χ1ψi2(t) (t ≥ 0).

If k ≥ 1 and (13.kok) then ϕ∗ ∈ Ckb (cl W0,Uα2 ).

In order to show the existence of a C1− manifold with the properties statedin Theorem 11 we proceed as follows. For fixed γ ∈ [α, β + 1[ we introduce theBanach space G0 := C0

b(cl W0,Uγ2 ) equipped with the supremum norm ||ϕ0||0 :=

supξ∈cl W0

|ϕ0(ξ)|γ . Let

Φ0 := ϕ0 ∈ G0 : ||ϕ0||0 ≤ r2, Lip (ϕ0) ≤ χ1.

Note that Φ0 is a closed subset of G0.


We introduce the two-point boundary value problems

u+Au = f(u) (18a.kok)

π1u(ϑ) = ξ, π2u(0) = ϕ0(π1u(0)) (18b.kok)

on [0, ϑ] with ξ ∈ cl W0, ϑ > 0, and ϕ0 ∈ Φ0. Showing that (18.kok) has a uniquesolution U0(·, ϑ, ξ, ϕ0) satisfying

U0(t, ϑ, ξ, ϕ0) ∈ Q (t ∈ [0, ϑ])

for any ϑ > 0, ξ ∈ cl W0, ϕ0 ∈ Φ0, we can define the G0(ϑ) : Φ0 → G0 by

(G0(ϑ)ϕ0)(ξ) = π2U0(ϑ, ϑ, ξ, ϕ0) (ϑ ≥ 0, ξ ∈ cl W0, ϕ0 ∈ Φ0).

Using some properties of U0(t, ϑ, ξ, ϕ0) we can show that G0(ϑ) maps Φ0 intoitself and that G0(ϑ) is uniformly contractive for ϑ ≥ T0 and sufficiently largeT0. Hence there is a unique fixed-point ϕ∗0(ϑ) in Φ0 for ϑ ≥ T0. Showing theexistence of these fixed-points for all ϑ > 0 and showing their independence ofϑ we get the locally positive invariance of graph(ϕ∗0|W0).

The exponential tracking property can also be proved reducing it to theestimation of solutions of boundary value problems.

In order to show higher smoothness of ϕ0 assuming the spectral gap condition(13.kok), we shall use the fiber contraction principle [Van89,CLS92,Tem97]. Since Ck-smoothness for k ≥ 3 can be proved similarly to the C2-smoothness, we restrictus to k ≤ 2.

First let k = 2. Let the spectral gap condition (13.kok) be satisfied and letγ ∈]α, β+1[ be fixed. Applying the implicit function theorem one can show thatU0(t, ϑ, ·, ϕ0) is twice continuously differentiable for t ∈ [0, ϑ], ϑ > 0, ϕ0 ∈ Φ0.

We introduce

G1 := C0b(cl W0,L(Uγ1 ,U

γ2 )),

G2 := C0b(cl W0,L(Uγ1 × U

γ1 ,U

γ2 )).

G1, G2 are complete with respect to the norms || · ||1 and || · ||2 defined by

||ϕ1||1 := supξ∈cl W0

maxh∈cl B

Uγ1

(1)|ϕ1(ξ)h|γ ,

||ϕ2||2 := supξ∈cl W0

maxhi∈cl BU

γ1

(1)|ϕ2(ξ)(h1, h2)|γ

for ϕ1 ∈ G1, ϕ2 ∈ G2. Further we introduce the closed sets Φ1 := ϕ1 ∈ G1 :||ϕ1||1 ≤ χ1, Φ2 := G2.

One can show that for any ϑ > 0, ξ ∈ cl W0, (ϕ0, ϕ1, ϕ2) ∈ Φ0 × Φ1 × Φ2,h1, h2 ∈ U

γ1 there are a unique classical solution U1(·, ϑ, ξ, ϕ0, ϕ1, h1) of

u+ Au = Df(U(t))u,

π1u(ϑ) = h1, π2u(0) = ϕ1(π1U(0))π1u(0)(19.kok)

148 Norbert Koksch

on [0, ϑ] and a unique classical solution U2(·, ϑ, ξ, ϕ0, ϕ1, ϕ2, h1, h2) of

u+Au = Df(U(t))u +R1(t),

π1u(ϑ) = 0, π2u(0) = ϕ1(π1U(0))π1u(0) +R2

(20.kok)

on [0, ϑ] where U(t) = U0(t, ϑ, ξ, ϕ0),

R1(t) = D2f(U(t))(U1(t, ϑ, ξ, ϕ0, ϕ1, h1), U1(t, ϑ, ξ, ϕ0, ϕ1, h2)),

R2 = ϕ2(π1U(0))(U1(0, ϑ, ξ, ϕ0, ϕ1, h1), U1(0, ϑ, ξ, ϕ0, ϕ1, h2)).

We define G1(ϑ) : Φ0 × Φ1 → G1, G2(ϑ) : Φ0 × Φ1 × Φ2 → G2 by

(G1(ϑ)(ϕ0, ϕ1))(ξ, h1) = π2U1(ϑ, ϑ, ξ, ϕ0, ϕ1, h1),

(G2(ϑ)(ϕ0, ϕ1, ϕ2))(ξ, h1, h2) = π2U2(ϑ, ϑ, ξ, ϕ0, ϕ1, ϕ2, h1, h2)(21.kok)

for ϑ > 0, ξ ∈ cl W0, (ϕ0, ϕ1, ϕ2) ∈ Φ0×Φ1×Φ2, hi ∈ Uγ1 . There are T2 ≥ 0 and

closed Φj ⊂ Φj such that G0(T2), G1(T2)(ϕ0, ·), G2(T2)(ϕ0, ϕ1, ·) are uniformlycontractive selfmappings on Φ0, Φ1, Φ2 respectively, for (ϕ0, ϕ1) ∈ Φ0 × Φ1.Because of these contraction properties, the mapping G : Φ0 × Φ1 × Φ2 →Φ0 × Φ1 × Φ2 defined by

G(ϕ0, ϕ1, ϕ2) := (G0(T2)(ϕ0), G1(T2)(ϕ0, ϕ1), G2(T2)(ϕ0, ϕ1, ϕ2))

for (ϕ0, ϕ1, ϕ2) ∈ Φ0×Φ1×Φ2 has a unique fixed-point (ϕ∗0, ϕ∗1, ϕ∗2) ∈ Φ0×Φ1×Φ2.

Showing the continuity of G1(·, ϕ1), G2(·, ·, ϕ2) for (ϕ1, ϕ2) ∈ Φ1 × Φ2, the fibercontraction principle implies the attractivity of (ϕ∗0, ϕ∗1, ϕ∗2), i.e., the convergenceof the iterates

(ϕ(n)0 , ϕ

(n)1 , ϕ

(n)2 ) := Gn(ϕ0, ϕ1, ϕ2)

to (ϕ∗0, ϕ∗1, ϕ∗2) ∈ Φ0 × Φ1 × Φ2 for any (ϕ0, ϕ1, ϕ2) ∈ Φ0 × Φ1 × Φ2.

Choosing (ϕ0, ϕ1, ϕ2) = (0, 0, 0) we have

Dϕ(n)0 = ϕ

(n)1 , D2ϕ

(n)0 = ϕ

(n)2 (n ∈ N).

This and ϕ(n)0 → ϕ∗0, ϕ(n)

1 → ϕ∗1, ϕ(n)1 → ϕ∗1 imply

Dϕ∗0 = ϕ∗1, D2ϕ∗0 = ϕ∗2,

i.e., the C2-smoothness of graph(ϕ∗0|W0).

For k = 1 the proof proceeds similar to the case k = 2 where we use G :Φ0 × Φ1 → Φ0 × Φ1 defined by G(ϕ0, ϕ1) := (G0(T2)(ϕ0), G1(T2)(ϕ0, ϕ1)).

In order to study (18.kok), (19.kok), (20.kok) we shall develop and use comparison theo-rems for such boundary value problems. The main difficulties are here that thecomparison problem in R2

≥0 will be a nonlinear one (in order to get an almostsharp gap condition) and that the differential inequality in general holds only ina part of R2

≥0 (because of the nonequivalence of | · |α and | · |2α−β for α > β.)


2.3 Proof of Theorem 8

In order to apply Theorem 11, we have to determine numbers ri and a suitablemodification f of f satisfying Assumption 10.

Let the assumptions of Theorem 8 be satisfied.

First let ri = ri = ∞. In this case we can choose f = f and r1 = ∞. Remainsthe choice of r2 <∞ satisfying (17.kok).

Because of Assumption 2, there is a constant K0 with |f(u)|β ≤ K0 foru ∈ Uα. One can show that a any r2 > Λ−1+α−β

2 K0 satisfies (17.kok) since⟨−A1+βπ2u+Aβπ2f(u)|A2α−βπ2u

⟩≤ (−Λ2r2 +K0Λ

α−β2 )r2 < 0

for any u ∈ Q ∩ U1+β with |π2u|α ≥ r2. Thus Assumption 10 is satisfied.Theorem 11 implies the existence of an inertial manifold M = graph(ϕ) withϕ ∈ C1−

b (Uα1 , π2Uα) and Lip (ϕ) ≤ χ1.

Let graph(ϕ′) with ϕ′ ∈ C1−b (Uα1 ,U

α2 ) be another inertial manifold with

Lip (ϕ′) ≤ χ1. Choosing r2 > max||ϕ||, ||ϕ′||, Λ−1+α−β2 K0, Theorem 11 implies

ϕ = ϕ′. Thus Theorem 8 is proved in the case ri =∞.

Let now ri < ri <∞. Let ri with ri < ri < ri be arbitrary.In order to construct the function f let b ∈ C∞(−∞,∞) be a bump function

with the following properties: b(w) = 0 for w ≤ 0, b(w) = 1 for w ≥ 1,Db(w) ≥ 0.We introduce fi ∈ C∞(Uα,Uα) defined by

fi(u) := b

(|πiu|2α − r2

i

r2i − r2

i

)πiu (u ∈ Uα).

Then for any u ∈ Uα we have

πifj(u) = 0 if i 6= j, fi(u) = 0 if |πiu|α ≤ ri, fi(u) = πiu if |πiu|α ≥ ri. (22.kok)

Further ⟨Aαπih|AαπiDfi(u)h

⟩≥ 0 (u, h ∈ Uα).

Applying the mean value theorem to the scalar-valued function

τ 7→⟨Aαπih|Aαπifi(u + τh)

⟩we obtain ⟨

Aαπih|Aαπi[fi(u+ h)− fi(u)]⟩≥ 0 (u, h ∈ Uα)

and hence⟨A2α−βπih|Aβπi[fi(u+ h)− fi(u)]

⟩≥ 0 (u ∈ Uα, h ∈ U2α−β). (23.kok)

150 Norbert Koksch

There is µ1 > 0 satisfying −Λ1r2 − γ1r + µ1r

2 ≥ 1 for r ∈ [r1, r1]. Furtherlet µ2 := (1

4 − Λ−12 γ2

2 + 1)/r2.Now we are in position to introduce f : Uα → Uβ defined by

f(u) := f(u) + µ1f1(u)− µ2f2(u) (u ∈ Uα)

satisfying Assumption 10: The inequalities (16.kok) follows from (23.kok) and (4.kok). Forany u ∈ Q ∩U1+β with |π1u|α ≥ r1, we have⟨

−A1+βπ1u+Aβπ1f(u)|A2α−βπ1u⟩≥ 1

by choice of µ1 and hence the first inequality in (17.kok). Further one can show⟨−A1+βπ2u+Aβπ2f(u)|A2α−βπ2u

⟩≤ −1

for any u ∈ Q ∩ U1+β with |π2u|α ≥ r2. Thus the second inequality in (17.kok) issatisfied, too.

Applying Theorem 11 to the prepared evolution equation (15.kok) we get anoverflowing invariant manifold M

∗= graph(ϕ∗) with the properties stated in

this theorem. Let ϕ := ϕ∗|BUα1(r1) and M := graph(ϕ). Then Lip (ϕ) ≤ χ1.

Because of (22.kok), we have

f(u) = f(u) (u ∈ Q).

Therefore, the manifold M∩Q is locally positively invariant with respect to (1.kok).Let u0 ∈ Q with S(t)u0 = S(t)u0 ∈ Q for t ≥ 0. By means of Theorem 11 thereis u0 ∈M∗ with S(t)u0 ∈ Q and

|πi[S(t)u0 − S(t)u0]|α ≤ |π2u0 − ϕ∗(π1u0)|αψi2(t)

for t ≥ 0. Thus

dist(S(t)u0,M)≤ |π2S(t)u0 − ϕ(π1S(t)u0)|α≤ |π2S(t)u0 − π2S(t)u0|α + |ϕ∗(π1S(t)u0)− ϕ∗(π1S(t)u0)|α≤ |π2u0 − ϕ∗(π1u0)|α(χ1ψ

12(t) + ψ2

2(t))

for t ≥ 0 such that (11.kok) follows.If Q = Q = Uα, the exponential attracting property follows directly from

Theorem 11.Let Q ⊂ Q have positive distance to ∂Q and let u0 satisfy S(t)u0 = S(t)u0 ∈ Q

for t ≥ 0. By means of Theorem 11 there is u0 with S(t)u0 ∈M∗ for t ≥ 0 and

|πi[S(t)u0 − S(t)u0]|α ≤ |π2u0 − ϕ∗(π1u0)|αψi2(t)

for t ≥ 0. Using these inequalities the existence of T ≥ 0 follows with S(t)u0 ∈ Q

for t ≥ T . Let u0 := S(T )u0. Then S(t)u0 ∈M∩Q for t ≥ 0 and the inequalities(12.kok) follow.

The smoothness properties of ϕ follow directly from Theorem 11. Thus The-orem 8 is proved. ut


3 Some Comparison Theorems for Two-point BoundaryValue Differential Inequalities

Let the assumptions of Theorem 11 be satisfied.The following Lemmas 12, 13, 14 give a connection between solutions or the

difference of solutions of the boundary value problems (18.kok), (19.kok), (20.kok) and asolution v ∈ C([0, ϑ],R2

≥0 of the boundary value differential inequality

v1(t) ≥ (−Λ1 − %)v1(t)− γ1|v(t)| −A1 for a.e. t ∈ [0, ϑ],

v2(t) ≤ (−Λ2 − %)v2(t) + γ2|v(t)| +A2 if v(t) ∈ V+(%,A2),

v1(ϑ) ≤ B1, v2(0) ≤ χ1v1(0) +B2

(24.kok)

where A1, A2, B1, B2 are nonnegative numbers, % > −Λ2, and

V+(%,A2) := v ∈ R2≥0 : −2(−Λ2 − %)v2 > γ2|v|+A2.

For a compact time interval T let min T (max T) denote the lower (upper)boundary point of T.

The main goal of this section is to develop Theorem 16 and 17 for the com-parison of solutions v ∈ C([0, ϑ],R2

≥0) of (24.kok) with solutions w ∈ C([0, ϑ],R2≥0)

of the boundary value problem

w1(t) = (−Λ1 − %)w1(t)− γ1|w(t)| − a1,

w2(t) = (−Λ2 − %)w2(t) + γ2|w(t)| + a2,(t ∈ T), (25a.kok)

w1(max T) = b1, w2(min T) = χ1w1(min T) + b2 (25b.kok)

where ai = Ai, bi = Bi, T = [0, ϑ], or with solutions w ∈ C([0, ϑ],R2≥0) of the

boundary value differential inequality

w1(t) ≤ (−Λ1 − %)w1(t)− γ1|w(t)| − a1,

w2(t) ≥ (−Λ2 − %)w2(t) + γ2|w(t)| + a2

(t ∈ T),

w1(max T) ≥ b1, w2(min T) ≥ χ1w1(min T) + b2

(26.kok)

where ai = Ai, bi = Bi, % ∈ R, T = [0, ϑ]. In an intermediate step we shallcompare solutions v ∈ C(T,R2

≥0) of

v1(t) ≥ (−Λ1 − %)v1(t)− γ1|v(t)| − a1,

v2(t) ≤ (−Λ2 − %)v2(t) + γ2|v(t)| + a2

(t ∈ int T),

v1(max T) ≤ b1, v2(min T) ≤ χ1v1(min T) + b2.

(27.kok)

with solutions w and w of (25.kok) or (26.kok), respectively.

152 Norbert Koksch

Lemma 12. Let u1 : [0, ϑ] → Q be a solution of the boundary value problem(18.kok) and let u2 : [0, ϑ] → Q be a solution of (18a.kok). Then v ∈ C([0, ϑ],R2

≥0)defined by v(t) = (|π1[u1(t)−u2(t)]|α, |π2[u1(t)−u2(t)]|α) satisfies the boundarydifferential inequality (24.kok) with % = 0, A1 = A2 = 0, B1 ≥ |π1u2(ϑ) − ξ|α,B2 ≥ |π2u2(0)− ϕ0(π1u2(0))|α.

Proof. 1. First we want to show⟨−A1+βπ1[u1 − u2] +Aβπ1[f(u1)− f(u2)]|A2α−βπ1[u1 − u2]

⟩≥ −Λ1|π1[u1 − u2]|2α − γ1|u1 − u2|α|π1[u1 − u2]|α

(28.kok)

for any u1, u2 ∈ Q ∩ U1+β . Moreover, we will show⟨−A1+βπ2[u1 − u2] +Aβπ2[f(u1)− f(u2)]|A2α−βπ2[u1 − u2]

⟩≤ −Λ2|π2[u1 − u2]|2α + γ2|u1 − u2|α|π2[u1 − u2]|α

(29.kok)

for any u1, u2 ∈ Q ∩ U1+β with

(|π1[u1 − u2]|α, |π2[u1 − u2]|α) ∈ V+(0, 0). (30.kok)

Let u1, u2 ∈ Q∩U1+β be arbitrary. For shortness let u∆ := u1−u2. Inequality(28.kok) follows directly from (16.kok) and |π1u∆|2α−β ≤ Λα−β1 |π1u∆|α.

Further (16.kok) implies

(−A1+βπ2u∆ +Aβπ2[f(u1)− f(u2)]|A2α−βπ2u∆)

≤ −Λ1−2α+2β2 |π2u∆|22α−β + γ2Λ

β−α2 |u∆|α|π2u∆|2α−β .

Thus (29.kok) is shown if α = β.Let now α > β. Since

τ 7→ −Λ1−2α+2β2 τ2 + γ2Λ

β−α2 |u∆|ατ

is monotonously decreasing for τ ≥ 12Λ−1+α−β2 γ2|u∆|α, we can use the estimate

|π2u∆|2α−β ≥ Λα−β2 |π2u∆|α

in order to get (29.kok) if

Λα−β2 |π2u∆|α ≥12Λ−1+α−β

2 γ2|u∆|α

i.e. if (30.kok) holds.

2. Now let u1, u2 be solutions of (18a.kok) with the properties as required in thelemma and let v as defined in the lemma. Futher let u∆ = u1 − u2. Then

vi(t)vi(t) = 12ddt |πiu∆|2α

=⟨−A1+βπiu∆ +Aβπi[f(u1(t)) − f(u2(t))]|A2α−βπiu∆

⟩.


Using (28.kok), (29.kok) we find v1(t)v1(t) ≥ −Λ1(v1(t))2 − γ1|v(t)|v1(t) for a.e. t > 0and v2(t) ≤ −Λ2v

2(t) + γ2|v(t)| for t > 0 with v(t) ∈ V+(0, 0).

3. We have v1(ϑ) = |π1u2(ϑ)− ξ|α ≤ B1 and

v2(0) = |[ϕ0(π1u1(0))− ϕ0(π1u2(0))] + [ϕ0(π1u2(0))− π2u2(0)]|α≤ χ1|π1[u1(0)− u2(0)]|α + |ϕ0(π1u2(0))− π2u2(0)|α≤ χ1v

1(0) +B2.

Thus Lemma 12 is proved. ut

Similarly to Lemma 12 one can prove the following two lemmas.

Lemma 13. Let k ≥ 1 and let U ∈ C([0, ϑ],Q). Let u : [0, ϑ]→ Uα be a solutionof (19.kok) on [0, ϑ] with ϕ1 ∈ Φ1, h1 ∈ Uα1 . Then v ∈ C([0, ϑ],R2

≥0) defined byv(t) = (|π1u(t)|α, |π2u(t)|α) satisfies (24.kok) with % = 0, A1 = A2 = 0, B1 ≥ |h1|α,B2 = 0.

Lemma 14. Let k ≥ 1 and let U ∈ C([0, ϑ],Q). Let u : [0, ϑ]→ Uα be a solutionof (20.kok) on [0, ϑ] with ϕ1 ∈ Φ1.

If R1 = 0 then v(t) = (|π1u(t)|α, |π2u(t)|α) satisfies (24.kok) with Ai = % = 0,B1 = 0, B2 ≥ |R2|α.

If |R1(t)| ≤ Ke%(t−ϑ), |R2| ≤ Ke−%ϑ, K > 0 then v ∈ C([0, ϑ],R2≥0) defined

by v(t) = K−1e−%(t−ϑ)(|π1u(t)|α, |π2u(t)|α) satisfies (24.kok) with % = %, A1 =Λα−β1 , A2 = Λα−β2 , B1 = 0, B2 = 1.

Now let ai, bi nonnegative numbers, % ∈ R and let T be a compact timeinterval. We introduce the cone KT := C(T,R2

≥0) in the Banach space C(T,R2)equipped with the norm || · || defined by

||w|| := maxt∈T|w(t)|.

For v1 and v2 belonging to KT we say v1 ≤ v2 if and only if v2 − v1 ∈ KT . Wesay v1 v2 if v2 − v1 belongs to the interior of KT . If v1 ≤ v2 and v1 6= v2 thenwe say v1 < v2. Note that KT is a closed and normal cone. Here the normality ofthe cone means the semi-monotony of the norm, i.e. there is a number M suchthat ||v1|| ≤M ||v2|| for any v1, v2 ∈ KT with v1 ≤ v2.

We introduce the nonlinear but homogene, isotone and completely continuousintegral operator LT,% : KT → KT defined by

(L1T,%w)(t) :=

max T∫t

e(t−τ)(−Λ1−%)γ1|w(τ)| dτ

(L2T,%w)(t) :=

t∫min T

e(t−τ)(−Λ2−%)γ2|w(τ)| dτ + e(t−min T)(−Λ2−%)χ1w1(min T)

154 Norbert Koksch

for w ∈ KT , and the function q(T, %, a1, a2, b1, b2) ∈ KT defined by

(q1(T, %, a1, a2, b1, b2))(t) :=max T∫t

e(t−τ)(−Λ1−%)a1 dτ + e(t−max T)(−Λ1−%)b1,

(q2(T, %, a1, a2, b1, b2))(t) :=t∫

min T

e(t−τ)(−Λ2−%)a2 dτ + e(t−min T)(−Λ2−%)b2

for t ∈ T. Then the fixed-point problem

LT,%w + q(T, %, a1, a2, b1, b2) = w (w ∈ KT) (31.kok)

is equivalent to the two-point boundary value problem (25.kok) in KT . A functionv ∈ KT is called lower solution of (31.kok) if v ≤ LT,%v + q(T, %, a1, a2, b1, b2).Analogously, a function v ∈ KT is called upper solution of (31.kok) if LT,%v +q(T, %, a1, a2, b1, b2) ≤ v. One can show that a solution w ∈ KT of (26.kok) is anupper solution of (31.kok) and that a solution v ∈ KT of (27.kok) is a lower solution of(31.kok).

Lemma 15. Let ai ≥ 0, bi ≥ 0, % ∈ R and let T be a compact time interval. Ifv ∈ KT is a solution of (27.kok) then

v ≤ w∗ ≤ w,

where w∗ ∈ KT is the unique solution of (25.kok) in KT and w ∈ KT is a solutionof (26.kok).

Proof. Let v be a solution of (27.kok), i.e. a lower solution of (31.kok).

1. We show that there is a solution w∗ of (31.kok) with v ≤ w∗. For it we introducew0 ∈ int KT defined by

w0(t) = e−%(t−min T)ψ2(t−min T) (t ∈ T).

Note that w0 is a solution of (25a.kok) with ai = bi = 0 and graph(w0) ⊂ Ψ2. Sincew1

0(max T) > 0 and w20(min T) = χ2 > χ1 = χ1w

10(min T) we have

q0 := LT,%w0 − w0 0.

There is η > 0 with v ≤ ηw0 and q ≤ ηq0. Setting w := ηw0 we have w ∈ KT

and

v ≤ w, LT,%w + q ≤ w.

For shortness let L : KT → KT be defined by Lw := LT,%w + q for w ∈ KT .Since v is a lower solution of (31.kok), the isotony of LT,% implies

v ≤ Lv ≤ Lw ≤ w.

The sequence (Lkw)k∈N is monotone decreasing in the normal cone KT . Using[EL75, Theorem 3.1] we get the convergence of (Lkw)k∈N to a solution w∗ ∈ KT

of (31.kok). Since v ≤ w we have v ≤ w∗, too.


2. Now we show the uniqueness of w∗. Assume there are two different solutionsw1, w2 of (31.kok). Then w1 and w2 are lower solutions of (31.kok). Proceeding as abovewe get the existence of a third solution w3 of (31.kok) with w1 ≤ w3, w2 ≤ w3.Therefore, without loss of generality, we can assume w1 < w2.

Let w∆ = w2−w1. Then w∆ > 0. Since w∆ = LT,%w2−LT,%w1 ≤ LTm,%w∆,w∆ is a lower solution of

w = LT,%w. (32.kok)

Thus is a solution w of (32.kok) such that w∆ ≤ w.Since (32.kok) is equivalent to the boundary value problem (25.kok) with ai = bi =

0, w is a solution of (25.kok) with ai = bi = 0. Since w(min T) belongs to theinvariant set Ψ1, the point w(max T) belongs to Ψ1, too. Since w1(max T) =0, this inclusion implies w(max T) = 0. By uniqueness of the solutions of thecorresponding initial value problem, we have w = 0 such that the contradictionw∆ = 0 follows. Thus w∗ is the unique solution of (31.kok) and hence of (25.kok) in KT .

3. Let w ∈ KT be a solution of (26.kok), i.e. let w be an upper solution of (31.kok). Since(Lkw)k∈N is monotonously decreasing and converging to a solution of (31.kok), theinequality w∗ ≤ w follows from the uniqueness of w∗. ut

Theorem 16. Let v ∈ K[0,ϑ] satisfy (24.kok) with Ai = % = 0. Then

v ≤ w ≤ w (33.kok)

where w ∈ K[0,ϑ] is the solution of (25.kok) and w ∈ K[0,ϑ] is a solution of (26.kok) withai = Ai, bi = Bi, T = [0, ϑ].

Proof. First we note that the existence and uniqueness of w as well as w ≤ wfollow from Lemma 15 with T = [0, ϑ], %0, ai = 0, bi = Bi since w is an uppersolution of (31.kok).

Studying the phase portrait of (8.kok) we find

−Λ2w2 + γ2|w| < 0 if w2 > χ1w

1, w ∈ R2≥0.

Hence V+ := v ∈ R2≥0 : v2 > χ1v

1 ⊂ V+(0, 0). Further we introduce V− :=v ∈ R2

≥0 : v2 < χ1v1.

1. Assume v(t) ∈ cl V+ for t ∈ [0, ϑ]. Then v is a solution of (27.kok) with T = [0, ϑ],% = 0, ai = 0, bi = Bi. The claim of the theorem follows directly from Lemma15.

2. Assume now there is a t ∈ [0, ϑ] with v(t) ∈ V−. We want show that there isa ϑ1 ∈ [0, ϑ] with

v(t) ∈ V+ for t ∈ [0, ϑ1[, v(t) ∈ cl V− for t ∈ [ϑ1, ϑ]. (34.kok)

Let ϑ1 be the first time point with v2(ϑ1) = χ1v1(ϑ1). Assume there are ϑ2 ∈

[ϑ1, ϑ[, ϑ3 ∈ ]ϑ2, ϑ[ with v2(ϑ2) = χ1v1(ϑ2) and v2(t) > χ1v

1(t) for t ∈ ]ϑ2, ϑ3].

156 Norbert Koksch

We set T = [ϑ2, ϑ3], b1 = v1(ϑ3), b2 = 0, ai = 0. Then (27.kok) holds with % = 0.Applying Lemma 15 we obtain v|[ϑ2, ϑ3] ≤ w where w ∈ K[ϑ2,ϑ3] is the solutionof (25.kok) on [ϑ2, ϑ3]. Since w(ϑ2) ∈ Ψ1, we have graph(w) ⊂ Ψ1. Hence

v2(ϑ3) ≤ w2(ϑ3) = χ1w1(ϑ3) = χ1v

1(ϑ3)

in contrary to the choice of ϑ2 and ϑ3. Therefore v(t) ∈ cl V− for t ≥ ϑ1 and(34.kok) is shown.

Applying Lemma 15 with T = [0, ϑ], % = 0, b1 = B1 + ε, b2 = B2, ai = 0,ε > 0, the existence of the unique solution wε ∈ K[0,ϑ] of (25.kok) follows. Moreover,Lemma 15 implies w ≤ wε.

Let δ = w1ε − v1. Then

v1(t) ≥ −Λ1v1(t)− γ1|v(t)| ≥ −Λ1v

1(t)− γ1

√(v1(t))2 + (χ1v1(t))2,

wε1(t) = −Λ1wε

1(t)− γ1|wε(t)| ≤ −Λ1wε1(t)− γ1

√(wε1(t))2 + (χ1wε1(t))2

for a.e. t ∈ [ϑ1, ϑ] and hence

δ(t) ≤ −Λ1δ(t) for a.e. t ∈ [ϑ1, ϑ] with δ(t) ≥ 0.

Since ε > 0, we have δ(ϑ) > 0. Assume we do not have δ(t) > 0 for anyt ∈ [ϑ1, ϑ]. Then there is τ ∈ [ϑ1, ϑ[ with δ(t) > 0 for t ∈ ]τ, ϑ] and δ(τ) = 0.Thus we have δ(t) ≤ −Λ1δ(t) for a.e. t ∈ [τ, ϑ] and δ(τ) = 0. This differentialinequality implies δ(t) ≤ 0 for t ∈ [τ, ϑ] in contrary to δ(ϑ) > 0. Thereforev1(t) ≤ wε

1(t) for t ∈ [ϑ1, ϑ]. Further v2(t) ≤ χ1v1(t) ≤ χ1wε

1(t) ≤ wε2(t) for

t ∈ [ϑ1, ϑ]. Hence v|[ϑ1, ϑ] ≤ wε|[ϑ1, ϑ] in K[ϑ1,ϑ]. Since w1ε(ϑ1) ≥ v1(ϑ1) and

w2ε(0) ≥ χ1w

1ε(0) + B2, we have v|[0, ϑ1] ≤ wε|[0, ϑ1] in K[0,ϑ] such that v ≤ wε

in K[0,ϑ] follows.Let w ∈ KT be the solution of (25.kok) with T = [0, ϑ], % = 0, ai = 1, b1 = 1,

b2 = 0. Then wε − w = εw such that wε → w as ε → 0. Since w ≤ w, theinequality (33.kok) follows. ut

Theorem 17. Let A1 = λα−βN , A2 = Λα−β2 . Then for any % ∈ ]%2, %1[ we have:

1. There is a unique nonnegative stationary point w0(%) of (25a.kok) with ai = Ai.

2. There are χ ∈ ]χ1, χ2[ and η(%) such that w0(%) ∈ K[0,ϑ] defined by w0(%)(t) =(η, ηχ) is a constant solution of (26.kok) with ai = Ai, b1 = 0, b2 = max1, w2

0(%),T = [0, ϑ].

3. If v ∈ K[0,ϑ] satisfies (24.kok) with B1 = 0, B2 = max1, w20(%) then v ≤ w0(%).

Proof. Let % ∈ ]%2, %1[ be fixed and let W1(w) = (−Λ1 − %)w1 − γ1|w| − A1,W2(w) = (−Λ2 − %)w2 + γ2|w|+A2.

1. In order to show the existence of w0(%) we note

W1(0, w2) < 0, D1W1(w1, w2) > −Λ1 − %1 − γ1 > 0, D2W1(w1, w2) < 0,


W2(w1, 0) > 0, D1W2(w1, w2) > 0, D2W2(w1, w2) < −Λ2 − %2 + γ2 < 0

for w > 0. Hence there are strongly increasing functions Ψi : R≥0 → R≥0 sat-isfying W2(η, Ψ1(η)) = 0 and W1(Ψ2(η), η) = 0 for η ≥ 0 and describing theisoclines w1 = 0 and w2 = 0 of (25a.kok), respectively. Thus there is exactly onepositive stationary point w0 = w0(%) of (25a.kok). This stationary point has an un-stable manifold graph(Ψ1) where Ψ1 : R≥0 → R≥0 is strongly increasing functionsatisfying Ψ1(η) > Ψ1(η) for η < w1

0 , and Ψ1(η) < Ψ1(η) for η > w10 .

2. Let B1 = 0, B2 = w20(%). We define pi : R≥0 → R by

p1(χ) := −Λ1 − %− γ1

√1 + χ2, p2(χ) := −Λ2 − %+ γ2

√1 + χ−2.

Then p1 is strongly concave and p2 is strongly convex with

p1(χ1) = p2(χ1) = %1 − % > 0, p1(χ2) = p2(χ2) = %2 − % < 0.

Thus there are χi ∈ ]χ1, χ2[ with χ2 < χ1 and pi(χi) = 0. Let χ ∈ ]χ2, χ1[ bearbitrary. Then

p1(χ) > 0, p2(χ) < 0, χ > χ1.

Therefore there is η > 0 such that w = (η, ηχ) satisfies

0 ≤ (−Λ1 − %)w1 − γ1|w| −A1,

0 ≥ (−Λ2 − %)w2 + γ2|w|+A2, w2 ≥ χ1w

1 +B2.

Thus w0(%) as defined in the theorem is a solution of (26.kok) with ai = Ai, bi = Bi,T = [0, ϑ].

3. Let B1 = 0, B2 = w20(%). By construction we have

w ∈ R2≥0 : w2 > Ψ1(w1) ⊂ V+(%,A2).

Let

V := w ∈ R2≥0 : w1 ≤ maxΨ2(w2), w1

0, V+ := w ∈ V : w2 ≥ Ψ1(w1).

Then v1(t) < 0 if t > 0 and v(t) 6= V+. Further V+ ⊂ V+(%,A2).Assume there is t1 ∈ [0, ϑ[ with v(t) 6∈ V. Because of v1(ϑ) = 0, there is a

t2 ∈ ]t1, ϑ[ with v(t) 6∈ V for t ∈ [t1, t2[ and

v1(t2) = w10 , v2(t2) ≤ w2

0 (35.kok)

or

v1(t2) = Ψ2(v2(t2)), v2(t2) > w20 . (36.kok)

If (35.kok) then there is τ ∈ [t1, t2] with v(t2) − v(t1) = v(τ)(t2 − t1) ≤ 0 incontrary to v(t2) > v(t1).

158 Norbert Koksch

If (36.kok) then v(t) 6= V+ implies the existence of t3 ∈ [t1, t2[ with v(t3) 6= V

and v(t) ∈ V+, i.e. with v2(t) ≤ 0 for t ∈ [t3, t2]. Thus there are τ1, τ2 withτi ∈ [t3, t2] and

v1(t2)− v1(t3) = v1(τ1)(t2 − t3) ≥ 0,

v2(t2)− v2(t3) = v2(τ2)(t2 − t3) ≤ 0

which would imply v(t1) ∈ V in contradiction to the choice of t3.Therefore v(t) ∈ V for t ∈ [0, ϑ].Let

V+ := V+ ∩ V, V− := V− ∩ V.

By the choice of B2, the solution w ∈ K[0,ϑ] of (25.kok) satisfies w(t) ∈ V+ fort ∈ [0, ϑ]. Thus we can proceed as in the proof of Theorem 16 in order to inferv ≤ w ≤ w0(%). ut

4 Proof of Theorem 11

4.1 Existence and Properties of G0(ϑ)

4.1.1 Uniqueness and Estimates of U0(·, ϑ, ξ, ϕ0). For ϕ0 ∈ Φ0 let

Wϑ(ϕ0) := π1S(ϑ)(ζ + ϕ0(ζ)) : ζ ∈W0 (ϑ ≥ 0).

Then for any ϕ0 ∈ Φ0, ϑ ≥ 0, ξ ∈ cl Wϑ(ϕ0) there is at least one solutionU0(·, ϑ, ξ, ϕ0) of the boundary value problem (18.kok).

Our goal is to prove that for any ϕ0 ∈ Φ0, ϑ > 0, ξ ∈ cl Wϑ(ϕ0) thereis at most one solution U0(·, ϑ, ξ, ϕ0) of the boundary value problem (18.kok) withmaximal existence interval satisfying

U0(t, ϑ, ξ, ϕ) ∈ Q (t ∈ [0, ϑ]). (37.kok)

Further we show some estimates which we need for G0(ϑ).

Lemma 18. There hold:

1. Let ui be solutions of (18a.kok) with ui(t) ∈ Q for t ∈ [0, T ] and with π1ui(ϑi) = ξi,π2ui(0) = ϕ0(π1ui(0)) where ϑi ∈ [ϑ, ϑ] ⊂ [0, T ], ϕ0 ∈ Φ0, ξi ∈ cl Wϑi(ϕ0). Thenthere is a constant K such that

|πi[u1(t)− u2(t)]|α ≤ (K|ϑ1 − ϑ2|+ |ξ1 − ξ2|α) maxΘ∈[ϑ,ϑ]

ψi1(t−Θ) (38.kok)

for t ∈ [0,maxϑ1, ϑ2].


2. Let ui be solutions of (18a.kok) with ui(t) ∈ Q for t ∈ [0, T ] and with π1ui(ϑ) = ξ,π2u1(0) = ϕ0(π1u1(0)), π2u2(0) = ϕ′0(π1u2(0)) where ϑ ∈ [0, T ], ϕ0, ϕ

′0 ∈ Φ0,

ξi ∈ cl Wϑ(ϕ0) ∩ cl Wϑ(ϕ′0). Then

|πi[u1(t)− u2(t)]|α ≤ψi2(t)χ2 − χ1

||ϕ0 − ϕ′0|| (t ∈ [0, ϑ]). (39.kok)

3. For any ϕ0 ∈ Φ0, ϑ > 0, and ξ ∈ Wϑ(ϕ0) there is at most one solutionU0(·, ϑ, ξ, ϕ0) of (18.kok) satisfying (37.kok).

Proof. 1. Let u1, u2 have the properties as required in the first claim. Withoutloss of generality we can assume ϑ1 ≥ ϑ2. Set ϑ := ϑ1, ξ := ξ1.

We have

|π1[u2(ϑ)− ξ]|α ≤ |π1[u2(ϑ1)− u2(ϑ2)]|α + |π1u2(ϑ2)− ξ|α.

Since −Aπ1 is a linear, bounded operator and since f maps bounded sets intobounded sets, there is a constant K with

K ≥ | −Aπ1u+ π1f(u)|α (u ∈ Q).

Because of |π1[u2(ϑ1)− u2(ϑ2)]|α ≤ K|ϑ1 − ϑ2|, π1u2(ϑ2) = ξ2, the estimate

|π1[u2(ϑ)− ξ]|α ≤ B1

follows where B1 := K|ϑ1−ϑ2|+ |ξ1−ξ2|α. Moreover |π2u2(0)−ϕ0(π1u2(0))|α =B2 := 0. Lemma 12 and Theorem 16 imply |πi[u1(t)−u2(t)]| ≤ wi1(t) for t ∈ [0, ϑ]where w1 ∈ K[0,ϑ] defined by w1(t) = (M |ϑ1 − ϑ2|+ |ξ1 − ξ2|α)ψ1(t− ϑ1) is thesolution of (25.kok) for these values of B1 and B2. Thus (38.kok) follows.

2. Let u1, u2 have the properties as required in the second claim. We have

|π1[u2(ϑ)− ξ]|α = B1 := 0.

Further |π2u2(0) − ϕ0(π1u2(0))|α = |ϕ′0(π1u2(0)) − ϕ0(π1u2(0))|α ≤ ||ϕ0 −ϕ′0||0 =: B2.

The function w2 : [0, ϑ]→ R2≥0 defined by

w2 := ψ2(t)(χ2 − χ1)−1||ϕ0 − ϕ′0||0 (t ∈ [0, ϑ])

is a solution of (26.kok) for ai = 0, bi = Bi, T = [0, ϑ]. Lemma 12 and Theorem 16imply |πi[u1(t)− u2(t)]| ≤ wi2(t) for t ∈ [0, ϑ], i.e. (39.kok).

3. Let ϕ0 ∈ Φ0, ϑ ∈ [0, T ], ξ ∈ Wϑ(ϕ0) be arbitrary. Assuming the existence oftwo different solutions of (18.kok) satisfying (37.kok) we obtain a contradiction to (38.kok)with ϑ1 = ϑ2 = ϑ = ϑ = ϑ, ξ1 = ξ2 = ξ. ut

160 Norbert Koksch

Because of (17.kok), there is a number T∗ > 0 with the following three properties

S(t)u0 ∈ Q (t ∈ [0, T∗]),

|π1S(t)u0|α > r1 (|π1u0|α = r1, t ∈ [0, T∗]) if r1 <∞,|π2S(t)u0|α < r2 (|π2u0|α = r2, t ∈ [0, T∗])

(40.kok)

for any u0 ∈ cl Q.

Lemma 19. Let T∗ > 0 satisfy (40.kok) and let ϕ0 ∈ Φ0. Then

cl W0 ⊂Wϑ(ϕ0) (ϑ ∈ ]0, T∗])

Proof. Let ϕ0 ∈ Φ0, ϑ ∈ ]0, T∗] be arbitrary. We define the continuous mappingH : [0, 1]× cl W0 → Uα1 by

H(τϑ, ζ) := π1S(τϑ)(ζ + ϕ0(ζ)) (ζ ∈ cl W0).

By definition of T∗ and Wϑ(ϕ0) and by means of Lemma 18 there is a uniqueU0(·, ϑ, ξ, ϕ0) for ξ ∈Wϑ(ϕ0). Hence we can define the inverseH−1(1, ·) ofH(1, ·)by

H−1(1, ξ) := π1U0(0, ϑ, ξ, ϕ0) (ξ ∈ cl Wϑ(ϕ0)).

Because of (38.kok) with ϑ = ϑ1 = ϑ2 = ϑ = ϑ, t = 0, ϕ0 = ϕ′0, the func-tion H−1(1, ·) is continuous, too. Thus H(1, ·) is a homeomorphism from W0

onto Wϑ(ϕ0). If r1 = ∞, i.e. W0 = Uα, the domain invariance theorem impliesWϑ(ϕ0) = W0.

If r1 <∞ then W0 is an open and bounded set. Because of (40.kok) we have

H(τ, ξ) : τ ∈ [0, 1], ξ ∈ ∂W0 ∩W0 = ∅.

Using an arbitrary base in the finite dimensional Banach space Uα1 , the homotopytheorem implies

deg(H(1, ·),W0, ξ) = deg(H(0, ·),W0, ξ) = deg(I,W0, ξ) = 1

for any ξ ∈ W0. Thus for any ξ ∈ W0 there exists ζ ∈ W0 with ξ = H(1, ζ).Therefore, cl W0 ⊆ cl Wϑ(ϕ0). Because of (40.kok), we have ∂Wϑ(ϕ0) ∩ cl W0 = ∅.

Since H(1, ·)|∂W0 is a bijection from ∂W0 onto ∂Wϑ(ϕ0) we have cl W0 ⊂Wϑ(ϕ0). ut

Lemma 20. Let T∗ > 0 satisfy (40.kok). Let U0(·, ϑ, ξ, ϕ0) be a solution of (18.kok)satisfying U0(t, ϑ, ξ, ϕ0) ∈ Q for t ∈ [0, ϑ] where ϑ ∈ [0, T∗], ξ ∈ Wϑ(ϕ0), ϕ0 ∈Φ0. Then

|π2U0(t, ϑ, ξ, ϕ0)|α ≤ r2 (t ∈ [0, ϑ]).

Proof. The claim follows from (40.kok) and |π2U0(0, ϑ, ξ, ϕ0)|α ≤ r2. ut


Lemma 21. Let T > 0 and let u be a solution of (18a.kok) with u(t) ∈ Q fort ∈ [0, T ]. Further let ϑ = T , ξ = π1u(T ) ∈ cl W0, ϕ0 ∈ Φ0 and let U0(·, ϑ, ξ, ϕ0)be a solution of (18.kok) satisfying U0(t, ϑ, ξ, ϕ0) ∈ Q for t ∈ [0, T ]. Then

|πi[u(t)− U0(t, ϑ, ξ, ϕ0)]|α ≤ |π2u(0)− ϕ0(π1u(0))|αψi2(t) (t ∈ [0, T ]).

Proof. The proof proceeds similarly to the proof of Lemma 18. ut

4.1.2 The Graph Transformation G0(ϑ). For simplicity let γ = α. Forϑ ≥ 0 let Φ0(ϑ) be the set of all ϕ0 ∈ Φ0 for which U0(·, ϑ, ξ, ϕ0) satisfiesU0(t, ϑ, ξ, ϕ0) ∈ Q for any t ∈ [0, ϑ] and any ξ ∈ cl W0. We define G0(ϑ) :Φ0(ϑ)→ G by

(G0(ϑ)ϕ0)(ξ) := π2U0(ϑ, ϑ, ξ, ϕ0) (ϕ0 ∈ Φ0(ϑ), ξ ∈ cl W0, ϑ ≥ 0).

Further let T∗ > 0 satisfy (40.kok).

Lemma 22. G0 possesses the following properties:

1. Φ0(ϑ) = Φ0, G0(ϑ)Φ0 ⊆ Φ0 for ϑ ≥ 0.

2. (ϑ, ϕ0) 7→ G0(ϑ)ϕ0 is continuous in (ϑ, ϕ0).

3. There are T0 > 0 and κ0(T0) ∈ ]0, 1[ such that

||G0(ϑ)ϕ0 −G0(ϑ)ϕ′0||0 ≤ κ0(T0)||ϕ0 − ϕ′0||0 (ϑ ≥ T0, ϕ0, ϕ′0 ∈ Φ0).

4. We have

G0(ϑ2)G0(ϑ1) = G0(ϑ1 + ϑ2) (ϑi ≥ 0). (41.kok)

Proof. 1. The first claim will be proved by induction. First we note that

Φ0(ϑ) = Φ0 (ϑ ∈ [0, T∗]) (42.kok)

follows from the definition of T∗ and from the Lemmata 18 and 19.Moreover, using Lemma 18 with t = ϑ1 = ϑ2 = ϑ = ϑ ∈ [0, T∗], the inequality

|(G0(ϑ)ϕ0)(ξ1)− (G0(ϑ)ϕ0)(ξ2)|α ≤ χ1|ξ1 − ξ2|α (ξi ∈ cl W0, ϕ0 ∈ Φ0)

follows. By means of Lemma 20 we have |(G0(ϑ)ϕ0)(ξ)|α ≤ r2 for any ξ ∈ cl W0,ϕ0 ∈ Φ0. Therefore G0(ϑ)Φ0 ⊆ Φ0 for ϑ ∈ [0, T∗].

Let now

Φ0(ϑ) = Φ0, G0(ϑ)Φ0 ⊆ Φ0 (ϑ ∈ [0,mT∗]) (43.kok)

for m = m0 ∈ N. We want show that (43.kok) holds for m = m0 + 1, too.Let ϕ0 ∈ Φ0 be arbitrary. Because of (43.kok), we have

ϕ(m0)0 := G0(m0T∗)ϕ0 ∈ Φ0.

162 Norbert Koksch

Because of (42.kok) for any (ϑ, ξ) ∈ [m0T∗, (m0 + 1)T∗]× cl W0, there is a unique

ξm0(ϑ, ξ, ϕ0) := π1U0(0, ϑ−m0T∗, ξ, ϕ(m0)0 ) ∈ cl W0.

Moreover, there is a unique

ξ0(ϑ, ξ) := π1U0(0,m0T∗, ξm0(ϑ, ξ), ϕ0) ∈ cl W0.

Thus S(·)(ξ0(ϑ, ξ) + ϕ0(ξ0(ϑ, ξ))) solves (18.kok). Since

S(t)(ξ0(ϑ, ξ) + ϕ0(ξ0(ϑ, ξ))) ∈ cl (BUα1(r1) + BUα2

(r2)) (t ∈ [0,m0T∗]),

we have

S(t)(ξ0(ϑ, ξ) + ϕ0(ξ0(ϑ, ξ))) ∈ Q (t ∈ [0, (m0 + 1)T∗]).

By means of Lemma 18 we have a unique U0(·, ϑ, ξ, ϕ0) in Q and hence

U0(t, ϑ, ξ, ϕ0) = S(t)(ξ0(ϑ, ξ) + ϕ0(ξ0(ϑ, ξ))) ∈ Q

for t ∈ [0, ϑ], (ϑ, ξ) ∈ [m0T∗, (m0 + 1)T∗]× cl W0, too. Applying Lemma 18 oncemore (with T = ϑ = ϑi = ϑ = ϑ, ξi = ξ, ϕ0 = ϕ′0) and by means of Lemma 19,the relation G0(ϑ)ϕ0 ∈ Φ0 follows. Thus (43.kok) is true for m = m0 + 1, too. Byinduction Φ0(ϑ) = Φ0, G0(ϑ)Φ0 ⊆ Φ0 follow for any ϑ ≥ 0.

2. The continuity properties of G0 follows from Lemma 18 with ξ1 = ξ2.

3. Let ϑ ≥ 0, ϕ0, ϕ′0 ∈ Φ0. Lemma 18 implies

||G0(ϑ)ϕ0 −G0(ϑ)ϕ0||1 ≤ κ0(ϑ)||ϕ0 − ϕ′0||1

where

κ0(ϑ) :=ψ2

2(ϑ)χ2 − χ1

= eϑ%2χ2

χ2 − χ1.

Because of (7.kok), there is a number T0 > 0 such that κ0(ϑ) ≤ κ0(T0) < 1 for anyϑ ≥ T0.

4. The solution u of (18a.kok) with initial value ξ+ (G0(ϑ2)G0(ϑ1)ϕ0)(ξ) at ϑ1 +ϑ2

satisfies (18b.kok) with ϑ = ϑ1 + ϑ2. Lemma 18 implies (41.kok). ut

Lemma 23. For any ϑ > 0 there is ϕ∗0 ∈ Φ0 being the unique fixed-point of

ϕ0 = G0(ϑ)ϕ0 (ϕ0 ∈ Φ0).

Moreover, ϕ∗0 is independent of ϑ.

Proof. Let ϑ ≥ T0. By means of the first three claims of Lemma 22 the operatorG0(ϑ) is a continuous, κ0(T0)-contractive self-mapping of the closed set Φ0 in


the Banach space G0. Thus for any ϑ ≥ T0 there is a unique fixed-point ϕ∗0(ϑ)of G0(ϑ) in Φ0.

Let m ∈ N \ 0, T ≥ T0 be arbitrary. Because of

G0(T/m)ϕ∗0(T ) = G0(T/m)G0(T )ϕ∗0(T ) = G0(T )(G0(T/m)ϕ∗0(T )

and the uniqueness of the fixed-point ϕ∗0T ofG0(T ), the point ϕ∗0(T ) is the uniquefixed-point of G0(T/m), too. Thus for any ϑ ≥ 0 there is a unique fixed-pointϕ∗0(ϑ) of G0(ϑ) in Φ0. Moreover

ϕ∗0(ϑ) = ϕ∗0(k

mϑ) (ϑ > 0, k,m ∈ N \ 0).

Using this property and the continuity of G0, i.e. the continuous dependence ofϕ∗0(ϑ) on ϑ, the independence of ϕ∗0(ϑ) of ϑ follows. Thus ϕ∗0 := ϕ∗0(T0) is theunique fixed-point of G0(ϑ) in Φ0 for any ϑ > 0. ut

4.2 Invariance and Exponential Tracking Properties of M∗

Let M∗ = graph(ϕ∗0|W0) where ϕ∗0 is the function as described in Lemma 23.Let u0 ∈ M∗ be arbitrary. There is τ > 0 such that π1S(ϑ)u0 ∈ W0 for

ϑ ∈ [0, τ ]. Thus there exists U0(·, ϑ, π1S(ϑ)u0, ϕ∗0) and we have

ϕ∗0(π1S(ϑ)u0) = (G0(ϑ)ϕ∗0)(π1S(ϑ)u0) = π2U(ϑ, ϑ, π1S(ϑ)u0, ϕ∗0) = π2S(ϑ)u0,

i.e. S(ϑ)u0 ∈M∗ for ϑ ∈ [0, τ ]. Thus M∗ is locally positively invariant.Because of (17.kok), the vector field of (15.kok) is pointing strictly outward and is

nonzero on the boundary ∂M∗ if ∂M∗ 6= ∅. Thus M∗

is overflowing invariant.

Now we shall prove the exponential tracking property. For it let u0 ∈ Q withπ1S(t)u0 ∈W0 for t ≥ 0. Further let τ > 0. Since S(mτ)u0 ∈W0, we may definethe sequence (ηm)∞m=0 by ηm = π1U0(0,mτ, S(mτ)u0, ϕ

∗0) for m ∈ N. Note that

ηm ∈W0.Applying Lemma 21 with T = mτ , ξ = S(mτ)u0, ϕ0 = ϕ∗0, u(t) = S(t)u0 for

[0, T ] we get

|πi[S(t)u0 − U0(t,mτ, S(mτ)u0, ϕ∗0)]|α ≤ |π2u0 − ϕ∗0(π1u0)|αψi2(t) (44.kok)

for t ∈ [0, ϑm]. Especially we have

|π1u0 − ηm|α ≤ |π2u0 − ϕ∗0(π1u0)|α.

Because of the compactness of cl W0 ∩ η ∈ Uα1 : |π1u0 − η|α ≤ |π2u0 −ϕ∗0(π1u0)|α, there is a subsequence (ηmj )

∞j=0 converging to some η ∈ cl W0. We

choose u(·, u0) = S(·)(η + ϕ∗0(η)). Let T > 0 and δ > 0 be arbitrary. Because ofthe continuous dependence on the initial data, there is j0 = j0(δ, T ) ∈ N suchthat mjτ ≥ T and

|πi[u(t, u0)− U0(t,mjτ, S(mjτ)u0, ϕ∗0)]|α ≤ δ|π2u0 − ϕ∗0(π1u0)|αψi2(t)

164 Norbert Koksch

for t ∈ [0, T ] and j ≥ j0. Combining this inequality with (44.kok) we find

|πi[S(t)u0 − u(t, u0)]|α ≤ (1 + δ)|π2u0 − ϕ∗0(π1u0)|αψi2(t) (t ∈ [0, T ])

for any T > 0 and any δ > 0 and hence, letting δ → 0, T →∞|πi[S(t)u0 − u(t, u0)]|α ≤ |π2u0 − ϕ∗0(π1u0)|αψi2(t) (t ≥ 0).

Moreover, we have u(t, u0) ∈ Q and π1u(t, u0) ∈ cl W0 for any t ≥ 0. Thusπ1u(0, u0) ∈W0 and u(t, u0) ∈M∗ for t ≥ 0.

4.3 Existence and Properties of G1(ϑ), G2(ϑ)

Let k ≥ 2 and let γ ∈]α, β + 1[ be fixed. For ϑ > 0 let Uϑ := C([0, ϑ],Uα) be theBanach space equipped with the norm || · || defined by ||u|| := max

t∈[0,ϑ]|π1u(t)|α +

maxt∈[0,ϑ]

|π2u(t)|α. Let Fϑ be the open set of all continuous functions u ∈ Uϑ with

π1u(0) ∈W0 and u(t) ∈ Q.For ϑ > 0, (ϕ0, ϕ1) ∈ Φ0 × Φ1, U ∈ Uϑ we introduce the integral operators

Fϑ,ϕ0(·, ·) : Fϑ ×Wϑ(ϕ0)→ Uϑ and P (ϑ, ϕ1, U) : Uϑ → Uϑ defined by

Fϑ,ϕ0(u, ξ)(t) =t∫ϑ

eπ1A(t−τ)π1f(u(τ)) dτ +t∫

0

e(t−τ)π2Aπ2f(u(τ)) dτ

+e(t−ϑ)π1Aξ + etπ2Aϕ0(π1u(0)),

(P (ϑ, ϕ1, U)u)(t) =t∫ϑ

eπ1A(t−τ)π1Df(U(τ))u(τ) dτ+

+t∫

0

e(t−τ)π2Aπ2Df(U(τ))u(τ) dτ + etπ2Aϕ1(π1U(0))π1u(0)

for (u, ξ) ∈ Fϑ ×Wϑ(ϕ0), t ∈ [0, ϑ]. Then the solution u = U0(·, ϑ, ξ, ϕ0) ∈ Fϑof (18.kok) is a fixed-point of Fϑ,ϕ0(·, ξ) in Fϑ and inversely. Moreover, a solutionu = U1(·, ϑ, ξ, ϕ0, ϕ1, h1) of (19.kok) is a solution of the fixed-point problem

u = P (ϑ, ϕ1, U0(·, ϑ, ξ, ϕ0))u +Q (45.kok)

with Q = Q1(ϑ, ξ, ϕ0, ϕ1, h1) defined by

Q1(ϑ, ξ, ϕ0, ϕ1, h1) = e(t−ϑ)π1Ah1,

and a solution u = U2(·, ϑ, ξ, ϕ0, ϕ1, ϕ2, h1, h2) of (20.kok) is a solution of (45.kok) withQ = Q2(ϑ, ξ, ϕ0, ϕ1, ϕ2, h1, h2) defined by

Q2(ϑ, ξ, ϕ0, ϕ1, ϕ2, h1, h2) =t∫ϑ

eπ1A(t−τ)π1D2f(U0(∗))(U1(τ, ϑ, ξ, ϕ0, ϕ1, h1), U1(τ, ϑ, ξ, ϕ0, ϕ1, h2)) dτ

+t∫

0

e(t−τ)π2Aπ2D2f(U0(∗))(U1(τ, ϑ, ξ, ϕ0, ϕ1, h1), U1(τ, ϑ, ξ, ϕ0, ϕ1, h2)) dτ

+etπ2Aϕ2(π1U0(0, ϑ, ξ, ϕ0))(U1(0, ϑ, ξ, ϕ0, ϕ1, h1), U1(0, ϑ, ξ, ϕ0, ϕ1, h2))

where U0(∗) stands for U0(τ, ϑ, ξ, ϕ0).


Lemma 24. The operator I − P (ϑ, ϕ1, U) : Uϑ → Uϑ is a linear homeomor-phism from Uϑ onto itself for ϑ > 0, ϕ1 ∈ Φ1, U ∈ Fϑ.

Proof. Let ϑ > 0, ϕ1 ∈ Φ1, U ∈ Fϑ be given. Obviously P (ϑ, ϕ1, U) is linear.Since γ > α one can show that P (ϑ, ϕ1, U) is completely continuous. Let u bea solution of u = P (ϑ, ϕ1, U)u. Then u is a solution of (19.kok) with h1 = 0, andthe Lemma 13 and Theorem 16 imply u = 0 since A1 = A2 = B1 = B2 = 0and w = 0 is the solution of (25.kok). Therefore, I − P (ϑ, ϕ1, U) is injective. SinceP (ϑ, ϕ1, U) is completely continuous, I − P (ϑ, ϕ1, U) is surjective, too. ThusI−P (ϑ, ϕ1, U) is a linear, continuous bijection from Banach space Uϑ onto itselfwhich has a continuous inverse by means of Banach’s Theorem. ut

Lemma 25. Let k = 2 and let ϑ > 0, ϕ0 ∈ Φ, ξ ∈ Wϑ(ϕ). If ϕ0 is C2 thenU0(·, ϑ, ξ, ϕ0) is C2 in ξ ∈ cl W0. Moreover,

D3U0(t, ϑ, ξ, ϕ0)h1 = U1(t, ϑ, ξ, ϕ0, Dϕ0, h1)

D3,3U0(t, ϑ, ξ, ϕ0)(h1, h2) = U2(t, ϑ, ξ, ϕ0, Dϕ0, D2ϕ0, h1, h2)

(46.kok)

for t ∈ [0, ϑ], ξ ∈ cl W0, h1, h2 ∈ Uα1 .

Proof. Let ϑ > 0 and let ϕ0 ∈ Φ0 be twice continuously differentiable.We note that Fϑ,ϕ0 belongs to C2(Fϑ ×Wϑ(ϕ0),Uϑ). Since D1Fϑ,ϕ0(U, ξ) =

P (ϑ, ϕ0, U), Lemma 24 implies that I − D1Fϑ,ϕ0(U, ξ) is a linear homeomor-phism of Uϑ into itself. Since U0(·, ϑ, ξ, ϕ0) solves u − Fϑ,ϕ0(u, ξ) = 0 we canapply the implicit function theorem in order to conclude the C2-smoothness ofU0(·, ϑ, ξ, ϕ0) in ξ ∈ Wϑ(ϕ0). Moreover, (46.kok) follows from the implicit functiontheorem. Since cl W0 ⊆Wϑ(ϕ0), the lemma is proved. ut

Similar to Lemma 22 but using some more technical estimates (since γ > α)one can show

Lemma 26. There are T2 > 0 and closed sets Φj ⊆ Φj with 0 ∈ Φj for j = 0, 1, 2such that:

1. G0(T2), G1(T2)(ϕ0, ·), G2(T2)(ϕ0, ϕ1, ·) are uniformly contractive on Φ0, Φ1,Φ2, respectively, for (ϕ0, ϕ1) ∈ Φ0 × Φ1.

2. G0(T2)Φ0 ⊆ Φ0, G1(T2)(Φ0 × Φ1) ⊆ Φ1, G2(T2)(Φ0 × Φ1 × Φ2) ⊆ Φ2.3. G1(T2)(·, ϕ1), G2(T2)(·, ·, ϕ2) are continuous for (ϕ1, ϕ2) ∈ Φ1 × Φ2.

Because of (46.kok), we have

DG0(T2)(ϕ0) = G1(T2)(ϕ0, Dϕ0), D2G0(T2)(ϕ0) = G2(T2)(ϕ0, Dϕ0, D2ϕ0)

for twice continuously differentiable ϕ0 ∈ Φ0. Choosing ϕ0 = 0 and applying thefiber contraction principle, the C2 smoothness of the manifold follows.

Thus Theorem 11 is proved. ut

166 Norbert Koksch

References

[CLS92] Chow, S.-N., Lu, K., Sell, G.R.: Smoothness of inertial manifolds. Journalof Mathematical Analysis and Applications 169 (1992) 283–312

[CFNT89] Constantin, P., Foias, C., Nicolaenko, B., Temam, R.: Integral Manifoldsand Inertial Manifolds for Dissipative Partial Differential Equations, vol-ume 70 of Applied Mathematical Sciences. Springer 1989

[EL75] Eisenfeld, J., Lakshmikantham, V.: Comparison principle and nonlinearcontractions in abstract spaces. Journal of Mathematical Analysis andApplications 49 (1975) 504–511

[FST88] Foias, C., Sell, G.R., Temam, R.: Inertial manifolds for nonlinear evolu-tionary equations. J. of Differential Equations 73 (1988) 309–353

[FST89] Foias, C., Sell, G.R., Titi, E.S.: Exponential tracking and approximationof inertial manifolds for dissipative nonlinear equations. J. of Dynamicsand Differential Equations 1 (1989) 199–244

[Hen81] Henry, D.: Geometric Theory of Semilinear Parabolic Equations, volume850 of Lecture Notes in Mathematics. Springer, 1981

[MPS88] Mallet-Parret, J., Sell, G.R.: Inertial manifolds for reaction diffussionequations in higher space dimension. J. Amer. Math. Soc. 1 (1988) 805–866

[Nin92] Ninomiya, H.: Some remarks on inertial manifolds. J. Math. Kyoto Univ.32 (1992) 667–688

[Rob93] Robinson, J.C.: Inertial manifolds and the cone condition. Dyn. Syst.Appl. 2 (1993) 311–330

[Rob96] Robinson, J.C.: The asymptotic completeness of inertial manifolds. Non-linearity 9 (1996) 1325–1340

[Rom94] Romanov, A.V.: Sharp estimates of the dimension of inertial manifoldsfor nonlinear parabolic equations. Russ. Acad. Sci., Izv., Math. 43 (1994)31–47

[Tem88] Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics andPhysics. Springer, New York, 1988

[Tem97] Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics andPhysics. 2nd ed. Springer, New York, 1988

[Van89] Vanderbauwhede, A.: Centre Manifolds, Normal Forms and ElementaryBifurcations, pages 89 – 169. Dynamics Reported, Volume 2. John Wiley& Sons, 1989

[Wig94] Wiggins, S.: Normally Hyperbolic Invariant Manifolds in Dynamical Sys-tems. Applied Mathematical Sciences 105. Springer, 1994



The Property (A) for a Certain Class

of the Third Order ODE

Monika Kovacova

Department of MathematicsFac. of Mechanical Engineering

Slovak Technical University, nam. Slobody 17,812 31 Bratislava, The Slovak Republic

Email: kovacova [email protected]

Abstract. We study oscillatory and non-oscillatory solutions of thethird order ODE

[g(t)(u′′(t) + p(t)u(t))]′ = f(t, u, u′, u′′), (∗)where g, p : [T,∞) → [0,∞) are bounded functions, g ≥ δ > 0. Thefunction f is assumed to be continuous and f(x1, x2, x3) · x1 ≤ 0.Many authors have consider ODE’s of the form (∗), where the main part,i.e. the term u′′ + pu is nonoscillatory. By contrast to these results weconsider here the case of the oscillatory kernel function u′′ + pu.The main goal is to show that any solution u of (∗) is either oscillatoryor it is a solution of the second order ODE u′′(t) + p(t)u(t) = β(t) withvanishing right hand side β ≥ 0, β(t)→ 0 as t→∞. In the latter caseall the derivatives u(n)(t) up to the second order tend to zero as t→∞,i.e. eq. (∗) has the property (A).The results are generalizations of these obtained by I. T. Kiguradze [1].

AMS Subject Classification. 34C10, 34C15

Keywords. The Property (A), Oscillatory Solutions ODE

1 Introduction

In this paper we consider a nonlinear third order differential equation in the form

(g(t) · [u′′(t) + p(t)u(t)])′ = f(t, u, u′, u′′). (1.kov)

Let T, g1, g2, p2 be positive constants and let

g : [T0,∞)→ (0,∞) belong to the class C1[T0,∞),0 < g1 ≤ g(t) ≤ g2 for all t ∈ [T0,∞), (2.kov)

p : [T0,∞)→ [0,∞) belong to the class C1[T0,∞),0 ≤ p(t) ≤ p2 for all t ∈ [T0,∞), (3.kov)

f : [T0,∞)×R3 → R is continuous function having the followingsign property

f(t, x1, x2, x3) · signx1 ≤ 0 for x1 6= 0. (4.kov)


168 Monika Kovacova

The main goal of this paper is to describe oscillatory and nonoscillatoryproperties of solution of ordinary differential equation (1.kov). This main result is adichotomy property saying that any solution u of equation (1.kov) is either oscillatoryor u together with its derivatives up to the second order tend to zero.

Several papers focused an aforementioned problem. An Oscillatory Criterionfor a Class of Ordinary Differential Equations [1] becomes one of the main ones.The author assumes that a left-sided operator u(n)(t) + u(n−2)(t) is oscillatoryat first. This assumption was considered as true, he searched necessary andsufficient conditions to fulfill that equation has a property A (B).

Several authors studied differential equation (1.kov), but they assumed that ope-rator u′′(t) + p(t)u(t), which forms an equational kernel is nonoscillatorical. Bycontrast to these results we consider here the case of oscillatory kernel functionu′′(t) + p(t)u(t).

2 Preliminaries

By a solution (proper solution) we mean a function u defined on an interval[T,∞) ⊂ [T0,∞), having a continuous third derivative and such that

sup|u(t)| : t > T > 0

for any t ∈ [T,∞) and u satisfies equation.By an oscillatory solution we mean a solution of (1.kov) having arbitrarily large

zeroes. Otherwise, a solution is said to be nonoscillatory.

3 Auxiliary lemmata

We begin with several auxiliary lemmata which are needed in order to provemain results in the main section. Let us consider the equation

y′′(t) + p(t)y(t) = r(t), (5.kov)

where p : [T,∞)→ [0,∞) and r : [T,∞)→ (0,∞) are continuous functions suchthat

p(t) ≤ p2 and 0 < r1 ≤ r(t) ≤ r2 for all t ∈ [T,∞), (6.kov)

where r1, r2, p2 are positive constants, with coefficients p, r satisfying (6.kov).

Lemma 1. Let y ∈ C2[T,∞) be a positive solution of differential equation (1.kov)and let r1, r2, p2 be positive constants which fulfill (6.kov) the conditions .

Let p0 > 0 be arbitrary large and put ε1 = r12p2

. Then for any δ > 0, small

enough, and 0 < ε ≤ min(

ε1·r12(1+2p0)2·r2+r1

, r1δ2

16

)the solution y has the following

property.

The Property (A) for a Certain Class of ODE 169

If we have 0 < y(t) < ε on some interval (t−, t+) andy(t−) = y(t+) = ε, (7.kov)

then y(t) ≥ ε for any t ∈ [t+, t+ + p0(t+ − t−)],t+ − t− ≤ δ. (8.kov)

Proof. Suppose that 0 < y(t) < ε for any t ∈ [t−, t+] and y(t−) = y(t+) = ε.Since y is a solution of (1.kov) we have

y′′(t) = r(t) − p(t) · y(t) ≥ r1 − p2ε1 =r1

2and y′′(t) ≤ r2 for each t such that 0 < y(t) ≤ ε1 (9.kov)

Put t0 = mint∈[t−,t+]

y(t).

Let us introduce the following auxiliary functions:

z(t) = y(t0) + y′(t0)(t− t0) +r1

2· (t− t0)2

2t,

w(t) = y(t0) + y′(t0)(t− t0) + r2 ·(t− t0)2

2, (10.kov)

and y(t) = y(t0) + y′(t0)(t− t0) + y′′(ξ) · (t− t0)2

2, ξ ∈ [t0, t].

According to (9.kov) and (10.kov) we have the estimate

z(t) ≤ y(t) ≤ w(t) provided that 0 < y(t) ≤ ε1. (11.kov)

170 Monika Kovacova

Then with regard to (10.kov) there exist t1, t2, t3 ∈ [T,∞) such that t0 > t3 andt1, t2 > t0 roots

w(t2) = ε1, z(t1) = ε, z(t3) = ε. (12.kov)

Furthermore, there exists t++ > t+ such that y(t++) ≥ εWe conclude from the definitions of the functions w(t), z(t) and (11.kov) that

t++ − t+ ≥ t2 − t1,t1 − t3 ≥ t+ − t−.

In what follows, we will prove that

t++ − t+ ≥ p0 · (t+ − t−).

As a consequence of this inequality we will obtain the statement (8.kov).Assume that 0 < ε ≤ ε1r1

r1+2r2(1+2p0)2 . Then

ε1 ≥ ε(

(1 + 2p0)2 · 2r2

r1+ 1)> 0. (13.kov)

As 0 < y(t0) = mint∈[t−,t+]

y(t) < ε we obtain

ε1 − y(t0)r2

≥ ε1 − εr2

≥ (1 + 2p0)2 · 2εr1≥ 2 · (1 + 2p0)2 · (ε− y(t0))

r1. (14.kov)

It easily follows from (10.kov) and (12.kov) that

(t2 − t1)2 = (ε1 − y(t0)) · 2r2, (15.kov)

(t1 − t0)2 = (ε− y(t0)) · 4r1. (16.kov)

Thereforeε1 − y(t0)

r2=w(t2)− y(t0)

r2=

r22 · (t2 − t0)2

r2.

With regard to (13.kov), (14.kov), (15.kov), (16.kov) we have

ε1 − y(t0)r2

≥ 2(1 + 2p0)2 · z(t1)− y(t0)r1

= 2(1 + 2p0)2 · r1

4r1· (t1 − t0)2.

Straightforward computations yield

12

(1 + 2p0)2 · (t1 − t0)2 ≤ 12

(t2 − t0)2,

(t1 − t0) · (1 + 2p0) ≤ t2 − t0,2p0 · (t1 − t0) ≤ t2 − t1,

p0(t+ − t−) ≤ (t1 − t3) · p0 ≤ t2 − t1 ≤ t++ − t+.


Hencet++ − t+ ≥ p0(t+ − t−).

Thus y(t) ≥ ε on [t+, t+ + p0(t+ − t−)].It remains to show the estimate

t+ − t− ≤ δ.

Due to (10.kov), (12.kov)

ε = z(t1) = y(t0) +r1

2(t1 − t0)2 ≥ r1

2(t1 − t0)2.

This is why t1 − t0 ≤√

2εr1

. Since

t+ − t− ≤ t1 − t3 = 2(t1 − t0) ≤ 2√

2√

ε

r1≤ 4 ·

√ε

r1

and ε < r1δ2

16 , we finally obtain

t+ − t− ≤ δ.

Lemma 2. Assume that

(i) h ∈ C[T,∞), h(t) > 0,∞∫T

h(t)dt = +∞.

(ii) There exists δ > 0 and the sequence

T ≤ t−1 < t+1 < t−2 < t+2 < · · · < t−k < t+k →∞

with the property

t+k − t−k ≤ δ,

t−k+1 − t+k ≥ t

+k − t

−k .

(iii) (a) Either there is h0 > 0 such that h(t) ≥ h0 > 0 on [T,∞)(b) or h is a nonincreasing function on [T,∞),

such that h(t)→ 0 as t→∞and there is k0 > 1 such that h(t) ≤ k0 · h(t+ δ) for all t ≥ T .

172 Monika Kovacova

Denote

S =∞⋃k=1

[t+k , t−k+1], Sc =

∞⋃k=1

(t−k , t+k ).

Then ∫S

h(t)dt = +∞.

Proof.S ∪ Sc = (t−1 ,∞) ⊂ [T,∞).

With respect to (iii) the proof splits into two parts.

The case (a) If µ(Sc) <∞ ( µ is the Lebesgue measure) then µ(S) =∞.If µ(Sc) =∞ then according to (ii) we again obtain µ(S) =∞.Since h(t) ≥ h0 > 0 we may conclude

∫S

h(t)dt =∞.

The case (b) If∫Sch(t)dt <∞ then clearly

∫S

h(t)dt =∞

because S ∪ Sc = (t−1 ,∞), h(t) ∈ C[T,∞) and∞∫T

h(t)dt =∞.

On the other hand suppose that∫Sch(t)dt =∞.

Choose δ > 0 sufficiently small and p0 > 0 sufficiently large. Let ε > 0 satisfythe assumptions of Lemma 1. Obviously, there is a sequence

T ≤ t−1 < t+1 < t−2 < t+2 < · · · < t−k < t+k →∞

such that

y(t) < ε for t ∈ (t−k , t+k ),

y(t) ≥ ε for t ∈ [t+k , t−k+1], k = 1, 2, . . . .

With regard to Lemma 1 we may conclude t+k − t−k ≤ δ. Then for any t ∈ [t−k , t

+k ]

we have h(t) ≤ k0 · h(t+ δ) ≤ k0 · h(t+ (t+k − t−k )) and thus

t+k∫t−k

h(t)dt ≤t+k∫t−k

h(t+ (t+k − t−k ))dt = k0 ·

t+k +(t+k−t−k )∫

t+k

h(u)du ≤ k0 ·

t−k+1∫t+k

h(t)dt.

Hence+∞ =

∫Sc

h(t)dt ≤∫S

h(t)dt.

It completes the proof of Lemma 2.


Lemma 3. Let

(g(t) · [u′′(t) + p(t)u(t)])′ = f(t, u, u′, u′′),

where p(.), g(.) and f(.) fulfill the following conditions:

(i) p(.), g(.) fulfill conditions (2.kov) a (3.kov).(ii)

f(t, x1, x2, x3) · signx1 ≤ 0, x1 6= 0, (17.kov)f(t, x1, x2, x3) · signx1 ≤ −h(t) · w(|x1|), (18.kov)

where h(.) fulfill on [T,∞) assumption (i), (ii), (iii) from Lemma 2.(iii) Let w : [0,∞)→ [0,∞) be a nonincreasing function such that

w(0) = 0, w(s) > 0 for all s > 0. (19.kov)

Then any proper solution of equation (1.kov) on [T,∞) is either oscillatory or thereexists β(.) ≥ 0 such that lim

t→∞β(t) = 0 and u(.) is a solution of equation

u′′(t) + p(t)u(t) = β(t) · sign u(t). (20.kov)

Proof. Let u be a nonoscillatory solution. We will show the existence of a functionβ as stated in Lemma 3. According to (4.kov), u solves (1.kov) iff −u does. Therefore,without loss of generality we may assume that

u(t) > 0 for all t ∈ [T,∞) (21.kov)

Denote

α(t) := g(t) · (u′′(t) + p(t)u(t)). (22.kov)

Then from (17.kov) and (21.kov) we see α′(t) = f(t, u(t), u′(t), u′′(t)) ≤ 0 for all t ≥ T ,is nonincreasing function on [T,∞). We will consider three distinct cases:

(i) limt→∞

α(t) < 0,

(ii) limt→∞

α(t) > 0,

(iii) limt→∞

α(t) = 0.

In the case (i) we have:Let there exist T1 > T such that α(t) ≤ −ε < 0 for all t ∈ [T1,∞). Then

u′′(t) + p(t)u(t)(22.kov)=

α(t)g(t)

(2.kov)

≤ −εg2

< 0.

According to (3.kov), (21.kov) we have p(.), u(.) > 0 on [T1,∞). Hence u′′(t) ≤ −εg2< 0

for all t ∈ [T1,∞) and so there is T2, T2 ≥ T1 such that u(t) < 0 for t ≥ T2 . Acontradiction.

174 Monika Kovacova

In the case (ii)u(.) is the solution of equation (1.kov). Let us mark:

u′′(t) + p(t)u(t) =α(t)g(t)

=: r(t) for all t ∈ [T,∞).

We know that α(.) is a nonincreasing function on [T,∞).Take α1 := lim

t→∞α(t) > 0 and α2 := α(T ). Then according to the definition

of function α we have α2 ≥ α(t) ≥ α1 > 0 for any t ≥ T and therefore

0 <α1

g2≤ α(t)g(t)

≤ α2

g1⇒ 0 < r1 ≤ r(t) ≤ r2,

which means that the function r(.) satisfies assumptions (5.kov), (6.kov) of Lemma 1with constants r1 = α1

g2and r2 = α2

g1.

According to Lemma 1, for any δ > 0 sufficiently small and any p0 > 0sufficiently large there is ε > 0 and a sequence t−1 < t+1 < t−2 < t+2 . . .→∞ suchthat

t+k − t−k ≤ δ,

y(t) < ε for all t ∈ (t−k , t+k ),

y(t) ≥ ε for all t ∈ [t+k , t−k+1], k = 1, 2, . . . .

Thus u(t) ≥ ε⇒ α′(t) ≤ −h(t)w(ε) on [t+k , t−k+1].

And α(t) ≤ α(t+k ) −t∫t+k

h(t)w(ε)dt for any t ∈ [t+k , t−k+1]. This yields the

following estimates.

α(t) ≤ α(t+1 )− w(ε)

t∫t+1

h(t)dt,

α(t−2 ) ≤ α(t+1 )− w(ε)

t−2∫t+1

h(t)dt,

α(t−3 ) ≤ α(t+2 )− w(ε)

t−3∫t+2

h(t)dt ≤ α(t−2 )− w(ε)

t−3∫t+2

h(t)dt

≤ α(t+1 )− w(ε)[ t−2∫t+1

h(t)dt+

t−3∫t+2

h(t)dt].


And in general:

α(t−k+1) ≤ α(t+1 ) + w(ε)[ t−2∫t+1

h(t)dt+

t−3∫t+2

h(t)dt+ · · ·+

t−k+1∫t+k

h(t)dt].

Hence

limk→∞

α(t−k+1) ≤ α(t+1 )− w(ε)[ ∫S

h(t)dt]→ −∞

because∫Sh(t)dt =∞ (see the Lemma 2), a contradiction.

This way we have excluded the cases (i) and (ii). Thus the case (iii) mustoccur, i.e.

limt→∞

α(t) = 0 .

Finally, if we put β(t) = α(t)g(t) for all t ≥ T , then we have β(t) ≥ 0 (α is

nonincreasing function ) and limt→∞

β(t) = 0 and the proof of Lemma 3 follows.

4 Main Theorems

Theorem 4. Let u be a solution of equation (1.kov)

(g(t)[u′′(t) + p(t)u(t)])′ = f(t, u, u′, u′′),

where p(.), g(.) and f(.) fulfill conditions (2.kov), (3.kov), (17.kov), (18.kov) and (19.kov).Let further,

u′′(t) + p(t)u(t) = β(t) on interval [T,∞), (23.kov)

where

u ∈ C2[T,∞), u(t) > 0 for all t ≥ T,β ∈ C2[T,∞), β(t) ≥ 0 for all t ≥ T

(24.kov)

and

limt→∞

β(t) = 0. (25.kov)

If u is a nonoscillatory solution of (1.kov), then

limt→∞

u(t) = 0.

Proof. It is sufficient to prove

lim inft→∞

u(t) = 0 = lim supt→∞

u(t).

176 Monika Kovacova

At first we show that

lim inft→∞

u(t) = 0 (26.kov)

We proceed by contradiction.If (26.kov) is not valid, then according to (24.kov) and (25.kov) suppose that there is

α > 0 and T1 ≥ T such that u(t) ≥ α for each t ≥ T1. Thus

u′′(t) = β(t) − p(t)u(t) ≤ β(t)− p1α for all t ∈ [T1,∞).

As limt→∞

β(t) = 0 , there exists T2 ≥ T1,

u′′(t) ≤ −p1

2· α < 0 for t ≥ T2

and therefore u(t) < 0 on [T3,∞), where T3 ≥ T2. This is a contradiction becauseu is positive in [T,∞).

Now we show that

lim supt→∞

u(t) = 0. (27.kov)

Again we will proceed by contradiction. Suppose that (27.kov) is not true. Thentwo cases can occur:

(i) lim supt→∞

u(t) =∞,

(ii) There is ε such that 0 < ε < lim supt→∞

u(t) < 2ε.

First we exclude the case (i).If lim sup

t→∞u(t) = +∞, then there is a sequence tk∞k=1 such that u(tk)→∞

for tk → ∞ and simultaneously with regard to the previous part of the prooflim inft→∞

u(t) = 0.

Thus there exist t∗, t∗∗ ∈ [T1,∞), T1 ≥ T such that

u′(t∗) = 0, u′′(t∗) > 0,u′(t∗∗) = 0, u′′(t∗∗) < 0.

Then according to (23.kov) we have

u′′(t)u′(t) + p(t)u(t)u′(t) = β(t)u′(t) in [T1,∞)

and after the integration

t∗∗∫t∗

u′′(t)u′(t)dt+

t∗∗∫t∗

p(t)u(t)u′(t)dt =

t∗∗∫t∗

β(t)u′(t)dt.


Because

t∗∗∫t∗

p1u(t)u′(t)dt ≤t∗∗∫t∗

p(t)u(t)u′(t)dt ≤t∗∗∫t∗

p2u(t)u′(t)dt,

we get

12

[u′(t∗∗)2︸︷︷︸=0

− u′(t∗)2︸︷︷︸=0

] +p1

2· (u(t∗∗)− u(t∗)) ≤ ε[u(t∗∗)− u(t∗)]. (28.kov)

Due to the assumption limt→∞

β(t) = 0, therefore there exists T1; T1 ≥ T such

that β(t) < ε for all t ≥ T1.According to inequality (28.kov) we have

p1

2[u(t∗∗)2 − u(t∗)2] ≤ ε[u(t∗∗)− u(t∗)],

u(t∗∗) ≤ u(t∗∗) + u(t∗) ≤ 2εp1

and this is a contradiction to (i)Now consider the case (ii).Suppose that there exists ε > 0 such that

0 < ε < lim supt→∞

u(t) < 2ε. (29.kov)

Let us choose β0 such that

0 < β0 <p1ε

4. (30.kov)

178 Monika Kovacova

According to assumptions (24.kov), (29.kov) there exists T1 ≥ T such that for allt ≥ T1

β(t) ≤ β0, u(t) ≤ 2ε. (31.kov)

We assume u(T1) > 0 thus according to (26.kov) there exists T2 ≥ T1 such that

u(T2) < min(u(T1), ε) (32.kov)

and with aspect to (29.kov) we have T4 ≥ T2 with the property u(T4) > ε. Thus

u(T2)(32.kov)< ε < u(T4) and we can find T3 such that

T2 ≤ T3 ≤ T4, u(T3) > ε, u′(T3) > 0. (33.kov)

Let

t0 = inft ≥ T1, u′(τ) ≥ 0 for all τ ∈ [t, T3]. (34.kov)

Since u(.) is continuous (33.kov) implies the inequality t0 < T3.If t0 < T1, then according to (34.kov), u(.) is nondecreasing in [T1, T3], what is

a contradiction to u(T2) ≤ u(T1), which follows from (32.kov).We have T1 ≤ t0 < T3. According to definition of t0, u′(t0) = 0. Then

u′′(t0) = limδ→0+

u′(t0+δ)−u′(t0)δ ≥ 0. Using (25.kov), (31.kov) and (21.kov) we obtain

0 ≤ u′′(t0) = β(t0)− p(t0)u(t0) ≤ β0 − p1u(t0),

u(t0) ≤ β0

p1≤ ε

4. (35.kov)

According to definition (34.kov) we have

u′(t) ≥ 0 for all t ∈ [t0, T3]. (36.kov)

And by (35.kov) we have u′(t0) < 0, u(T3) > ε. Then there exists t1, t1 ∈ (t0, T3)such that u(t1) = ε

2 . Hence we can obtain for u′′(t) the inequality on interval[t0, t1].

u′′(t)(24.kov)= β(t) − p(t)u(t)

(25.kov),(ii)

≤ β0 on [t0, t1].

By (36.kov) it follows that u′′(t) · u(t) ≤ β0 u′(t). And integrating we get

t1∫t0

u′′(t)u′(t)dt ≤t1∫t0

β0 u′(t)dt,

12

[u′(t1)]2 − 12

[u′(t0)]2 ≤ β0(u(t1)− u(t0)),

clearly12

[u′(t1)]2 ≤ β0ε

2. (37.kov)


If t ∈ [t1, T3] then we can obtain from (24.kov), (25.kov), (30.kov)

u′′(t) = β(t)− p(t)u(t) ≤ β0 − p1u(t1) ≤ −p1 · ε4

.

Since u′(t) ≥ 0 on [t1, T3] we have

T3∫t1

u′′(t)u′(t)dt ≤T3∫t1

−εp1

4u′(t)dt.

Thus

12

[u′(t1)]2 ≥ ε · p1

4[u(T3)− u(t1)] ≥ ε · p1

4ε

2.

And according to (37.kov)

ε · β0

2≥ 1

2(u′(t1))2 ≥ ε2p1

8,

which implies β0 ≥ εp14 .

The last inequality gives a contradiction to (ii). So

lim inft→∞

u(t) = lim supt→∞

u(t) = limt→∞

u(t) = 0.

Theorem 5. Let u be a solution of equation (1.kov)

(g(t)[u′′(t) + p(t)u(t)])′ = f(t, u, u′, u′′),

where p(.), g(.) and f(.) fulfill conditions (2.kov), (3.kov), (17.kov), (18.kov) and (19.kov).Then equation (1.kov) has the property A, so every proper solution of (1.kov) is

either oscillatory or it converges with its derivatives to zero as t→∞.

Proof. We proceed by contradiction.Let u(.) be a nonoscillatory solution. According to Lemma 3 and Theorem 4

we have limt→∞

u(t) = 0. Statement (iii) in Lemma 3 gives us that limt→∞

u′′(t) = 0.

So we need only to prove that u′(t)→ 0 for t→∞.Let lim

t→∞u′(t) 6→ 0, then lim sup

t→∞|u′(t)| ≥ A > 0. Thus in any neighbourhood

of ∞ we can find t0 such that |u′(t0)| ≥ A2 > 0.

We have limt→∞

u(t) = 0, limt→∞

u′′(t) = 0. Then we take 0 < ε < A6 such that

|u′′(t)| ≤ ε, |u(t)| ≤ ε in [t0 − 1, t0 + 1].Then on interval [t0 − 1, t0 + 1] we get the following inequalities (t0 is suffi-

ciently great):

u′(t)− u′(t0) = u′′(t0 + ξ(t− t0))(t − t0),|u′(t)| = |u′(t0)|︸︷︷︸

≥A2

− |u′′(t0 + ξ(t− t0))|︸︷︷︸≤ε

·|t− t0|︸︷︷︸≤1

,

180 Monika Kovacova

and therefore

|u′(t)| ≥ A

2− ε for all t ∈ [t0 − 1, t0 + 1].

Farther |u(t0)| ≤ ε and hence

u(t) = u(t0) + u′(t0 + ξ(t− t0))(t− t0),|u(t)| ≥ |u′(t0 + ξ(t− t0)| · |t− t0| − |u(t0)|,

|u(t)| ≥ (A

2− ε)|t− t0| − ε for all t ∈ [t0 − 1, t0 + 1].

We put t = t0 + 1 . Then |u(t0 + 1)| ≥ (A2 − ε) − ε = A2 − 2ε. Since we

took 0 < ε < A6 , we get |u(t0 + 1)| > ε, what is a contradiction to assumption

|u(t0 + 1)| ≤ ε.Thus lim

t→∞u′(t) = 0 and so equation (1.kov) has the property A.

References

[1] Kiguradze, I. T. , An oscillation criterion for a class of ordinary differential equa-tions , J. Diff. Equations, 28(1992), 207–219



On Factorization

of Fefferman’s Inequality

Miroslav Krbec1 and Thomas Schott2

1 Institute of Mathematics, Academy of Sciences of the Czech Republic,Zitna 25, 115 67 Prague 1, Czech Republic

Email: [email protected] Fakultat fur Mathematik und Informatik, Friedrich-Schiller-Universitat,

Ernst-Abbe-Platz 1–4, 07740 Jena, GermanyEmail: [email protected]

Abstract. This paper is concerned with conditions for a weight functionV in order that(∫

B

u2(x)V (x) dx

)1/2

≤ c(∫B

(∇u(x))2 dx

)1/2

, u ∈W 1,20 (B),

where B is a ball in RN . This inequality has found wide applicationsin many areas of analysis and this has been the reason for an effort toobtain various conditions, either sufficient or necessary and sufficient.Here we survey some of them and we also present a method, using de-composition of imbeddings between Sobolev and Lorentz-Orlicz spaces(and/or their weak counterpart). We state sufficient conditions in termsof a membership of the weight function V in Lorentz-Orlicz spaces andpay an attention to the so called ‘size condition’ in order to discuss ap-plications to the strong unique continuation property for |∆u| ≤ V |u| indimensions 2 and 3.

AMS Subject Classification. 46E35, 46E30 35J10

Keywords. Limiting imbeddings, Orlicz spaces, Orlicz-Lorentz spaces,strong unique continuation property

1 Introduction

Fefferman’s inequality [F]( ∫RN

u2(x)V (x) dx)1/2

≤ c( ∫RN

(∇u(x))2 dx

)1/2

, u ∈ W 1,2(RN ), (1.krb)

has turned out to be a very powerful tool to handle many topical problems inthe PDEs including the strong unique continuation property (the SUCP in thesequel), distribution of eigenvalues and so on.


182 Miroslav Krbec and Thomas Schott

After making a short trip into the history, when we recall some of the mostimportant results, our concern will be to establish efficient and manageable con-ditions for the function V , guaranteeing validity of a local version of (1.krb), thatis,

(∫B

u2(x)V (x) dx)1/2

≤ c(∫B

(∇u(x))2 dx

)1/2

, u ∈W 1,20 , (2.krb)

where B is a bounded domain in RN , say, a ball, |B| = 1. We shall use a naturalidea of a decomposition of the imbedding in (2.krb) into an imbedding of W 1,2

0

into a suitable target space and an imbedding from this target into L2(V ); weinvoke imbedding theorems for the Sobolev space W 1,2

0 — the classical Sobolevtheorem and a refinement in terms of Lorentz spaces in the role of target spacesin the dimension N ≥ 3, and the limiting imbedding theorem due to Brezis-Wainger [BW] (see also [Zi], Lemma 2.10.5) in the dimension N = 2, whichcan be viewed as an analogous refinement of Trudinger’s celebrated limitingimbedding [T]. The method suggested for proving (2.krb) is a kind of a generator of n-dimensional Hardy inequalities or, alternatively, of weighted imbeddings W 1,2

0 →L2(V ): general results of this nature will appear elsewhere. It is rather surprisingthat working with superpositions of imbeddings we do not lose much and thatcombining our conditions for validity of (2.krb) with the conditions for the SUCP inChanillo and Sawyer [CS] we recover or generalize some of known results aboutthe strong unique continuation property for |∆u| ≤ V |u| in dimensions 2 and3. In fact all the above imbeddings of the Sobolev spaces are sharp in the scaleof spaces considered and the same is true for the weighted imbeddings. In thelatter case we shall use only Holder’s inequality, nevertheless, we actually useconditions which are necessary as well.

2 Recent history — a partial survey

Let us start with an observation that the theory of weighted imbeddings is byno means complete; only special problems have been fully solved. For instancethe particular type of power weights has been considered in [OK] — powers ofdistance to the boundary of the domain in question (that is, power type weightsafter flattening the boundary using local coordinates). Passing to more generalweights, a natural idea is to apply what is known for the behaviour of Rieszpotentials in weighted spaces since (1.krb) follows for a weight function V providedthe boundedness of the Riesz potential of order 1 from the Lebesgue space L2

into the weighted Lebesgue space L2(V ) has been established. Let us observethat one of the peculiarities of the inequality (2.krb) is that the powers at both sidesare the same. Necessary and sufficient conditions have been found for the case ofimbeddings of W 1,p into Lq(V ), see Adams’ inequality in [A] and Maz’ya [Ma],when p < q. If p = q = 2 and N ≥ 3, then a necessary and sufficient condition

On Factorization of Fefferman’s Inequality 183

is due to Kerman and Sawyer [KeSa]; it reads∫RN

(∫Q

V (y)|x− y|N−1

dy

)2

dx ≤ K∫Q

V (x) dx (3.krb)

for all dyadic cubes Q ⊂ RN , with a constant K independent of Q. This condi-tion uses local potentials in an intrinsic way since it hangs on Sawyer’s theoremon two weight inequalities for the maximal function from [Sa] and on the good-λ-inequality due to Muckenhoupt and Wheeden [MW]; the latter giving a linkbetween an inequality for the corresponding Riesz potential and for the associ-ated fractional maximal function. The condition (3.krb) can sometimes be hard toverify since it involves the local potential of V , or, alternatively, the fractional in-tegral of V . Hence various sufficient conditions, including those preceding [KeSa]are of importance.

The celebrated Fefferman’s paper [F] gave a sufficient condition, which wedescribe in the following. Let us recall the definition of the Fefferman-Phongclass Fp, 1 ≤ p ≤ N/2. A function V belongs to Fp if

‖V ‖Fp = supx∈RNr>0

r2

(1

|B(x, r)|

∫B(x,r)

|V (y)|p dy)1/p

<∞.

Let us first formulate the basic result in the framework of the classes Fp.

Theorem 1 (Fefferman [F]). Let N ≥ 3, 1 < p ≤ N/2, and V ∈ Fp. Then(1.krb) holds.

A particularly fine and elegant proof of (1.krb) was given by Chiarenza andFrasca [CF].

It is worth observing that Fp2 ⊂ Fp1 for 1 ≤ p1 ≤ p2 ≤ N/2, and plainlyFN/2 = LN/2. Provided that we restrict ourselves to balls B(x, r) with radiussmaller than some ε0 > 0 in the above definition the result can be identifiedwith the Morrey space Lp,N−2p. We recall that, for 0 < λ ≤ N and 1 ≤ p <∞,the Morrey space Lp,λ is the collection of all V ∈ Lploc such that

‖V ‖Lp,λ = supx∈RN

0<r≤r0

r−λ/p( ∫B(x,r)

|V (y)|p dy)1/p

<∞.

Inserting a ‘hat function’, that is, u(x) = (r− |x|)χB(0,r), x ∈ RN , into (1.krb) isa standard way how to show that the weight V must belong to the Morrey spaceL1,1 in order (1.krb) holds. Nevertheless, as is well known this is not sufficient. ThusFefferman’s theorem gives a sufficient condition in terms the Morrey spaces andfurther investigation shows that the situation near L1,N−2 is of rather delicatenature. Observe also that when passing to various refined conditions, then theconstant C in (1.krb) can depend on suppu; this is quite sufficient for relevantapplications.


For f ∈ L1loc, let us denote

η(f, ε) = supx∈RN

∫|x−y|≤ε

|f(y)||x− y|N−2

dy.

The Stummel-Kato class is defined by

S = f ; η(f, ε) <∞ for all ε and η(f, ε) 0 as ε 0.

A variant of the Stummel-Kato class, sometimes denoted by S is defined as

S = f ; η(f, ε) <∞ for all ε > 0.

Restriction of these spaces to a domain in RN , say, Ω can be done in an obviousway, namely, by considering χΩV instead of V in the above definitions.

It will be useful to give relations between the spaces considered up to now.They are discussed e.g. in Zamboni [Za], Di Fazio [DiF] (the first inclusion in (i)),Piccinini [Pi] (the statement in (iii) below) and Kurata [K]; the last quoted au-thor considers also other variants of the Stummel-Kato class to get a backgroundtailored for more general elliptic operators.

Proposition 2. The following statements are true:

(i) L1,λ ⊂ S ⊂ S ⊂ L1,N−2, λ > N − 2.(ii) LN/2,∞ ⊂ Fp for every 1 ≤ p < N/2, where the former space denotes the

weak LN/2 space (the Marcinkiewicz space).(iii) For each p ≥ 2 and each 0 < λ < n, there exists a function f ∈ Lp,λ \ Lq

for every q > p.(iv) For every sufficiently small p > 1 there exists a function f ∈ Fp \ LN/2,∞.(v) S(Ω) ⊂ F1(Ω), and LN/2(Ω) is incomparable with S(Ω).

Let us observe that (ii) gives a sufficient condition for the validity of (1.krb) interms of another scale of function spaces, namely, of the weak Lebesgue spaces.We shall come to use of more general Lorentz spaces later in this paper.

Employing the class S, it is possible to prove (see [Za]):

Theorem 3. Let V ∈ S. Then for every r > 0 there is Cr depending only onη(V, r) and N such that∫

RN

u2(x)V (x) dx ≤ Cr∫RN

|∇u(x)|2 dx

holds for every u ∈ C∞0 supported in B(0, r).


A reader can find further results in Chang, Wilson and Wolff [CWW], whoconsider a certain Orlicz variant of Morrey spaces. An interesting Orlicz spacestype refinement of the well-known Adams’ inequality [A], has recently appearedin Ragusa and Zamboni [RZ].

The inequalities (1.krb), (2.krb) and further weighted imbeddings certainly deservefurther study aimed at obtaining necessary and sufficient conditions or to get asclose as possible to them; at the same time it is desirable that these conditionsare described in a manageable way.

3 The size condition and some applications

For the sake of applications we shall pay a special attention the so called ‘small-ness condition’ or the ‘size condition’ (see (5.krb) below), playing a important rolein the study of the strong unique continuation property. We shall restrict our-selves to a differential inequality arising from the Schrodinger operator, namely,|∆u| ≤ V |u|.

Let us recall that a locally integrable function u is said to have a zero ofinfinite order at x0 if

limr→0+

r−k∫

|x−x0|<r

|u(x)|2 dx = 0

for all k = 1, 2, . . . . If every solution of a given differential equation, with a zeroof infinite order, vanishes identically, then the corresponding operator is said tosatisfy the strong unique continuation property (the SUCP). As to non-analyticsetting of the problem let us recall that in 1939 Carleman [C] proved that theoperator −∆ + V has the strong unique continuation property provided V ∈L∞loc, that is, he showed that under this assumption a solution of the equation−∆u+V (x)u = 0 with a zero of infinite order vanishes identically. There is a lotof results concerning the SUCP, with various assumptions on the potential Vand also on coefficients in the case of a more general elliptic operator in question.Here we shall go along the lines of sufficient conditions in terms of integrabilityof the potential with no apriori assumptions on its pointwise behaviour.

Let us first recall Jerison and Kenig [JK], Stein [St], where the SUCP isproved for V ∈ LN/2loc or for V locally small in the Marcinkiewicz space LN/2,∞,N ≥ 3, and Pan [Pa] with the pointwise growth condition V (x) ≤ M/|x|2,N ≥ 2, and without the size conditions for V .

Wolff [W] has constructed counterexamples for N = 3 and N = 2, showingthat the assumption about the local smallness of the imbedding norm in (1.krb)cannot be removed in general. For N = 2 there is the result due to Gossez andLoulit [GL] with the sufficient condition V ∈ L1 logL for the SUCP.

Theorem 4 (Wolff [W]). The following statements are true:

(1) There exists a function u : R3 → R1, smooth and not identically zero, vanish-ing at infinite order at the origin and such that |∆u| ≤ V |u| with V ∈ L3/2,∞.


(2) There exists a function u : R2 → R1, smooth and not identically zero, van-ishing at infinite order at the origin and such that |∆u| ≤ V |u| with V ∈ L1.

Chanillo and Sawyer [CS] considered the classes Fp for p > (N − 1)/2 andproved the SUCP for potentials V which have locally small Fp-norm in the sensethat

lim supr→0

‖V χB(y,r)‖Fp ≤ ε(p,N) for all y ∈ RN , (4.krb)

where ε(p,N) is a sufficiently small constant. Since LN/2,∞ ⊂ Fp for all p < N/2(see Proposition 2) this gives a result for V in a larger class than in [JK], [St],however, with the size constraint, this time in the Fp class; again the value ofthe constant appearing in the size condition is not specified.

If N ≤ 3, then a condition for the SUCP in terms of the local smallness ofthe constant C in (1.krb) appears; more specifically:

Theorem 5 (Chanillo, Sawyer [CS]). Let us assume that N = 2 or N = 3and that Ω is a bounded open and connected subset of RN . Let T (V ) denote theimbedding in (1.krb). If

lim supr→0+

‖T (V χB(x,r))‖ ≤ ε (5.krb)

with a sufficiently small ε > 0 for all x ∈ Ω, then any solution u ∈ W 2,2loc of the

inequality |∆u| ≤ V |u| in Ω has the SUCP.

It turns out that the size condition can be effectively verified in some cases.We shall consider the scale of Lorentz spaces in the dimension 3, and for N = 2we present a general theorem, including [GL] as a special case. Proofs will appearelsewhere (see [KrSc]).

We shall need some basic facts from the Orlicz, Lorentz-Zygmund and Orlicz-Lorentz spaces theory. Let us agree that all the spaces in the sequel will beconsidered on a ball B ⊂ RN with the unit measure, N ≥ 2, or on the interval(0, 1); we shall usually omit the appropriate symbol for the domain since it willbe clear from the context.

We shall also need a finer scale of spaces, which includes Orlicz spaces ina rather same manner as Lorentz spaces include Lebesgue spaces. We refer toMontgomery-Smith [M-S].

Let us recall that an even and convex function Φ : R → [0,∞) such thatlimt→0

Φ(t) = limt→∞

1/Φ(t) = 0 is called a Young function. A general reference for

the (non-weighted) theory of Orlicz spaces is [KR], more general modular spacesare subject of [Mu].

Let Φ and Ψ be Young functions. For a function g even on R1 and positiveon (0,∞) let us put

g(t) =

1/g(1/t), t > 0,g(−t), t < 0,g(0), t = 0.


Let V be a weight in B and let f∗V denote the non-increasing rearrangement off with respect to the measure V (x) dx. An Orlicz-Lorentz space LΦ,Ψ (V ) is theset of all measurable f on B for which the Orlicz-Lorentz functional

‖f‖Φ,Ψ ;V = ‖f∗V Φ Ψ−1‖Ψ

= infλ > 0;

∞∫0

Ψ

(f∗V (Φ(Ψ−1(t)))

λ

)dt ≤ 1

(6.krb)

is finite. A measurable function f defined on B belongs to a weak Orlicz (orOrlicz-Marcinkiewicz) space LΦ,∞(V ) if its Orlicz-Marcinkiewicz functional

‖f‖Φ,∞;V = supξ>0

Φ−1(ξ)f∗V (ξ) (7.krb)

is finite. If V ≡ 1, we shall simply write LΦ,Ψ and LΦ,∞ instead of LΦ,Ψ(1) andLΦ,∞(1), resp.

For brevity and in accordance with a general usage we shall often use onlythe major part of a Young function (that is, functions equivalent to the Youngfunction in question in a neighbourhood of infinity) in symbols for spaces.

Let us observe that LΦ,Φ = LΦ, the Orlicz space. If Φ(t) = |t|p and Ψ(t) = |t|q,then LΦ,Ψ = Lp,q, the Lorentz space, LΦ,∞ = Lp,∞, the Marcinkiewicz space;analogously for the weighted variants.

Special cases of the Orlicz-Lorentz spaces are the Lorentz-Zygmund spaces,that is, logarithmic Lorentz spaces, investigated by Bennett and Rudnick [BR].For 0 < p, q ≤ ∞ and α ∈ R1, the Lorentz-Zygmund space Lp,q(logL)α consistsof functions f with the finite functional

‖f‖Lp,q(logL)α =( 1∫

0

[t1/p(log(e/t))αf∗(t)]qdt

t

)1/q

, for q <∞,

‖f‖Lp,∞(logL)α = sup0<t<1

t1/p (log(e/t))α f∗(t), for q =∞

(we put t1/∞ = 1). It is easy to check that these spaces increase with decreasingp, increasing q and decreasing α.

Note that later we shall also need the spaces of the form Lexp tr′,tr , where

1/r+ 1/r′ = 1. It turns out that they coincide (see [EdKr]) with spaces charac-terized by the integral condition (used e.g. in [BW] and in [Zi], Lemma 2.10.5)

1∫0

(f∗(t)

log(e/t)

)rdt <∞,

which equal to L∞,r(logL)−1 in the [BR] notation. Also, the Zygmund spaceL logL equals to L1,1 logL and it is nothing but Lt log t,t log t.


Remark 6. We recall that Lp1,q1(logL)α1 ⊂ Lp2,q2(logL)α2 if any of the follow-ing conditions holds:

(i) p1 > p2;(ii) p1 = p2, q1 > q2, and α1 + 1/q1 > α2 + 1/q2;

(iii) p1 = p2 <∞, q1 ≤ q2, and α1 ≥ α2;(iv) p1 = p2 =∞, q1 ≤ q2, and α1 + 1/q1 ≥ α2 + 1/q2

(see [BR], Theorems 9.1 and 9.3 and 9.5).

Remark 7. According to the limiting imbedding theorem due to Brezis andWainger [BW] we have, for N = 2,

W 1,20 → L∞,2(logL)−1. (8.krb)

The latter space, as was observed above, is the Orlicz-Zygmund space Lexp t2,t2 ,a space smaller than Lexp t2 = Lexp t2,exp t2 , and this interpretation of the targetspace in (8.krb) gives a natural analogue to the (sublimiting) imbeddings of Sobolevspaces into Lebesgue spaces and their Lorentz refinements.

4 Decomposition of imbeddings

Let us recall our agreement that for the sake of simplicity we shall supposethat the domain B is a ball, |B| = 1. We shall usually omit the symbol of thedomain. We are seeking for sufficient conditions for (2.krb) and (5.krb); we shall evenfind a condition stronger than (5.krb), namely,

limδ→0

supA⊂B|A|<δ

‖T (V χA)‖ = 0. (9.krb)

First we shall separately consider the scale of Lorentz spaces.

Theorem 8 ([KrSc]). Let N ≥ 3. and V ∈ LN/2,r, N/2 ≤ r < ∞. Then (2.krb)and (9.krb) hold.

We shall pass to Lorentz-Zygmund spaces and present a theorem, establishinga general sufficient condition for (2.krb) and various sufficient conditions for (9.krb); letus observe that the situation is not straightforward since three parameters canchange. The first parameter will be kept fixed, equal to 1; its changes lead tochanges too big for the fine tuning we need.

Theorem 9 ([KrSc]). Let N = 2.

(1) The inequality (2.krb) holds provided V ∈ L1,∞(logL)2.


(2) Let V ∈ L1,s(logL)β, where either

0 < s ≤ 1, β ≥ 1, (10.krb)

or

1 < s <∞, β ≥ 2− 1/s, (11.krb)

or

s =∞, β > 2. (12.krb)

Then (2.krb) and (9.krb) hold.

Remark 10. The proofs of Theorems 8 and 9 can be carried out making useof the refined Sobolev imbedding W 1,2 → L2N/(N−2),s for N ≥ 3 and of therefined limiting imbedding in (8.krb) for N = 2 together with conditions (necessaryand sufficient) for the imbeddings of weighted Orlicz-Lorentz spaces, taking,moreover, care about the quantitative behaviour of norms of the imbeddings.The details can be found in [KrSc].

Remark 11. The space L1,∞(logL)2 can be identified with the Orlicz-Marcinkie-wicz space Lt log2 t,∞ and L1,s(logL)β , 0 < s < ∞, with Lt logβ t,ts . This can bechecked easily. Indeed, considering for instance V ∈ L1,∞(logL)2, that is, if wehave sup

0<t<1t(log(e/t))2V ∗(t) < ∞, then F−1(t) = t(log(e/t))2 near the origin,

hence F (ξ) ∼ ξ(log(e/ξ))2 for large values of ξ.

By way of applications we give a sufficient condition for the SUCP, relyingon the SUCP theorem in [CS] invoked earlier.

Corollary 12 ([KrSc]). The following statements are true:

(1) Let N = 3. Let V ∈ L3/2,r, 3/2 ≤ r < ∞. Then the inequality |∆u| ≤ V |u|has the SUCP in W 2,2

loc ∩W1,20 .

(2) Let N = 2. Let V ∈ L1,s(logL)β, where s and β satisfy any of the conditions(10.krb)–(12.krb). Then the inequality |∆u| ≤ V |u| has the SUCP in W 2,2

loc ∩W1,20 .

Remark 13. The statement in (1) actually says that the size condition fromStein [St] is fulfilled under the given conditions.

If V ∈ L1,s(logL)β , where s and β satisfy either (10.krb) or (12.krb), then V ∈L1,1(logL)1 and we recover the SUCP theorem due to Gossez and Loulit [GL].Concerning (11.krb) one can construct functions, which show that L1,1(logL)1 andL1,s(logL)2−(1/s) are incomparable for 1 < s <∞.

Indeed, if V (α, . ), 0 < α ≤ 1, is such that

V ∗(α, t) =1t

(log(e/t))−2 (log (log(e/t)))−α , for t small,


then V (α, . ) /∈ L1,1(logL)1 and if s > 1/α, then V (α, . ) ∈ L1,s(logL)2−(1/s).On the other hand, if V (τ, . ), 0 < τ < 1, is such that V ∗(τ, t) = χ(0,τ)(t), then

‖V (τ, . )‖L1,1(logL)1 = τ(2 − log τ), 0 < τ < 1.

Going through some calculation one can check that

limτ→0

‖V (τ, . )‖sL1,s(logL)2−(1/s)

‖V (τ, . )‖sL1,1(logL)1

=∞.

Therefore L1,1(logL)1 is not continuously imbedded into L1,s(logL)2−(1/s) andby the closed graph theorem we get L1,1(logL)1 6⊂ L1,s(logL)2−(1/s).

The first author was in part supported by Grant No. 201/96/0431 of GACR. At the same time the first author would like to thank for a DAAD supportwhich made possible a visit at the FSU Jena in autumn 1997, when this paperhas been finished.

References

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[BR] C. Bennett and K. Rudnick, On Lorentz-Zygmund spaces, DissertationesMath. (Rozprawy Mat.) CLXXV (1980), 1–72.

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[BW] H. Brezis and S. Wainger, A note on limiting cases of Sobolev embeddings andconvolution inequalities, Comm. Partial Diff. Equations 5 (1980), 773–789.

[C] T. Carleman, Sur un probleme d´unicite pour les systemes d’equations auxderivees partielles a deux variables independantes, Ark. Mat. Astron. Fys.26(B) 1939, 1–9.

[CWW] S. Y. A. Chang, J. M. Wilson and T. H. Wolff, Some weighted norm inequalitiesconcerning the Schrodinger operator, Comment. Math. Helvetici 60 (1985),217–246.

[CS] S. Chanillo and E. Sawyer, Unique continuation for ∆ + V and the C. Fef-ferman-Phong class, Trans. Amer. Math. Soc. 318 (1990), 275–300.

[CW] S. Chanillo and R. L. Wheeden, Lp estimates for fractional integrals andSobolev inequalities with applications to Schrodinger operator, Comm. PartialDiff. Equations 10(9) (1985), 1077-1116.

[CFG] F. Chiarenza, E. Fabes and N. Garofalo, A remark on a paper by C. Fefferman,Proc. Amer. Math. Soc. 98 (1986), 415–425.

[CF] F. Chiarenza and M. Frasca, A remark on a paper by C. Fefferman, Proc.Amer. Math. Soc. 108 (1990), 407–409.

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[DiF] G. Di Fazio, Holder continuity of solutions for some Schrodinger equations,Rend. Sem. Mat. Univ Padova 79 (1988), 173–183.


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[FP] C. Fefferman and D. H. Phong, Lower bounds for Schrodinger operator,Journees “Equations aux derivees partielles”, Saint-Jean-de Monts, 7–11 juin1982.

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A Time Periodic Solution

of the Navier-Stokes Equations

with Mixed Boundary Conditions

Petr Kucera

Department of Mathematics, Faculty of Civil Engineering, CTU,Thakurova 7, 166 29 Prague, The Czech Republic

Email: [email protected]: http://mat.fsv.cvut.cz/

Abstract. We study qualitative properties of the system of time-peri-odic Navier-Stokes equations and continuity equation with the Dirichletboundary condition on the fixed wall and the natural boundary conditionon the input and on the output.

AMS Subject Classification. 35Q10, 58E35

Keywords. Navier-Stokes equations, mixed boundary conditions

1 Description of the Domain

We suppose that Ω ⊂ Rn, where n = 2 or n = 3, Ω is a bounded domain,∂ Ω ∈ |C0,1. Further, we suppose that Ω = Γ1 ∪ Γ2, where Γ1 and Γ2 areclosed (not necessarily connected) sets such that measn−1(Γ1 ∩ Γ2) = 0 andmeasn−1(Γ1) > 0.

The domain Ω corresponds to a channel filled up by a fluid. Γ1 is a fixed wallof the channel and Γ2 involves the input and the output of the channel.

2 Classical Formulation of the Problem

Let T > 0 be a positive number. (0, T ) denotes the time interval, Q = Ω×(0, T ),eij(u) (for 1 ≤ i, j ≤ n) denotes ∂ui/∂xj + ∂uj/∂xi.

The problem we will deal with can be classically formulated as follows:

∂u

∂t− ∂

∂xj(ν · eij(u)) + uj ·

∂ui∂xj

+∂P∂xi

= gi in Q, i = 1, . . . , n, (1.kuc)

divu = 0 in Q, (2.kuc)u = 0 in Γ1 × (0, T ), (3.kuc)

−P · ni + ν · eij(u) · nj = σi in Γ2 × (0, T ), (4.kuc)


http://mat.fsv.cvut.cz/

194 Petr Kucera

u(x, 0) = u(x, T ) in Ω, (5.kuc)u(., 0) = 0 on Γ1. (6.kuc)

Here u is the velocity, P is the pressure, ν denotes the viscosity, g is a bodyforce, σ is a prescribed vector function on Γ2 and n = (n1, . . . ,nn) is the outernormal vector. The problem (1.kuc)–(6.kuc) will be called the time-periodic Navier-Stokes problem with the mixed boundary conditions. We suppose that ν is apositive constant in the whole paper.

The used Dirichlet boundary condition expresses a non-slip behaviour of thefluid on the fixed walls of the channel. The condition (4.kuc) means that we prescribea normal component of the stress tensor on Γ2. The Navier-Stokes equations withcondition (4.kuc) were already treated in the works [1]–[6].

3 Some Function Spaces and Their Properties

To formulate the problem (1.kuc)–(6.kuc) weakly, we shall need some function spaces.Let us denote

E(Ω) = ϕ ∈ [C∞(Ω)]n; div ϕ ≡ 0, supp ϕ ∩ Γ1 ≡ ∅.

The Banach spaces V k,p, resp. V 0,q, is defined as the closure of E(Ω) in the normof the space [W k,p(Ω)]n, resp. [Lp(Ω)]n, where k > 0 (it need not be an integer)and 1 ≤ q ≤ ∞. For simplicity, we denote the space V 0,2 by the symbol H .

Both the spaces V 1,2 and H are Hilbert spaces with the scalar products

((ψ, φ))1,2 =∫Ω

eij(ψ) · eij(φ) d(Ω)

resp.

((ψ, φ))0,2 =∫Ω

ψi · φi d(Ω).

The symbol 〈., .〉 denotes the duality between elements from (V 1,2)∗ and V 1,2.It is obvious that V 1,2, H and (V 1,2)∗ are three Hilbert spaces, which satisfy

the following conditions

V 1,2 →→ H →→ (V 1,2)∗

and H coincides with the interpolation [V 1,2, (V 1,2)∗]1/2. Moreover, ifu ∈ L2(0, T, V 1,2), u′ ∈ L2(0, T, (V 1,2)∗), then u ∈ C([0, T ];H) and

||u||L∞(0,T ;H) ≤ c · (||u||L2(0,T ;V 1,2) + ||u′||L2(0,T ;(V 1,2)∗)),

where c = c(Ω).If X is a Banach space then (X )∗ will denote its dual and Lp(0, T ;X ), 1 <

p < ∞, will be the linear space of all measurable functions from the interval(0, T ) into X such that ∫ T

0

||u(t)||pX dt <∞.

A Time Periodic Solution of Navier-Stokes Equations 195

Let X and Y be the following Banach spaces:

X = u;u′ ∈ L2(0, T, V 1,2), u′′ ∈ L2(0, T, (V 1,2)∗), u(0) = u(T ) ∈ V 1,2,

u′(0) = u′(T ) ∈ H,||u||X = ||u||L2(0,T ;V 1,2) + ||u′||L2(0,T ;V 1,2) + ||u′′||L2(0,T ;(V 1,2)∗),

Y = f ; f ∈ C([0, T ], (V 1,2)∗), f ′ ∈ L2(0, T, (V 1,2)∗), f(0) = f(T ) ∈ (V 1,2)∗,||f ||Y = ||f ||L2(0,T ;(V 1,2)∗) + ||f ′||L2(0,T ;(V 1,2)∗).

4 Weak Formulation of the Problem

The weak formulation of the problem (1.kuc)–(6.kuc) will be based on an operator equa-tion. Therefore we define operators S, B and N at first.

The operator S from X to Y is defined by the equation

〈S(u), v〉 = ((u′, v))0,2 + ν · ((u, v))1,2

for every v ∈ V 1,2 and almost every t ∈ (0, T ).

b(ϕ, ψ, φ) will denote trilinear form on V 1,2 × V 1,2 × V 1,2 such that

b(ϕ, ψ, φ) =∫Ω

ϕj ·∂ψi∂xj· φi d(Ω).

It can be easily verified that b(ϕ, ψ, φ) satisfies the following estimate

|b(ϕ, ψ, φ)| ≤ c · ||ϕ||V 1,2 · ||φ||V 1,2 · ||ψ||V 1,2 , (7.kuc)

where c = c(Ω).Integrating by parts and using the theorems about imbedding the space

[W kp(Ω)]n into the space Lq(∂Ω) the following estimates are verified:

|b(ϕ, ψ, φ)| ≤ c · ||ϕ||V 1,2 · ||ψ||V

78 ,2· ||v||V 1,2 , (8.kuc)

|b(ϕ, ψ, φ)| ≤ c · ||ϕ||V

78 ,2· ||ψ||V 1,2 · ||v||V 1,2 . (9.kuc)

The symbols ϕ and ψ will sometimes also denote functions of the variable twith values in V 1,2.

B will be operator from X into Y defined by the equation

〈B(u), v〉 = b(u, u, v)

for every v ∈ V 1,2 and almost every t ∈ (0, T ).

196 Petr Kucera

Finally, operator N from X into Y is defined by the equation

N (u) = S(u) + B(u).

A function u ∈ X will be called a weak solution to the time-periodic Navier-Stokes problem with the right hand side f if

N (u) = f.

Notice that

〈f, v〉 =∫Ω

gi · vi d(Ω) +∫Γ2

σi · vi d(∂Ω).

5 The Local Diffeomorphism Theorem

Suppose that u0 and f0 are such elements of X and Y that

N (u0) = f0.

(This means that u0 is a weak solution to the time-periodic Navier-Stokes prob-lem with the right hand side f0.) Our further aim is to investigate the solvabilityof the equation N (u) = f with f from some neighbourhood of point f0 in Y . Tosolve this problem, we will use the following very important theorem (the LocalDiffeomorphism Theorem).

Theorem 1. Let X , Y be Banach spaces, F be a mapping from X into Y be-longing to C1 in some neighbourhood V of point u0. If F ′(u0) is one-to-onefrom X onto Y and continuous, then there exists a neighbourhood U of point u0,U ⊂ V and a neighbourhood W of point f(u0), W ⊂ Y, so that F is one-to-onefrom U to W .

6 The Frechet Derivative of Operator N

It is obvious that if there exists a point u ∈ X in which the operator N sat-isfies the assumption of the Local Diffeomorphism Theorem then the equationN (u) = f is “locally solvable” (i.e. solvable in some neighbourhood of u). It isclean that N ∈ C1(X). Further, need to express the Frechet derivative of oper-ator N at point u and to find out, whether it is one-to-one. We will express theFrechet derivative of N by means of operators K and G.

K is the bilinear operator from X ×X into Y defined by the equation

〈K(u), v〉 = b(u,w, v) + b(w, u, v)

for every v ∈ V 1,2 and for almost every t ∈ (0, T ).


The operator G from X ×X into Y is given by the equation

G(u,w) = S(w) +K(u,w).

It is possible to prove there exists a constant c = c(Ω) so that

||b(u,w, .)||Y ≤ c · ||u||X · ||w||X .

Theorem 2. Let u ∈ X. Then the operator G(u, .) is the Frechet derivative ofN at point u and G ∈ C1(X ×X,Y ).

Proof. It is possible to prove for arbitrary u,w ∈ X following estimate

||b(u,w, .)||Y ≤ c · ||u||X · ||w||X ,

where c = c(Ω). Therefore and from the estimate

||N (u + w) −N (u)− G(u,w)||Y = ||b(w,w, .)||Y ≤ c · ||w||2X ,

we get

lim||w||X→0

||N (u + w)−N (u)− G(u,w)||Y||w||X

= 0.

So G(u, .) is the Frechet derivative of N at point u. The smoothness of G followsimmediately from its definition. The proof is complete.

7 Local Properties of Operator N

We have proved that the operator G(u, .) has the form

G(u, .) = S(.) +K(u, .)

in the previous section. Further we will prove that operator S is a one-to-onelinear operator from X onto X and K(u, .) is a compact linear operator from Xinto Y . So the operator G(u, .) is the sum of a one-to-one operator and a compactoperator. Operators of this form have properties which will be used later.

Lemma 3. S is a linear continuous one-to-one operator from X onto Y .

Proof. The linearity and continuity of S are obvious. Next we prove that S isan operator from X onto Y . The form ((., .))1,2 is V 1,2-elliptic. Then there existsw ∈ L2(0, T, V 1,2) ∩ C([0, T ];H), so that w′ ∈ L2(0, T, (V 1,2)∗), the equation

d

dt((w(t), v))0,2 + ν · ((w(t), v))1,2 = 〈f ′, v〉

holds for every v ∈ V 1,2 andw(0) = w(T ).

198 Petr Kucera

Then there exists ω0 ∈ V 1,2 so that for every v ∈ V 1,2 holds

ν · ((ω0, v))1,2 = 〈f, v〉 − ((w(0), v))0,2 .

Let

u(t) = ω0 +∫ t

0

w(s) ds, t ∈ (0, T ).

Then u ∈ X and S(u) = f . Thus we have proved that S is from X onto Y . Letus suppose that S(u) = 0. Then u = 0. The proof is complete.

Lemma 4. Let u ∈ X. Then K(u, .) is a linear compact operator from X into Y .

Prior to the proof we recall a result from [7, Lemma 4.5]. Denote

Z = u;u ∈ L2(0, T, V 1,2), u′ ∈ L2(0, T, (V 1,2)∗)

with the norm||u||Z = ||u||L2(0,T ;V 1,2) + ||u′||L2(0,T ;(V 1,2)∗)

(u′ is the Schwartz derivative in the sence of imbedding V 1,2 → H → (V 1,2)∗).Then

Z →→ L2(0, T ;V78 ,2) (10.kuc)

Proof. Let wk ⊂ X be a bounded set in X . Using (10.kuc) we get w ∈ X such that

w′k → w′ in L2(0, T ;V78 ,2) (11.kuc)

andwk(0)→ w(0) in V

78 ,2

Combining it with (11.kuc) we get

wk → w in L∞(0, T ;V78 ,2). (12.kuc)

Note that

||K(u,wk)−K(u,w)||Y =

=||b(u,wk − w, .)||L2(0,T ;(V 1,2)∗) + ||b(wk − w, u, .)||L2(0,T ;(V 1,2)∗) ++||b(u,w′k − w′, .)||L2(0,T ;(V 1,2)∗) + ||b(u′, wk − w, .)||L2(0,T ;(V 1,2)∗) ++||b(w′k − w′, u, .)||L2(0,T ;(V 1,2)∗) + ||b(wk − w, u′, .)||L2(0,T ;(V 1,2)∗)

(13.kuc)

We estimate the third and fourth additive terms. Let v ∈ V 1,2. Use (8.kuc) to getthe estimate

|b(u(t), w′k(t)− w′(t), v)| ≤ c · ||u(t)||V 1,2 · ||w′k(t)− w′(t)||V

78 ,2· ||v||V 1,2 .


Therefore

||b(u(t), w′k(t)− w′(t), .)||(V 1,2)∗ ≤ c · ||u(t)||V 1,2 · ||wk(t)− w(t)||V

78 ,2

for almost all t ∈ (0, T ), c = c(Ω). It follows that

||b(u,w′k − w′, .)||L2(0,T ;(V 1,2)∗) ≤ c · ||u||L∞(0,T ;V 1,2) · ||w′k − w′||L2(0,T ;V78 ,2)

. (14.kuc)

Similarly, we get

|b(u′(t), wk(t)− w(t), v)| ≤ c · ||u′(t)||V 1,2 · ||wk(t)− w(t)||V

78 ,2· ||v||V 1,2

and therefore

||b(u′(t), wk(t)− w(t), .)||(V 1,2)∗ ≤ c · ||u′(t)||V 1,2 · ||wk(t)− w(t)||V

78 ,2

for almost all t ∈ (0, T ), c = c(Ω). It follows

||b(u′, wk − w, .)||L2(0,T ;(V 1,2)∗) ≤ c · ||u′||L2(0,T ;V 1,2) · ||wk − w||L∞(0,T ;V78 ,2)

. (15.kuc)

The same way we prove

||b(u,wk − w, .)||L2(0,T ;(V 1,2)∗) ≤ c · ||u||L2(0,T ;(V 1,2)∗) · ||wk − w||L∞(0,T ;V78 ,2)

,

(16.kuc)

||b(wk − w, u, .)||L2(0,T ;(V 1,2)∗) ≤ c · ||u||L2(0,T ;(V 1,2)∗) · ||wk − w||L∞(0,T ;V78 ,2)

,

(17.kuc)

||b(w′k − w′, u, .)||L2(0,T ;(V 1,2)∗) ≤ c · ||u||L∞(0,T ;V 1,2) · ||w′k − w′||L2(0,T ;V78 ,2)

(18.kuc)

and

||b(wk − w, u′, .)||L2(0,T ;(V 1,2)∗) ≤ c · ||u′||L2(0,T ;(V 1,2)∗) · ||wk − w||L∞(0,T ;V78 ,2)

.

(19.kuc)

From (11.kuc)–(19.kuc) we get

||K(u,wk)−K(u,w)||Y → 0.

The proof is complete.

The operator G(u, .) is the sum of a one-to-one operator and a compact op-erator. The operators of this form are widely treated in mathematical literatureand we can apply their known properties to prove the following theorem.

Theorem 5. Let u ∈ V 1,2. Then the following statements are equivalent:(a) G(u, .) is an injective operator .(b) G(u, .) is an operator onto (V 1,2)∗.Moreover, if the statements (a)–(b) are satisfied at point u then there exists

an open neighbourhood U of point u in X and an open neighbourhood W of pointN (u) in Y such that N is a one-to-one operator from U onto W .

200 Petr Kucera

References

[1] R. Glowinski, Numerical methods for nonlinear variational problems, Springer Ver-lag, Berlin-Heidelberg-Tokio-New York, 1984.

[2] S. Kracmar, J. Neustupa, Global existence of weak solutions of a nonsteady varia-tional inequalities of the Navier-Stokes type with mixed boundary conditions. Proc.of the conference ISNA’92, (1992), Part III, 156–157.

[3] S. Kracmar, J. Neustupa, Modelling of flows of a viscous incompressible fluidthrough a channel by means of variational inequalities. ZAMM 74, 6 (1994), 637–639.

[4] S. Kracmar, J. Neustupa, Some Initial Boundary Value Problems of the Navier-Stokes Type with Mixed Boundary Conditions. Proc. of the seminar NumericalMathematics in Theory and Practice, Pilsen, (1993), 114–120.

[5] S. Kracmar, J. Neustupa, Simulation of Steady Flows through Channels by Vari-ational Inequalities. Proc. of the conference Numerical Modelling in ContinuumMechanics, Prague 1994, (1995), 171–174.

[6] R. Rannacher, Numerical analysis of the Navier-Stokes equations. Proceedings ofconference ISNA’92, part I. (1992), 361–380.

[7] P. Kucera, Z. Skalak, Local solutions to the Navier-Stokes equations with mixedboundary conditions. Submitted to Acta Applicandae Mathematicae.



Numerical Solution of Compressible Flow

Maria Lukacova-Medvid’ova

Department of Mathematics,Faculty of Mechanical Engineering, Technical University Brno

Technicka 2, 616 00 Brno, Czech RepublicInstitute of Analysis and Numerics

Otto-von-Guericke University MagdeburgPSF 4120, 39 106 Magdeburg, Germany

Email: [email protected]@mathematik.uni-magdeburg.de

Abstract. The main feature of the equations describing the motion ofthe viscous compressible flows, i.e. the Navier-Stokes equations, is thecombination of dominating convective parts with the diffusive effects.These equations will be numerically solved by the combined finite vol-ume — finite element method via operator inviscid-viscous splitting. Themain idea of the method is to discretize nonlinear convective terms withthe aid of the finite volume scheme, whereas the diffusion terms are dis-cretized by piecewise linear conforming triangular finite elements. Thenonlinear convective terms can also be solved by the method of char-acteristics. Numerical solution obtained by latter method is truly mul-tidimensional and independent of the mesh character. We will presentresults of numerical experiments for some well-known test problems.

AMS Subject Classification. 65M12, 65M60, 35K60, 76M10, 76M25

Keywords. Compressible Navier-Stokes equations, nonlinear convec-tion-diffusion equation, finite volume schemes, finite element method, nu-merical integration, truly multidimensional schemes, evolution-Galerkinschemes

1 Formulation of the problem

We consider gas flow in a space-time cylinder QT = Ω × (0, T ), where Ω ⊂ R2

is a bounded domain representing the region occupied by the fluid and T > 0.By Ω and ∂Ω we denote the closure and boundary of Ω, respectively.

The complete system of viscous compressible flow consisting of the continuityequation, Navier-Stokes equations and energy equation can be written in theform

∂w

∂t+

2∑i=1

∂fi(w)∂xi

=2∑i=1

∂Ri(w,∇w)∂xi

in QT . (1.luk)


202 Maria Lukacova-Medvid’ova

Here

w = (w1, w2, w3, w4)T = (ρ, ρv1, ρv2, e)T, (2.luk)w = w(x, t), x ∈ Ω, t ∈ (0, T ),fi(w) = (ρvi, ρviv1 + δi1p, ρviv2 + δi2p, (e+ p) vi)T,

Ri(w,∇w) = (0, τi1, τi2, τi1v1 + τi2v2 + k∂θ/∂xi)T,

τij = λdiv v δij + µ

(∂vi∂xj

+∂vj∂xi

), i, j = 1, 2.

From thermodynamics we have

p = (γ − 1) (e− ρ|v|2/2), e = ρ(cvθ + |v|2/2). (3.luk)

We use the standard notation: t — time, x1, x2 — Cartesian coordinates in R2,ρ — density, v = (v1, v2) — velocity vector with components vi in the directionsxi, i = 1, 2, p — pressure, θ — absolute temperature, e — total energy, τij —components of the viscous part of the stress tensor, δij — Kronecker delta, γ > 1— Poisson adiabatic constant, cv — specific heat at constant volume, k — heatconductivity, λ, µ — viscosity coefficients. We assume that cv, k, µ are positiveconstants and λ = − 2

3µ. We neglect outer volume force. The functions fi arecalled inviscid (Euler) fluxes and are defined in the set D = (w1, . . . , w4) ∈R4; w1 > 0. The viscous terms Ri are obviously defined in D × R8. (Due tophysical reasons it is also suitable to require p > 0.)

System (1.1), (1.3) is equipped with the initial conditions

w(x, 0) = w0(x), x ∈ Ω (4.luk)

(which means that at time t = 0 we prescribe, e. g., ρ, v1, v2 and θ) and boundaryconditions: The boundary ∂Ω is divided into several disjoint parts. By ΓI , ΓOand ΓW we denote inlet, outlet and impermeable walls, respectively, and assumethat

(i) ρ = ρ∗, vi = v∗i , i = 1, 2, θ = θ∗ on ΓI , (5.luk)

(ii) vi = 0, i = 1, 2,∂θ

∂n= 0 on ΓW ,

(iii)2∑i=1

τijni = 0, j = 1, 2,∂θ

∂n= 0 on ΓO.

Here ∂/∂n denotes the derivative in the direction of unit outer normal n =(n1, n2) to ∂Ω; w0, ρ∗, v∗i and θ∗ are given functions.

Let us note that nothing is known about the existence and uniqueness of thesolution of problem (1.1), (1.3)–(1.5). Some solvability results for system (1.1) &(1.3) were obtained either for small data or on a very small time interval undersimple Dirichlet boundary conditions (for reference, see e. g., [3, Par. 8.10]).

We do not take care of the lack of theoretical results and deal with the numeri-cal solution of the above problem. Since the viscosity µ and heat conductivity

Numerical Solution of Compressible Flow 203

k are small, we treat the diffusion terms on the right hand side of (1.1) as aperturbation of the inviscid Euler system and conclude that a good method forthe solution of viscous flow should be based on a sufficiently robust scheme forinviscid flow simulation. Therefore, we will split the complete system (1.1) intoinviscid and viscous part:

∂w

∂t+

2∑i=1

∂fi(w)∂xi

= 0, (6.luk)

∂w

∂t=

2∑i=1

∂Ri(w,∇w)∂xi

(7.luk)

and discretize them separately. First we will pay attention to the inviscid flowproblem.

2 Numerical solution of the Euler equations

In what follows we will describe some numerical methods for solving the Eulerequations system (6.luk). The first part of this section will be devoted to the finitevolume methods, in the second part we will briefly describe truly multidimen-sional methods based on the method of characteristics, the so-called evolutionGalerkin schemes. In the third part we present some numerical experiments forthe Euler equations system.

It is easy to realize that fj ∈ C1(D;R4

)for j = 1, 2. Thus, we can apply the

chain rule to the function fj (w) and obtain a first order quasilinear system ofPDE’s

∂w

∂t+

2∑j=1

Aj (w)∂w

∂xj= 0, (8.luk)

where Aj (w) = Dfj(w)Dw are Jacobi matrices of fj (w) , j = 1, 2.

Definition 1. Let us consider general first order system of type (8.luk). The systemis said to be hyperbolic, if for arbitrary vectors w ∈ D and n = (n1, n2) ∈ R2

the matrix

P (w,n) =2∑j=1

nj Aj (w)

has four real eigenvalues λi = λi (w,n) , i = 1, . . . , 4, and is diagonizable, i.e.there exists a nonsingular matrix T = T (w,n) , s.t.

T−1 · P · T = D (w,n) = diag(λ1, λ2, λ3, λ4)

Theorem 2. The system of Euler equations (8.luk) is hyperbolic.The eigenvalues of the matrix P (w,n) are

λ1 = λ2 = n1v1 + n2v2, λ3 = λ1 + a|n|, λ4 = λ1 − a|n|.

Here a is a local speed of sound, i.e. a =√

kpρ .


2.1 Finite volume schemes

The above properties of the Euler equations allow us to construct efficientnumerical schemes for the solution of inviscid flow. We will carry out the dis-cretization of system (6.luk) with the use of the finite volume method (FVM) whichis very popular because of its flexibility and applicability.

Let Th be a triangulation of the domain Ωh which is a polygonal approxima-tion of the domain Ω . The so-called dual finite volume partition of Ωh will bedenoted by Dh = Dii∈J , J is a suitable index set. Moreover, it holds

Ωh =⋃i∈J

Di. (9.luk)

The dual finite volumes will be constructed in the following way: Join thecentre of gravity of every triangle T ∈ Th, containing the vertex Pi, with the cen-tre of every side of T containing Pi. If Pi ∈ ∂Ωh, then we complete the obtainedcontour by the straight segments joining Pi with the centres of boundary sidesthat contain Pi. In this way we get the boundary ∂Di of the finite volume Di.(See Figure 1.) Dual finite volume meshes were successfully used in a number ofworks. See, e.g., [1], [8].

HHHH

H

HHHHHHH

rCCCCCCCC

```` r

CCCCCCQQQQ

r

HHHHHHH

aaaaa

((JJ

JJ JJ

BB AA

AA

DDPi

@@

Pj

DjDi

Pk

Dk

Figure 1

If for two different finite volumes Di and Dj their boundaries contain acommon straight segment, we call them neighbours. Then we write

Γij = ∂Di ∩ ∂Dj = Γji. (10.luk)

The index set of all neighbours for the dual volume Di will be denoted byS(i). Furthermore, we introduce the following notation: |Di| = area of Di,nij =(n1ij , n2ij) = unit outer normal to ∂Di on Γij , `ij = length of Γij , and considera partition 0 = t0 < t1 < . . . of the time interval (0, T ) and set τk = tt+1 − tkfor k = 0, 1, . . . .


The finite volume method reads

wk+1i = wki −

τk|Di|

∑j∈S(i)

g(wki , wkj , nij) `ij , (11.luk)

Di ∈ Dh (i. e., i ∈ J), k = 0, 1, . . . .

To derive (11.luk) we integrate (6.luk) over every set Di×(tk, tk+1), use Green’s the-orem, the approximation of the exact solution by a piecewise constant functionwith values wki on Di × tk and the approximation of the flux∫

Γij

2∑r=1

fr(w)nr dS

of the quantity w through the segment Γij in the direction nij with the aid ofthe so-called numerical flux g(wki , w

kj , nij) calculated from wki , w

kj and nij .

In order to ensure the stability of the scheme (11.luk) the so-called CFL stabilitycondition has to be fulfilled

τk|Di|

maxj∈S(j)

maxs=1,...,4

λs(wk,nij) ≤ CFL ∀j ∈ J, (12.luk)

where CFL ∈ (0, 1]. In literature one can find a lot of numerical flux func-tions, e.g. Steger-Warming, Osher-Solomon, Van-Leer, Vijayasundaram numeri-cal fluxes. For references, see e.g., [3].

We do not discuss now the question of implementation of the boundary con-ditions. They have to be prescribed in such a way that the hyperbolic characterof the equations is taking into account. For more details the reader is referred to[4]. In the approach described about only the piecewise constant approximationis considered. Nevertheless, also the higher order schemes, using for examplediscontinuous piecewise linear approximate functions, can be constructed. Thedetails can be found, e.g., in [4].

2.2 Evolution Galerkin methods

Although in the recent years the most commonly used methods for hyperbolicproblems are the finite volume methods, it turns out that in special cases thisapproach leads to structural deficiencies in the solution (see, e.g., [7], [13]). Thisis due to the fact that the finite volume methods are based on a quasi dimensionalsplitting using one-dimensional Riemann solvers.

The evolution Galerkin method, first considered by Morton et al. in [2] forscalar hyperbolic equation, combines the theory of characteristics for hyperbolicproblems with the finite element ideas. The initial function is transported alongthe characteristic cone and then projected onto a finite element space.

Let E(t) be the exact evolution operator for our hyperbolic problem (8.luk), i.e.

w(·, tk+1) = E(∆t)w(·, tk), (13.luk)


then the evolution Galerkin scheme reads:

wk+1 = PhE∆wk, (14.luk)

where E∆ is an approximate evolution operator and Ph is a projection onto afinite element space. It can be shown that the method is unconditionally stableand the accuracy can be increased by increasing the order of the approximatespace and the accuracy of the approximate evolution operator. Using differentapproximate evolution operators E∆ and projections Ph one obtains a class ofthe evolution Galerkin methods.

The approach described can be fully exploited for simple problems, e.g. thelinear hyperbolic system of wave equation (see Lukacova, Morton and Warnecke[9], [10], [11]). More details about the application of this method to the Eulerequations can be found in the works of Fey [7] and Ostkamp [13].

2.3 Numerical experiments

(1) Flow through the GAMM channel (10 % circular arc in the channel of width1 m) for air, i. e. γ = 1.4, and inlet Mach number M := |v|

a = 0.67 was solvedby the Vijayasundaram higher order scheme applied on the dual mesh over atriangular grid. In Figure 2 the basic grid and dual mesh, respectively, are shown.Our aim was to obtain a steady state solution with the aid of the time marchingprocess for tk →∞. After 10000 time iterations the stability of the solution upto 10−5 was achieved. Figure 3 shows Mach number isolines and entropy isolines.We can see a sharp shock wave which is resolved very well.

Figure 2: Triangular mesh in the GAMM channel and the dual mesh

Figure 3: Mach number isolines and entropy isolines


(2) Two-dimensional Sod’s problem. Now we will show a comparison of thesolution obtained by the finite volume method and by the evolution Galerkinscheme. It will be showed that some symmetry structures are better preserved bythe truly multidimensional evolution Galerkin method than by the finite volumescheme.

The computational domain is the square [-1,1] × [-1,1]. To ensure the CFLstability condition, the CFL number is taken 0.8. We choose periodical boundaryconditions and the following initial data

ρ = 1, u = 0, v = 0, p = 1 if |x| ≤ 0.4 (15.luk)ρ = 0.125, u = 0, v = 0, p = 0.1 otherwise.

In Figure 4 the first picture on the left hand side shows the isolines of pressurefor the solution computed by the evolution Galerkin scheme at time T = 0.2 forquadrilateral grid with 200 × 200 grid cells. The symmetry of the data can beobserved very well. The resolution of the flow phenomena is the same in alldirections and information is moving in infinitely many directions in a circularmanner. However this is not the case for the finite volume method. In the nexttwo pictures of Figure 4 the isolines of pressure for the solution computed bythe Osher-Solemn finite volume scheme on the quadrilateral mesh (middle) andon the dual mesh (right hand side) are plotted.

Figure 4: Evolution Galerkin schemeand the Osher-Solomon FVM on the quadrilateral mesh and the dual mesh

3 Discretization of the complete system of theNavier-Stokes equations

In this section we will describe the combined finite volume – finite elementmethod which is used for the discretization of the Navier-Stokes equations (1.1).Let us note that we now use only the finite volume method in order to discretizethe Euler equations, however also other possibilities are open (cf. Section 2.2Evolution Galerkin methods).


First of all we will describe the finite element discretization of the purely vis-cous system (7.luk) equipped with initial conditions (1.4) and boundary conditions(1.5). We use conforming piecewise linear finite elements. This means that thecomponents of the state vector are approximated by functions from the finitedimensional space

Xh =ϕh ∈ C(Ωh); ϕh|T is linear for each T ∈ Th

.

Further, we set Xh = [Xh]4 and

a) V h = ϕh = (ϕ1, ϕ2, ϕ3, ϕ4) ∈Xh, ϕi = 0 on the part of ∂Ωh approxi-mating the part of ∂Ω where wi satisfies the Dirichlet condition

b) W h = wh ∈Xh; its components satisfy the Dirichlet boundaryconditions following from (1.5).

Multiplying (7.luk) considered on time level tk by any ϕh ∈ V h, integrating overΩh, using Green’s theorem, taking into account the boundary conditions (1.5)and approximating the time derivative by a forward finite difference, we obtainthe following explicit scheme for the calculation of an approximate solution wk+1

h

on the (k + 1)-th time level

a) wk+1h ∈W h, (16.luk)

b)∫Ωh

wk+1h ϕh dx =

∫Ωh

wkh ϕh dx−

−τk∫Ωh

2∑s=1

Rs(wkh, ∇wkh)∂ϕh∂xs

dx ∀ϕh ∈ V h.

The integrals are approximated by a numerical quadrature, called mass lump-ing, using the vertices of triangles as integration points:∫

T

F dx ≈ 13|T |

3∑i=1

F (P iT ) (17.luk)

for F ∈ C(T ) and a triangle T = T (P 1T , P

2T , P

3T ) ∈ Th with vertices P iT , i =

1, 2, 3. The numerical integration yields

wk+1i = wki −

τk|Di|

2∑s=1

∑T∈Th

|T |Rks∣∣∣T

∂ϕmi∂xs

∣∣∣T, (18.luk)

where i ∈ J, k = 0, 1, . . . , m = 1, . . . , 4, and ϕmi is a basis function from V h

having the only non-zero component on the m-th position; namely ϕi ∈ Xh,which corresponds to the vertex Pi.

Now we combine the finite volume scheme (11.luk) with the finite element scheme(18.luk). The resulting finite volume – finite element operator splitting scheme hasthe following form:


w0i =

1|Di|

∫Di

w(x, y, 0),

wk+1/2i = wki −

τk|Di|

∑j∈S(i)

g(wki , wkj , nij) `ij ,

wk+1i = w

k+1/2i − τk

|Di|

2∑s=1

∑T∈Th

|T |Rk+1/2s

∣∣∣T

∂ϕmi∂xs

∣∣∣T,

i ∈ J, m = 1, . . . , 4, k = 0, 1, . . . .

(19.luk)

The above scheme can be applied only under some stability conditions. Inthe case of explicit discretization of the viscous terms we have to consider notonly (12.luk) but also the additional stability condition in the form

34h

ρ

τk|T | max(µ, k) ≤ CFL, T ∈ Th, (20.luk)

where h is the length of the maximal side in Th and ρ = minT∈Th ρT , ρT =radius of the largest circle inscribed into T .

Concerning the theoretical results we are able to prove the convergence andthe error estimates for the combined finite volume – finite element method. Theseresults are obtained for one scalar nonlinear convection – diffusion equation.The convergence was proved by Feistauer, Felcman and Lukacova in [5] and byLukacova in [12]. Using the piecewise constant approximate functions in the finitevolume step and the piecewise linear approximation in the finite element stepit is possible to show that the method is of first order, see Feistauer, Felcman,Lukacova and Warnecke [6].

3.1 Computational Results

Viscous flow through the GAMM channel for γ = 1.4, µ = 1.72·10−5 kg m−1 s−1,λ = −1.15·10−5 kg m−1 s−1, k = 2.4·10−2 kg m s−3K−1, cv = 721.428 J ·kg·K−1

and the inlet Mach number M = 0.67 was computed by scheme (19.luk). In Figure 5Mach number isolines are drawn. Here we can see boundary layer at the walls,shock wave, wake and interaction of the shock with boundary layer.

Figure 5: Mach number isolines of viscous flow


Acknowledgements. This research was supported under the Grant No. 201/97/0153 of the Czech Grant Agency and the DFG Grant No. Wa 633/6-1 of DeutscheForschungsgemeinschaft. The numerical experiments for the EG schemes werecomputed with the code based on the original work of S. Ostkamp. The authorgratefully acknowledges these supports.

References

[1] P. Arminjou, A. Dervieux, L. Fezoui, H. Steve and B. Stoufflet, Non-oscillatoryschemes for multidimensional Euler calculations with unstructured grids, in:J. Ballmann, R. Jeltsch, Eds., Nonlinear Hyperbolic Equations-Theory, Computa-tion Methods, and Applications, Notes on Numerical Fluid Mechanics 24 (Vieweg,Braunschweig, 1989) 1–10.

[2] P. N. Childs and K. W. Morton, Characteristic Galerkin methods for scalar con-servation laws in one dimension. SIAM J. Numer. Anal., 27 (1990), 553–594.

[3] M. Feistauer, Mathematical Methods in Fluid Dynamics, Pitman Monographs andSurveys in Pure and Applied Mathematics 67 (Longman Scientific & Technical,Harlow, 1993).

[4] M. Feistauer, J. Felcman and M. Lukacova-Medvid’ova, Combined finite element –finite volume solution of compressible flow, Journal of Comput. and Appl. Math.,63 (1995), 179–199.

[5] M. Feistauer, J. Felcman, and M. Lukacova-Medvid’ova, On the convergence of acombined finite volume – finite element method for nonlinear convection - diffusionproblems, Num. Methods for Part. Diff. Eqs., 13 (1997), 1–28.

[6] M. Feistauer, J. Felcman, M. Lukacova-Medvid’ova and G. Warnecke, Error esti-mates of a combined finite volume - finite element method for nonlinear convection- diffusion problems, Preprint 27, (1996), University of Magdeburg, submitted toSIAM J. Numer. Anal.

[7] M. Fey, Ein echt mehrdimensionales Verfahren zur Losung der Eulergleichungen,Dissertation, ETH Zurich,1993.

[8] L. Fezoui and B. Stoufflet, A class of implicit schemes for Euler simulations withunstructured meshes, J. Comp. Phys, 84 (1989), 174–206.

[9] M. Lukacova-Medvid’ova, K. W. Morton, and G. Warnecke, The evolution Galer-kin schemes for hyperbolic systems in two space dimensions, in preparation.

[10] M. Lukacova-Medvid’ova, K. W. Morton, and G. Warnecke, The second orderevolution Galerkin schemes for hyperbolic systems, in preparation.

[11] M. Lukacova-Medvid’ova, K. W. Morton, and G. Warnecke, On the evolutionGalerkin method for solving multidimensional hyperbolic systems, to appear inProceedings of the Second European Conference on Numerical Mathematics andAdvanced Applications, ENUMATH’97, 1998.

[12] M. Lukacova-Medvid’ova, Combined finite element-finite volume method (conver-gence analysis), to appear in Comm. Math. Univ. Carolinae 1997.

[13] S. Ostkamp, Multidimensional characteristic Galerkin schemes and evolution op-erators for hyperbolic systems, Math. Meth. Appl. Sci., 20 (1997), 1111–1125.



Structure of Distribution Null-Solutions to

Fuchsian Partial Differential Equations

Takeshi Mandai

Faculty of Engineering, Gifu University,Yanagido 1-1, Gifu 501-11, JapanEmail: [email protected]

WWW: http://www.gifu-u.ac.jp/~mandai

Abstract. We give a structure theorem for distribution null-solutionsto Fuchsian partial differential equations in the sense of M. S. Baouendiand C. Goulaouic. We assume neither that the characteristic exponentsare real-analytic nor that the characteristic exponents do not differ byinteger.

AMS Subject Classification. 35D05, 35A07, 35C20

Keywords. Fuchsian partial differential operator, null-solutions, regu-lar singularity, characteristic exponent (index)

1 Introduction

Consider a Fuchsian partial differential operator with weight ω := m− k in thesense of M. S. Baouendi and C. Goulaouic ([1]) :

P = tk∂mt +k∑j=1

aj(x)tk−j∂m−jt +∑l<m

∑|α|≤m−l

bl,α(t, x)td(l)∂lt∂αx ,

0 ≤ k ≤ m, d(l) := max 0, l−m+ k + 1 , (t, x) ∈ R× Rn .

(1.man)

When m = k (ω = 0), M. Kashiwara and T. Oshima ([5], Definition 4.2) calledsuch an operator “an operator which has regular singularity in a weak sensealong Σ0 := t = 0 .”

In the following two categories (coefficients, data, solutions) :

(a) functions real-analytic in (t, x),(b) functions real-analytic in x and of class C∞ in t,

M. S. Baouendi and C. Goulaouic showed the following results:

A. The unique solvability of the characteristic Cauchy problems (Cauchy-Kova-levsky type theorem, Nagumo type theorem).


http://www.gifu-u.ac.jp/~mandai

212 Takeshi Mandai

B. The uniqueness in a wider class of solutions (Holmgren type theorem).

H. Tahara ([8], and so on) also showed similar results to A and B in thecategory of

(c) functions of class C∞ in (t, x),

for “Fuchsian hyperbolic operators”, which are, roughly speaking, operators be-ing weakly hyperbolic in t > 0 and satisfying “Levi conditions”.

In all cases, it easily follows that there exist no sufficiently smooth null-solutions. Here, a distribution u near (t, x) = (0, 0) is called a null-solution forP , if Pu = 0 near (0, 0) and if (0, 0) ∈ suppu ⊂ Σ+ := t ≥ 0 , where suppudenotes the support of u.

K. Igari ([4]) showed the existence of a distribution null-solution under a weakadditional condition in Case (a). This solution is real-analytic in x. The author([6]) showed the existence of a distribution null-solution under no additionalconditions in Case (a),(b),(c). This solution is also real-analytic (Case (a),(b))or of class C∞ (Case (c)) in x.

The aim of this study is to make the structure of all solutions belonging tothese classes as clear as possible. We consider the case (b) for simplicity. Namely,the coefficients of P are of class C∞ in t and real-analytic in x.

Many problems about Fuchsian partial differential equations have been con-sidered by many authors. Almost all of them, however, have some assumptionson the characteristic exponents (indices), especially the one that the character-istic exponents do not differ by integer. In this study, we assume neither thatthe characteristic exponents (indices) are real-analytic in x nor that the charac-teristic exponents do not differ by integer.

Notation:

(i) The set of all integers (resp. nonnegative integers) is denoted by Z (resp.N).

(ii) The real part of a complex number z is denoted by Rez.

(iii) Put ϑ := t∂t and (λ)l :=∏l−1j=0(λ− j) for l ∈ N.

(iv) For a domain Ω in Cn, we denote by O(Ω) the set of all holomorphicfunctions on Ω. For a complete locally convex topological vector space X ,we put O(Ω;X) := f ∈ C0(Ω;X) | 〈φ, f〉X ∈ O(Ω) for every φ ∈ X ′ ,where X ′ is the dual space of X and 〈·, ·〉X denotes the duality between X ′

and X . Note that if Ω is a domain in Cl and D is a domain in Rn, thenO(Ω;C∞(D)) = C∞(D;O(Ω)).

(v) The space of test functions on an open interval I of R is denoted by D(I)and the space of distributions by D′(I). The space of rapidly decreasingC∞ functions is denoted by S(R) and the space of tempered distributionsby S ′(R). The duality between each pair of these spaces is denoted by 〈·, ·〉.More generally, for a complete locally convex topological vector space X ,

Structure of Distribution Null-Solutions 213

the space of all X-valued distributions is denoted by D′(I;X), which is de-fines as L(D(I), X), where L(X,Y ) denotes the space of all continuous lin-ear mappings from X to Y (See [7]). Note that D′(I;O(Ω)) = O(Ω;D′(I)).Put D′+(I;X) := f ∈ D′(I;X) | f(t) = 0 in X for t < 0 . Also, forN ∈ N put

CN+ (I;X) : = f ∈ CN (I;X) | f(t) = 0 in X for t < 0 ,

C−N+ (I;X) : = ∂Nt (f) ∈ D′+(I;X) | f ∈ C0+(I;X) .

(vi) For z ∈ C with Rez > −1, we put

tz+ :=tz (t > 0)0 (t ≤ 0) ,

which is a locally integrable function of t with holomorphic parameter z,and hence belongs to D′+(R;O( z ∈ C | Rez > −1 )). By ∂t(tz+) = ztz−1

+ ,this distribution tz+ is extended to z ∈ C \ −1,−2, . . . meromorphicallywith simple poles at z = −1,−2, . . . ([3]).

(vii) For a commutative ring R, the ring of polynomials of λ with the coefficientsbelonging to R is denoted by R[λ]. The degree of F (λ) ∈ R[λ] is denotedby degλ F .

2 Review of some Results for Ordinary DifferentialEquations

In this section, we review some results for ordinary differential equations with aregular singularity at t = 0, which will help us to understand our result.

Consider

P = tk∂mt +k∑j=1

ajtk−j∂m−jt +

m∑l=0

bl(t)td(l)∂lt ,

where 0 ≤ k ≤ m, aj ∈ C, and bl ∈ C∞(−T0, T0). Namely, P is an operator witha regular singularity at t = 0 having C∞ coefficients.

Put

C[P ](λ) = C(λ) := t−λ+ωP (tλ)|t=0 = (λ)m +k∑j=1

aj(λ)m−j ∈ C[λ] ,

which is called the indicial polynomial of P . A root of C(λ) = 0 is called acharacteristic exponent (index) of P . We can decompose C as

C(λ) = (λ)ω C(λ − ω) ,

where C[P ](λ) = C(λ) := (λ)k +k∑j=1

aj(λ)k−j ∈ C[λ].

Let C(λ) =∏dl=1(λ−λl)rl , where d ∈ N, rl ≥ 1, and (λ1, . . . , λd) are distinct.

214 Takeshi Mandai

2.1 Formal Solutions

First, we consider the solutions in the space F of formal series of the form

u(t) = tρ∞∑j=0

tjqj∑ν=0

aj,ν(log t)ν ,

where ρ ∈ C, qj ∈ N, and aj,ν ∈ C. Note that we have only to consider tωPinstead of P in this space, since Pu = 0 if and only if tωPu = 0. Thus, weassume ω = 0 (k = m) without loss of generality.

Theorem 1. Assume k = m and put KerF P := u ∈ F | Pu = 0 . Forevery l with 1 ≤ l ≤ d and for every p with 1 ≤ p ≤ rl, there exists vl,p =

tλl(log t)p−1 +∞∑j=1

tλl+jqj∑ν=0

aj,ν(log t)ν ∈ KerF P , where aj,ν ∈ C. Further, these

m(= r1 + · · · + rd) solutions make a base of KerF P . Especially, there holdsdim KerF P = m.

Remark 2. (1) If the coefficients of P are holomorphic in a neighborhoodof 0, then this formal solution converges in O(R(B \ 0)) for some domain Bincluding 0, where R(V ) denotes the universal covering of V .(2) If rl = 1 for every l and if λl do not differ by integer, then we can takeqj = 0 for every j ∈ N, that is, the solutions never include the terms with log t.

2.2 Solutions in D′+Next, we consider the solutions of Pu = 0 in D′+(−T0, T0). In this case, we can

not reduce to the case where ω = 0, since Pu = 0 is not equivalent to tωPu = 0.Put

G(z) = G(z; t) :=tz+

Γ (z + 1)∈ D′+(R;O(C)) , (2.man)

G(j)(z) := ∂jz(G(z)) ∈ D′+(R;O(C)) . (3.man)

Note that ∂ht G(z) = G(z − h) (h ∈ N) and that G(−d) = ∂dt (G(0)) = δ(d−1)(t)for d = 1, 2, . . . .

Theorem 3. Put KerD′+ P := u ∈ D′+(−T0, T0) | Pu = 0 . For every lwith 1 ≤ l ≤ d and for every p with 1 ≤ p ≤ rl, there exists ul,p ∈ KerD′+ P

satisfying ul,p ∼ G(p−1)(λl + ω) +∞∑j=1

qj∑ν=0

aj,νG(ν)(λl + ω + j). Here, ∼ means

that for every N ∈ N, there holds ul,p−G(p−1)(λl+ω)−N∑j=1

qj∑ν=0

aj,νG(ν)(λl+ω+

j) ∈ CdReλle+ω+N+ (−T0, T0), where dae denotes the smallest integer M satisfying

M ≥ a ∈ R.


Further, these k (= r1+· · ·+rd) solutions make a base of KerD′+ P . Especially,there holds dim KerD′+ P = k. (Cf. Similarly, we have dim KerD′ P = m+ k.)

Example 4. (1) Consider P = (ϑ − d + 1)∂t = ∂t(ϑ − d), where d ∈ N andd ≥ 1. We have m = 2, k = 1, ω = 1, and C(λ) = λ(λ − d). We have KerF P =KerF (tP ) = Span1, td, KerD′+ P = Spantd+, KerD′ P = Spantd+, td, 1.(2) Consider P = (ϑ + d+ 1)∂t = ∂t(ϑ + d), where d ∈ N and d ≥ 1. We havem = 2, k = 1, ω = 1, and C(λ) = λ(λ + d). We have KerF P = KerF (tP ) =Span1, t−d, KerD′+ P = Spanδ(d−1), KerD′ P = Spanδ(d−1), 1, (t+ i0)−d.(3) Consider P = (ϑ − d + 1)2∂t = ∂t(ϑ − d)2, where d ∈ N and d ≥ 1.We have m = 3, k = 2, ω = 1, and C(λ) = λ(λ − d)2. We have KerF P =KerF (tP ) = Span1, td, td log t, KerD′+ P = Spantd+, td+ log t+, KerD′ P =Spantd+, td+ log t+, 1, td, td log(t+ i0).(4) Consider P = (ϑ + 1)∂t = ∂tϑ. We have m = 2, k = 1, ω = 1, and C(λ) =λ2. KerF P = KerF (tP ) = Span1, log t, KerD′+ P = Spant0+, KerD′ P =Spant0+, 1, log(t+ i0).

Remark 5. As in §2.1, if rl = 1 for every l and if λl do not differ by integer,then we can take qj = 0 for every j.

2.3 Rough Statement of our Result

We want to prove a similar fact for Fuchsian partial differential equations. Weshall show that

(O0)k ∼= (KerD′+ P )(0,0) , (4.man)

constructing the isomorphism (invertible linear map) rather concretely, where(. . . )0 and (. . . )(0,0) denote the spaces of all germs. Namely, O0 := indlim0∈Ω⊂CnO(Ω) and (KerD′+ P )(0,0) := indlimT>0;0∈Ω⊂Cn KerD′+(−T,T ;O(Ω)) P .

We shall state in a little more detail. Consider an operator (1.man) with aj ∈O(Ω0) and bl,α ∈ C∞(−T0, T0;O(Ω0)), where T0 > 0 and Ω0 is a domain in Cnincluding 0. Define

C(x;λ) := (λ)m +k∑j=1

aj(x)(λ)m−j = t−λ+ωP (tλ)|t=0 ,

which is also called the indicial polynomial of P , and a root λ of C(x;λ) = 0 iscalled a characteristic exponent of P . The indicial polynomial can be decomposedas

C(x;λ) = (λ)ω C(x;λ − ω) ,

where

C(x;λ) := (λ)k +k∑j=1

aj(x)(λ)k−j .

216 Takeshi Mandai

Let C(0;λ) =∏dl=1(λ− λl)rl , where d ∈ N, rl ≥ 1, and (λ1, . . . , λd) are distinct.

Rough Statement of our result is the following.

Theorem 6. There exist T > 0 and a subdomain Ω of Ω0 including 0 suchthat for every l with 1 ≤ l ≤ d and for every p with 1 ≤ p ≤ rl, there existsa continuous linear map ul,p from O(Ω0) to D′+(−T, T ;O(Ω)) satisfying thefollowing.

For every a ∈ O(Ω0), there holds P (ul,p[a]) = 0 and

ul,p[a]|x=0 ∼ a(0)G(p−1)(λl + ω) +∞∑h=1

qh∑ν=0

ah,νG(ν)(λl + ω + h)

for some qh ∈ N and ah,ν ∈ C.Conversely, every solution u ∈ D′+(−T0, T0;O(Ω0)) of Pu = 0 is represented

by∑

l,p ul,p[al,p] for some al,p ∈ O(Ω) (1 ≤ l ≤ d, 1 ≤ p ≤ rl) in a neighborhoodof (0, 0).

Further, if there exists (l, p) such that al,p 6≡ 0, then (0, 0) ∈ suppu, that is,u is a null-solution.

In the result by T.Mandai ([6]) stated in Introduction, he constructed asolution corresponding to the solution ul,0 for a root λl satisfying that C(0;λl +j) 6= 0 for j = 1, 2, . . . .

The major difficulty of the proof is the fact that a root of C(x;λ) = 0 is notnecessarily holomorphic in x, and this becomes a bigger difficulty when thereexist two roots with integer difference, as suggested from the case of ordinarydifferential equations.

3 Preliminaries

In this section, we give some lemmas and propositions needed for the proof ofour result.

For each 1 ≤ l ≤ d, we take a domain Dl in C enclosed by a simple closedcurve Γl such that the following three conditions hold.(a) λl ∈ Dl (1 ≤ l ≤ d).(b) Dl ∩Dl′ = ∅ if l 6= l′.(c) if j ∈ N and if λl′ − j ∈ Dl, then λl′ − j = λl.(This is equivalent to “ λl′ − j ∈ C | 1 ≤ l′ ≤ d, j ∈ N ∩Dl = λl for everyl”. Also these are equivalent to “C(0;λ + j) 6= 0 for every λ ∈

⋃dl=1(Dl \ λl )

and for every j ∈ N”. ) Note that λl′ − j ∈ C | 1 ≤ l′ ≤ d, j ∈ N is a discreteset and hence we can take such Γl.

There exist a domain Ω in Cn including 0, and monic polynomials El(x;λ) ∈O(Ω)[λ] (1 ≤ l ≤ d) such that(d) C(x;λ) =

∏dl=1 El(x;λ),

(e) El(0;λ) = (λ− λl)rl (1 ≤ l ≤ d),


(f) for 1 ≤ l ≤ d, if El(x;λ) = 0 and x ∈ Ω, then λ ∈ Dl,(g) C(x;λ + j) 6= 0 for every x ∈ Ω, every λ ∈

⋃dl=1 Γl, and every j ∈ N.

Further, by reducing Dl and Ω if necessary, we can take ε ≥ 0 and Ll ∈ Z(1 ≤ l ≤ d) such that(h) If C(x;λ) = 0 and if x ∈ Ω, then Reλ − ε 6∈ Z. Further, Dl ⊂ λ ∈ C |Ll + ε < Reλ < Ll + ε+ 1 .

Definition 7. For 1 ≤ l ≤ d, j ∈ N, and for φ ∈ O(Ω × Γl), put

Hl,j [φ](t, x) :=1

2πi

∫Γl

φ(x; ζ)El(x; ζ)

G(ζ + j; t) dζ ∈ D′+(R;O(Ω)) .

Also for 1 ≤ p ≤ rl, put

wl,p(t, x) :=(rl − p)!rl!

Hl,ω[∂pζEl](t, x) . (5.man)

Note that wl,p(t, 0) =1

(p− 1)!G(p−1)(λl + ω).

Remark 8. If we fix x = x0, then wl,p(t, x0) =∑

j,k:finite cj,kG(k)(µj + ω),

where cj,k ∈ C, and µj are the roots of C(x0;λ) = 0.

Proposition 9. (1) wl,p(·, x)l,p is a base of KerD′+ C(x;ϑ)∂ωt for every fixedx ∈ Ω.(2) If u ∈ D′+(R;O(Ω)) satisfies C(x;ϑ)∂ωt u = 0 for every x ∈ Ω, then u =∑

l,p al,p(x)wl,p(t, x) for some al,p ∈ O(Ω). The point is the holomorphy of al,p.

Example 10. Consider P = ϑ2 − x = E1(x;ϑ), where d = 1(= l), r1 = 2, andω = 0. We have

w1,1 =12H1,0[2ζ] =

12

12πi

∫Γ1

2ζζ2 − xG(ζ; t) dζ =

12G(√x; t) +G(−

√x; t),

w1,2 =12H1,0[2] =

12

12πi

∫Γ1

2ζ2 − xG(ζ; t) dζ =

G(√x; t)−G(−√x; t)

2√x

.

Proposition 11. (1) ∂ht Hl,j [φ] = Hl,j−h[φ].(2) thHl,j [φ] = Hl,j+h[(ζ + j + h)hφ].(3) For F (x;λ) ∈ O(Ω)[λ], there holds F (x;ϑ)Hl,j [φ] = Hl,j [F (x; ζ + j)φ].(4) ∂xνHl,j [φ] = Hl,j [Lν(φ)], where

Lν(φ)(x; ζ) := (∂xνφ)(x; ζ) − (∂xνEl)(x; ζ)El(x; ζ)

φ(x; ζ) .

Proposition 12. Hl,j [φ] ∈ Cj+Ll+ (R;O(Ω)).

218 Takeshi Mandai

For 1 ≤ l ≤ d and 1 ≤ p ≤ rl, we can construct an asymptotic solution ofPu = 0 in the form of

u = a(x)wl,p(t, x) +∞∑h=1

Hl,ω+h[Sh(a)](t, x) ,

where Sh = Sl,p,h is a continuous linear map from O(Ω) to O(Ω × Γl) of theform

Sh(a)(x; ζ) =∑|α|≤mh

sh,α(x; ζ)∂αx a(x) ,

where

sh,α = sl,p,h,α ∈1∏h

j=0 C(x; ζ + j)mh×O(Ω ×Dl) ,

for some mh ∈ N. Further, there exists qh ∈ N and ah,ν ∈ C (h ≥ 1; 0 ≤ ν ≤ qh)such that

u(t, 0) ∼ a(0)1

(p− 1)!G(p−1)(λl + ω) +

∞∑h=1

qh∑ν=0

ah,νG(ν)(λl + ω + h) .

4 Detailed Statement of our Result

Now, we can state our result in a full detail.Let Ω be a subdomain of Ω0 including 0 and T ∈ (0, T0).

Theorem 13. There exist T ′ ∈ (0, T ) and a subdomain Ω′ of Ω including0 such that for every l with 1 ≤ l ≤ d and for every p with 1 ≤ p ≤ rl,the following holds: There exists a continuous linear map ul,p from O(Ω) toCLl+ω+ (−T ′, T ′;O(Ω′)) such that for every a ∈ O(Ω), there holds

(i) P (ul,p[a]) = 0.(ii) ul,p[a](t, x) ∼ a(x)wl,p(t, x) +

∑∞h=1Hl,ω+h[Sh(a)](t, x),

Theorem 14. If u ∈ D′+(−T, T ;O(Ω)) satisfies Pu = 0, then there exists asubdomain Ω′ of Ω including 0 and there exists a unique al,p ∈ O(Ω′) such thatu =

∑l,p ul,p[al,p] in a neighborhood of (0, 0). Further, if there exists (l, p) such

that al,p 6≡ 0, then (0, 0) ∈ suppu, that is, u is a distribution null-solution forP .

These two theorems imply that (O0)k ∼= (KerD′+ P )(0,0), as we already stated.We can also show that (O0)m+k ∼= (KerD′ P )(0,0), similarly.

We can prove Theorem 13 by realizing the asymptotic solution constructedin the previous section.

We shall give a sketch of a proof of Theorem 14 in the next section.


5 Sketch of the Proof of Theorem 14

First, we give a sketch of the uniqueness of al,p.We have taken ε ≥ 0 such that if C(x;λ) = 0 (x ∈ Ω), then Reλ− ε 6∈ Z. We

have also taken Ll ∈ Z such that if x ∈ Ω and if El(x;λ) = 0, then Ll + ε <Reλ < Ll + ε+ 1.

Definition 15. For L ∈ Z, we put

W(N)L (−T, T ;X) :=

⊎Ns=0 ϑ

s∂|L|t (tε × C0

+(−T, T ;X)) (L ≤ 0)⊎Ns=0 ϑ

s(tL+ε × C0+(−T, T ;X)) (L ≥ 0)

.

Note that

W(N)L (−T, T ;O(Ω)) ⊂W (N)

L−1(−T, T ;O(Ω))

and

W(N)L (−T, T ;O(Ω)) ⊂W (N+1)

L (−T, T ;O(Ω)).

Take χ(t) ∈ D(−T, T ) with χ(t) = 1 near t = 0. Then, we have the following.

Hl,j [φ] ∈ W (0)Ll+j

, (6.man)

if v ∈W (N)L , then 〈v, χ(t)e−t/ρ〉t = o(ρL+ε+1) (ρ→ +0) . (7.man)

If we fix an arbitrary x, then there exists aj,k ∈ C such that

〈wl,p, χ(t)e−t/ρ〉t =∑j,k:finite aj,kρ

µj+ω+1(log ρ)k + o(ρ∞)

( = o(ρLl+ω+ε+1)) (ρ→ +0) ,(8.man)

where µj are the roots of C(x;λ) = 0, since 〈G(ν)(λ), e−t/ρ〉t = ρλ+1(log ρ)ν .From these, we can show that if

∑l,p ul,p[al,p] = 0, then al,p = 0 for every

(l, p).

Next, we give a sketch of the existence of al,p for a given solution u.

Proposition 16. (i) D′+(−T , T ;O(Ω)) ⊂⋃L∈ZW

(0)L (−T, T ;O(Ω)),

if T > T .(ii)

t×W (N)L (−T, T ;O(Ω)) ⊂

W

(N+1)L+1 (−T, T ;O(Ω)) (L ≤ −1)

W(N)L+1(−T, T ;O(Ω)) (L ≥ 0)

,

∂t(W(N)L (−T, T ;O(Ω))) ⊂

W

(N)L−1(−T, T ;O(Ω)) (L ≤ 0)

W(N+1)L−1 (−T, T ;O(Ω)) (L ≥ 1)

,

ϑ(W (N)L (−T, T ;O(Ω))) ⊂W (N+1)

L (−T, T ;O(Ω)).

220 Takeshi Mandai

(iii) For sufficiently large L, there holds

KerD′+(−T,T ;O(Ω)) P ∩W (N)L (−T, T ;O(Ω)) = 0

for every N ∈ N.(iv) For every g ∈ W (N)

L (−T, T ;O(Ω)), there exists v ∈ W (N)L+ω(−T, T ;O(Ω))

such that C(x;ϑ)∂ωt v = g.(v) For 1 ≤ l ≤ d and 1 ≤ p ≤ rl, there holds wl,p ∈ W

(0)Ll+ω

(−T, T ;O(Ω)).Further, for every L ∈ Z and every N ∈ N, there holds

KerD′+(−T,T ;O(Ω)) C(x;ϑ)∂ωt ∩W(N)L (−T, T ;O(Ω))

= Spanwl,p | 1 ≤ l ≤ d, Ll + ω ≥ L, 1 ≤ p ≤ rl .

From these, we can show the existence of al,p as follows.Let u ∈ D′+(−T, T ;O(Ω)) and Pu = 0. By (i) and by reducing T , there

exists L ∈ Z such that u ∈ W (0)L (−T, T ;O(Ω)). Putting P = C(x;ϑ)∂ωt + R, we

have C(x;ϑ)∂ωt u = −Ru ∈ W (m)L−ω+1(−T, T ;O(Ω)) by (ii). By (iv), there exists

v ∈ W(m)L+1(−T, T ;O(Ω)) such that u − v ∈ KerD′+(−T,T ;O(Ω)) C(x;ϑ)∂ωt . Since

u− v ∈ W (m)L (−T, T ;O(Ω)), there exists al,p[0] ∈ O(Ω) (1 ≤ l ≤ d, Ll + ω ≥ L,

1 ≤ p ≤ rl) such that

u− v =∑

1≤l≤d, Ll+ω≥L, 1≤p≤rl

al,p[0]wl,p ,

by (v). Put u[1] := u −∑

l,p ul,p[al,p[0]], then we have P (u[1]) = 0 and we can

show u[1] ∈ W (m)L+1(−T, T ;O(Ω)).

Similarly, we get u[2] := u[1] −∑l,p ul,p[al,p[1]] ∈ KerD′+(−T,T ;O(Ω)) P ∩

W(2m)L+2 (−T, T ;O(Ω)), for some al,p[1] ∈ O(Ω) by reducing Ω and T . By repeat-

ing this argument, we get u[N ] ∈ KerD′+(−T,T ;O(Ω)) P ∩ W (Nm)L+N (−T, T ;O(Ω))

that can be written as u[N ] = u −∑

l,p ul,p[al,p] for some al,p ∈ O(Ω). By (iii),we get u[N ] = 0, and hence u can be written as u =

∑l,p ul,p[al,p] for some

al,p ∈ O(Ω).

The research was supported in part by Grant-in-Aid for Scientific Research(Nos.07640191, 08640189), Ministry of Education, Science and Culture (Japan).

References

[1] M. S. Baouendi and C. Goulaouic, Cauchy problems with characteristic initial hy-persurface, Comm. Pure Appl. Math., 26 (1973), 455–475.

[2] A. Bove, J. E. Lewis, and C. Parenti, Structure properties of solutions of someFuchsian hyperbolic equations, Math. Ann., 273 (1986), 553–571.


[3] I. M. Gel’fand and G. E. Shilov, Generalized functions, Volume 1 : Properties andoperations, Academic Press, 1964, Transl. by E. Saletan.

[4] K. Igari, Non-unicite dans le probleme de Cauchy caracteristique — cas de type deFuchs, J. Math. Kyoto Univ., 25 (1985), 341–355.

[5] M. Kashiwara and T. Oshima, Systems of differential equations with regular singu-larities and their boundary value problems, Ann. of Math. (2), 106 (1977), 145–200.

[6] T. Mandai, Existence of distribution null-solutions for every Fuchsian partial dif-ferential operator, J. Math. Sci., Univ. Tokyo, 5 (1998), 1–18.

[7] L. Schwartz, Theorie des distributions a valeurs vectorielles, Ann. Inst. Fourier(Grenoble), 7 (1957), 1–141.

[8] H. Tahara, Singular hyperbolic systems, III. On the Cauchy problem for Fuchsianhyperbolic partial differential equations, J. Fac. Sci. Univ. Tokyo Sect. IA Math.,27 (1980), 465–507.



A Functional Differential Equation

in Banach Spaces

Nasr Mostafa

Faculty of Mathematics and Computer ScienceA. Mickiewicz University, Matejki 48/49

60–769 Poznan, PolandEmail: [email protected]

Permanent address: Department of MathematicsFaculty of Science, Suez Canal University

Ismailia, Egypt

Abstract. In this paper we prove the existence of pseudo-solution andweak solution for the Cauchy problem x′ = Fx, x(0) = x0, t ∈ [0, a].

AMS Subject Classification. 34G20

Keywords. Functional differential equation, existence theorem, weaksolution, pseudo-solution

The study of the Cauchy problem for differential and functional differentialequations in a Banach space relative to the strong topology has attracted muchattention in recent years. However a similar study relative to the weak topologywas studied by many authors, for example, Szep [11], Mitchell and Smith [9],Szufla [12], Kubiaczyk [6,7], Kubiaczyk and Szufla [8], Cichon [1], Cichon andKubiaczyk [2], and others.

Let E be a Banach space, E∗ the dual space. We set Bb(x0) = x ∈ E :‖x−xo‖ ≤ b, (b > 0). We denote by C(I, E) the space of all continuous functionfrom I to E, and by (C(I, E), w) the space C(I, E) with the weak topology. Put

B = x ∈ C(J,E) : x(J) ⊂ Bb(xo), ‖x(t) − x(s)‖ ≤M |t− s|, for t, s ∈ J ,

note that B is nonempty, closed, bounded, convex and equicontinuous, whereJ = [0, h], h = min

a, bM

and M > 0 is a constant.

We deal with the Cauchy problem:

x′ = Fx, x(0) = x0, t ∈ I = [0, a], (1.mos)

in the case of F being an bounded operator of Volterra type from B into P (I, E)(the space of all Pettis integrable functions on I).

Let us introduce the following definitions.


224 Nasr Mostafa

Definition 1. F is said to be of Volterra type if for x1, x2 ∈ B and for anyso > 0 the equality x1(t) = x2(t) for t < so implies (Fx1)(t) = (Fx2)(t) fort ≤ so.

Now fix x∗ ∈ E∗, and consider

(x∗x)′(t) = x∗((Fx)(t)), t ∈ I. (1′.mos)

Definition 2. A function x : I −→ E is said to be a pseudo-solution of theCauchy problem (1.mos) if it satisfies the following conditions:

(i) x(·) is absolutely continuous,(ii) x(0) = xo,

(iii) for each x∗ ∈ E∗ there exists a negligible set A(x∗) (i.e., mes (A(x∗)) = 0),such that for each t 6∈ A(x∗),

x∗(x′(t)) = x∗((Fx)(t)) .

Here ′ denotes a pseudoderivative (see Pettis [10]).

In other words, by a pseudo-solution of (1.mos) we will mean an absolutely continuousfunction x(·), with x(0) = xo, satisfying (1′.mos) a.e. for each x∗ ∈ E∗.

Definition 3. A function r : [0,∞) −→ [0,∞) is said to be a Kamke functionif it satisfies the following conditions:

(i) r(0) = 0,(ii) u(t) ≡ 0 is the unique solution of the integral equation

z(t) =∫ t

0

r(z(s))ds , t ∈ I .

Lemma 4 ([9]). Let H ⊂ C(I, E) be a family of strongly equicontinuous func-tions. Then

βc(H) = supt∈I

β(H(t)) = β(H(I)) ,

where βc(H) denote the measure of weak noncompactness in C(I, E) and thefunction t→ β(H(t)) is continuous.

Now suppose that:

(∗) For each strongly absolutely continuous function x : J :−→ E, (Fx)(·) isPettis integrable, F (·) is weakly-weakly sequentially continuous, then theexistence of a pseudo-solution of (1.mos) is equivalent to the existence of asolution for

x(t) = xo +∫ t

0

(Fx)(s)ds , (2.mos)

where the integral is in the sense of Pettis (see [10]).

A Functional Differential Equation in Banach Spaces 225

Theorem 5. Let F be a bounded continuous operator of Volterra type from Binto P (I, E) and under the assumption (∗) and

β(⋃(Fx)[J ] : x ∈ X

)≤ r(β(X)) , (3.mos)

holds for every subset X of B, where r is a non-decreasing Kamke function andβ is the measure of weak noncompactness. Then the set S of all pseudo-solutionsof the Cauchy problem (1.mos) on J is non-empty and compact in (C(J,E), w).

Proof. Put

Tu(t) = xo +∫ t

0

Fu(s)ds , t ∈ I, u ∈ B ,

where the integral is in the sense of Pettis.By our assumptions the operator T is well defined and maps B into B.Using Lebesgue’s dominated convergence theorem for the Pettis integral

(see [4]), we deduce that T is weakly sequentially continuous.Suppose that V = Conv(x ∪ T (V )) for some V ⊂ B. We will prove that V

is relatively weakly compact, thus Theorem 1 in [7] is satisfied.From the definition of B and Lemma 4 it follows that the function v : t →

β(V (t)) is continuous on J .For fixed t ∈ J , divide the interval [0, t) into m parts:

0 = to < t1 < · · · < tm = t, where ti = it/m , i = 0, 1, 2, . . . ,m .

Put

V ([ti−1, ti]) = u(s) = u ∈ V, ti−1 ≤ s ≤ ti .

By Lemma 4 and the continuity of v there is si ∈ [ti−1, ti] such that

β(V ([ti−1, ti])) = supβ(V (s)) : ti−1 ≤ s ≤ ti = v(si) . (4.mos)

On the other hand, by the mean value theorem we obtain

Tu(t) = xo +m−1∑i=0

∫ ti+1

ti

Fu(s)ds ∈ xo +m−1∑i=0

(ti+1 − ti)ConvFu([ti, ti+1])

for each u ∈ V . Therefore

TV (t) ⊂ xo +m−1∑i=0

(ti+1 − ti)ConvF ([V ])([ti, ti+1]) .

226 Nasr Mostafa

By (4.mos) and the corresponding properties of β it follows that

β(T (V )(t)) ≤ β(xo +m−1∑i=0

(ti+1 − ti)ConvF ([V ])([ti, ti+1])) ≤

≤m−1∑i=0

(ti+1 − ti)β(F (V )([ti, ti+1])) ≤

≤m−1∑i=0

(ti+1 − ti)r(β(V [ti, ti+1])) ≤

≤m−1∑i+0

(ti+1 − ti)r(β(V (si)) , for some si ∈ [ti, ti+1]

=m−1∑i=0

(ti+1 − ti)r(v(si)) .

By letting m→∞, we have

β(T (V (t)) ≤∫ t

0

r(v(s))ds . (5.mos)

Since V = Conv(x ∪ T (V )) we have β(V (t)) ≤ β(T (V (t))) and in view of (5.mos),it follows that v(t) ≤

∫ t0r(v(s))ds for t ∈ J .

Hence applying now a theorem on differential inequalities (cf. [5]) we getv(t) = β(v(t)) = 0.

By Lemma 4, V is relatively weakly compact.So, by Theorem 1 in [7], T has a fixed point in B which is actually a pseudo-

solution of (1.mos).As S = T (S), by repeating the above argument with V = S we can show

that S is relatively compact in (C(J,E), w).Since T is weakly sequentially continuous on S(J)

ω, S is weakly sequentially

closed. By Eberlein-Smulian Theorem [3], S is weakly compact.

Remark 6. One can easily prove that the integral of a weakly continuous func-tion is weakly differentiable with respect to the right endpoint of the integrationinterval and its derivative equals the integral at the same point (see [6], Lemma2.3). In this case a pseudo-solution is, actually, a weak solution. Moreover, insome classes of spaces our pseudo-solutions are also strong C-solutions (in sep-arable Banach spaces, for instance).

References

[1] Cichon, M., Weak solutions of differential equations in Banach spaces. Discuss.Math. — Diff. Inclus. 15 (1995), 5–14.

A Functional Differential Equation in Banach Spaces 227

[2] Cichon, M. and Kubiaczyk, I., On the set of solutions of the Cauchy problem inBanach spaces. Arch. Math. 63 (1994), 251–257.

[3] Edwards, R. E., Functional Analysis. Holt Rinehart and Winston, New York 1965.[4] Geitz, R. F., Pettis integration . Proc. Amer. Math. Soc. 82 (1981), 81–86.[5] Hartman, P., Ordinary Differential Equations. New York 1964.[6] Kubiaczyk, I., A functional differential equation in Banach spaces. Demonstratio

Math. 15 (1982), 113–129.[7] Kubiaczyk, I., On a fixed point theorem for weakly sequentially continuous map-

pings. Discuss. Math. — Diff. Inclus. 15 (1995), 15–20.[8] Kubiaczyk, I. and Szufla, S., Kneser’s theorem for weak solutions of ordinary

differential equations in Banach spaces. Publ. Inst. Math. 32 (1982), 99–103.[9] Mitchell, A. R. and Smith, C., An existence theorem for weak solutions of differen-

tial equations in Banach spaces. pp. 387–404 in, Nonlinear Equations in AbstractSpaces, ed. by V. Lakshmikantham 1978.

[10] Pettis, B. J., On integration in vector spaces. Trans. Amer. Math. Soc. 44 (1938),277–304.

[11] Szep, A., Existence theorem for weak solutions of ordinary differential equationsin reflexive Banach spaces. Studia Sci. Math. Hungar. 6 (1971), 197–203.

[12] Szufla, S., Kneser’s theorem for weak solutions of ordinary differential equationsin reflexive Banach spaces. Ibid. 26 (1978), 407–413.



On the Limit Cycle of the van der Pol Equation

Kenzi Odani

Department of Mathematics, Aichi University of Education,Igaya-cho, Kariya-shi, Aichi 448-8542, Japan.

Email: [email protected]: http://www.auemath.aichi-edu.ac.jp/

Abstract. In the paper, we estimate the amplitude (maximal x-value)of the limit cycle of the van der Pol equation

x = y − µ(x3/3− x), y = −x

from above by ρ(µ) < 2.3439 for every µ 6= 0. The result is an improve-ment of the author’s previous estimation ρ(µ) < 2.5425.

AMS Subject Classification. 34C05, 58F21

Keywords. Van der Pol equation, limit cycle, amplitude

1 Introduction

We are interested in the limit cycle (isolated periodic orbit) of the Lienard equa-tion:

x = y − F (x), y = −g(x). (L.oda)

The following is our result.

Theorem A. Suppose that Lienard equation satisfies the following conditions:(1) F, g are of class C1 and odd; (2) g(x) has the same sign as x; (3) F has apositive zero β such that F (x) < 0 on (0, β) and > 0 on (β,∞); (4) there are twopiecewise differentiable, continuous mappings φ, ψ : [0, β]→ [β,∞) such that

(i) −φ′(x)g(φ(x))F (φ(x)) ≥ −g(x)F (x), (ii) −φ′(x)f(φ(x)) ≥ −f(x),(iii) ψ′(x)g(ψ(x))F (ψ(x)) ≥ −g(x)F (x), (iv) ψ′(x)f(ψ(x)) ≥ f(x),(v) ψ′(x)g(ψ(x)) ≤ g(x), (vi) φ(0) ≤ ψ(β),

where f = F ′. Then it has a periodic orbit in the strip |x| < ψ(β).

The above theorem is effective to estimate the amplitude (maximal x-value) ofthe limit cycle of the van der Pol equation:

x = y − µ(x3/3− x), y = −x. (vdP.oda)

We know that the van der Pol equation has a unique limit cycle for every µ 6= 0;see [O] for example. The following is an application of Theorem A.


http://www.auemath.aichi-edu.ac.jp/

230 Kenzi Odani

Theorem B. The amplitude ρ(µ) of the limit cycle of the van der Pol equationis estimated by ρ(µ) < 2.3439 for every µ 6= 0.

The upper bound 2.3439 is better than previous results, namely, 2.8025 of Al-sholm [A] and 2.5425 of the author [O]. Due to a computer experiment, we expectthat the amplitude ρ(µ) < 2.0235 for every µ 6= 0. So Theorem B is not a sharpresult in comparison with it. We give the result of the experiment in Section 4.

2 Proof of Theorem A

We consider an orbit γ which starts from a point on the left half of the curvey = F (x) and reaches to the right half of it. Then we can regard the y-coordinateof γ as a function of x, that is, y = y(x). In the proof of Theorem A, we use thefollowing notation:

v1(x) = y(x)− F (x), v2(x) = y(−x) + F (x). (1.oda)

Then the functions v1, v2 must satisfy the following differential equations:

dv1

dx= −g(x)

v1− f(x),

dv2

dx= −g(x)

v2+ f(x). (2.oda)

By the definition of γ, we know that v1(x), v2(x) ≥ 0 on [0, ψ(β)].

Proof (of Theorem A). We assume that the orbit γ starts from the curve y =F (x) at x = −ψ(β), that is, v2(ψ(β)) = 0. We want to prove that the orbit γ getsacross the curve at the left-hand side of x = ψ(β). To prove it by a contradiction,we assume that v1(x) is defined on [0, ψ(β)].

By using (i), we know that φ′(x) < 0 on (0, β). So by using (ii), we calculateas follows:

d

dx

(v1(x) − v1(φ(x))

)= − g(x)

v1(x)+φ′(x)g(φ(x))v1(φ(x))

− f(x) + φ′(x)f(φ(x)) ≤ 0. (3.oda)

By integrating it on [x, β], we obtain that

v1(x)− v1(φ(x)) ≥ v1(β) − v1(φ(β)) = y(β)− y(φ(β)) + F (φ(β)) > 0 (4.oda)

because y(x) is strictly decreasing on [−φ(β), φ(β)].On the other hand, by using (iv), (v), we calculate as follows:

d

dx

(v2(x) − v2(ψ(x))

)= − g(x)

v2(x)+ψ′(x)g(ψ(x))v2(ψ(x))

+ f(x)− ψ′(x)f(ψ(x))

≤ g(x)v2(x)v2(ψ(x))

(v2(x)− v2(ψ(x))

). (5.oda)

On the Limit Cycle of the van der Pol Equation 231

By integrating it on [x, β], we obtain that

v2(x)− v2(ψ(x)) ≥(v2(β) − v2(ψ(β))

)exp(−∫ β

x

g(u)duv2(u)v2(ψ(u))

)> 0. (6.oda)

We can easily confirm the following equality:

d

dx

( 12y(x)2 +

∫ x

0

g(u)du)

= − g(x)F (x)y(x)− F (x)

. (7.oda)

By integrating it on [0, ψ(β)], we obtain that

12

(y(ψ(β))2 − y(−ψ(β))2

)= −

∫ ψ(β)

0

g(x)F (x)v1(x)

dx−∫ ψ(β)

0

g(x)F (x)v2(x)

dx. (8.oda)

By using (i), (4.oda), we calculate the first term of (8.oda) as follows:

≤ −∫ β

0

g(x)F (x)v1(x)

dx−∫ φ(0)

φ(β)

g(x)F (x)v1(x)

dx

= −∫ β

0

g(x)F (x)v1(x)

dx+∫ β

0

φ′(x)g(φ(x))F (φ(x))v1(φ(x))

dx < 0. (9.oda)

On the other hand, by using (iii), (6.oda), we calculate the second term of (8.oda) asfollows:

≤ −∫ β

0

g(x)F (x)v2(x)

dx−∫ ψ(β)

ψ(0)

g(x)F (x)v2(x)

dx

= −∫ β

0

g(x)F (x)v2(x)

dx−∫ β

0

ψ′(x)g(ψ(x))F (ψ(x))v2(ψ(x))

dx < 0. (10.oda)

By combining (8.oda), (9.oda), (10.oda), we obtain that

y(ψ(β))2 < y(−ψ(β))2 = F (ψ(β))2. (11.oda)

It is in contradiction with v1(ψ(β)) ≥ 0. So the function v1(x) does not definedon [0, ψ(β)], that is, the orbit γ gets across the curve y = F (x) at the left-handside of x = ψ(β). Thus the orbit γ winds toward inside. On the other hand, everyorbit near the origin winds toward outside. Hence the equation has a periodicorbit in the strip |x| < ψ(β). ut

3 Proof of Theorem B

In the proof of Theorem B, we use the following functions:

P (x) :=f(x)g(x)

= µ(x− 1

x

), Q(x) :=

f(x)g(x)F (x)

=3(x2 − 1)x4 − 3x2

. (12.oda)

By checking the derivatives, we know that the function P is strictly increasing on(0,∞) and that the function Q is strictly decreasing on (0,

√3) and on (

√3,∞).

232 Kenzi Odani

Proof (of Theorem B). We can assume without loss of generality that µ > 0because the transformation (x, y, t, µ) → (x,−y,−t,−µ) preserves the form ofthe equation. We first define φ(x) by the following algebraic equation:∫ φ

x

uF (u)du =µ

15(φ5 − 5φ3 − x5 + 5x3) = 0. (13.oda)

Of course, φ(√

3) =√

3. By differentiating it, we obtain that

− φ′(x)φ(x)F (φ(x)) + xF (x) = 0. (14.oda)

Since φ′(x) < 0 on [0,√

3], the mapping φ is strictly decreasing (orientationreversing) on it. Since the function −Q(φ(x)) + Q(x) is strictly decreasing on(0,√

3), it has a unique zero ξ1 in (0,√

3). A computer experiment indicates thatξ1 ≈ 0.6941, ξ2 := φ(ξ1) ≈ 2.2043. By substituting φ′(x) from (14.oda) and by thedefinition of ξ1, we obtain that

φ′(x)f(φ(x)) − f(x) = −xF (x)(−Q(φ(x)) +Q(x)

)≤ 0 (15.oda)

on [ξ1,√

3]. Since (15.oda) does not hold on [0, ξ1), the definition (13.oda) is valid onlyon [ξ1,

√3].

On the interval [0, ξ1), we define φ(x) by the following algebraic equation:∫ φ

ξ2

f(u)du+∫ ξ1

x

f(u)du

=µ

3(φ3 − 3φ− x3 + 3x− ξ3

2 + 3ξ2 + ξ31 − 3ξ1) = 0. (16.oda)

By differentiating it, we obtain that

φ′(x)f(φ(x)) − f(x) = 0 (17.oda)

on [0, ξ1). By substituting φ′(x) from (17.oda) and by the definition of ξ1, we obtainthat

−φ′(x)φ(x)F (φ(x)) + xF (x) = − xF (x)Q(φ(x))

(−Q(φ(x)) +Q(x)

)≥ 0 (18.oda)

on [0, ξ1). Hence the mapping φ satisfies (i), (ii) of Theorem A.We first define ψ(x) by the following algebraic equation:∫ ψ

θ2

uF (u)du+∫ x

θ1

uF (u)du =µ

15(ψ5 − 5ψ3 + x5 − 5x3 + 4

√6) = 0, (19.oda)

where θ1, θ2 :=√

2∓√

3 = (√

3∓ 1)/√

2. Of course, ψ(θ1) = θ2. By differenti-ating it, we obtain that

ψ′(x)ψ(x)F (ψ(x)) + xF (x) = 0. (20.oda)


Since ψ′(x) > 0 on [0,√

3], the mapping ψ is strictly increasing (orientationpreserving) on it. Since the function Q(ψ(x)) + Q(x) is strictly decreasing on(0,√

3), it has a unique zero η1 in (0,√

3). A computer experiment indicatesthat η1 ≈ 1.3784, η2 := ψ(η1) ≈ 2.2006. By substituting ψ′(x) from (20.oda) and bythe definition of η1, we obtain that

ψ′(x)f(ψ(x)) − f(x) = −xF (x)(Q(ψ(x)) +Q(x)

)≥ 0 (21.oda)

on [0, η1]. Since (21.oda) does not hold on (η1,√

3], the definition (19.oda) is valid onlyon [0, η1].

On the interval (η1,√

3], we define ψ(x) by the following algebraic equation:∫ ψ

η2

f(u)du−∫ x

η1

f(u)du

=µ

3(ψ3 − 3ψ − x3 + 3x− η3

2 + 3η2 + η31 − 3η1) = 0. (22.oda)

By differentiating it, we obtain that

ψ′(x)f(ψ(x)) − f(x) = 0 (23.oda)

on (η1,√

3]. By substituting ψ′(x) from (23.oda) and by the definition of η1, weobtain that

ψ′(x)ψ(x)F (ψ(x)) + xF (x) =xF (x)Q(ψ(x))

(Q(ψ(x)) +Q(x)

)≥ 0 (24.oda)

on (η1,√

3]. Hence the mapping ψ satisfies (iii), (iv) of Theorem A.To prove (v), we prepare the mapping χ(x) :=

√x2 + 2

√3 . By the proof of

Example 2 of [O], we obtain that

F (χ(x)) ≥ −F (x) (25.oda)

on [0,√

3]. By combining (20.oda) and (25.oda), we obtain that

χ′(x)χ(x)F (χ(x)) ≥ −xF (x) = ψ′(x)ψ(x)F (ψ(x)). (26.oda)

By integrating it on [x, θ1], we obtain that∫ ψ(x)

χ(x)

uF (u)du ≥ 0 (27.oda)

on [0, θ1]. Since uF (u) > 0 on (√

3,∞), we obtain that ψ(x) ≥ χ(x) on [0, θ1].So we obtain that

F (ψ(x)) ≥ F (χ(x)) ≥ −F (x) on [0, θ1]. (28.oda)

To prove the same inequality as (28.oda) on (θ1, η1], we consider the minimum ofthe function F (ψ) + F (x) under the restriction (19.oda). We denote by ψ0, x0 the

234 Kenzi Odani

variables which attain the minimum. To find the minimum, we consider thefollowing function:

Λ(ψ, x) = F (ψ) + F (x)− λ(∫ ψ

θ2

uF (u)du+∫ x

θ1

uF (u)du). (29.oda)

By the Lagrange’s method of indeterminate coefficients, we obtain that

Λψ.(ψ0, x0) = f(ψ0)− λψ0F (ψ0) = 0, (30.oda)Λx(ψ0, x0) = f(x0)− λx0F (x0) = 0. (31.oda)

By the first equality, we obtain that λ > 0. So we obtain that

F (ψ(x)) + F (x) ≥ F (ψ0) + F (x0) = (1/λ)(P (ψ0) + P (x0)

)≥ (1/λ)

(P (θ2) + P (θ1)

)= 0 (32.oda)

on (θ1, η1]. By substituting ψ′(x) from (20.oda) and by using (28.oda) and (32.oda), we obtainthat

x− ψ′(x)ψ(x) =x

F (ψ(x))

(F (ψ(x)) + F (x)

)≥ 0 (33.oda)

on [0, η1]. On the other hand, by substituting ψ′(x) from (23.oda), we obtain that

x− ψ′(x)ψ(x) =x

P (ψ(x))

(P (ψ(x)) − P (x)

)≥ 0 (34.oda)

on (η1,√

3]. Hence the mappings φ, ψ satisfy all the conditions of Theorem Aexcept (vi).

A computer experiment indicates that φ(0) ≈ 2.3439, ψ(√

3) ≈ 2.3233. Sowe must replace ψ by the following mapping:

ψ(x) :=√ψ(x)2 − ψ(β)2 + φ(0)2 . (35.oda)

Of course, ψ(β) = φ(0). Moreover, we can calculate as follows:

ψ′(x)ψ(x) = ψ′(x)ψ(x) ≤ x, (36.oda)

ψ′(x)ψ(x)F (ψ(x)) = ψ′(x)ψ(x)F (ψ(x))≥ ψ′(x)ψ(x)F (ψ(x)) ≥ −xF (x), (37.oda)

ψ′(x)f(ψ(x)) = ψ′(x)ψ(x)P (ψ(x)) ≥ ψ′(x)ψ(x)P (ψ(x))= ψ′(x)f(ψ(x)) ≥ f(x). (38.oda)

Hence the mappings φ, ψ satisfy all the conditions of Theorem A. ut


4 A Conjecture

Since the limit cycle of the van der Pol equation is unique, its amplitude ρ(µ)is a continuous function of the parameter µ 6= 0. In [L], the following facts areproved:

ρ(µ)→ 2 as µ→ 0, ρ(µ)→ 2 as µ→∞. (39.oda)

More precisely, it is proved in [H] that ρ(µ) = 2+(7/96)µ2+O(µ3) for sufficientlysmall µ > 0 and in [C] that ρ(µ) = 2+(0.7793 · · · )µ−4/3+o(µ−4/3) for sufficientlylarge µ > 0.

By a computer experiment, we have the following table.

µ ↓ 0 0.1 1.0 2.0 3.0 3.2ρ ↓ 2 2.00010 2.00862 2.01989 2.02330 2.02341µ 3.3 3.4 4.0 5.0 10 ↑ ∞ρ 2.02342 2.02341 2.02296 2.02151 2.01429 ↓ 2

We calculate the amplitude ρ of the above table by using the Runge-Kuttamethod with a step size 2−20. In comparison with the above table, we realizethat Theorem B is not a sharp result. So we want to pose the following conjecture.

Conjecture. The amplitude ρ(µ) of the limit cycle of the van der Pol equationis estimated by 2 < ρ(µ) < 2.0235 for every µ 6= 0.

However, to estimate the amplitude is a very difficult problem. An attempt toestimate the amplitude is done by Giacomini and Neukirch [GN].

Acknowledgement. The author wishes to thank Professors K. Shiraiwa andK. Yamato for reading the manuscript.

References

[A] P. Alsholm, Existence of limit cycles for generalized Lienard equation, J. Math.Anal. Appl. 171 (1992), 242–255.

[C] M. L. Cartwright, Van der Pol’s equation for relaxation oscillation, in “Contri-butions to the Theory of Non-linear Oscillations II”, S. Lefschetz, ed., Ann. ofMath. Studies, vol. 29, Princeton Univ. Press, 1952, pp. 3–18.

[GN] H. Giacomini and S. Neukirch, On the number of limit cycles of Lienard equa-tion, Physical Review E, to appear.

[H] W. T. van Horssen, A perturbation method based on integrating factors, SIAMJ. Appl. Math., to appear.

[L] S. Lefschetz, “Differential Equations: Geometric Theory”, 2nd Ed., Interscience,1963; reprint, Dover, New York, 1977.

[O] K. Odani, Existence of exactly N periodic solutions for Lienard systems, Funk-cialaj Ekvacioj 39 (1996), 217–234.

[Y] Y.-Q. Ye et al., “Theory of Limit Cycles”, Transl. of Math. Monographs, vol.66, Amer. Math. Soc., 1986. (Eng. transl.)

[Z] Z.-F. Zhang et al., “Qualitative Theory of Differential Equations”, Transl. ofMath. Monographs, vol. 102, Amer. Math. Soc., 1992. (Eng. transl.)



Lp Solutions of Non-linear Integral Equations

Alejandro Omon Arancibia1 and Manuel Pinto Jimenez2

1 Departamento de Ingenierıa Matematica, Universidad de Chile,Casilla 170, correo 3, Santiago, Chile

Email: [email protected] Departamento de Matematicas, Facultad de Ciencias, Universidad de Chile,

Casilla 653, Santiago, ChileEmail: [email protected]

Abstract. We study nonlinear Volterra integral operators of first andsecond kind on unbounded domains. We get bounded and Lp solutions onall [0,∞) as domain with Schauder’s fixed point theorem over unboundedsets.

AMS Subject Classification. 45B05, 45D05, 47H15

Keywords. Volterra integral equation of first and second kind, Schau-der’s fixed point theorem, Arzela-Ascoli and Frechet-Kolmogorov com-pactness theorems

1 Introduction

We wish to find solutions x(t) for the following nonlinear problems of Volterra’skind:

g(t) =

t∫0

F (t, s, x(s)) ds, t ≥ 0, (1.omo)

x(t) = x0(t) +

t∫0

k(t, s, x(s)) ds, t ≥ 0. (2.omo)

General existence results can be found in [2], [3] essentially using standard tech-niques of functional analysis. On first kind equation (1.omo), which is very difficultfor its implicit character, very few methods have been implemented on its study.

In this article for both equations we look for bounded and Lp solutions with1 ≤ p < +∞, when easily checkable conditions are imposed to functions g, F, x0

and k. The main technique is based in compactness method which do not en-sure uniqueness. The uniqueness problem in nonlinear integral equations is veryinteresting but also not too touched; in [2] and [3] there are some interesting re-sults. For the Lipschitz situation where the Banach contraction theorem is used,there are important results in [5] and [6].


238 Alejandro Omon and Manuel Pinto

In equation (2.omo) (of second kind) the results are not hard to extend to theFredholm integral equation of second kind, namely

x(t) = x0(t) +

∞∫0

k(t, s, x(s)) ds (2’.omo)

For our purpose we need a compactness criterion over not bounded subsets ofthe whole real axis which are given in the first and second lemmas, and are awell known generalization of the Arzela-Ascoli theorem and Frechet-Kolmogorovtheorem [1].

2 Bounded Solutions

Initially, consider the equation (1.omo). Assuming that g is differentiable and Fhas partial derivative with respect to the first variable (that will be denotedFt = ∂F

∂t ), then differentiating (1.omo) we obtain:

g′(t) = F (t, t, x(t)) +

t∫0

Ft(t, s, x(s)) ds. (3.omo)

Let us denote C the Banach space of continuous and bounded functions over[0,∞), normed by the supremum over all [0,∞). Now let us define the operatorT: C → C, such that given any x in C

Tx(t) = g(t) + Fx(t) + Kx(t), t ≥ 0, (4.omo)

where

g(t) = g′(t)− F (t, t, 0)−t∫

0

Ft(t, s, 0) ds,

F x(t) = x(t) + F (t, t, 0)− F (t, t, x(t)),

Kx(t) = −t∫

0

(Ft(t, s, x(s)) − Ft(t, s, 0)) ds.

With this definition any solution of equation (1.omo) satisfies the problem

Tx = x.

For this approach we will need the following definition and lemma:

Definition 1. Let f : [0,∞) × [0,∞) × Cn → Cn. We say that f(t, s, u) ist-locally equicontinuous with respect to s and u if

∀ε > 0 ∃δ > 0 s.t. |t1 − t2| < δ ⇒ |f(t1, s, u)− f(t2, s, u)| < ε,

uniformly on s over compact sets and u over bounded sets.

Lp Solutions of Non-linear Integral Equations 239

Lemma 2. Given A ⊂ C bounded, locally equicontinuous and equiconvergent,then A is relatively compact on C.

Theorem 3. Let F = F (t, s, u) : [0,∞) × [0,∞) × Cn → Cn be continuouson each variable, derivable with respect to the first one ; Ft continuous on eachvariable and t-locally equicontinuous with respect to s and u. Assume

I. a) The functional F is equicontinuous over any M ⊂ C bounded.b) sup

t∈[0,∞)

|F (t, t, 0)| < +∞.

c) There exists a : [0,∞) → R+ ∪ 0 bounded, continuous, such thata(t)→ 0 as t→∞ and verifying

|x+ F (t, t, 0)− F (t, t, x)| ≤ a(t)|x| ∀x ∈ Cn, ∀t ∈ [0,∞).

II. There exists a function L : [0,∞)× [0,∞)→ R+ ∪ 0 such that

a)

t∫0

L(t, s)ds is continuous, and it goes to zero as t →∞,

b) |Ft(t, s, u)− Ft(t, s, 0)| ≤ L(t, s)|u| ∀t, s ∈ [0,∞), ∀u ∈ Cn.

III. a) supt∈[0,∞)

t∫0

|Ft(t, s, 0)| ds <∞,

b) supt∈[0,∞)

[a(t) +

t∫0

L(t, s)ds] < 1.

IV. g′ ∈ C and g(0) = 0.

Then, there exists a solution x ∈ C of first kind equation (1.omo).

Proof. To prove the theorem we use a fixed point approach, showing first thatthe operator T , given by (4.omo) is well defined in C; let us take any x ∈ C, then byI.c) and II.b):

|Tx(t)| ≤ |g(t)|+ |F x(t)| + |Kx(t)|

≤ |g(t)|+ [a(t) +

t∫0

L(t, s) ds]‖x‖∞.

Then Tx(·) is bounded. Moreover, Tx(·) is continuous for all x ∈ C, indeed: g(·)

is continuous (g′, F , and

t∫0

|Ft(t, s, 0)|ds are continuous) ; Fx(·) is continuous


(x, F (·, ·, 0), F (·, ·, ·) are continuous) ; Kx(·) is continuous, because

|Kx(t)− Kx(t′)| ≤t∫

0

|Ft(t, s, x(s)) − Ft(t′, s, x(s))| ds ≤

t∫0

|Ft(t, s, 0)− Ft(t′, s, 0)| ds+

t∫t′

L(t′, s)ds‖x‖∞

and Ft is t-locally equicontinuous and

t∫t′

L(t′, s)ds−→t→t′

0. Then, Tx(·) is well

defined and continuous.Secondly, using Lemma 2 we prove that T is a compact operator. Let M ⊆ Cbounded, i.e., ∀x ∈M , ‖x‖∞ ≤ R <∞, we will show that:

i) T (M) is bounded in C ;ii) T (M) is locally equicontinuous in C ;

iii) T (M) is equiconvergent.

i) T (M) is bounded. As the functions g(.), a(.) and L(.,.) are bounded, givenx ∈M , we use I.b), I.c), II.b) and III.b), and we get that:

|Tx(t)| ≤ |g(t)|+ |F x(t)|+ |Kx(t)| ≤

|g(t)|+ [a(t) +

t∫0

L(t, s) ds]R ≤ |g(t)|+R.

ii) T (M) is equicontinuous. First g is continuous over [0,∞) because g’, F (., ., 0)and Ft(., s, 0) are continuous on [0,∞). Moreover, F x is equicontinuous onbounded subsets of C. With these considerations, given [a, b] compact in[0,∞) and t1 ≤ t2 on [a, b], then the equicontinuity of T (M) follows easilyfrom

|Tx(t1)− Tx(t2)| ≤ |g(t1)− g(t2)|+ |F x(t1)− F x(t2)|+t2∫t1

|Ft(t2, s, 0)| ds

+

t1∫0

|Ft(t1, s, x(s))− Ft(t2, s, x(s))| ds

+

t2∫t1

|Ft(t2, s, x(s))− Ft(t2, s, 0)| ds

≤ |g(t1)− g(t2)|+ |F x(t1)− F x(t2)|+t2∫t1

|Ft(t2, s, 0)| ds


+

t1∫0

|Ft(t1, s, x(s))− Ft(t2, s, x(s))| ds +R

t2∫t1

L(t2, s) ds.

iii) T (M) is equiconvergent, because

|Tx(t)− g(t)| ≤ |F x(t)|+ |Kx(t)| ≤ Ra(t) +

t∫0

L(t, s) ds −→t→∞

0.

So, by Lemma 2, T (M) is relatively compact in C. From Schauder’s fixed pointtheorem ∃x ∈ C such that x = T x. Integrating this last fixed point equation andas g(0) = 0, implies that x is a solution of equation (1.omo). ut

Now, consider the equation of second kind:

x(t) = x0(t) +

t∫0

k(t, s, x(s)) ds, t ≥ 0, (2.omod)

where k : [0,∞)× [0,∞)×Cn → Cn is continuous in s and x. We can formulatenow the following

Theorem 4. Assume that the function k : [0,∞)× [0,∞)×Cn → Cn is t-locallyequicontinuous and satisfies

I. supt∈[0,∞)

t∫0

|k(t, s, 0)|ds <∞.

II. There exists L : [0,∞)× [0,∞)→ R+ ∪ 0 such that:a) ∀(t, s, u) ∈ [0,∞)× [0,∞)× Cn with t ≥ s,

|k(t, s, u)− k(t, s, 0)| ≤ L(t, s)|u|.

b)

t∫0

L(t, s)ds is continuous and converges to 0 as t→∞.

c) supt∈[0,∞)

t∫0

L(t, s)ds < 1.

Then for all x0 ∈ C, equation (2.omod) has a solution x ∈ C.

Proof. Consider the operator T : C → C defined as

Tx(t) = x0(t) +

t∫0

k(t, s, 0)ds+

t∫0

(k(t, s, x(s))− k(t, s, 0)) ds.

Proceeding as in the previous Theorem, from Schauder’s fixed point theoremwe obtain that there exists x ∈ C such that x = T x, and then a solution ofequation (2.omod). ut


3 Lp Solutions

Now, we will find Lp[0,∞) solutions to equation (2.omod) with 1 ≤ p < ∞. For thewhole section, q will be the Holder conjugate of p, i.e., 1

p + 1q = 1. We will need

the next definition and lemma for our result over Lp spaces.

Definition 5. The function k(t, s, u) is t-locally Lp equicontinuous if givena, b, c, d ∈ R+, such that a ≤ b and c ≤ d, then

b∫a

d∫c

|k(t+ h, s, x(s))− k(t, s, x(s))|q dsdt−→h→0

0,

with x on bounded subsets of Lp[0,∞).

Lemma 6. A ⊂ Lp[0,∞) bounded will be relatively compact if:

a) The restriction A|[a,b] where [a, b] ⊂ [0,∞) is a compact interval, satisfy theLp-equicontinuity of the translations (Frechet-Kolmogorov criterion).

b) Equiconvergence: there exists u in Lp[0,∞) such that

∞∫t

|x(s)−u(s)|p ds→ 0

as t →∞ uniformly for x ∈ A.

Now, our next result is

Theorem 7. Assuming k is t-locally Lp equicontinuous, and

I. There exist a function L : [0,∞)× [0,∞)→ R+ ∪ 0 such that:

a)

∞∫0

( ∞∫0

Lq(t, s) ds) pq

dt <∞,

b) supt∈[0,∞)

∞∫0

( t∫0

Lq(t, s)ds) pq

dt < 1,

c) ∀(t, s, u) ∈ [0,∞)× [0,∞)× Cn with t ≥ s,

|k(t, s, u)− k(t, s, 0)| ≤ L(t, s)|u|.

II.

t∫0

|k(t, s, 0)| ds ∈ Lp[0,∞).

Then, given x0 ∈ Lp[0,∞), equation (2.omod) has a solution on Lp[0,∞).


Proof. Let T : Lp[0,∞)→ Lp[0,∞) defined by

Tx(t) = x0(t) +

t∫0

(k(t, s, x(s)) − k(t, s, 0)) ds,

where

x0(t) = x0(t) +

t∫0

k(t, s, 0) ds.

Clearly x0 is in Lp, and there exists a constant c ≥ 0, such that

|Tx(t)|p ≤ c|x0(t)|p +

( t∫0

|k(t, s, x(s))− k(t, s, 0)|ds)p

and using hypothesis I.b) and Holder’s inequality we get that

|Tx(t)|p ≤ c|x0(t)|p +

( t∫0

Lq(t, s) ds) pq ‖x‖pLp

.

Then by hypothesis I.a) and II.), we have that Tx ∈ Lp[0,∞). Now we want tosee that T is a compact operator from Lp[0,∞) to Lp[0,∞). To this end, we useLemma 6. Let us take M ⊆ Lp[0,∞) bounded, i.e., ∀x ∈ M , ‖x‖Lp ≤ R < ∞,then we must prove that T (M) is relatively compact.

First, by I.a), I.b), II. and the last inequality T (M) is bounded. Moreover,we have

a) Equicontinuity in the translations. Given [a, b] ⊂ [0,∞) compact, there existsa constant c ≥ 0 such that

|Tx(t+ h)− Tx(t)|p ≤ c|x0(t+ h)− x0(t)|p +

[ t+h∫t

|k(t+ h, s, x(s))− k(t+ h, s, 0)| ds]p

+[ t∫

0

|k(t+ h, s, x(s))− k(t, s, x(s))| ds]p

≤ c|x0(t+ h)− x0(t)|p +

( t+h∫t

Lq(t+ h, s) ds) pq ‖x‖pLp

+[ t∫

0

|k(t+ h, s, x(s))− k(t, s, x(s))| ds]p

.


Then, from Holder’s inequality, we have

b∫a

|Tx(t+ h)− Tx(t)|pdt ≤ c b∫a

|x0(t+ h)− x0(t)|p dt+

Rpb∫a

( t+h∫t

Lq(t+ h, s)ds) pq

dt

+ bpb∫a

b∫0

|k(t+ h, s, x(s))− k(t, s, x(s))|q dsdt.

and then, due to x0 ∈ Lp, the t-equicontinuity of k, and the integrability ofL(., .), we obtain the equicontinuity in the translations.

b) Finally, the Lp-equiconvergence follows from I.a) and I.b) because:

|Tx(t)− x0(t)|p ≤( t∫

0

Lq(t, s)ds) pq( t∫

0

|x(t)|p dt)

≤ Rp( t∫

0

Lq(t, s) ds) pq

and∞∫t

|Tx(t)− x0(t)|p dt ≤ Rp∞∫t

( t∫0

Lq(t, s) ds) pq

dt.

Thus, T is a compact operator from Lp[0,∞) to Lp[0,∞) and Schauder’sfixed point theorem implies there exist x ∈ Lp[0,∞) satisfying T x = x, andthen x is an Lp solution of equation (2.omod). ut

As an example of the first theorem consider a function F (., ., .) as follows:F (t, s, u) =

(1 + (t− s)

)/4e−tu+ f(t, s), such that

supt∈[0,∞)

|f(t, t)| <∞, supt∈[0,∞)

t∫0

∣∣∣∂f∂t

(t, s)∣∣∣ds <∞.

The function a = 0 satisfies I.c). Moreover, conditions II.a) and II.b) are fulfilledsince

|Ft(t, s, x)− Ft(t, s, 0)| ≤ |1 + (t− s)|/4e−t|x|,and

t∫0

|1 + (t− s)|/4e−t ds→ 0


as t → ∞. Then for any function g that satisfies g(0) and g′ in C, theorem 3implies that the equation

g(t) =

t∫0

((1 + (t− s))/4e−tx(s) + f(t, s)

)ds

has a continuous and bounded solution.

Acknowledgement. Fondecyt 1980835Catedra Presidencial , res. 031 , 21-09-1996

References

[1] Brezis, H., Analyse Fonctionnelle, Masson, 1993 (5e tirage)[2] Corduneanu, C., Integral Equations and Applications, Cambridge Univ. Press, 1991[3] Krasnoselskii, M. A., Topological Methods in the Theory of Nonlinear Equations,

Pergamon Press, 1964[4] Rejto, P., Taboada, M., Unique solvability of nonlinear Volterra equations in

weighted spaces, J. Math. Anal. Appl. 167 (1992), 368–381[5] Rejto, P., Taboada, M., Weighted resolvent estimates for Volterra operators on

unbounded intervals, J. Math. Anal. Appl. 160 (1991), 223–235[6] Willet, D., Nonlinear vector integral equations as contraction mappings, Arch. Ra-

tional Mech. Anal. 15, No 1 (1964), 79–86



Rothe’s Method for Degenerate Quasilinear

Parabolic Equations

Volker Pluschke

Department of Mathematics and Computer Science,Martin-Luther-University Halle-Wittenberg, P.O.Box,

06 099 Halle, GermanyEmail: [email protected]

WWW: http://www.mathematik.uni-halle.de/~analysis/pluschke

Abstract. In the contribution we state local existence of a weak solutionu to a degenerate quasilinear parabolic Dirichlet problem. Degenerationoccurs in the coefficient g(x, t, u) ≥ 0 in front of the time derivative,which is not assumed to be bounded below and above, resp., by positiveconstants. The nonlinear coefficients and the right-hand side are definedwith respect to u only in a neighbourhood of the initial function.

The quasilinear parabolic problem is approximated by linear ellipticproblems by means of semidiscretization in time (Rothe’s method). Weobtain L∞-estimates for the approximations and uniform convergence toa Holder continuous weak solution. An essential tool for this are esti-mates of the first order derivatives uniformly for all subdivisions in thespace L∞([0, T ], Lν(G)) with certain ν > 2.

AMS Subject Classification. 35K65, 65M20, 35K20

Keywords. Degenerate equations, Rothe’s method, L∞-estimates

1 Introduction

In this contribution we formulate a local existence result for the parabolic initialboundary value problem

g(x, t, u)ut +A(t, u)u = f(x, t, u) in QT , (1.plu)u(x, t) = 0 on Γ, (2.plu)u(x, 0) = U0(x) x ∈ G, (3.plu)

where

A(t, v)u = −N∑

i,k=1

∂

∂xk

(aik(x, t, v)

∂u

∂xi

)+

N∑i=1

ai(x, t, v)∂u

∂xi, (4.plu)

The paper is an overview article summarizing the results of [8] and [9].

http://www.mathematik.uni-halle.de/~analysis/pluschke

248 Volker Pluschke

which we obtain by means of semidiscretization in time (Rothe’s method). Herewe denote by G ⊂ RN , N ≥ 2, a simply connected, bounded domain with bound-ary ∂G ∈ C1, I = [0, T ], QT = G× I, Γ = ∂G× I.

In the following we give an overview on the results of the author’s papers [8]and [9]. In [8] the non-degenerated quasilinear problem (1.plu)–(3.plu) is investigatedwhere g ≡ 1. The aim of the paper [9] is to deal with the case where the coefficientg(·, t, u) may degenerate (i.e. g = 0 or g = ∞) on some sets St,u ⊂ G withmeas(St,u) = 0 (there A = A(t) is linear).

In both papers it is supposed that the nonlinear coefficients and the right-hand side f are defined only in a neighbourhood

MR(U0) = (x, t, u) ∈ RN+2 : x ∈ G, t ∈ I, |u − U0(x)| ≤ R

of the initial function for some given R > 0. In order to have convergence of theRothe approximations to a solution we have to ensure that the approximationsbelong to the ball

BR(U0) = u ∈ C(G) : ‖u− U0‖C(G) ≤ R.

This holds for small time t ∈ I = [0, T ] due to L∞-estimates which are derivedby means of the technique of Moser [7] combined with recursive estimates due toAlikakos [1]. Because of the nonlinear coefficient g we only can apply this tech-nique to the semidiscrete problem if we have uniform boundedness of the discretetime derivative δuj in L∞(I, Lν(G)) with sufficiently large ν > 2. In standardliterature on Rothe’s method (cf. Kacur [2], Chapter 2) such an estimate of δujis derived under the assumption of monotonicity of the nonlinear operator A.Moreover, one obtains this estimate for ν = 2 only. Without assumption of fullmonotonicity and with nonlinear coefficients in general one only obtains an es-timate in L2(I, L2(G)) (cf. e.g. Kacur [4], Lemma 2.7, where a similar problemwith nonlinear coefficient of ut is treated). We use Lp-theory with p > 2, power-type test functions, interpolation arguments, and nonlinear Gronwall lemma toderive the desired a priori estimate. Moreover, degeneration forces to work inweighted Lebesgue spaces.

These strong a priori estimates also yield strong convergence results for theapproximates despite of weak regularity of the data (Lebesgue data). We obtainuniform convergence of the Rothe functions in Holder space with respect to spaceand time variables.

2 Preliminaries and result

We use standard notations of the function and evolution spaces, resp. (cf. [5]).By ‖ · ‖p, ‖ · ‖1,p, and ‖ · ‖0,λ we denote the norms in Lp(G), W 1,p

0 (G), andCλ(G), respectively. Lp,g(G) denotes the weighted Lebesgue space with norm‖u‖p,g = (

∫G g|u|p dx)1/p for nonnegative g ∈ L1(G). 〈·, ·〉 is the duality between

Lp(G) and Lp′(G) (1/p+ 1/p′ = 1). For t ∈ I and v ∈ C(G) the operator A(t, v)from (4.plu) generates a bilinear form on W 1,p

0 (G)×W 1,p′

0 (G) denoted by A(t,v)(·, ·).

Degenerate Parabolic Equations 249

First we formulate the complete assumptions which we fix throughout thepaper.

Assumptions. For given R > 0 let g, aik ai, and f be Caratheodory functionsdefined on MR(U0). Let further r1, r2, r3, µ1, µ2, µ3, µ4, ν1, ν, σ, κ be realnumbers fulfilling the relations 2 ≤ κ < ∞, r1 > N , r2 > N(κ−1)

2κ−N , r3 > N2 ,

N2 < µ1 ≤ ν1 = σ

σ+1κ, µi ≤ ν < NκN−2 (i = 2, 3, 4), Nκ

κ−2 < µ2, Nκ2κ−2 < µ3,

Nκ2κ+N−2 < µ4, σ > 1.

Then we suppose for arbitrary t, t′ ∈ I and u, u′ ∈ BR(U0)

(i) U0 ∈o

W1r1(G) , A(0, U0)U0 ∈ L1(G);

(ii) g(·, t, u) : I × BR(U0) → Lr2(G) is bounded in Lr2(G) and fulfils the Lip-schitz condition‖g(·, t, u)− g(·, t′, u′)‖µ1 ≤ l1 (|t− t′|+ ‖u− u′‖ν1).

Furthermore, g(x, t, u) ≥ 0 for all (x, t, u) ∈MR(U0) and1/g(·, t, u) : I × BR(U0)→ Lσ(G) is bounded in Lσ(G).

(iii) aik(·, t, u) : I × BR(U0) → C(G) and ai(·, t, u) : I × BR(U0) → L∞(G) arebounded mappings which fulfil the Lipschitz conditions‖aik(·, t, u)− aik(·, t′, u′)‖µ2 ≤ l2 (|t− t′|+ ‖u− u′‖ν)‖ai(·, t, u)− ai(·, t′, u′)‖µ3 ≤ l3 (|t− t′|+ ‖u− u′‖ν)

as well as ellipticity condition (a > 0)∑i,k aik(x, t, v) ξiξk ≥ a ξ2 for all (x, t, v) ∈MR(U0) and ξ ∈ RN .

(iv) f(·, t, u) : I × BR(U0) → Lr3(G) is bounded in Lr3(G) and fulfils the Lip-schitz condition‖f(·, t, u)− f(·, t′, u′)‖µ4 ≤ l4 (|t− t′|+ ‖u− u′‖ν).

(v) It holds the compatibility condition(f(·, 0, U0)−A(0, U0)U0

)/g(·, 0, U0) ∈ Lκ,g(·,0,U0)(G) .

We remark that the coefficient g may not only decay to zero on some sets(this decay is governed by the assumption 1/g ∈ L∞(I, Lσ(G)) but it also mayhave singularities because it belongs to the Lebegue space L∞(I, Lr2(G)). Thisis equivalent to some degeneration of ellipticity of the operator A.

In order to solve the problem by semidiscretization in time (Rothe’s method)we subdivide the time interval I by points tj = jh (h > 0, j = 0, . . . , n) andreplace (1.plu)–(3.plu) by the time discretized problem (in weak formulation)

〈gj δuj, v〉+Aj(uj, v) = 〈fj , v〉 ∀v ∈o

W1p′(G) , (1j .plu)

uj = 0 on ∂G , (2j .plu)

u0 = U0 , (30.plu)

250 Volker Pluschke

j = 1, . . . , n, where δuj := (uj −uj−1)/h, gj := g(x, tj , uj−1), fj := f(x, tj , uj−1),and Aj(·, ·) := A(tj ,uj−1)(·, ·). This is a set of linear elliptic boundary value prob-lems to determine the approximate uj if uj−1 is already known. However, we donot know if uj ∈ BR(U0), i.e. whether the data of (1j+1), (2j+1) are well-definedbecause of the local assumptions. Therefore we define global extensions

ψR(x, t, u) =

ψ(x, t, u) for (x, t, u) ∈ MR(U0)ψ(x, t, U0(x) +R sign (u − U0(x)) otherwise

and replace g, aik, ai, and f by gR, aRik, aRi , and fR, respectively. By Lemma 3we state that uj ∈ BR(U0) for tj ≤ T , hence we omit the superscript R.

For sufficiently small fixed h now we can solve these elliptic boundary valueproblems applying the Lax-Milgram theorem (after an interpolation procedureto deal with the weighted norm with weight gj) and a regularity theorem (cf. [6,Theorem 5.5.4’]).

Lemma 1. Let assumptions (i)–(iv) be fulfilled. Then there are h0 > 0, r > Nsuch that for 0 < h ≤ h0 the problem (1j .plu), (2j .plu), (30.plu) has a unique solution

uj ∈o

W1r(G), j = 1, . . . , n.

Especially, the embedding theorem implies continuity of uj .By interpolation with respect to time we obtain the Rothe functions

un(x, t) =tj − th

uj−1(x) +t− tj−1

huj(x) , t ∈ [tj−1, tj ]

and

un(x, t) =

uj(x) if t ∈ (tj−1, tj ],U0(x) if t ≤ 0.

Our result is the following

Theorem 2. Suppose assumptions (i)–(v). Then the following assertions hold:

a) There is an interval I = [0, T ] such that problem (1.plu)–(3.plu) has a unique weak so-

lution u with u(·, t) ∈ BR(U0) for any t ∈ I fulfilling for all v ∈ L1(I ,o

W1r′(G)∩

L%′(G)) (%′ = r′2κ′) the relation∫

I

〈g(·, t, u)ut , v〉 dt+∫I

A(t,u)(u , v) dt =∫I

〈f(·, t, u) , v〉 dt

and initial condition (3.plu).b) The solution u belongs to the spaces

u ∈ Cα(QT ) ∩ L∞(I ,o

W1r(G)) ∩W 1

∞(I , Lν1(G))

for some r > N and α > 0. Moreover, ut ∈ Lκ(I , Ls(G)) for s < NκN−2 and

g(·, ·, u)ut ∈ L∞(I , L%(G)) with % = r2κr2+κ−1 .


c) The solution u is pointwise uniformly approximated by the Rothe functionsun with further convergence properties

un −→ u in Cα(QT )

un, un −→ u in L∞(I , Cλ(G)) (λ < 1−N/r)

un, un−∗ u in L∞(I ,o

W1r(G))

unt −∗ ut in L∞(I , Lν1(G))

as n tends to infinity.d) It holds an error estimate

supt∈I‖un(·, t)− u(·, t)‖ν1 ≤ c h 1/2

n .

The r > N may be explicitly given in terms of the Lebesgue exponents fromthe assumptions. Furthermore, because of uniform boundedness of the approxi-mations in L∞(I ,

o

W1r(G)) an interpolation inequality yields an error estimate in

Holder space, too,

supt∈I‖un(·, t)− u(·, t)‖0,λ ≤ c h (1−λ−N/r)/2

n , 0 < λ < 1−N/r .

3 A priori estimates for the approximations

In this final section we sketch some steps of the proof of Theorem 2. For thedetails compare [8] and [9].

In order to prove uj ∈ BR(U0) we have to estimate zj := uj − U0 in L∞(G).Therefore, we rewrite (1j .plu) into

〈gj δzj , v〉+Aj(zj , v) = 〈fj , v〉 −Aj(U0, v) ∀v ∈o

W1r′(G). (5.plu)

We use the Moser iteration technique [7]. The idea of this technique is toestimate ‖z‖p for arbitrary p ≥ p0 and then pass with p to infinity. Since ‖z‖p→‖z‖∞ (cf. [2, Theorem 2.11.4]) one obtains an estimate in L∞(G). In our case,because of degeneration, we have to work with the weighted norm ‖z‖p,g. Butwe have the same property ‖z‖p,g → ‖z‖∞ as p → ∞, i.e. the influence of thedegeneration vanishes in the limit. In order to obtain an estimate of the weightedLp,g-norm for arbitrary p we test (5.plu) with v = |zj |p−2zj and obtain after somemanipulations

‖zj‖ pp,gj+1− ‖zj−1‖ pp,gj + ch ‖wj‖ 2

1,2

≤ ch ‖zj‖ pp + cph ‖fj‖r3‖zj‖p−1r′3(p−1) + cph ‖U0‖1,r0‖wj‖1,2‖zj‖ (p−2)/2

s

+ cph (1 + ‖δuj‖ν1) ‖zj‖ pµ′1p , (6.plu)

252 Volker Pluschke

where wj := |zj |(p−2)/2zj . The last item on the right-hand side appears sincethe item ‖zj‖ pp,gj arising from (5.plu) on the left-hand side must be replaced by‖zj‖ pp,gj+1

. Hence, for our L∞-estimate we need uniform boundedness of ‖δuj‖ν1

(cf. Lemma 3). Then we have to estimate the unweighted norms of zj on theright-hand side by weighted norms occuring on the left-hand side. For this weuse the Nirenberg-Gagliardo interpolation inequality

‖w‖s ≤ C ‖w‖ θ1,2 ‖w‖ 1−θ1

for some θ ∈ (0, 1), s < 2N/(N − 2). This enables us to insert the weight bymeans of Cauchy-Schwarz’ inequality

‖wj‖1 = ‖zj‖ p/2p/2 ≤ ‖1/gj+1‖ 1/21 ‖zj‖ p/2p,gj+1

. (7.plu)

Summing up the inequalities (6.plu) for j = 1, . . . , i we would then come to anestimate of the weighted norm ‖zi‖p,gi+1 . However, in our case it is not possibleto hold the bounds depending on p uniformly bounded as p → ∞. Therefore,for the limit process we use a recursive approach due to Alikakos [1]. Since1/g ∈ Lσ(G) with σ > 1 is supposed we may even obtain the weighted norm‖zj‖λp,gj+1 with λ < 1 on the right-hand side of (7.plu). Then we derive an recursiveestimate of the form

maxtj≤t‖zj‖ pp,gj+1

≤ cpc t(

maxtj≤t‖zj‖ pλp,gj+1

+ maxtj≤t‖zj‖ β(p)p

λp,gj+1

)which is investigated for the special sequence pk = λ−kp0. Passing to the limitk →∞ this yields an estimate of ‖zj‖∞ in terms of ‖zj‖p0,gj+1 for fixed p0. Afterestimation of this norm for fixed p0 we obtain

Lemma 3. Let be ‖δuj‖ν1 ≤ C for j = 1, . . . , n independent of the subdivision.Then there are constants c, γ > 0 such that

max0≤tj≤t

‖uj − u0‖∞ ≤ c tγ .

Obviously, since uj ∈ C(G) due to Lemma 1 we have uj ∈ BR(U0) for alltj ∈ I := [0, T ] if we fix T for given R > 0 by cT γ = R.

Remark 4. In [3, Theorem 4.17] J. Kacur proves a L∞-estimate for quasilinearequations without the assumption ‖δuj‖ ≤ C of our Lemma 3. The reason isthat the degeneration in [3] corresponds to the case g(x, t, s) = b′(s), i.e. the itemconcerning the time derivative is written in the form b(u)t. This is not possible inour case. Moreover, we have weaker regularity of the data with respect to x. Onthe other hand, the technique in [3] allows stronger degeneration with respectto u.

The next task is to check the assumption of Lemma 3. One obtains an es-timate of δuj by forming the difference (1j.plu)–(1j−1.plu) and testing the resultingrelation with an appropriate test function (cf. [2, Chapters 2.1, 2.2]). However,


we run into problems since we have no full monotonicity of the nonlinear op-erator Au := A(t, u)u unlike in [2, Example 2.2.17]. Testing (1j.plu)–(1j−1.plu) withv = |δuj |κ−2δuj in order to estimate the weighted norm ‖δuj‖κ,gj we are forcedto deal with an item

ch (1 + ‖δuj−1‖ν) ‖uj‖1,r‖ωj‖1,2‖δuj‖ (κ−2)/2s (8.plu)

(ωj = |δuj |(κ−2)/2δuj) arising from(Aj − Aj−1

)(uj , v) on the right-hand side.

Hence, we have to estimate the space-like derivative ‖uj‖1,r in order to obtainan estimate of the discrete time derivative. This is possible by means of a prioriestimates for elliptic equations like (1j .plu) is. However, we are not able to splitgj δuj into gj

ujh and gj

uj−1h ,resp., and then to use estimates of the solution uj

of the elliptic equation with right-hand side fj + gjuj−1h since we need a priori

bounds uniformly with respect to h > 0. Hence we write (1j.plu) in the form

Ajuj = fj − gj δuj =: Fj

where we obtain from Lp-theory for elliptic equations (cf. [6, Theorem 5.5.5’] theestimate

‖uj‖1,r ≤ c (‖Fj‖ρ + ‖uj‖1) ≤ c(

1 +j∑i=1

‖δui‖ν1

). (9.plu)

The constant c now is independent of the subdivision since the coefficients ofthe elliptic operator Aj are uniformly bounded. Inserting this estimate into (8.plu)we notice that the total power of ‖δuj‖ on the right-hand side of the result-ing estimate is κ + 1 while we have the power κ on the left-hand side, only.This seems to contradict the intention to obtain boundedness of ‖δuj‖ by theseestimations. However, after some very technical manipulations, we are able toapply a nonlinear discrete version of the Gronwall lemma (cf. Willett, Wong [10,Theorem 4]) to obtain at least a local bound for small tj . Since the unweightednorm ‖δuj‖ν1 may be estimated by the weighted norm ‖δuj‖κ,gj we obtain

Lemma 5. Suppose assumptions (i)–(v). Then for h ≤ h0 there is a time in-terval [0, T ∗] such that the estimate

‖δuj‖ν1 ≤ C1 ∀tj ∈ [0, T ∗]

holds independent of the subdivision.

By means of (9.plu) this lemma also yields boundedness of the space-like derivatives.

Lemma 6. For all h ≤ h0 the estimate

‖uj‖1,r ≤ C2 ∀tj ∈ [0, T ∗]

holds independent of the subdivision.

The time T ∗ is a bound for the length of our local existence interval I. If T > T ∗

for the T choosen after Lemma 3 we have to fix I := [0, T ∗].The a priori estimates from Lemma 3, 5, and 6 now provide the tools to prove

the convergence results of Theorem 2.

254 Volker Pluschke

References

[1] N. D. Alikakos, Lp-bounds of solutions of reaction-diffusion equations, Comm.Part. Diff. Equ., 4 (8) (1979), 827–868.

[2] J. Kacur, Method of Rothe in Evolution Equations, B. G. Teubner Verlagsges.,Leipzig, 1985.

[3] J. Kacur, On a solution of degenerate elliptic-parabolic systems in Orlicz-Sobolevspaces II, Math. Z., 203 (1990), 569–579.

[4] J. Kacur, Solution to strongly nonlinear parabolic problems by a linear approxima-tion scheme, Comenius University, Faculty of Mathematics and Physics, PreprintNo. M1-94 (1994).

[5] A. Kufner, O. John, S. Fucık, Function Spaces, Noordhoff Intern. Publ., Leyden/Academia, Prag, 1977.

[6] C. B. Morrey, Jr., Multiple Integrals in the Calculus of Variations, Springer,Berlin–Heidelberg–New York, 1966.

[7] J. Moser, A new proof of De Giorgi’s theorem concerning the regularity problemfor elliptic differential equations, Comm. Pure and Appl. Math., 13 (3) (1960),457–468.

[8] V. Pluschke, Local Solutions to quasilinear parabolic equations without growthrestrictions, Z. Anal. Anwend., 15 (1996), 375–396.

[9] V. Pluschke, Rothe’s method for parabolic problems with nonlinear degeneratingcoefficient, Martin-Luther-University Halle, Dept. of Math., Report No. 14 (1996).

[10] D. Willett, J. Wong, On the discrete analogues of some generalizations of Gron-wall’s inequality, Monatshefte fur Math., 69 (1965), 362–367.



A Posteriori Error Estimates for a Nonlinear

Parabolic Equation?

Karel Segeth

Mathematical Institute, Academy of Sciences, Zitna 25,CZ-115 67 Praha 1, Czech Republic


Abstract. A posteriori error estimates form a reliable basis for adaptiveapproximation techniques for modeling various physical phenomena. Theestimates developed recently in the finite element method of lines forsolving a parabolic differential equation are simple, accurate, and cheapenough to be easily computed along with the approximate solution andapplied to provide the optimum number and optimum distribution ofspace grid nodes.The contribution is concerned with a posteriori error estimates needed forthe adaptive construction of a space grid in solving an initial-boundaryvalue problem for a nonlinear parabolic partial differential equation bythe method of lines. Under some conditions, it adds some more state-ments to the results of [2] in the semidiscrete case. Full text of the con-tribution will appear as a paper [4].

AMS Subject Classification. 65M15, 65M20

Keywords. A posteriori error estimate, nonlinear parabolic equation,finite element method, method of lines

1 A Nonlinear Model Problem

The principal ideas of semidiscrete a posteriori error estimation for nonlinearparabolic partial differential equations can be demonstrated with the help of asimple initial-boundary value one-dimensional model problem. We consider thenonlinear equation

∂u

∂t(x, t)− ∂

∂x

(a(u)

∂u

∂x(x, t)

)+ f(u) = 0, 0 < x < 1, 0 < t ≤ T,

for an unknown function u(x, t) with the homogeneous Dirichlet boundary con-ditions

u(0, t) = u(1, t) = 0, 0 ≤ t ≤ T,? This work was supported by Grant No. 201/97/0217 of the Grant Agency of the

Czech Republic.


256 Karel Segeth

and the initial condition

u(x, 0) = u0(x), 0 < x < 1.

In the above formulae, T > 0 is a fixed number and a, f , and u0 are smoothfunctions. Let

0 < µ ≤ a(s) ≤M, s ∈ R,and let a and f satisfy the global Lipschitz conditions

|a(r) − a(s)| ≤ L|r − s|,|f(r)− f(s)| ≤ L|r − s|, r, s ∈ R.

We employ the usual L2(0, 1) inner product to introduce the weak solutionu(x, t) ∈ H1([0, T ], H1

0 (0, 1)) of the model problem by the identity(∂u∂t, v)

+(a(u)

∂u

∂x,∂v

∂x

)+ (f(u), v) = 0

holding for almost every t ∈ (0, T ] and all functions v ∈ H10 , and the identity(

a(u0)∂u

∂x,∂v

∂x

)=(a(u0)

∂u0

∂x,∂v

∂x

)holding for t = 0 and all functions v ∈ H1

0 .

2 Discretization

Finite element solutions of the model problem are constructed from this weakformulation, too. We first introduce a partition

0 = x0 < x1 < · · · < xN−1 < xN = 1

of the interval (0, 1) into N subintervals (xj−1, xj), j = 1, . . . , N , and then put

hj = xj − xj−1, j = 1, . . . , N, and h = maxj=1,...,N

hj .

We further use the notation

(v, w)j =∫ xj

xj−1

v(x)w(x) dx

for the L2(xj−1, xj) inner product.We construct the finite dimensional subspace SN,p0 ⊂ H1

0 with a piecewisepolynomial hierarchical basis of degree p ≥ 1 in the following way. We put

SN,p0 =V | V ∈ H1

0 , V (x) =N−1∑j=1

Vj1ϕj1(x) +N∑j=1

p∑k=2

Vjkϕjk(x),

A Posteriori Error Estimates for a Nonlinear Parabolic Equation 257

where ϕj1 are the usual piecewise linear shape functions of the finite elementmethod,

ϕj1(x) = (x− xj−1)/hj , xj−1 ≤ x < xj ,

= (xj+1 − x)/hj+1, xj ≤ x ≤ xj+1,

= 0 otherwise,

while for k > 1,

ϕjk(x) =

√2(2k − 1)hj

∫ xj

xj−1

Pk−1(y) dy, xj−1 ≤ x ≤ xj ,

= 0 otherwise

are bubble functions with Pk being the kth degree Legendre polynomial scaledto the subinterval [xj−1, xj ] (see, e.g., [5]).

We say that a function U(x, t) is the semidiscrete finite element approximatesolution of the model problem if it belongs, as a function of the variable t, intoH1([0, T ], SN,p0 ), if the identity(∂U

∂t, V)

+(a(U)

∂U

∂x,∂V

∂x

)+ (f(U), V ) = 0

holds for each t ∈ (0, T ] and all functions V ∈ SN,p0 , and if the identity(a(u0)

∂U

∂x,∂V

∂x

)=(a(u0)

∂u0

∂x,∂V

∂x

)holds for t = 0 and all functions V ∈ SN,p0 . The procedure for constructing theapproximate solution

U(x, t) =N−1∑j=1

Uj1(t)ϕj1(x) +N∑j=1

p∑k=2

Ujk(t)ϕjk(x)

described above is the method of lines. It transforms the solution of the origi-nal initial-boundary value problem for a parabolic partial differential equationinto an initial value problem for a system of ordinary differential equations forthe unknown functions Ujk(t) that, in practice, is solved by proper numericalsoftware.

3 A Posteriori Semidiscrete Error Indicators

Lete(x, t) = u(x, t)− U(x, t)

be the error of the semidiscrete approximate solution. We employ the finitedimensional subspace

SN,p+10 =

V | V ∈ H1

0 , V (x) =N∑j=1

Vjϕj,p+1(x)

258 Karel Segeth

of piecewise polynomial bubble functions of degree p + 1 equal to zero at thegrid points xj to construct error indicators. We say that a function E(x, t) =EPN ∈ H1([0, T ], SN,p+1

0 ) is a parabolic nonlinear a posteriori semidiscrete errorindicator if the identities(∂E

∂t, V

)j

+(a(U + E)

∂E

∂x,∂V

∂x

)j

= −(f(U + E), V )j −(∂U∂t, V)j−(a(U + E)

∂U

∂x,∂V

∂x

)j

hold for j = 1, . . . , N , all t ∈ (0, T ] and all functions V ∈ SN,p+10 , and if the

identities (a(u0)

∂E

∂x,∂V

∂x

)j

=(a(u0)

∂(u0 − U)∂x

,∂V

∂x

)j

hold for j = 1, . . . , N , t = 0 and all functions V ∈ SN,p+10 . Note that the special

choice of the bubble function space SN,p+10 results in an uncoupled system of

ordinary differential equations. On each interval (xj−1, xj), the error indicator

E(x, t) =N∑j=1

Ej(t)ϕj,p+1(x)

is computed independently of the other intervals. The indicator thus has a localcharacter and its computation is rather cheap.

When E is neglected in the argument of the functions a and f we say thata function E(x, t) = EPL ∈ H1([0, T ], SN,p+1

0 ) is a parabolic linear a posteriorisemidiscrete error indicator if the identities(∂E∂t, V)j

+(a(U)

∂E

∂x,∂V

∂x

)j

= −(f(U), V )j −(∂U∂t, V)j−(a(U)

∂U

∂x,∂V

∂x

)j


identities (a(u0)

∂E

∂x,∂V

∂x

)j

=(a(u0)

∂(u0 − U)∂x

,∂V

∂x

)j

hold for j = 1, . . . , N , t = 0 and all functions V ∈ SN,p+10 . The practical com-

putation of the linear error indicator is thus easier.The task to compute error indicators can be simplified if the derivative ∂E/∂t

is neglected. The corresponding a posteriori semidiscrete error indicator is thencalled (linear or nonlinear) elliptic indicator since the resulting uncoupled alge-braic system does not depend on t. Moreover, the practical computation of suchan elliptic indicator need not be carried out for each t but only when required.We thus say that the function E(x, t) = EEN that maps [0, T ] into SN,p+1

0 is anelliptic nonlinear a posteriori semidiscrete error indicator if the identities(a(U + E)

∂E

∂x,∂V

∂x

)j

= −(f(U + E), V )j −(∂U∂t, V)j−(a(U + E)

∂U

∂x,∂V

∂x

)j



identities (a(u0)

∂E

∂x,∂V

∂x

)j

=(a(u0)

∂(u0 − U)∂x

,∂V

∂x

)j

hold for j = 1, . . . , N , t = 0 and all functions V ∈ SN,p+10 .

Finally, we say that the function E(x, t) = EEL that maps [0, T ] into SN,p+10

is an elliptic linear a posteriori semidiscrete error indicator if the identities(a(U)

∂E

∂x,∂V

∂x

)j

= −(f(U), V )j −(∂U∂t, V)j−(a(U)

∂U

∂x,∂V

∂x

)j


identities (a(u0)

∂E

∂x,∂V

∂x

)j

=(a(u0)

∂(u0 − U)∂x

,∂V

∂x

)j

hold for j = 1, . . . , N , t = 0 and all functions V ∈ SN,p+10 .

To assess properties of the above semidiscrete a posteriori error indicators,we introduce the quantity

Θ =‖E‖1‖e‖1

called the effectivity index of the respective error indicator. The norm used isthe H1(0, 1) norm. Then we can prove the following statement.

Theorem 1. Let u(x, t) ∈ H10 be smooth, let U(x, t) ∈ SN,p0 and E ∈ SN,p+1

0 .Let the norm of the difference between the semidiscrete solution U and its ellipticprojection is a nondecreasing function of t. Let the same hold for the norm ofthe difference between the error indicator E and its elliptic projection.

Moreover, let‖e‖1 ≥ Chp.

Thenlimh→0

Θ = 1

holds for ΘPN, ΘPL, and ΘEL.

The exact assumptions as well as a complete proof will be published in [4].

4 A Numerical Example

We present numerical results obtained by the finite element method of linesfor a nonlinear parabolic initial-boundary value problem (a reaction-diffusionmodel) with a simple grid adjustment procedures described in [1] (Fig. 1) and[3] (Fig. 2). Both the procedures are based on the equidistribution of error. Theexample fully confirms the above statement.

260 Karel Segeth

The differential equation solved is

∂u

∂t− ∂2u

∂x2−D(1 + α− u) exp(−δ/u) = 0,

D = Rexp δαδ

, 0 < x < 1, 0 < t ≤ 0.6,

α = 1, δ = 20, R = 5,

with the boundary conditions

∂u

∂x(0, t) = 0, u(1, t) = 1, 0 < t ≤ 0.6,

and the initial condition

u(x, 0) = 1, 0 < x < 1.

We used piecewise linear shape functions, i.e. p = 1, for the computation ofthe solution U and required a very small error bound in the integration of thecorresponding system of ordinary differential equations by a standard differentialsystem solver.

The model describes a single step reaction of a reacting mixture of temper-ature u in a region 0 < x < 1. Further, α is the heat release, δ is the activationenergy, D is called the Damkohler number, and R is the reaction rate. For smalltimes, the temperature gradually increases from unity with a “hot spot” formingat x = 0. At a finite time, ignition occurs and the temperature at x = 0 jumpsrapidly from near 1 to near 1+α. A sharp flame front then forms and propagatestowards x = 1 with velocity proportional to 1

2 exp(αδ)/(1+α). In real problems,α is about unity and δ is large. The flame front thus moves exponentially fastafter ignition. The problem reaches a steady state once the flame propagates tox = 1.

The trajectories of nodes of the partition of interval (0, 1) as constructed bythe two procedures mentioned are shown in Figs. 1 and 2. The grid is rather slowand is unable to follow the dynamics of the problem properly. The integrationwith respect to t requires small time steps and is expensive during this rapidtransience as the solution changes rapidly along the grid node trajectories. Theproblem is very difficult and yet adaptive grid methods are capable of finding asolution with relative ease.

References

[1] Adjerid, S., Flaherty, J. E. : A moving finite element method with error estimationand refinement for one-dimensional time dependent partial differential equations.SIAM J. Numer. Anal. 23 (1986) 778–796

[2] Moore, P. K. : A posteriori error estimation with finite element semi- and fullydiscrete methods for nonlinear parabolic equations in one space dimension. SIAMJ. Numer. Anal. 31 (1994) 149–169


Fig. 1. Trajectories of nodes constructed by the procedure of [1]

[3] Segeth, K. : A grading function algorithm for space grid adjustment in the methodof lines. Software and Algorithms of Numerical Mathematics 10. (Proc. of SummerSchool, Cheb 1993.) Plzen, University of West Bohemia (1993) 139–152

[4] Segeth, K. : A posteriori error estimation with the finite element method of linesfor a nonlinear parabolic equation in one space dimension (to appear)

[5] Szabo, B., Babuska, I. : Finite Element Analysis. New York, J. Wiley & Sons 1991

262 Karel Segeth

Fig. 2. Trajectories of nodes constructed by the procedure of [3]



The Solvability Conditions of the Infinite

Trigonometric Moment Problem with Gaps

G. M. Sklyar1 and I. L. Velkovsky2

1 Dept. of Math. Analysis, Kharkov State UniversitySvoboda sqr. 4, 310077, Kharkov, Ukraine

Email: [email protected] Reiterstrasse 11,

93 053, Regensburg, GermanyEmail: [email protected]

Abstract. The infinite Markov trigonometric moment problem with pe-riodic gaps is considered. The precise analytical description of the solv-ability set of the problem is given. The introduced approach is based oninvestigation of the special subclass of the Caratheodory function classcorresponding to given periodic law.

AMS Subject Classification. 42A70

Keywords. Markov trigonometric moment problem, periodic gaps, pe-riodic law, Caratheodory function class

Let p be a natural number and M = m0, . . . ,mν , where

0 ≤ m0 < m1 < m2 < · · · < mν < p,

is the subset of the set 0, 1, . . . p− 1.

Definition 1. We will refer to the sequence M = mk∞k=0, which is the p-pe-riodic extension of M to the set N ∪ 0, i.e.

0 ≤ m0 < m1 < · · · < mν < p ≤ mν+1 = m0 + p < · · ·· · · < m2ν+1 = mν + p < · · · ,

as a p-periodic law generated by M.If from l ∈ M implies p − l ∈ M , we will say, that the p-periodic law is a

symmetric one.

Let M = mk∞k=0 be a p-symmetric periodic law. Consider the infiniteMarkov trigonometric moment problem of the form:

θ∫0

eimktf (t) dt = smk , |f(t)| ≤ 1, t ∈ (0, θ), k = 0, 1, 2, . . . , (1.skl)


264 G. M. Sklyar and I. L. Velkovsky

where 0 < θ < 2πp .

Our first goal is to give conditions for the complex sequence smk∞k=0 =amk + ibmk

∞k=0 (if 0 ∈M then b0 = 0) to be a moment one. That means that

there exists at least one measurable function f satisfying moment equalities (1.skl).As it is known, the classical trigonometric moment problem (mk = k) is

closely connected with Caratheodory coefficient problem [1] and based on thetechnique of the Caratheodory functions [2].

Remind that for the class C of Caratheodory functions one can write:

C := F : F is holomorphic, ReF (z) > 0 for |z| < 1 .

Further we need the following theorem describing properties of certain functionsfrom this class:

Theorem 2. Let T =N⋃j=1

Tj be a collection of nonintersecting intervals Tj =(τj , τ

′j

)⊂ [0, 2π] . Then the following statements are equivalent to each other:

i) A function F (z) ∈ C is holomorphic for z = eiτ and ImF (eiτ ) = 0, τ ∈ T.ii) The following representation holds:

F (z) = |F (0)| exp

i

4

∫[0,2π]\T

eit + z

eit − zϕ (t) dt

,

−1 ≤ ϕ(t) ≤ 1, t ∈ [0, 2π]\T.

iii) Two functions

F±(z) = F (z) ·(

N∏j=1

eiτ′j − z

eiτj − z · e− i

2

N∑j=1

(τ ′j−τj))± 1

2

belong to the class C .

Introduce the subclass of the Caratheodory function class associated with ap-periodic law.

Definition 3. For the subclass C(M) corresponding to the periodic law M wecall the set of functions F(z), satisfying the following conditions:

i) F ∈ C.

ii) F is holomorphic and real on the arc z = eiτ , where τ ∈(

2πp (p− ν) , 2π

).

iii) Power series for the function ln(F (z)|F (0)|

)is of the form:

ln(F (z)|F (0)|

)=∞∑k=0

ρkzmk , |z| < 1.

The Solvability Conditions 265

Further we give a multiplicative description of the class C(M) .Introduce the polynomial rM (z) of the form:

rM (z) =q∏

k=1

(1− e−γkp z

)=

p−1∑l=0

rlzl,

where ep is a primitive root of unity, of order p,

Γ = γ1, . . . , γq = 0, 1, . . . , p− 1 \M.

Note that r0 = 1, rl = 0, l > q = p− ν − 1.Besides, if M is a symmetric law then rk are real, k = 0, 1, . . . , p− 1.

Theorem 4. A function F (z) ∈ C(M), where M is a p-periodic symmetric law,iff

F (z) = |F (0)| exp

i

4

2πp∫

0

q∑l=0

rlei(t+

2πp l) + z

ei(t+2πp l) − z

ϕ (t) dt

,

where −µ ≤ ϕ (t) ≤ µ, µ−1 = max|rl| , l = 0, q

.

Let a sequence smk∞k=0 be a moment one for the problem (1.skl). Complete

then the definition of the function f(t) by f(t) = 0, t ∈(

0, 2πp

). Next consider

the function ϕ(t), t ∈ (0, 2π) , of the form:

ϕ(t) = µrl f

(t− 2π

pl

), t ∈

(2πpl,

2πp

(l + 1))

= ∆l, l = 0, 1, . . . p− 1.

Note that

|ϕ(t)| ≤ 1, t ∈ (0, 2π) ,|ϕ(t)| ≤ µ, t ∈ ∆0;

Next consider the complex function

F (z) = exp

i

4

2π∫0

eit + z

eit − zϕ (t) dt

. (2.skl)

Note that |F (0)| = 1. Hence due to Theorem 4 obtain that F (z) ∈ C(M). Besidesϕ(τ) ≡ 0, τ ∈

(θ, 2π

p

), then the function F (z) is holomorphic and real on the

arc z = eiτ , τ ∈(θ, 2π

p

)(Theorem 2). Applying Theorem 2 once more and

considering T =(θ, 2π

p

)∪(

2πp (q + 1), 2π

), we obtain that F±(z) ∈ C, where

F±(z) = F (z) ·(ep − zeiθ − z ·

1− zeq+1p − z

· exp− i

2

(2πp

(p− q)− θ))± 1

2

. (3.skl)

266 G. M. Sklyar and I. L. Velkovsky

Let

F±(z) = α± +∞∑j=1

α±j zj , |z| < 1. (4.skl)

Express now coefficients α±, α±j , j = 1, 2, . . . of the expansion via elements ofthe moment sequence smk , k = 0, 1, . . .

Let

lnF±(z) =∞∑j=0

s±j zj , (5.skl)

then

s±j =iµ2 (2− δ0j)rM (ejp)sj ± nj , j ∈M,

±nj , j /∈M,(6.skl)

where

n0 =i

4

(2πp

(p− q)− θ),

nj =e−iθ + e−

2πp (q+1)i − e− 2π

p i − 1j

,

(7.skl)

One can see from (4.skl) and (6.skl) that:

α±j = exp s±0 ,

A±j =

∣∣∣∣∣∣∣∣α±1 2α±2 · · · jα±jα± α±1 · · · α±j−1

. . . . . . . . . . . . . . . . . .0 0 · · · α±1

∣∣∣∣∣∣∣∣ = j(α±)ns±j . (8.skl)

Thus, taking account of (6.skl) we can regard (8.skl) as recurrent expressions of thecoefficients α±, α±j , j = 1, 2, . . . via smk , k = 0, 1, 2 . . .

Now we are able to formulate the main result of the paper.

Theorem 5. A sequence smk∞k=0 is a moment one for the Markov trigono-

metric moment problem (1.skl) associated with the p-symmetric periodic law M iff

A±n ≥ 0, n = 0, 1, 2, . . .

where— the coefficients α±, α±j ,j = 1, 2, . . . are expressed via sequence

s±j∞j=0

by means of (8.skl),— s±j , j = 0, 1, . . . are of the form (6.skl) ,— nj , j = 0, 1, 2, . . . are defined from (5.skl),

— A±n , n = 0, 1, . . . are symmetric matrices such that A±n =(α±k−j

)nk,j=1

,

where α±0 = α± + α± = 2Reα±, α±−j = α±j , j = 1, 2, . . .

The Solvability Conditions 267

References

[1] Caratheodory, C., Uber Variabilitatsbereich der Koefficienten von Potenzreichen,die gegebene Werte nicht annehmen, Math. Annalen, 64 (1907)

[2] Akhiezer, N. I., Krein, M. G., Uber Fourierische Reichen beschrankter summierbarerFunctionen und ein neues Extremumproblem, Common. Soc. Math., Kharkov, 9(1934)



Elliptic Equations with Decreasing

Nonlinearity I :Barrier method for Decreasing Solutions

Tadie

Matematisk Institut, Universitetsparken 52100 Copenhagen, Denmark


Abstract. In this note, we establish existence theorems for positive andclassical solutions of the problem (Ea.tad) below using a barrier method.Moreover we show that the existence of such solutions can be obtainedfrom the sole existence of a supersolution or of a subsolution of theequation.

AMS Subject Classification. 35J70, 35J65 34C10

Keywords. Quasilinear elliptic, integral operators, fixed points theory

1 Introduction

Let f ∈ C1([0,∞)× (0,∞)) be such that

f1) ∀r ≥ 0, f(r, .)+ := max0, f(r, .) ∈ C1((0,∞)) and non increasing;f2) ∀S, T > θ > 0, if f(r, S), f(r, T ) > 0 then ∃k1(θ), k2(θ) > 0 such that|f(r, T )− f(r, S)| ≤ k1(θ)f2(r, k2(θ) )|T − S|; f2(., S) := |∂f(., S)/∂S|.

For a > 1, p ∈ (1, 2] and Dpau := (ra|u′|p−2u′)′, consider in R+ the problem

Ea(u) := Dpau+ raf(r, u)+ = 0; u(0) > 0; u′(0) = 0. (Ea.tad)

Definition 1. Let M be a positive number, finite or not. Let IM := [0,M) andw, v ∈ C1(IM ) be piecewise C2 be non increasing.

1) v will be said to be a supersolution (subsolution) of the problem (Ea.tad) inIM if Ea(v) ≥ 0 (Ea(v) ≤ 0) almost everywhere in IM ;

2) w and v will be said to be Ea-compatible in IM ifi) Ea(w) ≤ 0 ≤ Ea(v) a.e. in IM ,ii) 0 < w ≤ v and w′ ≤ v′ ≤ 0 in IM ,iii) ∀r ∈ IM , f(r, .) > 0 and decreasing in [w(r), v(r)].


270 Tadie

For a non-increasing positive φ ∈ C1(IM ) define

Φ(r) = Tφ(r) := φ(0)−∫ r

0

dt

∫ t

0

(s/t)af(s, φ)ds1/(p−1)

. (T)

Definition 2. A non increasing (respectively decreasing) positive supersolutionv (resp. subsolution w) of (Ea.tad) in IM will be said to be Ea-compatible if Tv andv (resp. w and Tw) are Ea-compatible in IM .

In the sequel super- and subsolutions are supposed to be C1 and piecewise C2

in the corresponding domains. Also for ease writing, under the integral signs wewill write f(., .) for f(., .)+. The main results are the following:

Theorem 3. If there are w and v which are Ea-compatible in IM , then (Ea.tad)has a solution u ∈ C2(IM ) such that w ≤ u ≤ v in IM .

Theorem 4. Assume that there is a non increasing (resp. decreasing) positivesupersolution v (resp. subsolution w) which is Ea-compatible in IM .Then (Ea.tad) has a decreasing solution u ∈ C2(IM ) such that Tv ≤ u ≤ v (resp.w ≤ u ≤ Tw) in IM .

Theorem 5. 1) Assume that there are w and v which are Ea-compatible in[0,∞) such that ∫ ∞

0

1 + sp−1f(s, w)ds <∞. (1.tad)

Then (Ea.tad) has a solution u ∈ C2([0,∞)) such that w ≤ u ≤ v in [0,∞).2) Assume that there is a non increasing (resp. decreasing) positive supersolutionv (resp. subsolution w) Ea-compatible in R+.Then (Ea.tad) has a positive decreasing solution u ∈ C2([0,∞)) such that it holdsTv ≤ u ≤ v (resp. w ≤ u ≤ Tw) in [0,∞).

Theorem 6. 1) Assume that there are w and v which are Ea-compatible in[0,∞) with ∫ ∞

0

sf(s, w)1/(p−1) <∞. (2.tad)

Then (Ea.tad) has a solution u ∈ C2([0,∞)) such that w ≤ u ≤ v.2) Assume that there is a non increasing positive supersolution v of (Ea.tad) in[0,∞) such that

i) V (r) = Iv(r) :=∫∞r dt

∫ t0 (s/t)af(s, v)ds1/(p−1) satisfies (2.tad);

ii) V and v are Ea-compatible in [0,∞).

Then (Ea.tad) has such a solution u with V ≤ u ≤ v.Similarily if there is a decreasing positive subsolution w such that w and Iw areEa-compatible in [0,∞) and which satisfies (2.tad), then (Ea.tad) has such a solution uwith w ≤ u ≤W := Iw.

Barrier Method 271

2 Proof of the theorems

2.1 Preliminaries

Let Cf (M) := φ ∈ C(IM ) | f(r, φ) > 0 ∀r ∈ IM and b := 1/(p− 1). For someA = φ(0), define on Cf (M) the operator T by

Φ(r) := Tφ(r) := A−∫ r

0

dt

∫ t

0

(s/t)af(s, φ)dsb. (3.tad)

Then DpaΦ+ raf(r, φ) = 0 in IM, Φ(0) = A, Φ′(0) = 0 and Φ′ ≤ 0.

From [5], as b ≥ 1, ∀t ≤M , with s∗ := max1, s,

|Φ(t)| ≤ p− 1a+ 1− p

∫ t

0

sp−1∗ f(s, φ)ds

b; (4.tad)

|Φ′(t)| ≤ 1t∗

∫ t

0


b. (5.tad)

As Φ′′(t) = −b∫ t

0 (s/t)af(s, φ)dsb−1f(t, φ)− at

∫ t0 (s/t)af(s, φ)ds,

|Φ′′(t)| ≤ b∫ t

0

(s/t)af(s, φ)dsb−1

f(t, φ) +a

t

∫ t

0

f(s, φ)ds. (6.tad)

Thus TCf(M) ⊂ C2(IM ) and for φ ∈ Cf (M),

|Tφ|C2([0,M ] ≤ C2M (φ) := A+

a

a+ 1− p

∫ M

0


b+

+ b(a+ 1)|f(., φ)|C(IM )

∫ M

0

f(s, φ)dsb−1

. (7.tad)

Lemma 7. Let w, v be those in Theorem 3 and define

EM (w, v) := φ ∈ C1(IM ) | w ≤ φ ≤ v; w′ ≤ φ′ ≤ v′ inIM.

Then with A ∈ [w(0), v(0)], TEM (w, v) ⊂ EM (w, v) ∩ C2(IM ) .

Proof. Let V := Tv and W := Tw; then in IM

w ≤W ≤ V ≤ v and w′ ≤W ′ ≤ V ′ ≤ v′.

In fact, as V ′, v′ ≤ 0, DpaV −Dp

av = (ra|v′|p−1−|V ′|p−1)′ ≤ 0 whence |v′|p−1 ≤|V ′|p−1 or V ′ ≤ v′ ≤ 0. Because V (0) ≤ v(0) we then have V ≤ v in IM .Similarily we have w′ ≤ W ′ and w ≤ W in IM . Also in the same way, w ≤ vand W (0) = V (0) imply that W ′ ≤ V ′ and W ≤ V . If φ ∈ EM (w, v) thenf(r, v) ≤ f(r, φ) ≤ f(r, w) in IM , hence Φ := Tφ satisfies

W ≤ Φ ≤ V and W ′ ≤ Φ′ ≤ V ′.

272 Tadie

Corollary 8. Let v (w) be a non increasing (decreasing) positive supersolution(subsolution) which is Ea-compatible in IM .Then TEM (v) ⊂ EM (v) ∩ C2(IM ), where EM (v) ≡ EM (Tv, v) (TEM (w) ⊂EM (w) ∩ C2(IM ), where EM (w) ≡ EM (w, TW )).

Proof. In the light of Lemma 7, it is enough to notice that V := Tv (W := Tw)is a subsolution (supersolution) of (Ea.tad) in IM .

Lemma 9. Let w and v be as in Theorem 3. Then, T : EM (w, v) −→ C1(IM )is continuous and TEM (w, v) is equicontinuous in C1(IM ).

Proof. The continuity follows from the fact that for φ, ψ ∈ EM (w, v) and | |rdenoting the norm in C([0, r]),

|(|(Tφ)′|p−1 − |(Tψ)′|p−1)(t)| ≤ k1(θ)|φ − ψ|r∫ r

0

(s/r)af2(s, k2(θ)),

where φ, ψ > θ > 0 in IM is assumed (see f2) ) and a similar bound for |Tφ−Tψ|is obtained easily. The equicontinuity in C1 follows from (7.tad).

2.2 Proof of Theorems 3 and 4

Lemma 7 and Lemma 9 imply that T has a fixed point in EM (w, v) by theSchauder-Tychonoff’s fixed point theorem [2]; (6.tad)–(7.tad) imply that the fixed pointis in C2(IM ). In the same way Corollary 8 and Lemma 9 imply that T has sucha fixed point in EM (v) (EM (w)).


We prove 1) only as 2) and 3) would be simple readaptations.If (1.tad) holds, then V := Tv and W := Tw are in E(w, v) ∩C2([0,∞)). With (1.tad),(4.tad)–(7.tad) imply that ∀φ ∈ E(w, v) := E∞(w, v),

|Tφ|C2(IM ) ≤ C2∞(w) ∀M > 0. (8.tad)

Let (Mk)k∈N be an increasing sequence such that Mk ∞ and (uk := uMk) the

corresponding solutions in Ik := IMk. uk is extended by uk := Tuk ∈ C2(R+),

say, which satisfies (8.tad) and Ea(uk) = 0 in Ik, uk(0) = A. By means of theSchauder-Tychonoff’s fixed point theorem, such a required solution is an induc-tive limit of the (uk) ([3]).


Define this time the inverse operator of (Ea.tad) in IM , K := KM on Cf (M) by

Φ(r) = Kφ(r) :=∫ M

r

dt

∫ t

0

(s/t)af(s, φ)dsb.

Barrier Method 273

From Jensen’s inequality (1/t)∫ tosaf(s, φ)dsb ≤ (1/t)

∫ t0saf(s, φ)bds and

simple integrations by parts, as in (4.tad)–(7.tad), ∀t ∈ IM ,

b(a− 1)Φ(t) ≤∫ M

0

sbf(s, φ)bds := IbM (φ); |Φ′(t)| ≤ (1/t)Ibt (φ)

and (6.tad) holds for this case.If necessary, we replace f by f1 := λf such that

[(p− 1)/(a− 1)]∫ ∞

0

sf1(s, w)1/(p−1)ds < v(0) in (2.tad);

the required solution will be u(r) := u1(µr) for some suitable µ = µ(λ), u1 beingobtained with f1. So, without major difficulties the proof of this Theorem followsthe same steps as that of Theorem 5.

References

[1] Istratescu, V. I., Fixed point theory. Math. and its Appli., Reidel Publ. 1981.[2] Kufner, A. et al., Function Spaces. Noordhoff 1977.[3] Tadie, Weak and classical positive solutions of some elliptic equations in Rn, n ≥ 3:

radially symmetric cases. Quart. J. Oxford 45 (1994), 397–406.[4] Tadie, Semilinear ODE (part 2): Positive solutions via super-sub-solutions method.

Proc. Prague Math. Conf. 1996, 320–323.[5] Yasuhiro, F., Kusano, T. and Akio, O., Symmetric positive entire solutions of second

order quasilinear degenerate elliptic equations. Arch. Rat. Mech. Anal. 127 (1994),231–254.



Elliptic Equations with Decreasing

Nonlinearity II :Radial Solutions for Singular Equations

Tadie

Matematisk Institut, Universitetsparken 52100 Copenhagen, Denmark


Abstract. By means of the super-sub-solutions method from [3], theexistence of decreasing solutions of some singular elliptic equations willbe established.

AMS Subject Classification. 35J70, 35J65, 34C10

Keywords. p-Laplacian, integral equations

1 Introduction

Let f ∈ C1([0,∞);R+) with f(r) > 0 ∀r ≥ 0 and f(r) ' r−θ at ∞ for someθ > 0. For some a > 1 and p ∈ (1, 2], assume that

f) ∃b ∈ (0, a+ 1− p]; for w(t) := (1 + t)−b/(p−1) , some γ > 0 and

ψ(r) := f(r)w(r)−γ ,∫ ∞

0

sb+p−1ψ(s)ds <∞.

In this note, we investigate the existence of positive and decreasing solutionsu ∈ C2 := C2([0,∞)) of

Qu ≡(ra|u′|p−2u′)′ + raF νq (r, u)+ = 0, u′(0) = 0,

where q > 0, Fνq(r, u) := f(r)u−γ − νuq, ν ≥ 0,

or Fνq(r, u) := νf(r)u−γ + uq, ν > 0.

(Q.tadd)

For a = n − 1, n ∈ N, such u is a radial solution in Rn of the p-Laplacianequations div(|∇u|p−2∇u) + F νq (|x|, u)+ = 0.For a positive and decreasing function φ, define

Φ(r) = Tφ(r) := φ(0)−∫ r

0

dt

∫ t

0

(s/t)aF νq (s, φ)+ds

1/(p−1)

.


276 Tadie

Given such a function φ, the following result from [3] will be used:

assume that∫ ∞

0

(1 + sp−1)F νq (s, φ)+ds <∞; (φ.tadd)

if ∀r ≥ 0 Qφ ≥ 0 (≤ 0 respectively) and F νq (r, .) is positive and decreasingin [Φ(r), φ(r)] ( [φ(r), Φ(r)] respect.), then (Q.tadd) has a decreasing solution u ∈C2([0,∞)) such that Φ ≤ u ≤ φ (φ ≤ u ≤ Φ respect.) in [0,∞).The main results are the following:

Theorem 1 (Uniqueness). Assume that ∀r ≥ 0 t 7→ F νq (r, t)+ is decreasingin t > 0. Then

a) ∀b ≥ 0, if it exists the decreasing solution ub ∈ C1 of (Q.tadd) such thatlim∞ ub = b is unique;

b) ∀R > 0, if it exists the decreasing solution u ∈ C1([0, R)) of (Q.tadd) such thatu(R) = 0 is unique.

Theorem 2 (Existence). Suppose that for some γ > 0 and b ∈ (0, a+ 1− p]∫ ∞0

sb+p−1f(s)(1 + s)bγ/(p−1) <∞. (1.tadd)

1) Then, the equation

(ra|u′|p−2u′)′ + raf(r)u(r)−γ = 0 (2.tadd)

has a unique positive and decreasing solution u ∈ C2 := C2([0,∞)) suchthat

u ≤ C r−b/(p−1) (u ' r−b/(p−1) if b = a+ 1− p) at ∞;

2) if also q > maxp(p− 1)/b,−γ + θ(p− 1)/b,i) there is ν0 > 0 depending only on f such that for ν ∈ (0, ν0 ]

(ra|v′|p−2v′)′ + raf(r)v(r)−γ − νv(r)q+ = 0 (3.tadd)

has a unique decreasing and positive solution v ∈ C2; if in additionq > (p− 1)(b + p)/b, then v(r) ≤ C r−b/(p−1) at ∞;

ii) there is ν1 > 0 depending only on f such that ∀ν > ν1

(ra|U ′|p−2U ′)′ + raνf(r)U−γ + U q = 0 (4.tadd)

has a positive and decreasing solution U such that U ≤ C r−b/(p−1)

at ∞.

2 Preliminaries

Definitions and notations:µ := 1/(p − 1); m := µb, b ∈ (0, a + 1 − p]; w(r) := (1 + r)−m;

∫v(s) :=∫

v(s)ds; ψ(r) := f(r)w(r)−γ ; t∗ := max1, t and Dpau := (ra|u′|p−2u′)′.

Elliptic Equations 277

2.1 Properties of some integrals

Define for t ≥ 0

J(t) :=∫ ∞t

(∫ r

0

(sr

)aψ(s)

)µ. (5.tadd)

We normalized f so that

Ψ1 :=∫ 1

0

(∫ r

0

ψ

)µ+

1m

(∫ ∞0

sb+p−1ψ

)µ≤ 1. (6.tadd)

Lemma 3. If∫ ∞0

sb+p−1ψ(s) <∞ or 0 < γ < (p− 1)(θ − b− p)

b, (7.tadd)

where b ∈ (0, a+ 1− p], then ∀t ≥ 0

(p− 1)a+ 1− p

(∫ 1

0

saψ

)µ≤ J(t) ≤ Ψ1 t

−m∗ ; (8.tadd)

b = a+ 1− p =⇒ mJ(t) ≥ t−m∫ 1

0

saψ(s)dsµ

∀t > 1; (9.tadd)

|J(t)′| ≤(∫ 1

0

ψ

)µ+(∫ ∞

0

sb+p−1ψ

)µt−m−1∗ ; (10.tadd)

|J(t)′′| ≤ (a+ 1)µ|J(t)′|(µ−1)/µ|ψ|∞, (11.tadd)

where (7.tadd) is not necessary for the lower bound in (8.tadd).

Proof. We have

J(t) =∫ ∞t

r−m−1

r−a+b+p−1

∫ r

0

saψ

µ≤∫ ∞t

r−m−1

(∫ ∞0

sb+p−1ψ

)µon one hand and

J(t) ≤∫ 1

0

(∫ r

0

ψ

)µ+∫ ∞

1

r−m−1

(∫ ∞0

sb+p−1ψ

)µon the other hand; the RHS of (8.tadd) then follows from integrations by parts . Fort ≤ 1,

J(t) ≥∫ ∞

1

(r−a

∫ r

0

saψ

)µ≥(∫ 1

0

saψ

)µ ∫ ∞0

r−aµdr

and for t > 1 ,

J(t) ≥(∫ 1

0

saψ

)µ ∫ ∞t

r−aµdr.

278 Tadie

We thus get the LHS of (8.tadd).If b = a+ 1− p, J(t) ≥ (

∫ 1

0saψ)µ

∫∞tr−m−1dr and (9.tadd) follows.

For t > 1, as a > b+ p− 1,

0 ≤ −J(t)′ ≤(t−b+1−p

∫ t

0

sb+p−1ψ

)µ≤ t−m−1

(∫ ∞0

sb+p−1ψ

)µ.

For t ≤ 1 |J(t)′| ≤ (∫ 1

0ψ)µ and (10.tadd) is obtained.

For (11.tadd),

J(r)′′ = −µr−a

∫ r

0

saψ

µ−1−ar−a−1

∫ r

0

saψ(s) + ψ(r)

hence from

|J(r)′′| ≤ µ(a+ 1)|ψ|∞(r−a

∫ ∞0

saψ

)µ−1

(11.tadd) follows.

Lemma 4. Under the assumptions (6.tadd)–(7.tadd)

(ra|U ′|p−2U ′)′ + raψ(r) = 0; r ≥ 0 (12.tadd)

has a decreasing and positive solution U ∈ C2([0,∞)) such that

U(r) ≤ (1 + r)−b/(p−1) ∀r ≥ 0. (13.tadd)

Proof. It is easy to verify that U = J where J is defined in (5.tadd) satisfies (12.tadd).Then (8.tadd)–(11.tadd) complete the proof.


Let u and v be two such solutions with u > v > 0 in some [0, R).As they are decreasing, from the equations, in [0, R)

ra(|v′|p−1 − |u′|p−1)′ = raF νq (r, v)− F νq (r, u) > 0

with ra(|v′|p−1 − |u′|p−1)|r=0 = 0, whence |v′| > |u′| or v′ < u′ ≤ 0 in (0, R).This implies that u(r)− v(r) > u(0)− v(0) whenever v(r) > 0.


In the lights of the super-sub-solutions methods established in [3], it suffices foreach case to find an appropriate sub- or supersolution of the problem.1) The function U in Lemma 4 is a supersolution of (2.tadd) as

ψ(r) = f(r)(1 + r)bγ/(p−1) ≤ f(r)U(r)−γ .

Elliptic Equations 279

The estimate for the case b = a+ 1− p follows from (9.tadd).2) i) The solution v, say, obtained in 1) satisfies v(r) ≤ (1 + r)−b/(p−1).

F (r, v) = vqf(r)v−(γ+q) − ν ≥ vqf(r)(1 + r)b(γ+q)/(p−1) − ν.

So, as f(r) > 0 everywhere, there is ν0 := infr>0[f(r)(1 + r)b(q+γ)/(p−1)] suchthat if ν ≤ ν0, then F (r, v) := f(r)v−γ − νvq ≥ 0 and ∂vF (r, v) ≤ 0.Then v is a suitable subsolution of (3.tadd) as the condition (1.tadd) of Theorem 5 of [3]is guaranteed by q > maxp(p− 1)/b,−γ + θ(p− 1)/b (see (φ.tadd) ).If in addition q > (b+ p)(p− 1)/b, then V (r) :=

∫∞r

(∫ t

0(s/t)aF (s, v)ds)µdt is a

supersolution of the equation with rb/(p−1)V (r) bounded.ii) For G(r, φ) := νf(r)φ−γ + φq,

∂φG(r, φ) = qφ−1−γφq+γ − γνf(r)/q := qΦ−1−γΨν(r),

where Ψν(r) := (1 + r)−b(γ+q)/(p−1) − νγf(r)/q.If q > θ(p − 1)/b − γ and φ < (1 + r)−b/(p−1), then for some large R > 0there is Ψν(r) < 0; in this case there is ν1 := sup[0,R] qγ(1 + r)b(γ+q)/(p−1)f(r)such that ν > ν1 implies that G is decreasing in such positive φ. The solution vobtained in 1) is then a suitable supersolution of (4.tadd).

This work is dedicated to my late uncle Toam Chatue J.B., ( † on 14/08/1997).

References

[1] Furusho, Y. On decaying entire positive solutions of semilinear elliptic equations.Japan J. Math. 14 #1 (1988), 97–118.

[2] Kusano T. and Swanson C. A. Radial entire solutions of a class of quasilinear ellipticequations. J. Differential Equations 83 (1990), 379–399.

[3] Tadie Elliptic equations with decreasing nonlinearity I : Barrier method for decreas-ing radial solutions.

[4] Tadie Subhomogeneous and singular quasilinear Emden-type ODE. Preprint # 11,Series 1996, Copenhagen University.



Mathematical Models of Suspension Bridges:

Existence of Unique Solutions

Gabriela Tajcova

Department of Mathematics, Faculty of Applied Sciences,University of West Bohemia,

Univerzitni 22, 306 14 Plzen, Czech RepublicEmail: [email protected]

WWW: http://home.zcu.cz/~gabriela/

Abstract. In this paper, we try to explain two mathematical modelsdescribing a dynamical behaviour of suspension bridges such as TacomaNarrows Bridge. Our attention is concentrated on their analysis concern-ing especially the existence of a unique solution. Finally, we include aninterpretation of particular parameters and a discussion of known andobtained results. This paper is based on our diploma thesis which dealswith a qualitative study of dynamical structures of this type.

AMS Subject Classification. 35B10, 70K30, 73K05

Keywords. Nonlinear beam equation, jumping nonlinearities, periodicoscillations

1 Introduction and historical review

One of the most problematic and not fully explained areas of mathematical mod-elling involves nonlinear dynamical systems, especially systems with so calledjumping nonlinearity. It can be seen that its presence brings into the wholeproblem unexpected difficulties and very often it is a cause of multiple solutions.

An example of such a dynamical system can be a suspension bridge. Thenonlinear aspect is caused by the presence of supporting cable stays which re-strain the movement of the center span of the bridge in a downward direction,but have no influence on its behaviour in the opposite direction.

Our paper sets a goal to develop a simple model describing the behaviourof the suspension bridge, to make its analysis which means to determine underwhat conditions the existence of a unique stable solution is guaranteed, and tofind out safe parameters of the bridge constructions.

We do not try to model the bridge in its full complexity, but on the otherhand, we would like to avoid some over-simplifications. That is why we consideronly one dimensional model and neglect the torsional motion, but we do notsimplify the problem even more — e.g. by eliminating the space variable at all.


http://home.zcu.cz/~gabriela/

282 Gabriela Tajcova

As a result of this effort, we describe the behaviour of the suspension bridgeby one beam equation, or by a system of two coupled equations of “string-beam”type, respectively.

As a motivation of our interest, we can mention the event which changedradically the common view of these nonlinear dynamical systems.

On July 1, 1940, the Tacoma Narrows bridge in the state of Washington wascompleted and opened to traffic. From the day of its opening the bridge beganto undergo vertical oscillations, and it was soon nicknamed “Galloping Gertie”.As a result of its novel behaviour, traffic on the bridge increased tremendously.People came from hundreds of miles to enjoy riding over a galloping, rollingbridge. For four months, everything was all right, and the authorities in chargebecame more and more confident of the safety of the bridge. They were evenplanning to cancel the insurance policy on the bridge.

At about 7:00 a.m. of November 7, 1940, the bridge began to undulate per-sistently for three hours. Segments of the span were heaving periodically up anddown as much as three feet. At about 10:00 a.m., the bridge started suddenlyoscillating more wildly. At one moment, one edge of the roadway was twenty-eight feet higher than the other; the next moment it was twenty-eight feet lowerthan the other edge. At 10:30 a.m. the bridge began cracking, and finally, at11:00 a.m. the entire structure fell down into the river.

The federal report on the failure of the Tacoma Narrows suspension bridgepoints out that the essentially new feature of this bridge was its extreme flexibil-ity. Already, the Golden Gate bridge exhibited travelling waves, or in the BronxWhitestone Bridge, large amplitude oscillations were observed of such a mag-nitude to make a traveller seasick. But due to a combination of damping andreadjusted stays, they were not considered threatening to the structure.

As soon as the more flexible Tacoma Narrows bridge was built, it began toexhibit complex oscillatory motion with an order of magnitude higher than thatof earlier mentioned bridges. This resulted in a pronounced tendency to oscillatevertically, under widely differing wind conditions. The bridge might be quiet inwinds of forty miles per hour, and might oscillate with large amplitude in windsas low as three or four miles per hour. These vertical oscillations were standingwaves of different nodal types. They were not considered to be dangerous, and itwas expected that the bridge would be stabilized by a combination of the samedevices as in case of the Bronx Whitestone bridge.

The second type of oscillation was observed just before the collapse of thebridge. It was a pronounced torsional mode with some of the cables alternatelyloosening and tightening. Sometimes the oscillations even preferred one end ofthe bridge to the other. These phenomenons caused that a large portion of thecenter span fell into the river.

Subsequently, the entire structure was destroyed, and a new, much moreexpensive bridge of more conventional and less flexible design was built in itsplace.

Mathematical Models of Suspension Bridges 283

The first standard explanation (see e.g. M. Braun [4]) claims that the fre-quency of a periodic force caused by alternating trailing vortices just happenedto be very close to the natural frequency of the bridge, and caused the linearresonance. Thus, even though the magnitude of the forcing term was small, thiscould explain the large oscillations and eventual collapse of the bridge.

However, the federal report includes the following paragraph:

“It is very improbable that resonance with alternating vortices playsan important role in the oscillations of suspension bridges. First, it wasfound that there is no sharp correlation between wind velocity and os-cillation frequency, as is required in the case of resonance with vorticeswhose frequency depends on the wind velocity. Second there is no evi-dence for the formation of alternating vortices at a cross section similarto that used in the Tacoma bridge . . . It seems that it is more correctto say that the vortex formation and frequency is determined by theoscillation of the structure than that the oscillatory motion is inducedby the vortex formation.”

But the precise cause of the large-scale oscillations of suspension bridges has notbeen satisfactorily explained yet.

The aspect which distinguishes the suspension bridges is their fundamen-tal nonlinearity. As we have mentioned above, it is caused by the presence ofsupporting cable stays which restrain the movement of the center span in a down-ward direction, but have no influence on its behaviour in the opposite direction.

This type of nonlinearity, often called jumping or asymmetric, has given riseto the following principle:

Systems with asymmetry and large uni-directional loading tend to havemultiple oscillatory solutions: the greater the asymmetry, the larger thenumber of oscillatory solutions, the greater the loading, the larger theamplitude of the oscillations.

As we mentioned above, our paper tries to analyze such nonlinear dynamicalsystems and to bring some new pieces of information into this area.

First of all, we present two possibilities how to model suspension bridges —by a single beam and by a beam coupled with a vibrating string by nonlinearcables — and give a brief survey of known facts in this field.

Then we introduce our own results concerning existence and uniqueness oftime-periodic solutions of two chosen models. We use two different attitudes.The first one is based on the Banach contraction theorem which needs somerestrictions on the bridge parameters. The second one works in relatively greatergenerality but with an additional assumption of sufficiently small external forces.

In the end, we summarize our intention and results and make a short discus-sion where we compare our foundations with known facts.

We would like to emphasis that this paper is a short abstract of our diplomathesis [21] and that is why it does not contain proofs of the assertions statedhere.


2 Mathematical models and known results

One of the easiest ways how to model a behaviour of a suspension bridge is toconsider only one dimension. We do not have to take into account the othertwo dimensions because proportions of the bridge in these dimensions are verysmall in comparison with its length and so can be omitted (see Fig. 1). If wealso neglect the influence of the towers and side parts, we can use a model ofa simply supported one-dimensional beam.

center-span

road-bedside-span side-span

side-cableside-cable

cable stays

main cable

towers

6

? ?

663

QQQk

ZZZZZZ

>

+

QQQQQs

@@@@@R

@@@@@@@@R

Fig. 1. The main ingredients in a model of a one-dimensional suspension bridge.

2.1 Single beam equation

In the first idealization, the construction holding the cable stays can be taken asa solid and immovable object. Then we can describe the behaviour of the sus-pension bridge by a vibrating beam with simply supported ends. It is subjectedto the gravitation force, to the external periodic force (e.g. due to the wind) andin an opposite direction to the restoring force of the cable stays hanging on thesolid construction. Our system is illustrated on Fig. 2.

The displacement u(x, t) of this beam is described by nonlinear partial dif-ferential equation:

m∂2u(x, t)∂t2

+ EI∂4u(x, t)∂x4

+ b∂u(x, t)∂t

= −κu+(x, t) +W (x) + εf(x, t), (1.taj)

with the boundary conditions

u(0, t) = u(L, t) = uxx(0, t) = uxx(L, t) = 0,u(x, t+ 2π) = u(x, t), −∞ < t <∞, x ∈ (0, L).

(2.taj)


An immovable object

A bending beam with supported ends@@I 6

Nonlinear springs under tension

HHHHHHj

CCCW

Fig. 2. The simplest model of a suspension bridge — the bending beam withsimply supported ends, held by nonlinear cables, which are fixed on an immovableconstruction.

The meaning of particular parameters used in the equation is the following:

m mass per unit length of the bridge,E Young’s modulus,I moment of inertia of the cross section,b damping coefficient,κ stiffness of the cables (spring constant),W weight per unit length of the bridge,εf external time-periodic forcing term (due to the wind),L length of the center-span of the bridge.

As we can see from the equation (1.taj) and the boundary conditions (2.taj), weare describing vibrations of a beam of length L, with simply supported ends.Its deflection u(x, t) at the point x and at time t is measured in the downwarddirection. The first term in the equation represents an inertial force, the secondterm is an elastic force and the last term on the left hand side describes a viscousdamping. On the right hand side, we have the influence of the cable stays, thegravitation force and the external force due to the wind (we assume it to be time-periodic). The cable stays can be taken as one-sided springs, obeying Hooke’slaw, with a restoring force proportional to the displacement if they are stretched,and with no restoring force if they are compressed. This fact is described by theexpression κu+, where u+ = max0, u and κ is a coefficient, which characterizesthe stiffness of the cable stays.

We have not considered the inertial effects of the rotation motion (in a planexu) in the equation since they are usually omitted.

This model was introduced e.g. in a paper [16] by P. J. McKenna and A. C. La-zer and is used as a starting point for study of suspension bridges in the most ofcited works by the other authors. It does not describe exactly the behaviour ofa suspension bridge but on the other hand it is reasonably simple and applicable.


For further considerations, it would be useful to transform the equation (bymaking a change of the scale of the variable x and dividing by the mass m) tothe following form:

utt + α2uxxxx + βut + ku+ = W (x) + εf(x, t),u(0, t) = u(π, t) = uxx(0, t) = uxx(π, t) = 0, (3.taj)u(x, t+ 2π) = u(x, t), −∞ < t <∞, x ∈ (0, π),

where α2 = EIm

(πL

)4 6= 0 and β = bm > 0. (We use the same symbols for rescaled

W , ε and f .)

As for as the results, which are known for this model, we can mention thetheorem proved (by the degree theory) in [5] by P. Drabek.

It says that the problem (3.taj) has at least one solution for an arbitrary righthand side. Further, there is proved that in case that there is no external force (itmeans no wind), the bridge achieves a unique position (called the equilibrium)determined only by its weight W (x). Under some special assumptions on W (x),the paper [5] shows that in case of small external disturbances, there is alwaysa solution “near” to the equilibrium. If we assume that W (x) = W0 sinx anda periodic function f(x, t) is of a special form then there is another solutionwhich is in a certain sense “far” from this position.

Another known result concerns the case when the damping term is equal tozero. This was studied by W. Walter and P. J. McKenna in paper [19]. Underan additional assumption α = 1 they proved the theorem which says that ifW (x) ≡ W0 (positive constant) and f(x, t) is even and π-periodic in the timevariable t and symmetric in the space variable x about π

2 , then, if 0 < k < 3, theequation (3.taj) has a unique periodic solution of the period π, which correspondsto small oscillations about the equilibrium. If 3 < k < 15, the equation has inaddition another periodic solution with a large amplitude.

In other words, this theorem says that strengthening the stays, which meansincreasing the coefficient k, can paradoxically lead to the destruction of thebridge.

The similar result can be proved for the system of ordinary differential equa-tions which we obtain from the equation (3.taj) using the spatial discretization. Thetheorem proved in [1] by J. M. Alonso and R. Ortega says that if the condition

k < β2 + 2αβ

holds then there exists N0 ∈ N such that if N ≥ N0 then the discretization ofa suspension bridge equation has a unique bounded solution that is exponentiallyasymptotically stable in the large.

This result has a similar sense as the previous one — the more flexible thecable stays are, then the better the situation is and oscillations of the bridgecannot be too high.


2.2 “String-beam” system

Another possible but a little more complicated process is not to consider theconstruction holding the cable stays as an immovable object, but to treat it asa vibrating string, coupled with the beam of the roadbed by nonlinear cablestays (see Fig. 3).

6?

6

?

v(x, t)

u(x, t)

The cable represented by a vibrating string

The vibrating beam with supported ends

Nonlinear springsCCCCCCCO

SSSSSSSo

AAAU

AAAK

Fig. 3. A more complicated model of a one-dimensional suspension bridge — thecoupling of the main cable (a vibrating string) and the roadbed (a vibrating beam)by the stays, treated as nonlinear springs.

Instead of one equation, we have now a system of two connected equationsin the following form:

m1vtt − Tvxx + b1vt − κ(u− v)+ = W1 + εf1(x, t),

m2utt + EIuxxxx + b2ut + κ(u− v)+ = W2 + εf2(x, t),(4.taj)

with boundary conditions

u(0, t) = u(L, t) = uxx(0, t) = uxx(L, t) = v(0, t) = v(L, t) = 0,

where v(x, t) measures the displacement of the vibrating string representing themain cable and u(x, t) means — as in the previous section — the displacementof the bending beam standing for the roadbed of the bridge. Both functions areconsidered to be periodic in the time variable. The nonlinear stays connectingthe beam and the string pull the cable down, hence we have the minus sign infront of k(u − v)+ in the first equation, and hold the roadbed up, therefore weconsider the plus sign in front of the same term in the second equation.

We can transform both equations into a simpler form in the same way as inthe previous section. It means that we divide by the mass m1, and m2 respec-tively, and change the scale of the space variable x. Then we obtain


vtt − α21vxx + β1vt − k1(u− v)+ = W1 + εf1(x, t),

utt + α22uxxxx + β2ut + k2(u− v)+ = W2 + εf2(x, t),

u(0, t) = u(π, t) = uxx(0, t) = uxx(π, t) = v(0, t) = v(π, t) = 0,−∞ < t <∞, x ∈ (0, π),

(5.taj)

where α21 = T

m1

(πL

)2, α22 = EI

m2

(πL

)4, k1 = κm1

, k2 = κm2

, β1 = b1m1

and β2 = b2m2

.We use the same symbols as in the previous equations for the other transformedparameters.

We can find a description of this model again in A. C. Lazer and P. J. McKen-na [16], but these authors consider the right hand sides in a rather purer form.In the first equation, they neglect the weight of the string W1, and on the otherhand, in the second equation, they ignore the external force εf2(x, t). However,nobody (as far as we know) has treated this model in detail yet.

3 Application of Banach contraction principle

As we can see from the previous survey of known results, one of the problemsis to prove the existence of the solutions of particular models and find out theconditions, under which the solution is unique and stable. In particular, it meansthat we are looking for conditions which guarantee that the bridge cannot exhibitlarge-scale oscillations and cannot be destructed by any wind of an arbitrarypower. We have tried to clear up these problems with use of Banach contractionprinciple for both one-dimensional models — the first one considers the bridgeas a single beam supported by nonlinear springs, and the second one describesthe bridge as a beam coupled with a string by nonlinear cables.

3.1 The first case — a single beam

As we stated above, we model the suspension bridge as a one-dimensional beamwith simply supported ends, which is held by nonlinear springs hanging on an im-movable construction. This situation is described by the boundary value prob-lem (3.taj).

Let us denote Ω = (0, π) × (0, 2π) the considered domain, H = L2(Ω) theusual Hilbert space with the corresponding L2-norm

‖u(x, t)‖ =[∫

Ω

|u(x, t)|2dxdt] 1

2

and D the set of all smooth functions satisfying the boundary conditions fromequation (3.taj). Now we can generalize the notion of a classical solution by whichwe mean a continuous function with continuous derivatives up to the fourthorder with respect to x and up to the second order with respect to t in the set[0, π]× [0, 2π], satisfying the boundary value problem (3.taj), and define a so calledgeneralized solution of (3.taj).


Definition 1. A function u(x, t) ∈ H is called a generalized solution of theboundary value problem (3.taj) if and only if the integral identity∫

Ω

u(vtt + α2vxxxx − βvt) dxdt =∫Ω

(W + εf − ku+)v dxdt

holds for all v ∈ D.

Remark 2. We can extend the generalized solution u = u(x, t) by 2π-periodicityin t to (0, π) × R. So, any generalized solution can be regarded as a functiondefined on (0, π)× R.

Let us consider a complex Sobolev space H = H + iH . As the set

eint sinmx;n ∈ Z,m ∈ N

forms a complete orthogonal system in this space, each function u(x, t) can berepresented by Fourier series

u(x, t) =∞∑

n=−∞

∞∑m=1

unmeint sinmx. (6.taj)

Moreover, we have∑n

∑m

|unm|2 <∞, and u−nm = unm

(see J. Berkovits and V. Mustonen [3]).Let p, r ∈ Z+. If we use this Fourier interpretation, we can define the following

spaces

Hp,r = h ∈ H ;∞∑

n=−∞

∞∑m=1

(n2r +m2p)|hnm|2 <∞ (7.taj)

and the corresponding norm

‖h‖Hp,r =

( ∞∑n=−∞

∞∑m=1

(n2r +m2p)|hnm|2) 1

2

. (8.taj)

Then Hp,r equipped with the norm ‖ · ‖Hp,r is the Sobolev space. In particular,H0,0 = H .

First of all, we will treat the solvability of the linear equation

utt + α2uxxxx + βut − λu = h. (9.taj)

If we define a generalized solution of this equation in an analogous way as inDefinition 1, then the following lemma is a consequence of the expansion (6.taj)(cf. [2]).


Lemma 3. If unm and hnm are the corresponding Fourier coefficients of thefunctions u and h, then the equation (9.taj) has a generalized solution if and only if

(−n2 + α2m4 + iβn − λ)unm = hnm (10.taj)

holds for all n ∈ Z, m ∈ N.

If we denoteL(u) = utt + α2uxxxx + βut

the linear operator, and put

Nλ = (m,n) ∈ N× Z; α2m4 − n2 − λ = 0,

S = λ ∈ R; Nλ 6= ∅,σ = λ ∈ R; λ = α2q4, q ∈ N,

then σ is a set of eigenvalues of the operator L, and σ ⊂ S holds. Further, wecan rewrite the equation (9.taj) into a new form

L(u)− λu = h

and formulate the following theorem (for the proof see G. Tajcova [20]).

Theorem 4. Let λ ∈ R. Then for an arbitrary h ∈ H the equation (9.taj) hasa unique generalized solution u ∈ H if and only if

λ 6∈ σ.

If λ 6∈ σ, then there exists a mapping

Tλ : H → H, Tλ : h 7→ u

with the following properties:

(i) Tλ is linear and R(Tλ) ⊂ C(Ω);(ii) Tλ : Hp,r → Hp+2,r+1 and there exists a constant c > 0 such that for any

h ∈ Hp,r, p, r ∈ N ∪ 0, we have

‖u‖Hp+2,r+1 ≤ c‖h‖Hp,r ,

whenever u = Tλh;(iii) Tλ is compact from H into C(Ω) (and thus from H into H) and for its

norm we have

‖Tλ‖ ≤1

maxdist(λ, S), minβ, dist(λ, σ) =

=1

mindist(λ, σ), maxβ, dist(λ, S) .


Now we turn our attention to the equation (3.taj) and deal with its solvability.As zero is not an eigenvalue of the operator L, we can rewrite this equation

— in accordance with the previous paragraph — into an equivalent form

u = T0(−ku+ +W + εf). (11.taj)

Moreover, we have for the norm of the operator T0 the following estimate

‖T0‖ ≤ maxm∈N, n∈Z

1√β2n2 + (α2m4 − n2)2

≤ 1minα2, β = K0.

If we want to find out conditions for the existence of a unique solution, it issuitable to use the Banach contraction principle which reads as follows:

Let the operator G : H → H be a contraction, i.e. there exists c ∈ (0, 1)such that

‖G(u)−G(v)‖ ≤ c‖u− v‖ ∀u, v ∈ H.

Then there exists a unique u0 such that

G(u0) = u0.

In our case G(u) = T0(−ku+ +W + εf) and

‖G(u)−G(v)‖ = ‖T0(W + εf − ku+)− T0(W + εf − kv+)‖ == ‖T0(kv+ − ku+)‖ ≤≤ k‖T0‖‖v+ − u+‖ ≤≤ kK0‖v − u‖.

If we require the operator G to be a contraction, the condition

0 < kK0 < 1

must be satisfied, and thus

0 <k

minα2, β < 1.

Hence, if we put again k = κm , a sufficient condition for the existence of a unique

solution of our boundary value problem has a form

κ < m ·minα2, β. (12.taj)


3.2 The second case — the coupling of a beam and a string

In this part we complete the previous model by a movable main cable, whichholds the nonlinear cable stays and which is represented by a vibrating string.Our model is described by a coupled system of partial differential equations (5.taj)(see A. C. Lazer, P. J. McKenna [16])

If we introduce a new vector function

w =[vu

], (13.taj)

we can rewrite the system (5.taj) into the following matrix form[1 00 1

]︸︷︷︸

I

wtt +[

0 00 α2

2

]︸︷︷︸

A2

wxxxx +[−α2

1 00 0

]︸︷︷︸

A1

wxx +

+[β1 00 β2

]︸︷︷︸

B

wt + F(w) =[h1

h2

]︸︷︷︸

h

, (14.taj)

and thus

wtt + A2wxxxx + A1wxx + Bwt + F(w) = h, (15.taj)

where F(w) is a nonlinear vector function

F(w) =[−k1(u − v)+

k2(u − v)+

].

Moreover, we require the unknown function w(x, t) to be time-periodic andto satisfy the boundary conditions prescribed for a vibrating string in its firstcomponent, and the boundary conditions prescribed for a supported beam in itssecond component.

Let us denote

L(w) = wtt + A2wxxxx + A1wxx + Bwt.

Then L is a linear operator and the equation (15.taj) can be written in the followingway

L(w) = −F(w) + h. (16.taj)

The set of all real eigenvalues of the operator L has a form

σ = λ ∈ R; λ = α21m

2 ∨ λ = α22m

4, ∀m ∈ N.

Now, we have our system described by an operator equation which has a sim-ilar character as the operator equation for the single beam. It allows us to use thesame methods and formulate the analogous statements as in previous section.


We can again define the notion of generalized solution and use the Fourierrepresentation of the considered functions. Moreover, we can as well prove theexistence of the resolvent Tλ (see G. Tajcova [20]).

Theorem 5. Let λ ∈ R. Then for an arbitrary h ∈ H the equation Lw −λw = h has a unique solution w ∈ H if and only if

λ 6∈ σ.

If λ 6∈ σ then there exists the mapping

Tλ : H→ H, Tλ : h 7→ w

with the following properties:

(i) Tλ is linear and ImTλ ⊂ C(Ω)× C(Ω);(ii) Tλ : Hp,r ×Hp,r → Hp+1,r+1 ×Hp+2,r+1 and there exists a constant c > 0

such that for any h ∈ Hp,r × Hp,r, p, r ∈ N ∪ 0, we have

‖w‖Hp+1,r+1×Hp+2,r+1 ≤ c‖h‖Hp,r×Hp,r ,

whenever w = Tλh.(iii) Tλ is compact from H into C(Ω) × C(Ω) (and thus from H into H), and

for its norm we have an estimate

‖Tλ‖ ≤ max

maxm,n

1|Aλnm|

; maxm,n

1|Bλnm|

,

where Aλnm = −n2 + α21m

2 + iβ1n − λ,Bλnm = −n2 + α2

2m4 + iβ2n − λ.

As zero is not the eigenvalue of the operator L, we can define the operatorT0 and to estimate its norm as follows

‖T0‖ ≤ max

maxm,n

1|A0nm|

; maxm,n

1|B0nm|

.

Further,

maxm,n

1|A0nm|

= maxm,n

1| − n2 + α2

1m2 + iβ1n|

= maxm,n

1√β2

1n2 + (α2

1m2 − n2)2

≤

≤ 1minα2

1, β1,

maxm,n

1|B0nm|

= maxm,n

1| − n2 + α2

2m4 + iβ2n|

= maxm,n

1√β2

2n2 + (α2

2m4 − n2)2

≤

≤ 1minα2

2, β2.


Hence we finally obtain

‖T0‖ ≤ max

1minα2

1, β1;

1minα2

2, β2

=

=1

min α21, α

22, β1, β2

= K0. (17.taj)

If we use this operator T0, we can rewrite our equation (16.taj) in the equivalentform

w = T0(h− F(w)). (18.taj)

Since we want to prove its unique solvability, it is again suitable to apply theBanach contraction principle.

In our case G(w) = T0(h−F(w)). We have to verify, whether this operatoris a contraction:

‖G(w1)−G(w2)‖ = ‖T0(h− F(w1))−T0(h− F(w2))‖ == ‖T0‖‖F(w2)− F(w1)‖ ≤≤ ‖T0‖(k1 + k2)‖(u2 − v2)+ − (u1 − v1)+‖ ≤≤ ‖T0‖(k1 + k2)‖(u2 − v2)− (u1 − v1)‖ == ‖T0‖(k1 + k2)‖(u2 − u1)− (v2 − v1)‖ ≤≤ ‖T0‖(k1 + k2) [‖u2 − u1‖+ ‖v2 − v1‖] ≤≤ (k1 + k2)K0‖w2 −w1‖.

Hence it follows that the operator G is a contraction if the condition

0 < (k1 + k2)K0 < 1

holds. Equivalently,

k1 + k2 < minα2

1, α22, β1, β2

.

As we have k1 = κm1

, k2 = κm2, we obtain a condition of the existence of

a unique solution of the operator equation (16.taj) in the following form

κ <m1m2

m1 +m2min

α2

1, α22, β1, β2

. (19.taj)

Remark 6. The question left is whether the condition (19.taj) is stronger or weakerthan the condition

κ = m2k < m2 minα22, β2,

obtained by the same way for the bridge modelled only as a supported beam(i.e. by a scalar equation — see (12.taj)), and whether they have any practical sense.


4 General existence and uniqueness result

In the previous chapter, we proved the existence and uniqueness of the solu-tion, but the price we had to pay was a certain restriction on the coefficient κrepresenting the stiffness of the nonlinear cable stays. On the other hand, theadvantage was an arbitrary right hand side. Now we can convert the situationand prove the mentioned existence and uniqueness of the solution in a relativegenerality of the structure coefficient, but with some special assumptions on theexternal forcing terms.

We again pay our attention to two chosen mathematical models of suspensionbridges. The first one consists of the single beam equation and the second onerespects the coupling of the main cable and the roadbed — i.e. the string-beamsystem.

4.1 The first case — a single beam

We again consider the periodic-boundary value problem (3.taj) for the beam equa-tion which serves as a simple one-dimensional model of a suspension bridge.

Before we state our main result, we formulate some auxiliary assertions whichare necessary for its full understanding and which are proved by J. Berkovits,P. Drabek, H. Leinfelder, V. Mustonen and G. Tajcova in [2].

Proposition 7. Let u ∈ H and h ∈ H, h is independent of t. Then u is a uniquegeneralized solution of

utt + α2uxxxx + βut + ku+ = h(x) (20.taj)

if and only if the function u is independent of the variable t and u(x) = u(x, t)is a classical solution of the boundary value problem

αu(4) + ku+ = h(x) in (0, π),u(0) = u(π) = u′′(0) = u′′(π) = 0.

(21.taj)

Under even more special assumption that the right hand side is a constantfunction, we can prove some other properties of the generalized solution.

Proposition 8. Assume in (3.taj) that ε = 0 and W (x) ≡W0 (nonzero constant).Then the corresponding generalized solution u0 of (3.taj) is unique, positive, time-independent, symmetric with respect to the line x = π

2 and satisfies

(u0)x(0, t) > 0, (u0)x(π, t) < 0 (22.taj)

for every t ∈ R.

Remark 9. In particular, this means that the equation

utt + α2uxxxx + βut + ku+ = 0

has due to uniqueness only a trivial generalized solution for any k ∈ R.


Our main result is the following.

Theorem 10. Let ε ∈ R, k > 0, W (x) ≡ W0 > 0, f ∈ H1,1. Then there existsε0 > 0 such that for |ε| < ε0 the problem (3.taj) has a unique generalized solutionu ∈ H3,2. Moreover, this generalized solution is strictly positive in (0, π)× R.

The proof of this main result would be carried out in several steps. We knowthat there exists at least one generalized solution of the equation (3.taj) for anyright hand side (see P. Drabek [5]). Moreover, by Proposition 8, there existsa positive, time-independent solution u0(x, t) = u0(x) of the equation

utt + α2uxxxx + βut + ku+ = W0,

with u′0(0) > 0 and u′0(π) < 0.

Step 1. We prove that there exists a positive generalized solution u ∈ H3,2

of (3.taj) which is “close” to u0 from Proposition 8 with respect to the norm inH3,2.

Step 2. There is no other positive generalized solution of (3.taj) than u =u0 + uε.

Step 3. There is no other generalized solution of (3.taj) (changing signs) thanuε = u0 + uε if |ε| < ε0 and ε0 is small enough.

(For the complete proof see G. Tajcova [21] or J. Berkovits, P. Drabek, H. Lein-felder, V. Mustonen and G. Tajcova [2].)

4.2 The second case — the coupling of a beam and a string

Now, we can try to apply the previous ideas on the system of two coupledequations which model the suspension bridge as a simply supported beam anda string connected by nonlinear cable stays.

We work again with a periodic-boundary value problem (5.taj). We would liketo formulate a similar assertion as in the previous section, it means to proveunder some additional assumptions that if the weight of the bridge W1 andthe weight of the main cable W2 are constant and the external forces εf1(x, t)and εf2(x, t) are sufficiently small, then our problem (5.taj) has a unique solutionwhich is symmetric and strictly positive in its both components and close to thestationary solution. However, as it can be seen later, we are not able to overcomesome problems with regularity of the solution and thus we formulate statementswhich are more general and — in some sense — weaker.

Similar argument as that used in [2] enables us to prove the following asser-tion.


Proposition 11. Let u, v ∈ H and h1, h2 ∈ H, h1, h2 are independent of t.Then [v, u]T is a generalize solution of

vtt − α21vxx + β1vt − k1(u− v)+ =h1(x),

utt + α22uxxxx + β2ut + k2(u− v)+ =h2(x)

(23.taj)

if and only if the functions v, u are independent of the variable t and [v(x), u(x)]T

= [v(x, t), u(x, t)]T is a solution of the boundary value problem

−α1v′′ − k1(u− v)+ = h1(x),

α2u(4) + k2(u− v)+ = h2(x) in (0, π), (24.taj)

v(0) = v(π) = u(0) = u(π) = u′′(0) = u′′(π) = 0.

As for as the uniqueness of the solution, the following statement holds.

Proposition 12. Let k1, k2 > 0 and h1, h2 ∈ H, h1 and h2 are independentof t. Then (23.taj) has at most one generalized solution w0 = [v0, u0]T ∈ H whichis independent of t.

Remark 13. As a consequence of Propositions 11 and 12, we can state that forε = 0 and W1 = W2 = 0 (it means no loading), the nonlinear system (5.taj)

vtt − α21vxx + β1vt − k1(u− v)+ = 0,

utt + α22uxxxx + β2ut + k2(u − v)+ = 0,

with standard string-beam boundary conditions, has only a trivial solution.

Now, we have all auxiliary assertions to formulate the following theoremconcerning the general existence of a solution of the system (5.taj) for an arbitraryright hand side. The proof is based on the degree theory and is a direct analogyto the proof by P. Drabek in [5].

Theorem 14. Let ε ∈ R, k1, k2 > 0, W1(x), W2(x) ∈ L2(0, π), and f1(x, t),f2(x, t) ∈ H. Then the system (5.taj) has at least one generalized solution w =[v, u]T ∈ H.

Now, we can have a look at the case when the right hand sides are constantfunctions. It means that the corresponding solution is (according to Proposi-tion 11) a stationary solution and should express the equilibrium of the suspen-sion bridge.

By a detailed analysis of a linear system

−γ1v′′ − u+ v = h1, x ∈ (0, π),

γ2u(4) + u− v = h2, (25.taj)

v(0) = v(π) = u(0) = u(π) = u′′(0) = u′′(π) = 0

we can prove the following assertion.


Proposition 15. Assume in a boundary value problem (5.taj) that W1(x) ≡ W1

and W2(x) ≡ W2 are nonzero constants and ε = 0. Moreover, let the weightW2 is “large enough”. Then (5.taj) has a unique generalized solution w0 which ispositive, time-independent, symmetric with respect to the line x = π

2 in its bothcomponents and satisfies

u0(x, t) > v0(x, t) ∀(x, t) ∈ (0, π)× R

and(u0 − v0)x(0, t) > 0, (u0 − v0)x(π, t) < 0

for every t ∈ R.

Now, we can have a closer look at the solution of the system (5.taj) and its prop-erties. We would like — on the basis of the previous statements — to formulatethe analogy of Theorem 10.

However, the only thing we know is that there exists at least one generalizedsolution of the boundary value problem (5.taj) and, moreover, (see Proposition 15),that under some additional assumptions, there exists a symmetric, strictly pos-itive, time-independent solution w0 = [v0, u0]T of the system

vtt − α21vxx + β1vt − k1(u− v)+ = W1,

utt + α22uxxxx + β2ut + k2(u − v)+ = W2,

where W1 and W2 are positive constants, and the conditions

(u0 − v0)x(0, t) > 0, (u0 − v0)x(π, t) < 0,

hold.But we are not able to prove the existence and uniqueness of the solution of

the system (5.taj) which would be “close” to this w0. The obstacle is the fact thatwe have not manage to prove a better regularity that w ∈ H2,2×H3,2. It means(due to embedding theorems) that w ∈ C0,0 × C1,0. And this is not enough toguarantee the existence of the positive solution

w = w0 + wε

neither for ε sufficiently small.

5 Final remarks and discussion

In this chapter, we would like to clear up our results and compare them withknown facts mentioned in Chapter 2.

Our main effort was to determine sufficient conditions for the existence anduniqueness of the solution. Let us have a closer critical look at them.


5.1 The application of Banach contraction principle

First of all, we dealt with the uniqueness using the Banach contraction principle.The price we had to pay was a certain restriction on the magnitude of the stiffnessκ of the cable stays.

For the single beam model, it was in the roughest form (cf. (12.taj))

κ < m minα2, β. (26.taj)

The corresponding result for the string-beam model was (cf. (19.taj))

κ <m1m2

m1 +m2minα2

1, α22, β1, β2. (27.taj)

In Chapter 2, we mentioned two similar results. The first one was obtainedby W. Walter and P. J. McKenna in [19] for a non-damped single beam modelunder an additional assumption α = 1. It says that the solution of such a systemis unique in case that

0 < k < 3, (28.taj)

where k = κ/m.The second result was obtained by J. M. Alonso and R. Ortega in [1] for

a discrete system of ordinary differential equations derived from a damped singlebeam model using the spatial discretization by finite differences. It says againthat the solution of such a system is unique if

k < β2 + 2αβ. (29.taj)

We have again k = κ/m.

Obviously, all these results have a similar sense — the more flexible the cablestays are, the better the situation is, because the nonlinearity is less pronouncedand we have guaranteed the uniqueness of the solution.

We can make short discussion where we compare our result (26.taj) with (29.taj)derived by J. M. Alonso and R. Ortega, and the results for single beam with thatones for a string-beam model.

We ask whether the result (29.taj)

k < β2 + 2βα

by J. M. Alonso, R. Ortega is stronger or weaker than our relation (26.taj) whichcan be formulated as

k < minα2, β.

(In both cases, k = κ/m, where κ is the stiffness of the cable stays and m is themass of the bridge.)

We can make the following discussion.


(i) If the condition

α ∈(

0;β +√

2β⟩∪⟨

1− β2

;∞)

is satisfied, which means (in an equivalent form)

β ∈⟨√

2α− α;∞)∪ 〈1− 2α;∞) ,

(see Fig. 4), then the implication

k < minα2, β =⇒ k < β2 + 2αβ

holds and the result of J. M. Alonso and R. Ortega is stronger than (26.taj).

-

6β

α

AAAAAAAAAAAAAA

!!!!!!

!!!!!!

!!!!

!!!!!β =√

2α− α

β = 1− 2α

Fig. 4. The shaded region where the result of J. M. Alonso and R. Ortega isstronger than our condition.

(ii) But if the condition

α ∈⟨β +√

2β;1− β

2

⟩is satisfied, which again means

β ∈(

0;√

2α− α⟩∩ (0; 1− 2α〉 ,

(see Fig. 5), then the implication

k < β2 + 2αβ =⇒ k < minα2, β

holds and our result (26.taj) is stronger than that of J. M. Alonso and R. Ortega.


-

6β

α

AAAAAAAAAAAAAA

!!!!!!

!!!!!!

!!!!

β =√

2α− α

β = 1− 2α

Fig. 5. The shaded region where our condition is stronger than the result ofJ. M. Alonso and R. Ortega.

Remark 16.

1. By physical reason we take into account only positive values of the parame-ters α and β.

2. In particular, the previous discussion means that for sufficiently small α andβ in a certain relation, our result is stronger than the result published inJ. M. Alonso, R. Ortega [1].

3. The question left concerns the real values of bridge parameters.

Now, we can have a look at the conditions (26.taj) and (27.taj). It means to comparethe relation

κ < m2 minα22, β2,

obtained for the single beam model (we use the notation m = m2, α = α2,β = β2), with the condition

κ <m1m2

m1 +m2minα2

1, α22, β1, β2

concerning the string-beam model.We can expect that the mass of the main cable m1 will be considerably less

than the mass of the roadbed m2, and thusm1m2

m1 +m2' m1.

The damping coefficients β1 a β2 can be considered almost the same.The relation between α2

1 and α22 is still an open problem for us.

As for the real parameters of particular suspension bridges, we have foundin the paper [7] by A. Fonda, Z. Schneider and F. Zanolin the following values.


Tacoma Golden Gate Bronx-Whitestonem 8.5× 103 kg m−1 3.1× 104 kg m−1 1.6× 104 kg m−1

I 0.2 m4 5.3 m4 0.4 m4

L 855 m 1 280 m 700 m

The acceleration of gravity at earth’s surface and the steel’s modulus ofYoung are usually taken to be

g = 9.8 m s−2,

E = 2× 1011 kg m−1s−2.

However, we still have not found anything about the real values of the stiffnessof the cable stays k, of the inner tension T and the mass m1 of the main cable.

On the other hand, we succeeded to gain approximate values concerninga similar structure — a concrete suspension footbridge. The corresponding pa-rameters are as follows.

m1.= 256 kg m−1,

m2.= 7 300 kg m−1,

L.= 103 m,

E.= 30 000 MPa

I.= 1 m4

T.= 2 708 000 N

κ.= 4.5 105 kg m−1 s−2.

It means that

α21.= 9.8,

α22.= 3.5.

However, we still do not know anything about the damping coefficients β1, β2.

Unfortunately, on basis of this information, we can say that our condi-tions (26.taj), (27.taj) are too restrictive and cannot be satisfied in practice.

Remark 17. Another aspect we have not mentioned so far is the periodicity ofthe external force and of the solution with respect to the time variable. Weassume from the beginning that the period is equal to 2π. Of course, the realityis a little bit different and if we consider a different period, we can obtain newvalues of the mentioned parameters and the situation can change considerably.

5.2 The general existence and uniqueness

Now, we can make a short discussion about our main result of Chapter 4, whichis summed up in Theorem 10. It says that in case of constant weight and smallexternal force (e.g. due to the wind), the bridge stays in a unique position nearthe equilibrium.


This is surprisingly different from previous results obtained in this direction.

This problem was also studied under the special assumption that the weightof the bridge has a form W (x) = W0 sinx. Moreover, it was assumed that theexternal force f(x, t) = f(t) sinx, as well as the displacement of the bridgeu(x, t) = y(t) sinx have a similar character.

This assumptions lead to an ordinary differential equation and results ob-tained by A. C. Lazer and P. J. McKenna in [15], [16], [17], and by J. Glover,A. C. Lazer and P. J. McKenna in [10] are, roughly speaking, of the followingspirit:

Even if the external force is small enough, the system admits at least two(small amplitude and large amplitude, asymptotically stable) solutions.

These results are illustrated by several interesting numerical experiments (seee.g. A. C. Lazer, P. J. McKenna [16], J. Glover, A. C. Lazer, P. J. McKenna [10],A. Fonda, Z. Schneider, F. Zanolin [7], etc.).

However, from the practical point of view, the assumptions on W and f seemto be somewhat peculiar and it seems to be more natural to assume that theweight of the roadbed is constant along the bridge instead of having distribu-tion as a function W0 sinx. This, more natural situation is discussed in [19] byP. J. McKenna and W. Walter. However, also in this case the problem is notstudied in its full generality and some oversimplifications are made. First of all,the authors neglect the damping term. Second, the data as well as the solutionare considered in the space of functions with certain symmetries with respect toboth variables x and t.

The main result is summed up in [19] and says that under the assumptionsmentioned above, the external force sufficiently small, and 3 < k < 15, thenon-damped problem has at least two solutions.

Also this result supports the idea of multiple solutions of a single beam modelunder more general assumptions.

However, our Theorem 10 shows that the presence of nonzero damping inthe model changes the situation qualitatively and we get uniqueness result.

Moreover, our result describes that the problem is well-posed. If there is noexternal disturbance (no wind, no cars driving across the bridge, etc.) then thebridge achieves unique steady state position (the equilibrium) determined onlyby its weight W0. In the case of “small external disturbances” represented bythe term εf(x, t) there is always unique solution which is “near” the steady stateposition when the bridge is not disturbed. This fact illustrates the stability ofthe solution with respect to small perturbations given by εf(x, t).

Of course, there are still many open questions. We can expect that for a cer-tain critical value of parameter ε1 > 0 we have lack of uniqueness of the solutionwhen ε ≥ ε1. Another question concerns asymptotical stability of the uniquesolution.

Unfortunately, for the system of two coupled equations describing the motionof the main cable and of the roadbed, we are not able to obtain a similar result


as in the case of one single beam equation. The only thing we can prove is thegeneral existence of at least one solution for any right hand side.

The problem is the lack of regularity in the string equation and this fact isthe obstacle for the proof of uniqueness. Moreover, it does not allow us to statethat there is at least one solution which is “close” is some sense to the steadystate position (the equilibrium) determined only by the weight of the main cableand by the weight of the roadbed.

This unpleasant problem could be solved for example by the following way.We can put an additional small term

εvxxxx

into the string equation and thus modify the model little bit. The presence ofsuch a term can ensure that we obtain higher degree of regularity and, moreover,it expresses a relatively natural fact that the main cable has some stiffness andit is not only a simple string.

However, we have not considered this situation yet and thus we have no ideawhether adding this term into the model cannot cause some other troubles.

Another element, which could be added into the problem and which wouldhave a reasonable interpretation, is a certain “pretension”. It can be representede.g. by a function h(x) which would appear in the nonlinear terms. It means,that we could replace the term (u − v)+ with the term (u + h − v)+, or — incase of a single beam model — replace the term u+ with the term (u + h)+,respectively. Such a modification can cause that in case of no external force (nowind, no cars driving across the bridge), the beam representing the roadbedachieves a negative (or zero) position. This result would be more realistic sincethe real suspension bridges are never bent in a downward direction if they arein a steady state position.

This pretension was considered e.g. by A. Fonda, Z. Schneider and F. Zanolinin [7]. In this paper, the function h(x) was used in a form

h(x) = h sinπx

L

which allowed under some additional assumptions on the right hand side toeliminate the space variable x from the boundary value problem.

It would be interesting to include this term into the models considered in ourpaper as well and find out how it influences our results and whether it can drawthem near to the real behaviour of a suspension bridge.

Acknowledgements. The work was partially supported by the Grant Agencyof Czech Republic, Grant No. 201/97/0395, and by Ministry of Education, GrantNo. 429/1997.

References

[1] J. M. Alonso, R. Ortega: Global asymptotical stability of a forced Newtonian systemwith dissipation, J. Math. Anal. Applications 196 (1995), 965–966.


[2] J. Berkovits, P. Drabek, H. Leinfelder, V. Mustonen, G. Tajcova: Time-periodicoscillations in suspension bridges: existence of unique solution, in preparation.

[3] J. Berkovits, V. Mustonen: Existence and multiplicity results for semilinear beamequations, Colloquia Mathematica Societatis Janos Bolyai, Budapest (1991),49–63.

[4] M. Braun: Differential equations and their applications, Springer-Verlag, New York(1975).

[5] P. Drabek: Jumping nonlinearities and mathematical models of suspension bridges,Acta Math. Inf. Univ. Ostraviensis 2 (1994), 9–18.

[6] P. Drabek: Nonlinear noncoercive equations and applications, Z. Anal. Anwendun-gen 1 (1983), 53–65.

[7] A. Fonda, Z. Schneider, F. Zanolin: Periodic oscillations for a nonlinear suspensionbridge model, Journal of Computational and Applied Mathematics 52 (1994),113–140.

[8] S. Fucık: Nonlinear noncoercive problems, Conf. del Seminario di Mat. Univ. Bari(S. A. F. A. III), Bari (1978), 301–353.

[9] S. Fucık: Solvability of Nonlinear Equations and Boundary Value Problems, D. Rei-del Publishing Company, Dordrecht (1980).

[10] J. Glover, A. C. Lazer, P. J. McKenna: Existence and stability of large scale nonlin-ear oscillations in suspension bridges, J. Appl. Math. Physics (ZAMP) 40 (1989),172–200.

[11] D. Jacover, P. J. McKenna: Nonlinear torsional flexings in a periodically forcedsuspended beam, Journal of Comp. and Applied Mathematics 52 (1994), 241–265.

[12] P. Krejcı: On solvability of equations of the 4-th order with jumping nonlinearities,Cas. pest. mat. (Math. Bohemica) 108 (1983), 29–39.

[13] A. C. Lazer, P. J. McKenna: Fredholm theory for periodic solutions of some semi-linear P.D.Es with homogeneous nonlinearities, Contemporary Math. 107 (1990),109–122.

[14] A. C. Lazer, P. J. McKenna: A semi-Fredholm principle for periodically forced sys-tems with homogeneous nonlinearities, Proc. Amer. Math. Society 106 (1989),119–125.

[15] A. C. Lazer, P. J. McKenna: Existence, uniqueness, and stability of oscillations indifferential equations with asymmetric nonlinearities, Trans. Amer. Math. Society315 (1989), 721–739.

[16] A. C. Lazer, P. J. McKenna: Large-amplitude periodic oscillations in suspensionbridges: some new connections with nonlinear analysis, SIAM Review 32 (1990),537–578.

[17] A. C. Lazer, P. J. McKenna: Large scale oscillatory behaviour in loaded asymmetricsystems, Ann. Inst. Henri Poincare, Analyse non lineaire 4 (1987), 244–274.

[18] H. Leifelder, G. G. Simader: Schrodinger Operators with Singular Magnetic VectorPotentials, Math. Z. 176 (1981), 1–19.

[19] P. J. McKenna, W. Walter: Nonlinear Oscillations in a suspension bridge, Arch.Rational Mech. Anal. 98 (1987), 167–177.

[20] G. Tajcova: Mathematical models of suspension bridges, Preprint University ofWest Bohemia no. 94 (1997), to appear in Applications of Mathematics.

[21] G. Tajcova: Mathematical models of suspension bridges, Diploma thesis, Universityof West Bohemia, (1997) Pilsen.

[22] O. Vejvoda et al.: Partial Differential Equations — time periodic solutions, SijthoffNordhoff, The Netherlands (1981).



Numerical Analysis of High-Temperature Strains

and Stresses in Superalloys

Jirı Vala

J.Vala, software engineering, Zemedelska 10613 00 Brno, Czech Republic


Abstract. In [10] a new micromechanical approach to the prediction ofcreep flow in composites with perfect matrix/particle interfaces, mak-ing use of the nonlinear Maxwell viscoelastic model, taking into accounta finite number of discrete slip systems in the matrix, has been sug-gested; high-temperature creep in such composites is conditioned by thedynamic recovery of the dislocation structure due to slip/climb motionof dislocations along the matrix/particle interfaces. In this contributionthe proper formulation of the system of PDE’s generated by this modelis presented together with the overview of existence and uniqueness re-sults, based on the properties of Rothe sequences, applied in the originalnon-commercial FEM software CDS; complete proofs will be publishedin [15].

AMS Subject Classification. 73F10, 69B11.

Keywords. Viscoelasticity, high-temperature creep, interface diffusion,generation and recovery of dislocations, PDE’s of evolution, method ofdiscretization in time, Rothe sequences.

1 Introduction

The creep resistance of composites (metal reinforced by hard particles) dependsstrongly on the active diffusion processes on the interface between matrix andparticles. Since in standard software packages all such processes are neglected,the conventional time-dependent strain and stress analysis at high temperature(e.g. making use of some “user supplied material description” in the ABAQUSfinite element code as in [8]) gives no satisfactory results. To overcome thisdifficulty, in 1992–97 at the Institute of Physics of Materials of the Academyof Sciences of the Czech Republic in Brno the non-commercial PC softwareCDS (abbreviation for “creep with diffusion and sliding”) in C++ languagefor the study of high-temperature phenomena in materials consisting of severalphases has been developed; its functions have been demonstrated e.g. in [12]. Oneinteresting application of this software to the design of metal matrix composites(where the particles are added into the matrix during solidification) has been


308 Jirı Vala

published in [11]. In this contribution we pay attention to the case of superalloys(where the particles precipitate in the matrix during the heat treatment of thematerial).

The physical background of the processes of elastic deformation, slip in thematrix, deposition of dislocation on matrix/particle interfaces and recovery ofdislocation structure is discussed in details in [10] (making use of some ideasfrom [9]), including the confrontation with other approaches to similar problemsin the literature and software. The access of [10] leads to the formulation of 5types of equations (to these types we shall refer later):

A) equations of the principle of virtual displacement rates for materials consist-ing of two phases,

B) constitutive equations for stress components, making use of the serial Max-well model with one linear elastic and one nonlinear viscous parts,

C) compatibility conditions for geometrical configuration on the matrix/particleinterfaces,

D) equations for the kinetics of the displacement between the matrix and theparticle,

E) equations for the evolution of the dislocation density.

Our aim will be to formulate correctly this system of PDE’s of evolution and toverify its solvability and convergence of sequences of Rothe functions, using themethod of discretization in time.

The limited extent of this CD ROM text file (documenting the author’s com-munication at the 9th Equadiff Conference in Brno with the same title — see itsabstract [13]) does not admit to present neither detailed physical analysis norcomplete mathematical proofs of lemmas and theorems. For more informationabout physical processes taken into consideration we must refer to [10] and [14];proofs mentioned here can be found in [15].

2 Virtual configurations and stresses

To make the resulting system of equations as simple as possible, we shall for-mulate in an explicit way only the equations of type A) and B), containing twotypes of unknown abstract functions — actual stresses σ and actual displace-ment rates v mapping some closed time interval Θ with zero starting point intocorresponding function spaces V of virtual displacement rates and S of virtualstresses in a deformable body, respectively. The spaces V and S must be definedcarefully to include the equations of type C) — in other words, to respect bothsupport boundary conditions and the special slip structure of deformation withdiscontinuities in matrix/particle interface configuration.

In the 3-dimensional Euclidean space R3 let us consider an open set Ω,occupied by a deformable body, containing a finite number N + 1 of open setsΩ0 (creeping matrix) and Ω1, . . . , ΩN (mutually separated elastic particles) suchthat all intersections (bars denote closures in R3) of Ωi, Ωj are empty for anyi, j ∈ 1, . . . , N and Ω = Ω0 ∪ . . . ∪ ΩN holds and for each i ∈ 0, . . . , N thefollowing requirements are fulfilled:

Numerical Analysis of High-Temperature Strains 309

• the imbedding of W 12 (Ωi, R) into L2(Ωi, R) is compact (“Rellich theorem”,

see [6], p.204),• if i > 0 then the imbedding of W 1

2 (Ωi, R) into L2(Ψi, R) is compact (“tracetheorem”, following [4] see [6], p.222),• the condition of coerciveness of strains onΩi in sense of [7], p.75, is preserved;

the decomposition of a boundary ∂Ωi = Γi ∪ Ψi for Ψi = ∂Ωi \ ∂Ω and Γi =∂Ωi \ Ψi is used for simplicity. In the whole article we shall apply the standardnotation of Sobolev spaces from [6] and of spaces of Bochner integrable abstractfunctions from [1].

Following the physical considerations from [10] we shall assume that creepin Ω0 is active only in a finite number M of slip systems, characterized by slipdirections ak ∈ R3 and normals to slip planes ck ∈ R3 for k ∈ 1, . . . ,M. (It willbe useful to introduce tensors bk of order 2 such that bkij = 1

2 (aki ckj + akj c

ki ) for

every i, j ∈ 1, 2, 3 too.) Let Uik be the set of all boundaries of non-empty2-dimensional cuts of Ωi by parallel planes x · ck = ξ (centered dot symbolsfor scalar products in R3 are applied) for any ξ ∈ R, x ∈ R3 and certain i ∈1, . . . , N, k ∈ 1, . . . ,M. Let

Λ ∈N∏j=0

L∞(Γj , R3×3)

(is sense of Cartesian products) be given (not variable in time) support charac-teristics (in extremal cases locally represented by a regular matrix for a Dirichletboundary condition or by a zero matrix for a Neumann one). The evolution ofdeformation of Ω can be quantified using the the rate of displacement betweenthe corresponding points of Ω in actual and reference (initial) configurations. Inthis way, making use of the Sobolev space

V =N∏j=0

W 12 (Ωj , R3) ,

we can define the space of virtual displacement rates V as the completionof the set of all

v ∈N∏j=0

C∞(Ωj , R3)

with the properties

∀ i ∈ 1, . . . , N ∀ k ∈ 1, . . . ,M ∀Υ ∈ Uik ∀x, y ∈ Υ | (Dv(x)−Dv(y)).ak = 0

and (as a matrix multiplied by a vector in R3)

∀x ∈ ∂Ω | Λ(x)v(x) is a zero point of R3

in the norm of V ; Dv here has to be understood in the following sense: let x be apoint of Ψi for some i ∈ 1, . . . , N, then Dv(x) means the difference vi(x)−v0(x)

310 Jirı Vala

in sense of traces of functions v0 ∈W 12 (Ω0, R

3), vi ∈ W 12 (Ωi, R3) (see assumption

(ii)); similarly Λ and v on ∂Ω can be represented by corresponding traces of v0

or vi.The configuration changes, forced by external loads (that will be specified in

the proceeding section), are closely connected with the redistribution of stressesin time. Unfortunately, the constitutive relations between strains (and strainrates) and stresses, based on the elastic deformation and on the creep flow inthe matrix and modified by the matrix/particle interface diffusion, is not trivial;therefore (unlike the pure theory of elasticity) we have to introduce the spaceof virtual stresses

S =N∏j=0

L2(Ωj , R3×3sym) .

Evidently, V , V , and S are Hilbert spaces.

3 Equations of type A)

We shall assume that the strain and stress development in a deformable bodyin time is the consequence of superposition of the following 4 types of forces:

a) surface loads

γ ∈ W 22 (Θ,B) for B =

N∏j=0

L2(Γj , R3) ,

b) volume loads (body forces)

ϕ ∈ L2(Θ,H) for H =N∏j=0

L2(Ωj , R3) ,

c) prescribed strains (e.g. generated by temperature changes or support mo-tions)

ϑ ∈ W 12 (Θ,S) ,

d) forces caused by length changes of dislocation loops (for details see [10]) forparticular slip systems

η1, . . . , ηM ∈W 22 (Θ,Q) for Q =

N∏j=1

L2(Ψj , R) ,

recalling that W 12 (Θ,X) is a space of all abstract functions v ∈ L2(Θ,X) map-

ping the time interval Θ to certain Hilbert space X such that v ∈ L2(Θ,X) too(dotted symbols are used for time derivatives). In this classification a), b) andd) will be incorporated into the principle of virtual displacement rates, c) willappear in the constitutive equations in the next section.


In special Hilbert spaces we shall apply the brief notation for scalar products

(., .) in H , 〈., .〉 in B , [., .] in S , b., .c in Q

and for norms

|.|H in H , |.|B in B , |.|S in S , |.|Q in Q , ‖.‖ in V

(as usually, |.| are absolute values in R). For every k ∈ 0, . . . ,M let

qk ∈N∏j=1

L2(Ψj , R)

be the magnitudes of (a priori unknown) contact loads in ak directions on ma-trix/particle interfaces. Moreover, let

ρ ∈N∏j=0

L∞(Ωj , R+)

be the material density (needed for inertia forces ρv). Then we are ready toformulate the variational principle of virtual displacement rates in any timet ∈ Θ (for the sake of brevity the time variable t is not emphasized explicitly ifthere is no danger of misunderstanding)

∀ v ∈ V | (v, ρv) + [ε(v), σ] +⌊Dv.ak, qk

⌋= (v, ϕ) + 〈v, γ〉+

⌊Dv.ak, ηk

⌋(1.val)

where Einstein summation rule for indices k is applied (for k not repeated if willbe explicitly underlined in the same sense) and the symbol ε(.) is reserved forthe well-known small deformation tensor from the linearized theory of elasticity(see e.g. [7], p.33).

Now it is necessary to involve the equations of types D) and E). Let us define

S1 =N∏j=1

L2(Ωj , R) , T =N∏j=1

L∞(Ψj , R+) .

Let us select an arbitrary k-th slip system for k ∈ 1, . . . ,M. Let ρk (unlikethe material density ρ) be the dislocation density on Ψk corresponding to suchsystem. If in time t = 0 some dislocation density ρk0 ∈ Q is a priori known thenby [10] the values ρk(t) in every time t ∈ Θ can be computed from the relation

ρk(t) = ρk0 +∫ t

0

G(f(σ(t′) : bk),Dv(t′).ak) dt′

(a binary operator : here means the sum of products of all corresponding elementsof 2 tensors of order 2) where

G : Q×Q → Q

312 Jirı Vala

is some bounded mapping continuous in both variables and

f : S1 → Q

some compact and weakly continuous mapping; this is the equation of type E).(In general we cannot replace f by the standard trace operator, as no traces tosome elements from L2(Ωj , R) with j ∈ 1, . . . , N may exist.) Finally, for somemapping

β : Q → T

such that g1 ≤ β(ρ) ≤ g2 for any ρ ∈ Q with certain g1, g2 ∈ R+ (g1 ≤ g2)Lipschitz continuous with a constant κ ∈ R+ in sense

∀ ρ, ρ ∈ Q | |β(ρ)− β(ρ)|T ≤ κ |ρ− ρ|Q

the equations of type D) can be considered in form

qk = β(ρk)Dv.ak .

Substituting these expressions into (1.val) we receive

∀ v ∈ V | (v, ρv) + [ε(v), σ] +

+⌊

Dv.ak, β(ρk0 +

∫ t

0

G(f(σ(t′) : bk),Dv(t′).ak) dt′)

Dv.ak⌋

=

= (v, ϕ) + 〈v, γ〉+⌊Dv.ak, ηk

⌋. (2.val)

For the complete analysis of strain and stress distribution in time we need toknow initial values v0 ∈ V , σ0 ∈ S of abstract functions v and σ (their requiredproperties will be precised in the next section) and initial values v0, σ0 of theirderivatives too. It is easy to see that they cannot be chosen in an arbitrary way.Let us suppose that some ϕ? ∈ H exists that

∀ v ∈ V | (v, ϕ?) = [ε(v), σ0] + 〈v, γ(0)〉+⌊Dv.ak, β(ρk0)Dv0.a

k − ηk(0)⌋

holds. Then it is possible to set

v0 =ϕ(0)− ϕ?

%.

The analogy of the resulting relation

∀ v ∈ V | (v, ρv0) + [ε(v), σ0] +⌊Dv.ak, β(ρk0)Dv0.a

k⌋

=

= (v, ϕ(0)) + 〈v, γ(0)〉+⌊Dv.ak, ηk(0)

⌋(3.val)

with (2.val) is obvious.


4 Equations of type B)

The principle of virtual displacement rates from the preceding section assumesno special constitutive strain-stress relations (it is evident that σ has to be cal-culated using ε(v), but no such algorithm is available). Therefore it is necessaryto formulate them separately in the equations of type B). Equations presentedin this and following sections without additional comment should be valid on Ωifor any i ∈ 0, . . . , N; this will be not repeated.

We shall come out from the Maxwell viscoelastic model (for the classificationof rheological models see [5], p.143, variational principles are discussed in [16],p.593). For simplicity we shall suppose that there is no viscous component fori = 0. Then for any time t ∈ Θ we have

ε(v) = Aσ − ϑ (4.val)

where the a priori strains ϑ have been mentioned in the preceding section (astype c) loads) and

A ∈N∏j=0

L∞(Ωj , R(3×3)×(3×3))

are conventional stiffness characteristics from the Hooke law that are symmetri-cal in sense

∀ σ, σ ∈ S | [σ, Aσ] = [σ, Aσ]

and, moreover, positive definite in sense

∀ σ ∈ S | [σ, Aσ] ≥ α |σ|2S

where a constant α ∈ R+ must exist independently on the choice of σ ∈ S.For i 6= 0 the dominant creep flow cannot be ignored. The resulting creep

strain rate can be studied as a superposition of the strain rates corresponding toparticular slip systems; one its form of practical importance (based on the Nortonpower-law relation with the initial Orowan stress barrier) has been suggested in[10] and [15]. In general we shall assume the existence of some weakly continuousmapping

F :N∏j=0

L2(Ωj , R3×3sym) →

N∏j=0

L2(Ωj , R3×3sym)

such that a zero point of R3×3sym is mapped to itself and there exists such ζ ∈ R+

that for all σ, σ ∈ S|F (σ)− F (σ)|S ≤ ζ |σ − σ|S

holds. Using the mapping F we can write (as the generalization of (4.val)) for everytime t ∈ Θ the constitutive equations

ε(v) = Aσ + F (σ) − ϑ . (5.val)

314 Jirı Vala

Obviously the validity of (5.val) must be preserved also in case t = 0. Thisinfluences the choice of admissible initial status. Since we have accepted noassumption concerning σ0 yet, we can consider

σ0 = A−1ε(v0) +A−1ϑ(0)− F (σ0)

(A−1 is an inverse operator to A) which is nothing else the analogy of (5.val) intime t = 0

ε(v0) = Aσ0 + F (σ0)− ϑ(0) . (6.val)

Let us remark that (5.val) and (6.val) can be easily transformed into the form similarto (2.val) and (3.val)

∀ σ ∈ S | [σ, ε(v)]− [σ, Aσ]− [σ, F (σ)] = − [σ, ϑ]

for every t ∈ Θ and

∀ σ ∈ S | [σ, ε(v0)]− [σ, Aσ0]− [σ, F (σ0)] = − [σ, ϑ(0)] .

We can summarize that we have reached the following main problem (L indicesare used to emphasize the Lipschitz continuity):

Problem 1. For given v0 ∈ V , σ0 ∈ S and v0 ∈ V , σ0 ∈ S preserving (3.val) and (6.val)to find such v ∈ CL(Θ,H) ∩ L∞(Θ, V ) and σ ∈ CL(Θ,S) that v ∈ L∞(Θ,H)and σ ∈ L∞(Θ,S) and (2.val) and (5.val) are satisfied.

5 Existence and convergence results

Problem 1 can be studied using the standard technique of discretization in time.Let m be certain positive integer. In addition to a closed interval Θ from 0 tosome real constant (final time) τ we shall consider also partial time intervals Θsfrom (s− 1)h (open) to sh (closed) for h = τ/m and s ∈ 1, . . . ,m. Let X be aHilbert space (supplied with a norm |.|X). To arbitrary elements a0, . . . , am ∈ Xfor every t ∈ Θ we are able to set am(t) = am(t) = am(t) = a0 and for every non-zero t ∈ Θ and s ∈ 1, . . . ,m also am(t) = as, am(t) = as−1 and am(t) = as +(sh− t)δas; the notation δas = (as− as−1)/h is used here for simplicity. If somea ∈ L2(Θ,X) exists, it is natural similarly to am(t) to consider am(0) = a(0)and for every non-zero t ∈ Θ and s ∈ 1, . . . ,m also am(t) = a(sh); (in allsituations with no danger of misunderstanding we shall write a0 and as insteadof a(0) and a(sh) too).

Now we shall try to substitute (2.val) by

∀ v ∈ V | (v, ρvm) + [ε(v), σm] +

+⌊

Dv.ak, β(∫ t

0

G(f(σm(t′) : bk),Dvm(t′).ak) dt′)

Dvm.ak⌋

=

= (v, ϕm) + 〈v, γm〉+⌊Dv.ak, ηkm

⌋(7.val)


and (5.val) by

ε(vm) = Aσm + F (σm)− ϑm . (8.val)

This trick leads to the decomposition of an original system of PDE’s of evolution,step-by-step for s ∈ 1, . . . ,m, into m systems without any time variables —(7.val) yields

∀ v ∈ V | (v, ρδvs) + [ε(v), σs] +⌊Dv.ak, β(ρks−1)Dvs.ak

⌋= (9.val)

= (v, ϕs) + 〈v, γs〉+⌊Dv.ak, ηks

⌋with the notation

ρks = ρk0 + h

s∑r=1

G(f(σr : bk),Dvr.ak)

and (8.val) similarly

ε(vs) = Aδσs + F (σs)− ϑs . (10.val)

In this way we obtain a discretized problem:

Problem 2. For given v0, . . . , vs−1 ∈ V and σ0, . . . , σs−1 ∈ S to find such vs ∈ Vand σs ∈ S that (9.val) and (10.val) are satisfied.

We need to study certain limit procedure from (7.val) and (8.val) to (2.val) and (5.val).Using a special technique of approximation of nonlinear operators by sequences oflinear ones (compatible with the algorithms incorporated in CDS software) andstandard Lax-Milgram and Eberlein-Shmul’yan theorems (see [17], pp.134, 201)we receive:

Theorem 3. Problem 2 has always a solution.

The following “discrete version” (similar to [3], p.29) of Gronwall lemma isneeded frequently:

Lemma 4. Let m be an arbitrary positive integer and a0, a1, . . . , am some pos-itive real numbers. There exist a positive real number a? and a positive integerm? such that if m > m? then the relation

∀ s ∈ 1, . . . ,m | as ≤ a0 + h(a1 + . . .+ as)

implies as ≤ a? for any s ∈ 1, . . . ,m.

Let us select positive integers m ∈ 1, 2, . . . and s ∈ 1, . . . ,m again; theconstants c0, c1, c2, c in following Lemmas 5, 6, 7 and Theorem 8 do not dependon this choice. The (nearly trivial) consequence of (10.val) (often applied togetherwith the condition of coerciveness of strains) is:

316 Jirı Vala

Lemma 5. There exists c0 ∈ R+ such that

|ε(vs)|2S ≤ c0(

1 + |σs|2S + |δσs|2S).

The alternative setting v = vs, σ = σs and v = δvs, σ = δσs in (9.val) and (10.val)(with respect to (3.val) and (6.val) in the second case) leads to a couple of a prioriestimates:

Lemma 6. There exists c1 ∈ R+ such that

|vs|2H + |σs|2S ≤ c1(

1 + h

s∑r=1

|vr|2H + h

s∑r=1

|σr|2S + h

s∑r=1

|δσr|2S).

Lemma 7. If (3.val) and (6.val) are preserved then there exists c2 ∈ R+ such that

|δvs|2H + |δσs|2S ≤

≤ c2(

1 + |vs|2H + |σs|2S + h

s∑r=1

|vr|2H + h

s∑r=1

|σr|2S + h

s∑r=1

|δσr|2S).

Lemmas 4, 5, 6, 7 together with Theorem 3 guarantee:

Theorem 8. If (3.val) and (6.val) are preserved then there exists c ∈ R+ such that

max (‖vs‖ , |σs|S , |δvs|H , |δσs|S) ≤ c .

Let us construct all Rothe sequences needed in (7.val) and (8.val). Theorem 8 makes itpossible to analyze their boundedness and equicontinuity properties. From thetheory of Bochner integral (see [1], p.124, and [3], p.24), the Arzela-Ascoli theo-rem (see [17], p.125, and [3], p.24) and some other facts from the functional anal-ysis (e.g. Eberlein-Shmul’yan theorem) we are able to prove the (weak or strong)convergence of these sequences (or at least of their appropriate subsequences)to certain limits in spaces of abstract functions mapping Θ into correspondingSobolev spaces. (For more precise formulations see [15].) It can be verified thatthese limit coincide with v and σ from (2.val) and (5.val) and their time derivatives.This yields:

Theorem 9. Problem 1 has always a solution.

The study of uniqueness of solution of Problem 1 is based on the followinglemma, derived from (2.val) and (5.val) with help of the usual “continuous version” ofGronwall lemma (see [2], p.52, and [3], p.28):


Lemma 10. All couples (v, σ), (v, σ) of solutions of Problem 1 satisfy the equa-tion

12

(∆v, ρ∆v) +12

[∆σ,A∆σ] +∫ t

0

[∆σ(t′), F (σ(t′))− F (σ(t′))] dt′ +

+∫ t

0

⌊D∆v(t′).ak, β(ρk(t′))D∆v.ak

⌋dt′ =

=∫ t

0

⌊D∆v(t′).ak,

(β(ρk(t′))− β(ρk(t′))

)Dv(t′).ak

⌋dt′

in every time t ∈ Θ for ∆v = v − v and ∆σ = σ − σ.

(The new dislocation density ρk here is supposed to be defined analogously toρk with the same ρk0 for every k ∈ 1, . . . ,M). Let us introduce an additionalgrowth condition

∀ ς , ς ∈ S1 ∀ q, q ∈ Q | |G(f(ς), q)−G(f(ς), q)|Q ≤

≤ $(|ς − ς |S1

+ |q − q|Q). (11.val)

Then Lemma 10 yields the uniqueness result:

Theorem 11. If (11.val) is valid then Problem 1 has exactly one solution.

We can summarize: we have verified the solvability of problem of time evolu-tion of strain and stress distributions in superalloy composites under some phys-ically motivated assumptions and simplifications. This conclusion correspondswith the “reasonable” numerical results obtained for special material structuresfrom CDS software; some of them have been published in [10] and [14].

References

[1] H. Gajewski, K. Groger, K. Zacharias, Nichtlineare Operatorgleichungen und Ope-ratordifferentialgleichungen, Akademie Verlag, Berlin, 1974.

[2] M. Gregus, M. Svec M., V. Seda, Obycajne diferencialne rovnice, Alfa, Bratislava,1985.

[3] J. Kacur, Method of Rothe in evolution equations, Teubner Verlag, Leipzig, 1985.[4] J. Kacur, Solution to strongly nonlinear parabolic problem by a linear approx-

imation scheme, lectures at the Department of Mathematics of the Faculty ofMechanical Engineering, Technical University in Brno, 1996.

[5] V. Kolar, Nelinearnı mechanika, CSVTS, Ostrava, 1985.[6] V. G. Maz’ya, Prostranstva S. L. Soboleva, Izdatel’stvo Leningradskogo Univer-

siteta, Leningrad (St. Petersburg), 1985.[7] J. Necas, I. Hlavacek, Uvod do teorie pruznych a pruzne plastickych teles, SNTL,

Prague, 1983.[8] H. E. Petterman, H. J. Bohm, F. G. Rammerstorfer, An elasto-plastic constitutive

law for composite materials, in: Modelling in materials science and processing(M. Rappaz, M. Kedro, eds.), European Commission (COST 512 Action Manage-ment Committee), Brussels, 1996, 384–392.

318 Jirı Vala

[9] J. Svoboda, P. Lukas, Modelling of kinetics of directional coarsening in Ni-superalloys, Acta Materialia, 44 (1996), 2557–2565.

[10] J. Svoboda, J. Vala, Micromodelling of creep in composites with perfect ma-trix/particle interfaces, Modeling in Materials Science and Engineering, to appear.

[11] J. Vala, J. Svoboda, V. Kozak, J. Cadek, Modelling discontinuous metal matrixcomposite behavior under creep conditions: effect of diffusional matter transportand interface sliding, Scripta Metallurgica et Materialia, 30 (1994), 1201–1206.

[12] J. Vala, Programovy system CDS pro analyzu napjatosti a deformace vıcefazovychmaterialu, Proceedings of the Summer School Programs and Algorithms of Nu-merical Mathematics 8 in Janov nad Nisou (1996), 199–206.

[13] J. Vala, Numerical analysis of high-temperature strains and stresses in superalloys,Abstracts of the Equadiff 9 Conference in Brno (1997), 86.

[14] J. Vala, Micromechanical considerations in modelling of superalloy creep flow,Abstracts of the Conference Numerical Modelling in Continuum Mechanics 3 inPrague (1997), 69, and Proceedings, in print.

[15] J. Vala, On one mathematical model of creep in superalloys, Applications of Math-ematics, in print.

[16] K. C. Valanis, A gradient theory of finite viscoelasticity, Archives of Mechanics, 49(1997), 589–609.

[17] K. Yosida, Funkcional’nyj analiz, Mir, Moscow, 1967.



Asymptotic Behavior of Solutions of PartialDifference Inequalities

Patricia J. Y. Wong

Division of MathematicsNanyang Technological University

469, Bukit Timah Road, Singapore 259756Singapore


Abstract. We offer sufficient conditions for the oscillation of all solu-tions of the partial difference equations

y(m− 1, n) + β(m,n)y(m,n− 1) − δ(m,n)y(m,n) +

+ P (m,n, y(m+ k, n+ `)) = Q(m,n, y(m+ k, n+ `))

and

y(m− 1, n) + β(m,n)y(m,n− 1) − δ(m,n)y(m,n) +

+

τ∑i=1

Pi(m,n, y(m+ ki, n+ `i)) =

τ∑i=1

Qi(m,n, y(m+ ki, n+ `i)).

Several examples which dwell upon the importance of our results are alsoincluded.

AMS Subject Classification. 39A10

Keywords. Oscillatory solutions, partial difference equations

1 Introduction

The theory of difference equations, the methods used in their solutions, and theirwide applications have been and still are drawing numerous attention. In fact,in the last few years several monographs and hundreds of research papers havebeen written, e.g., see [1,2,3,4,5,6,13,14,15,16,17,19,20,22,23,24,25,26,27,28,29]and the references therein. In contrast, relatively few studies have been focusedon the qualitative theory of partial difference equations, for instance, refer to[7,8,9,10,11,12,18,30,31,32,33,34]. Partial difference equations are not less im-portant than difference equations - their significance is well illustrated in appli-cations involving population dynamics with spatial migrations, chemical reac-tions, control systems, combinatorics and also finite difference schemes [14,18,21].


320 Patricia J. Y. Wong

Hence, to further the qualitative theory of partial difference equations, in thispaper we shall consider the partial difference equations

y(m− 1, n) + β(m,n)y(m,n− 1)− δ(m,n)y(m,n) + P (m,n, y(m+ k, n+ `))

= Q(m,n, y(m+ k, n+ `)), m ≥ m0, n ≥ n0 (1.won)

and

y(m−1, n)+β(m,n)y(m,n−1)−δ(m,n)y(m,n)+τ∑i=1

Pi(m,n, y(m+ki, n+ì))

=τ∑i=1

Qi(m,n, y(m+ ki, n+ ì)), m ≥ m0, n ≥ n0, (2.won)

where k, `, ki, ì, 1 ≤ i ≤ τ are nonnegative integers, and β(m,n), δ(m,n) arefunctions of m and n such that for all large m and n,

β(m,n) ≥ β > 0 and δ(m,n) ≤ δ (> 0).

It is noted that δ(m,n) is not required to be positive eventually.Recently, Zhang and Liu [33] have discussed particular cases of (1.won) and (2.won),

namely,

y(m− 1, n) + y(m,n− 1)− y(m,n) + a(m,n)y(m+ k, n+ `) = 0 (3.won)

and

y(m− 1, n) + y(m,n− 1)− y(m,n) +τ∑i=1

ai(m,n)gi(y(m+ ki, n+ ì)) = 0,

(4.won)

where a(m,n), ai(m,n), 1 ≤ i ≤ τ are positive, and gi, 1 ≤ i ≤ τ are nonde-creasing functions with ugi(u) > 0 for all u 6= 0. Our results not only generalizeand extend their work, but also complement several other oscillation criteriagiven in [7,8,9,10,11,12,30,31,32,34].

By a solution of (1.won) ((2.won)), we mean a nontrivial double sequence y(m,n)satisfying (1.won) ((2.won)) for m ≥ m0, n ≥ n0. A sequence y(m,n) is eventuallypositive (negative) if y(m,n) > (<) 0 for all large m and n. A solution of (1.won)((2.won)) is said to be oscillatory if it is neither eventually positive nor negative, andnonoscillatory otherwise.

Throughout, with respect to equation (1.won) we shall assume that there existsa function f : R → R and double sequences p(m,n), p′(m,n), q(m,n),q′(m,n) such that

(A1) for u 6= 0, uf(u) > 0,f(u)u≥ γ ∈ (0,∞);

Partial Difference Inequalities 321

(A2) for u 6= 0,

p(m,n) ≤ P (m,n, u(m+ k, n+ `))f(u(m+ k, n+ `))

≤ p′(m,n),

q(m,n) ≤ Q(m,n, u(m+ k, n+ `))f(u(m+ k, n+ `))

≤ q′(m,n); and

(A3) lim supm,n→∞

[p(m,n)− q′(m,n)] > 0.

Further, with respect to equation (2.won) for each 1 ≤ i ≤ τ it is assumed thatthere exists a function fi : R→ R and double sequences pi(m,n), p′i(m,n),qi(m,n), q′i(m,n) such that

(B1) for u 6= 0, ufi(u) > 0,fi(u)u≥ γi ∈ (0,∞);

(B2) for u 6= 0,

pi(m,n) ≤ Pi(m,n, u(m+ ki, n+ ì))fi(u(m+ ki, n+ ì))

≤ p′i(m,n),

qi(m,n) ≤ Qi(m,n, u(m+ ki, n+ ì))fi(u(m+ ki, n+ ì))

≤ q′i(m,n); and

(B3) pi(m,n) > q′i(m,n) eventually.

The plan of the paper is as follows. In Section 2 we shall present some pre-liminary results which are needed later. The oscillation criteria for equations(1.won) and (2.won) are respectively offered in Sections 3 and 4. To illustrate the resultsobtained, five examples are discussed in Section 5.

2 Preliminaries

Lemma 1. Suppose that y(m,n) is an eventually positive solution of (1.won).Then, for all large m,n and all i ≥ 0,

y(m− 1, n) ≤ δy(m,n), y(m,n− 1) ≤ δ

βy(m,n), (5.won)(

1δ

)iy(m− i, n) ≤ y(m,n) ≤ δiy(m+ i, n) (6.won)

and (β

δ

)iy(m,n− i) ≤ y(m,n) ≤

(δ

β

)iy(m,n+ i). (7.won)


Remark 2. It is obvious that (5.won)–(7.won) also hold if y(m,n) is an eventually posi-tive solution of any one of the following: equation (2.won), or either of the inequalities

y(m− 1, n) + β(m,n)y(m,n− 1)− δ(m,n)y(m,n) + P (m,n, y(m+ k, n+ `))

≤ Q(m,n, y(m+ k, n+ `)), (8.won)

y(m−1, n)+β(m,n)y(m,n−1)−δ(m,n)y(m,n)+τ∑i=1

Pi(m,n, y(m+ki, n+`i))

≤τ∑i=1

Qi(m,n, y(m+ ki, n+ `i)), (9.won)

where m ≥ m0, n ≥ n0.

Throughout, we shall use the equation number (·)′ to denote (·) with theinequality sign(s) reversed.

Remark 3. By a similar argument, it can be shown that (5.won)′–(7.won)′ hold if y(m,n)is an eventually negative solution of any one of the following: (1.won), (2.won), (8.won)′ or(9.won)′.

Remark 4. Let y(m,n) be an eventually positive solution of either (1.won), (2.won),(8.won) or (9.won). If β ≥ δ and δ ≤ 1, then (5.won) implies that

y(m− 1, n) ≤ y(m,n) and y(m,n− 1) ≤ y(m,n), (10.won)

i.e., eventually positive solutions of (1.won), (2.won), (8.won) as well as of (9.won) are nondecreas-ing.

Remark 5. Let y(m,n) be an eventually negative solution of either (1.won), (2.won),(8.won)′ or (9.won)′. If β ≥ δ and δ ≤ 1, then from (5.won)′ we get (10.won)′, i.e., eventuallynegative solutions of (1.won), (2.won), (8.won)′ as well as of (9.won)′ are nonincreasing.

Lemma 6. The following identity holds

m∑i=m−k

n∑j=n−`

[y(i − 1, j) + βy(i, j − 1)− δy(i, j)]

= (1 + β − δ)m−1∑i=m−k

n−1∑j=n−`

y(i, j) + β

m−1∑i=m−k

y(i, n− `− 1) + (1− δ)m−1∑i=m−k

y(i, n)

+ (β − δ)n−1∑j=n−`

y(m, j) + βy(m,n− `− 1)− δy(m,n) +n∑

j=n−`y(m− k − 1, j).


3 Oscillation of equation (1.won)

For simplicity, we shall use the notation

µ(m,n) = p(m,n)− q′(m,n).

Further, letE = r > 0 | δ − rµ(m,n) > 0 eventually.

Theorem 7. Suppose that there exist integers M ≥ m0 and N ≥ n0 such that

supr∈E,m≥M,n≥N

r

γβ`min

δ`θ1/`, δkθ1/k

< 1 (11.won)

where

θ =k∏i=1

∏j=1

[δ − rµ(m + i, n+ j)]. (12.won)

Then,

(a) the inequality (8.won) has no eventually positive solution;(b) the inequality (8.won)′ has no eventually negative solution;(c) all solutions of equation (1.won) are oscillatory.

Corollary 8. Suppose that k, ` ≥ 1 and

lim infm,n→∞

1k`

k∑i=1

∑j=1

µ(m+ i, n+ j) >δk+`+1

γβ`max

kk

(1 + k)1+k,

``

(1 + `)1+`

=δk+`+1

γβ`αα

(1 + α)1+α, (13.won)

where α = mink, `. Then, the conclusion of Theorem 7 holds.

Theorem 9. Suppose that there exist integers M ≥ m0 and N ≥ n0 such thatif ` ≥ k,


r

γβ`

(δ

2

)k k∏i=1

[δ − rµ(m + i, n+ i)]×

∏j=k+1

[δ − rµ(m+ k, n+ j)] < 1; (14.won)


and if ` < k,


r

γβ`

(δ

2

)` ∏i=1

[δ − rµ(m + i, n+ i)]×

k∏j=`+1

[δ − rµ(m + j, n+ `)] < 1. (15.won)

Then, the conclusion of Theorem 7 holds.

Corollary 10. Suppose that

lim infm,n→∞

µ(m,n) = µ >νν

γβ`

(δ

2

)α(δ

1 + ν

)1+ν

, (16.won)

where α = mink, ` and ν = maxk, `. Then, the conclusion of Theorem 7holds.

Theorem 11. Suppose that there exist integers M ≥ m0 and N ≥ n0 such that


r

γβ`

k∏i=1

[δ − rµ(m+ i, n)]∏j=1

[δ − rµ(m + k, n+ j)] < 1. (17.won)


Corollary 12. Suppose that

µ(m,n) ≥ c > δk+`+1

γβ`(k + `)k+`

(k + `+ 1)k+`+1. (18.won)


4 Oscillation of equation (2.won)

Theorem 13. Suppose that for each 1 ≤ i ≤ τ,

lim infm,n→∞

pi(m,n) = pi, lim infm,n→∞

q′i(m,n) = q′i, pi > q′i; (19.won)

and

τ∑i=1

(pi − q′i)γiβì

δki+ì+1(αi + 1)αi+1

(2αi

)αi> 1, (20.won)

where αi = minki, ì, 1 ≤ i ≤ τ. Then,

(a) the inequality (9.won) has no eventually positive solution;


(b) the inequality (9.won)′ has no eventually negative solution;(c) all solutions of equation (2.won) are oscillatory.

Theorem 14. Suppose that for each 1 ≤ i ≤ τ,

lim supm,n→∞

τ∑s=1

m∑i=m−k′

n∑j=n−`′

[ps(i, j)− q′s(i, j)]γs1

δi+ks−m

(β

δ

)j+`s−n> w, (21.won)

where k′ = min1≤i≤τ ki, `′ = min1≤i≤τ `i, and

w = δ, β ≥ δ, δ ≤ 1,

= δk′+1, β ≥ δ, δ ≥ 1,

= δ

(δ

β

)`′, β ≤ δ, δ ≤ 1,

= δ

[(δ

β

)`′− 1 + δk

′

], β ≤ δ, δ ≥ 1, δ − β ≤ 1,

= δ

β + (δ − β − 1)δk

′+1

(δ − 1)(δ − β)

[(δ

β

)`′− 1

]+ δk

′

, β ≤ δ, δ ≥ 1, δ − β ≥ 1.


5 Examples

Example 15. Consider the partial difference equation

y(m− 1, n) +n+ 1n

y(m,n− 1)− n+ 1n− 1

y(m,n) +n+ 4n

y(m+ 6, n+ 4) = 0,

m ≥ 1, n ≥ 21. (22.won)

Here, k = 6, ` = 4,

β(m,n) =n+ 1n≥ 1 ≡ β

andδ(m,n) =

n+ 1n− 1

= 1 +2

n− 1≤ 1 +

220

= 1.1 ≡ δ.

Choosing f(u) = u, we have γ = 1. Further, since

P (m,n, y(m+ 6, n+ 4))f(y(m+ 6, n+ 4))

=n+ 4n

,Q(m,n, y(m+ 6, n+ 4))f(y(m+ 6, n+ 4))

= 0,

we may take

p(m,n) = p′(m,n) =n+ 4n

, q(m,n) = q′(m,n) = 0.


Thus, (A1)–(A3) are fulfilled.

Case (a) : Corollary 8

The left side of (13.won) is

lim infm,n→∞

124

6∑i=1

4∑j=1

n+ j + 4n+ j

= 1,

which is more than the right side (= 0.234).

Case (b) : Corollary 10

We find that

lim infm,n→∞

µ(m,n) = 1 >νν

γβ`

(δ

2

)α(δ

1 + ν

)1+ν

= 0.0101

and so (16.won) is satisfied.

Case (c) : Corollary 12

We have

µ(m,n) ≥ 1 ≡ c > δk+`+1

γβ`(k + `)k+`

(k + `+ 1)k+`+1= 0.1.

Hence, (18.won) is fulfilled.

Case (d) : Theorem 13

Here, τ = 1, p1 = 1 and q′1 = 0. The left side of (20.won) is 68.5, which is morethan 1.

Case (e) : Theorem 14

This is the case when β ≤ δ, δ ≥ 1, δ − β ≤ 1. We see that (21.won) holds as

lim supm,n→∞

m∑i=m−6

n∑j=n−4

j + 4j

1(1.1)i+6−m

1(1.1)j+4−n

=( 6∑i=0

11.1i

)( 4∑i=0

11.1i

)= 22.3 > w = 2.46.

Hence, it follows from Corollaries 8–12, Theorems 13 and 14 that equation (22.won)

is oscillatory. In fact, (22.won) has an oscillatory solution given by y(m,n) =(−1)m 1

n

.



y(m− 1, n) +n

n+ 1y(m,n− 1)− y(m,n) +

n

n+ 1y(m+ 4, n+ 3) = 0,

m ≥ 1, n ≥ 1. (23.won)

In this example,

β(m,n) =n

n+ 1≥ 1

2≡ β and δ(m,n) = 1 ≡ δ.

Taking f(u) = u, we have γ = 1. Subsequently, we may choose

p(m,n) = p′(m,n) =n

n+ 1, q(m,n) = q′(m,n) = 0.

Clearly, (A1)–(A3) are satisfied. Further,

limm,n→∞

µ(m,n) = 1 and µ(m,n) ≥ 12≡ c.

It can be checked that all the conditions of Corollaries 8–12, Theorems 13 and 14(the cases β ≤ δ, δ ≤ 1 or β ≤ δ, δ ≥ 1, δ − β ≤ 1) are fulfilled. Therefore, weconclude that (23.won) is oscillatory. In fact, (23.won) has an oscillatory solution givenby y(m,n) = (−1)mn .


y(m−1, n)+n+ 2n+ 1

y(m,n−1)− 12y(m,n)+

(n− 4)(n+ 2)2n(n+ 1)

y(m+2, n+1) = 0,

m ≥ 1, n ≥ 5. (24.won)

Here,

β(m,n) =n+ 2n+ 1

≥ 1 ≡ β and δ(m,n) =12≡ δ.

Choosing f(u) = u, we have γ = 1. Let

p(m,n) = p′(m,n) =(n− 4)(n+ 2)

2n(n+ 1), q(m,n) = q′(m,n) = 0.

Then, it follows that

limm,n→∞

µ(m,n) =12

and µ(m,n) ≥ (5− 4)(5 + 2)2(5)(5 + 1)

=760≡ c.

We check that all the conditions of Corollaries 8–12, Theorems 13 and 14 (thecase β ≥ δ, δ ≤ 1) are satisfied. Hence, all solutions of (24.won) are oscillatory. Onesuch solution is given by y(m,n) =

(−1)m 1

n+1

.



y(m−1, n)+2m+ 1m

y(m,n−1)− 32y(m,n)+

(3m− 2)(m+ 3)2m(m− 1)

y(m+3, n+4)

+(2m+ 1)(m− 2)(m+ 1)

2m2(m− 1)y(m+ 1, n+ 2) = 0, m ≥ 3, n ≥ 1. (25.won)

In this example, τ = 2, k1 = 3, `1 = 4, k2 = 1, `2 = 2,

β(m,n) =2m+ 1m

≥ 2 ≡ β and δ(m,n) =32≡ δ.

Taking f1(u) = f2(u) = u, we have γ1 = γ2 = 1. Let

p1(m,n) = p′1(m,n) =(3m− 2)(m+ 3)

2m(m− 1),

p2(m,n) = p′2(m,n) =(2m+ 1)(m− 2)(m+ 1)

2m2(m− 1),

qi(m,n) = q′i(m,n) = 0, i = 1, 2.

Then,

p1 =32, p2 = 1, q′1 = q′2 = 0.

It can be easily computed that the right side of (20.won) is more than 1.

Further, condition (21.won) also holds as

lim supm,n→∞

2∑s=1

m∑i=m−1

n∑j=n−2

[ps(i, j)− q′s(i, j)]1

δi+ks−m

(β

δ

)j+`s−n

= lim supm,n→∞

m∑i=m−1

n∑j=n−2

(3i− 2)(i+ 3)2i(i− 1)

1δi+3−m

(β

δ

)j+4−n

+ lim supm,n→∞

m∑i=m−1

n∑j=n−2

(2i+ 1)(i − 2)(i+ 1)2i2(i− 1)

1δi+1−m

(β

δ

)j+2−n

= 15.0 > w =94

(the case β ≥ δ, δ ≥ 1).

Hence, by Theorems 13 and 14 equation (25.won) is oscillatory. In fact, one suchsolution is given by y(m,n) =

(−1)n 1

m

.


y(m− 1, n) +n− 1n

y(m,n− 1)− n+ 1n

y(m,n) +n+ 1

2ny(m+ 2, n+ 1)

+12y(m+ 4, n+ 4) = 0, m ≥ 1, n ≥ 3. (26.won)


We have

β(m,n) =n− 1n≥ 2

3≡ β and δ(m,n) =

n+ 1n≤ 4

3≡ δ.

By letting f1(u) = f2(u) = u and

p1(m,n) = p′1(m,n) =n+ 1

2n, p2(m,n) = p′2(m,n) =

12,

qi(m,n) = q′i(m,n) = 0, i = 1, 2,

we check that the hypotheses of Theorem 13 are satisfied. Therefore, all solutionsof equation (26.won) are oscillatory. In fact, (26.won) has an oscillatory solution given byy(m,n) = (−1)m(n+ 1) .

It is, however, noted that this example does not fulfill the condition of The-orem 14 (the case β ≤ δ, δ ≥ 1, δ − β ≤ 1). This illustrates well the differencein nature of the criteria developed.

References

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Author Index

Anashkin Oleg, 1

Benes Michal, 17Blazhievskiy Andriy, 37Bobochko V. N., 43Bognar Gabriella, 53Brigadnov Igor A., 61

Cakoni Fioralba, 73Chrastina Jan, 83

El-Sayed Ahmed M. A., 93

Farkas Henrik, 101Francu Jan, 111

Gaiko Valery, 123

Hayashi Nakao, 131Hirata Hitoshi, 131

Karch Grzegorz, 137Koksch Norbert, 139Kovacova Monika, 167Krbec Miroslav, 181Kucera Petr, 193

Lukacova-Medvid’ova Maria, 201

Mandai Takeshi, 211Markush I. I., 43Mostafa Nasr, 223

Odani Kenzi, 229Omon Alejandro Arancibia, 237

Pinto Manuel Jimenez, 237Pluschke Volker, 247

Schott Thomas, 181Segeth Karel, 255Simon Peter L., 101Sklyar G. M., 263

Tadie, 269, 275Tajcova Gabriela, 281

Vala Jirı, 307Velkovsky I. L., 263

Wong Patricia J. Y., 319

Subject Index

34B05, 4334B15, 5334C05, 123, 22934C10, 93, 167, 269, 27534C15, 16734C23, 101, 12334C30, 13934E05, 4334E20, 4334G20, 139, 22334K05, 8334K20, 1

35A05, 5335A07, 21135A30, 8335A40, 1735B10, 28135B27, 11135B32, 10135C20, 21135D05, 131, 21135E15, 13135J10, 18135J55, 6135J65, 269, 27535J70, 269, 27535K15, 13735K20, 24735K22, 13935K55, 37, 13735K60, 20135K65, 24735L, 7335L10, 8335Q10, 19335Q55, 131

39A10, 93, 319

40G99, 37

42A70, 263

45B05, 23745D05, 237

46E30, 18146E35, 181

47A, 7347H15, 23747H20, 139

58E35, 19358F14, 101, 12358F21, 123, 229

65M12, 20165M15, 25565M20, 247, 25565M60, 201

69B11, 307

70K30, 281

73B27, 11173D, 7373E05, 61, 11173F10, 30773K05, 28173V20, 61

76B15, 13176M10, 20176M25, 201

80A22, 17

82C26, 17

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