Introduction - math.kit.eduschnaubelt/media/reduction.pdf · In his illuminating short paper [13],...

REDUCTION PRINCIPLE AND ASYMPTOTIC PHASE FORCENTER MANIFOLDS OF PARABOLIC SYSTEMS WITH

NONLINEAR BOUNDARY CONDITIONS

RUSSELL JOHNSON, YURI LATUSHKIN, AND ROLAND SCHNAUBELT

Abstract. We prove the reduction principle and study other attractivity

properties of the center and center-unstable manifolds in the vicinity of asteady-state solution for quasilinear systems of parabolic partial differential

equations with fully nonlinear boundary conditions on bounded or exterior

domains.

1. Introduction

In his illuminating short paper [13], K. Palmer proved a fundamental lemmasaying that any given solution of an ODE can be tracked by a solution on the centermanifold as long as the given solution stays in a small ball around an equilibrium.Magically, this simple assertion implies the existence of asymptotic phase as wellas an important Pliss Reduction Principle [14] saying that every compact invariantset in a small ball centered at the equilibrium is a graph over an invariant set forthe reduced ODE on the center manifold. Moreover, if the latter invariant set is(asymptotically) stable for the flow on the center manifold then the former invariantset is (asymptotically) stable for the full flow.

The objective of this paper is to give a generalization of these ODE results for afairly broad class of parabolic partial differential equations. Specifically, in the cur-rent paper we continue the work began in [9, 10], and prove Palmer’s FundamentalLemma, the Pliss reduction principle, the existence of asymptotic phase, and someother properties of the center and center-unstable manifolds in the vicinity of asteady-state solution for quasilinear systems of parabolic partial differential equa-tions with fully nonlinear boundary conditions on bounded or exterior domains.

We consider the equations

∂tu(t) +A(u(t))u(t) = F (u(t)), on Ω, t > 0,

Bj(u(t)) = 0, on ∂Ω, t ≥ 0, j = 1, · · · ,m, (1.1)

u(0) = u0, on Ω,

on a (possibly unbounded) domain Ω in Rn with compact boundary ∂Ω, wherethe solution u(t, x) takes values in CN . The main part of the differential equationis given by a linear differential operator A(u) of order 2m (with m ∈ N) whose

1991 Mathematics Subject Classification. Primary: 35B35, 35B40; Secondary: 35K50, 35K60.Key words and phrases. Parabolic system, initial–boundary value problem, invariant manifold,

attractivity, stability, center manifold reduction, maximal regularity.Supported by the MIUR (Italy) and the MCI (Spain), by the US National Science Foundation

under Grant NSF DMS-0754705 and the Research Board and Research Council of the Universityof Missouri, and by the Deutsche Forschungsgemeinschaft.

1

2 R. JOHNSON, Y. LATUSHKIN, AND R. SCHNAUBELT

matrix–valued coefficients depend on the derivatives of u up to order 2m− 1, andF is a general nonlinear reaction term acting on the derivatives of u up to order2m− 1. Therefore the differential equation is quasilinear. Our analysis focusses onthe fully nonlinear boundary conditions

[Bj(u)](x) := b(x, u(x),∇u(x), · · · ,∇mju(x)) = 0, x ∈ ∂Ω, j = 1, · · · ,m,

for the partial derivatives of u up to order mj ≤ 2m − 1. We assume mild localregularity of the coefficients and that the linearization at a given steady state u∗ isnormally elliptic and satisfies the Lopatinskii–Shapiro condition. For illustration,we give a simple example where N = 1 and m = 2 (see e.g. [3] or [9, §6] for thesystem case N > 1). In the case of the quasilinear heat equation with a nonlinearNeumann boundary condition

∂tu(t)− a(u(t))∆u(t) = f(u(t)), on Ω, t > 0,

b(∇u(t)) = 0, on ∂Ω, t ≥ 0,

u(0) = u0, on Ω,

we have to require that a, f ∈ C1(R), b ∈ C2(R) are real, and that there is a steadystate u∗ ∈ W 2

p (Ω) with a(u∗) ≥ δ > 0 and |b′(u∗) · ν| ≥ δ > 0 for the outer unitnormal ν. Fully nonlinear boundary conditions appear naturally in the treatmentof free boundary problems, see e.g. [7] or [15]. The equations (1.1) are a model casefor such problems.

Our fairly general setup is explained in the next section. The local existenceof solutions and the existence and properties of invariant manifolds for (1.1) havebeen studied in [9, 10], where we have also discussed related literature. To make thepresent paper readable independently of [9, 10], we quote several results from thesepapers in the next section. In particular, Theorem 2.7 taken from [10] says that thecenter manifold is locally exponentially attractive with a tracking solution if thereis no unstable spectrum and the flow on the center manifold is stable. Moreover,here and in our theorems in Section 3 we assume that the center (or the center–unstable) manifold is finite dimensional. Versions of Theorem 2.7 have been shownfor simpler boundary conditions in more abstract settings, see [11, §9.3] and also[4, 12, 16, 17, 18, 19]. We note that in [12] a version of Palmer’s lemma for ellipticproblems on infinite cylinders has been proved.

In Section 3, we establish the analogues for (1.1) of Palmer’s lemma in Lem-mas 3.1 and 3.4. The Pliss reduction principle is proved in Theorem 3.6 in thesetting of Theorem 2.7. Moreover, in Theorem 3.2 we show that any solution of(1.1) on R+ that stays in a small ball centered at the equilibrium u∗ already con-verges exponentially to a solution on the center–unstable manifold. Similarly, if theflow is stable on the center manifold, then any solution starting near the equilibriumeither leaves a certain neighborhood or it belongs to the center–stable manifold andconverges exponentially to a solution on the center manifold, see Theorem 3.5. Inthese two results, we allow for unstable spectrum in contrast to Theorem 2.7 of [10].A stronger version of Theorem 3.5 was shown in the recent paper [16] in an abstractsetting, but only for problems with linear boundary conditions and assuming thatthe center manifold consists of equilibria only.

Notation. We setDk = −i∂k = −i∂/∂xk and use the multi index notation. Thek–tensor of the partial derivatives of order k is denoted by ∇k, and we let ∇ku =(u,∇u, · · · ,∇ku). For an operator A on a Banach space we write dom(A), ker(A),

REDUCTION PRINCIPLE 3

ran(A), σ(A), and ρ(A) for its domain, kernel, range, spectrum, and resolventset, respectively. B(X,Y ) is the space of bounded linear operators between twoBanach spaces X and Y , and B(X) := B(X,X). A ball in X with the radiusr and center at u will be denoted by BX(u, r). For an open set U ⊂ Rn with(sufficiently regular) boundary ∂U or for a Banach space U , Ck(U) are the spacesof k–times continuously differentiable functions on U . We write BCk(U) for thespace of u ∈ Ck(U) such that u and its derivatives up to order k are boundedand have continuous extensions to ∂U . This space is endowed with the supnorm.For unbounded U , Ck0 (U) consists of u ∈ BCk(U) such that u and its derivativesup to order k vanish at infinity. Similar spaces are used on ∂U . By W k

p (U) wedenote the Sobolev spaces, see e.g. [1, Def.3.1], and by W s

p (U) the Slobodetskiispaces, see [1, Thm.7.48] or [20, Rem.4.4.1.2]. Finally, J ⊂ R is a closed intervalwith nonempty interior, c is a generic constant, and ε : R+ → R+ is a genericnondecreasing function with ε(r)→ 0 as r → 0.

2. Setting and preliminaries

In this section we recall the setting and results from [9, 10] needed in the sequel.Let Ω ⊂ Rn be an open connected set with a compact boundary ∂Ω of class C2m

and outer unit normal ν(x), where m ∈ N is given by (2.4) below. Throughout thispaper, we fix a finite exponent p with

p > n+ 2m. (2.1)

Let E = CN with B(E) = CN×N for some fixed N ∈ N. We put

X0 = Lp(Ω; CN ), X1 = W 2mp (Ω; CN ), X1−1/p = W 2m(1−1/p)

p (Ω; CN ),

and denote the norms of these spaces by | · |0, | · |1, and | · |1−1/p, respectively. (Wewarn the reader that in [9, 10] the latter norm was denoted by | · |p.) We set

Y0 = Lp(∂Ω; CN ), Yj1 = W 2mκjp (∂Ω; CN ), Yj,1−1/p = W 2mκj−2m/p

p (∂Ω; CN ),Y1 = Y11 × · · · × Ym1 , Y1−1/p = Y1,1−1/p × · · · × Ym,1−1/p

for j ∈ 1, · · · ,m, mj ∈ 0, · · · , 2m− 1 given by (2.4), and the numbers

κj = 1− mj

2m− 1

2mp. (2.2)

Here the Sobolev–Slobodetskii spaces on ∂Ω are defined via local charts, see The-orem 7.53 in [1] or Definition 3.6.1 in [20]. We observe that

X1 → X1−1/p → X0, Yj1 → Yj,1−1/p → Y0,

and also that

X1−1/p → C2m−10 (Ω; CN ), and Yj,1−1/p → C2m−1−mj (∂Ω; CN ) (2.3)

by (2.1), (2.2), and standard properties of Sobolev spaces, cf. [20, §4.6.1]. Our basicequations (1.1) involve the operators given by

[A(u)v](x) =∑|α|=2m

aα(x, u(x),∇u(x), · · · ,∇2m−1u(x))Dαv(x), x ∈ Ω,

[F (u)](x) =f(x, u(x),∇u(x), · · · ,∇2m−1u(x))), x ∈ Ω, (2.4)

[Bj(u)](x) =bj(x, (γu)(x), (γ∇u)(x), · · · , (γ∇mju)(x)), x ∈ ∂Ω,


for j ∈ 1, · · · ,m and functions u ∈ X1−1/p and v ∈ X1, where γ is the spatialtrace operator and the integers m ∈ N and mj ∈ 0, · · · , 2m−1 are fixed. In viewof (2.3), only continuous functions will be inserted into the nonlinearites. Thus wewill omit γ in Bj(u) and in similar expressions. We set B = (B1, · · · , Bm). Weassume throughout that the coefficients in (2.4) satisfy

(R) aα ∈ C1(E×En×· · ·×E(n2m−1);BC(Ω;B(E))) for α ∈ Nn0 with |α| = 2m,aα(x, 0) −→ aα(∞) in B(E) as x→∞, if Ω is unbounded,f ∈ C1(E × En × · · · × E(n2m−1);BC(Ω;E)),bj ∈ C2m+1−mj (∂Ω× E × En × · · · × E(nmj );E) for j ∈ 1, · · · ,m.

We will need one more degree of smoothness of the coefficients as recorded in thefollowing hypothesis:

(RR) aα ∈ C2(E×En×· · ·×E(n2m−1);BC(Ω;B(E))) for α ∈ Nn0 with |α| = 2m,f ∈ C2(E × En × · · · × E(n2m−1);BC(Ω;E)),bj ∈ C2m+2−mj (∂Ω× E × En × · · · × E(nmj );E) for j ∈ 1, · · · ,m.

For each k ∈ N0, we fix an order of the multi indices β ∈ Nn0 with |β| = k. We orderthe nk components of a k–tensor in the same way, thus using β as the label for thecomponent corresponding to β ∈ Nn0 with |β| = k. For a function w depending onz ∈ E(nk), we denote by ∂βw its partial derivative with respect to β-th argument.It is not difficult to see that

A ∈ C1(X1−1/p;B(X1, X0)) and F ∈ C1(X1−1/p;X0) (2.5)

with the locally bounded derivatives

[F ′(u)v](x) =2m−1∑k=0

∑|β|=k

ik (∂βf)(x, u(x),∇u(x), · · · ,∇2m−1u(x)) Dβv(x),

[A′(u)w]v(x) = A′(u)[v, w](x)

=∑|α|=2m

2m−1∑k=0

∑|β|=k

(∂βaα)(x, u(x), · · · ,∇2m−1u(x)) [∂βv(x), Dαw(x)]

for x ∈ Ω, u, v ∈ X1−1/p, and w ∈ X1, see formula (25) of [9] and the text beforeit. (Observe that (∂βaα)(x, z) : E2 → E is bilinear.) We further have

Bj ∈ C1(X1−1/p;Yj,1−1/p) ∩ C1(X1;Yj1), j ∈ 1, · · · ,m, (2.6)

with the locally bounded derivatives

[B′j(u)v](x) =mj∑k=0

∑|β|=k

ik (∂βbj)(x, u(x),∇u(x), · · · ,∇mju(x)) Dβv(x),

where x ∈ ∂Ω and u, v ∈ X1−1/p, resp. u, v ∈ X1. The continuous differentia-bility of Bj : X1−1/p → Yj,1−1/p was shown in Corollary 12 of [9], and Bj ∈C1(X1;Y1,1−1/p) can be proved by the arguments used in step (4) and (5) of theproof of Proposition 10 of [9], see in particular inequality (69) in [9]. We setB′(u) = (B′1(u), · · · , B′m(u)).

The symbols of the principal parts of the linear differential operators are thematrix–valued functions given by

A#(x, z, ξ) =∑|α|=2m

aα(x, z) ξα, Bj#(x, z, ξ) =∑|β|=mj

imj (∂βbj)(x, z) ξβ


for x ∈ Ω, z ∈ E × · · · × E(n2m−1) and ξ ∈ Rn, resp. x ∈ ∂Ω, z ∈ E × · · · × E(nmj )

and ξ ∈ Rn. We further set A#(∞, ξ) =∑|α|=2m aα(∞) ξα if Ω is unbounded.

We introduce the normal ellipticity and the Lopatinskii–Shapiro condition for A(u0)and B′(u0) at a function u0 ∈ X1−1/p as follows:

(E) σ(A#(x,∇2m−1u0(x), ξ)) ⊂ λ ∈ C : Reλ > 0 =: C+ and (if Ω is un-bounded) σ(A#(∞, ξ)) ⊂ C+, for x ∈ Ω and ξ ∈ Rn with |ξ| = 1.

(LS) Let x ∈ ∂Ω, ξ ∈ Rn, and λ ∈ C+ with ξ ⊥ ν(x) and (λ, ξ) 6= (0, 0). Thefunction ϕ = 0 is the only solution in C0(R+; CN ) of the ODE system

λϕ(y) +A#(x,∇2m−1u0(x), ξ + iν(x)∂y)ϕ(y) = 0, y > 0,

Bj#(x,∇mju0(x), ξ + iν(x)∂y)ϕ(0) = 0, j ∈ 1, · · · ,m.We refer to [3], [5], [6], and the references therein for more information concerningthese conditions. We can now state our basic hypothesis.

Hypothesis 2.1. Condition (R) holds, and (E), (LS) hold at a steady state u∗ ∈X1 of (1.1), i.e., A(u∗)u∗ = F (u∗) on Ω, B(u∗) = 0 on ∂Ω.

For the investigation of (1.1), we need several spaces of functions on J × Ω andJ × ∂Ω, where J ⊂ R is a closed interval with a nonempty interior. The base spaceand solution space of (1.1) are

E0(J) = Lp(J ;Lp(Ω; CN )) = Lp(J ;X0),

E1(J) = W 1p (J ;Lp(Ω; CN )) ∩ Lp(J ;W 2m

p (Ω; CN )) = W 1p (J ;X0) ∩ Lp(J ;X1),

respectively, equipped with the natural norms. We need the crucial embeddings

E1(J) → BC(J ;X1−1/p) → BC(J ;C2m−10 (Ω; CN )), (2.7)

see Theorem III.4.10.2 of [2] for the first and (2.3) for the second embedding. Wenote that the norm of the first embedding is uniform for intervals J of length greaterthan a fixed ` > 0. Observe that (2.7) implies that the trace operator γ0 at timet = 0 is continuous from E1(J) to X1−1/p if 0 ∈ J . The boundary data of ourlinearized equations will be contained in the spaces

Fj(J) = Wκjp (J ;Lp(∂Ω; CN )) ∩ Lp(J ;W 2mκj

p (∂Ω; CN ))

= Wκjp (J ;Y0) ∩ Lp(J ;Yj1), j ∈ 1, · · · ,m,

endowed with their natural norms, where F(J) := F1(J)× · · · × Fm(J). We have

Fj(J) → BC(J ;Yj,1−1/p) → BC(J × ∂Ω) and γ0 ∈ B(Fj(J), Yj,1−1/p) (2.8)

if 0 ∈ J , see [6, §3].For α, β ∈ R, we set eα(t) = eαt for t ∈ R and define the function eα,β by setting

eα,β(t) = eα(t) for t ≤ 0 and eα,β(t) = eβ(t) for t ≥ 0. Then we introduce theweighted spaces

Ek(R±, α) = v : eαv ∈ Ek(R±), F(R±, α) = v : eαv ∈ F(R±),Ek(α, β) = v : eα,βv ∈ Ek(R), F(α, β) = v : eα,βv ∈ F(R), (2.9)

where k = 0, 1, endowed with the canonical norms ‖v‖E0(R+,α) = ‖eαv‖E0(R+) etc.We also use the analogous norms on compact intervals J .

We assume that Hypothesis 2.1 holds. Due to (2.5) and (2.6), we can linearizethe problem (1.1) at the steady state u∗ ∈ X1 obtaining the operators defined by

A∗ = A(u∗) +A′(u∗)u∗ − F ′(u∗) ∈ B(X1, X0),


Bj∗ = B′j(u∗) ∈ B(X1−1/p, Yj,1−1/p) ∩ B(X1, Yj1).

We set B∗ = (B1∗, · · · , Bm∗). We further define the nonlinear maps

G ∈ C1(X1;X0) and Hj ∈ C1(X1−1/p;Yj,1−1/p) ∩ C1(X1;Yj1)

with G(0) = Hj(0) = 0 and G′(0) = H ′j(0) = 0(2.10)

for j ∈ 1, · · · ,m by setting

G(v) =(A(u∗)v −A(u∗ + v)v

)−(A(u∗ + v)u∗ −A(u∗)u∗ − [A′(u∗)u∗]v

)+(F (u∗ + v)− F (u∗)− F ′(u∗)v

),

Hj(v) = B′j(u∗)v −Bj(u∗ + v),

for v ∈ X1, resp. v ∈ X1−1/p. Again, we put H(v) = (H1(v), · · · , Hm(v)). Thecorresponding Nemytskii operators are denoted by

G(v)(t) = G(v(t)), Hj(v)(t) = Hj(v(t)), H(v)(t) = H(v(t))

for v ∈ Eloc1 (J) (which is the space of v : J → X0 such that v ∈ E1([a, b]) for all

intervals [a, b] ⊂ J).Theorem 14 of [9] shows that (1.1) generates a local semiflow on the solution

manifoldM = u0 ∈ X1−1/p : B(u0) = 0.

In particular, a function u0 is the initial value of the (unique) solution u ∈ E1([0, T ])of (1.1) for some T > 0 if and only if u0 ∈M. Setting v = u−u∗ and v0 = u0−u∗,we further see that u0 ∈M if and only if v0 ∈ X1−1/p and B∗v0 = H(v0) and thatu ∈ E1([0, T ]) solves (1.1) if and only if v ∈ E1([0, T ]) satisfies

∂tv(t) +A∗v(t) = G(v(t)) on Ω, a.e. t > 0,

Bj∗v(t) = Hj(v(t)) on ∂Ω, t ≥ 0, j ∈ 1, · · · ,m,v(0) = v0, on Ω.

(2.11)

Remark 2.2. Let Hypothesis 2.1 hold. Theorem 14(a) of [9] and (2.7) then implythe following facts: For each given T > 0, there is a radius ρ = ρ(T ) > 0 suchthat for every u0 = u∗ + v0 ∈ M with |v0|1−1/p ≤ ρ there exists a unique solutionu = u∗ + v of (1.1) on [0, T ], and |w(t)|1−1/p ≤ c‖v‖E1([0,T ]) ≤ c|v0|1−1/p forall t ∈ [0, T ] with constants c = c(T ) independent of u0 in this ball. Moreover,Proposition 15 of [9] and the Sobolev embedding imply that

|v(t)|1 ≤ c|v0|1−1/p for all t ∈ [T0, T ], ♦

where T > T0 > 0 and the constant may depend on T0.

We now recall some results from [9] regarding the solvability of the inhomoge-neous linear problem

∂tv(t) +A∗v(t) = g(t) on Ω, a.e. t ∈ J,B∗v(t) = h(t) on ∂Ω, t ∈ J,v(0) = v0, on Ω,

(2.12)

in weighted function spaces on the unbounded interval J ∈ R+,R−,R. We as-sume that Hypothesis 2.1 holds. (Actually, when dealing only with (2.12) we do nothave to assume that u∗ ∈ X1 is a steady state of (1.1).) We recall from Theorem 2.1of [6] that on a bounded interval J = [a, b] the boundary value problem obtainedby combining the first two lines of (2.12) with the initial condition v(a) = v0 has a


unique solution v ∈ E1([a, b]) if and only if g ∈ E0([a, b]), h ∈ F([a, b]), v0 ∈ X1−1/p,and B∗v0 = h(a). A solution v ∈ Eloc

1 (J) of (2.12) on J will be denoted byv = S(v0, g, h), where J ⊂ R is any closed interval containing 0. We stress thatthis notation incorporates the compatibility condition B∗v0 = h(0) because of thesecond line in (2.12) and the embeddings (2.7) and (2.8). Moreover, the solutionS(v0, g, h) is unique if J = R+, but uniqueness may fail on J = R−.

We define the operator A0 = A∗|kerB∗ with the domain

dom(A0) = u ∈ X1 : Bj∗u = 0, j = 1, . . . ,m .

It is known that −A0 generates an analytic semigroup on X0 which we denote byT (·). We need the extrapolation space X−1 of A0 defined as the completion ofX0 with respect to the norm |u0|−1 = |(µ+ A0)−1u0|0 for some fixed µ ∈ ρ(−A0).There exists an extension A−1 of A0 to X−1 which generates the analytic semigroupT−1(·) extending T (·) to X−1. We further employ the map

Π = (µ+A−1)N1 ∈ B(Y1, X1) (2.13)

where N1 ∈ B(Y1, X1) is the solution operator, N1 : ϕ 7→ u, of the elliptic boundaryvalue problem (µ+A∗)u = 0 on Ω, B∗u = ϕ on ∂Ω, see Proposition 5 of [9]. Thisproposition also gives a right inverse

N1−1/p ∈ B(Y1−1/p, X1−1/p) (2.14)

of B∗. Due to Proposition 6 of [9], the solution v ∈ Eloc1 (J) of the problem (2.12)

is given by the variation of constants formula

v(t) = T (t− τ)v(τ) +∫ t

τ

[T (t− s)g(s) + T−1(t− s)Πh(s)] ds (2.15)

for all t ≥ τ in J .In order to treat solutions of (2.11) or (2.12) on the intervals J = R±, we

assume that the (rescaled) semigroupeδtT (t)

t≥0

is hyperbolic for δ ∈ [δ1, δ2] forsome segment [δ1, δ2] ⊂ R (i.e., σ(−A0 +δ)∩iR = ∅). Let P be the (stable) spectralprojection for −A0 + δ corresponding to the part of σ(−A0 + δ) in the open lefthalfplane, and set Q = I − P . Then T (t) is invertible on QX0 with the inverseTQ(−t)Q, and ‖etδT (t)P‖, ‖e−tδTQ(−t)Q‖ ≤ ce−εt for t ≥ 0 and some ε > 0. IfeδT (·) is hyperbolic on X0, then eδT−1(·) is hyperbolic on X−1 with projectionsP−1 and Q−1 = I − P−1 being the extensions of P and Q, respectively. Moreover,Q−1 maps X−1 into dom(A0), and P leaves invariant X1−1/p, X1, and dom(A0).The projections commute with the semigroup and its generator as well as with theirextrapolations. (See [9, §2] for these facts and related references.)

When needed, we assume that T (·) has an exponential trichotomy, i.e., there isa splitting

σ(−A0) = σs ∪ σc ∪ σu with (2.16)max Reσs < −ωs < −ωc < min Reσc ≤ 0 ≤ max Reσc < ωc < ωu < min Reσu .

(If Ω is bounded, σ(−A0) is discrete and thus (2.16) automatically holds withσu ⊂ iR and arbitrarily small ωc = ωc.) We take numbers α ∈ [ωc, ωs] andβ ∈ [ωc, ωu] and denote by Pk the spectral projections for −A0 correspondingto σk, k = s, c, u. We set Pcs = Ps + Pc, Pcu = Pc + Pu, and Psu = Ps + Pu.Then the rescaled semigroups eαT (·) and e−βT (·) are hyperbolic on X0 with stable


projections Ps and Pcs, respectively. The restriction of T (t) to PkX0 yields a groupdenoted by Tk(t), t ∈ R, where k = c, u, cu.

When needed, we impose the following assumption which is weaker than (2.16):There exist positive numbers ωs, ωu, ωcu, ωcs > 0 such that at least one of thefollowing assertions holds:

σ(−A0) = σs ∪ σcu with max Reσs < −ωs < −ωcu < min Reσcu , (2.17)

σ(−A0) = σcs ∪ σu with max Reσcs < ωcs < ωu < min Reσu . (2.18)

We continue to use notation Pk for the spectral projections for −A0 correspondingto the sets σk, k ∈ s, cs, cu, u. Using standard facts, we see that PuX0 ⊂ PcuX0 ⊂dom(A0) and that on PcuX0 the norms in X0, X1−1/p and X1 are equivalent.Finally, we recall the notation X0

1−1/p = z0 ∈ X1−1/p : B∗z0 = 0 for the tangentspace at u∗ to the nonlinear phase space M = u0 ∈ X1−1/p : B(u0) = 0 for(1.1), and that P = I − N1−1/pB∗ projects X1−1/p onto X0

1−1/p, see (2.14) andremarks preceding Theorem 14 in [9]. In the theorems stated below, the invariantmanifolds are graphs over the corresponding spectral subspaces of X0

1−1/p, whereit is important to note that PcuX0 ⊂ dom(A0) ⊂ X0

1−1/p.We now recall the main result Theorem 4.2 of the paper [10] where one constructs

a local center manifold Mc and shows some of its basic properties. In particular,Mc is a C1–manifold in X1−1/p being tangent to PcX0 at u∗. It is given by functionsu∗ + v(0) for initial values v(0) of solutions v ∈ E1(α,−β) to a modified version of(2.11) on J = R which involves a nonlocal cutoff. This fact is not needed below,but it is described in detail in [10].

We assume that the spectrum of −A0 has the trichotomy decomposition de-scribed in (2.16), and recall that this assumption automatically holds if the spatialdomain Ω is bounded.

Theorem 2.3. Assume that Hypothesis 2.1 and (2.16) hold. Take any α ∈ (ωc, ωs)and β ∈ (ωc, ωu). There exists a radius ρ > 0 and maps φc ∈ C1(PcX0;PsuX1−1/p)and Φc ∈ C1(PcX0; E1(α,−β)) with bounded derivatives such that the followingassertions hold.

(a) We have φc(0) = 0 and φ′c(0) = 0. The local center manifold is given by

Mc :=u0 = u∗ + z0 + φc(z0) : z0 ∈ PcX0

∩BX1−1/p

(u∗, ρ)

=u0 = u∗ + v(0) : v = Φc(Pc(u0 − u∗))

∩BX1−1/p

(u∗, ρ). (2.19)

If u0 ∈ Mc, then the function v from (2.19) is given by v = Pcv + φc(Pcv). Itfurther solves the equation (2.11) (at least) for t ∈ [−3, 3], so that Mc ⊂ M. Thedimension of Mc is equal to dimPcX0.

(b) Let u0 ∈ Mc and v be given by (2.19). If the forward solution u of (1.1)exists and stays in BXp

(u∗, ρ) on [0, t1] for some t1 > 0, then u(t) = v(t)+u∗ ∈Mc

for 0 ≤ t ≤ t1. If the function u = v + u∗ stays in BXp(u∗, ρ) on [t0, 0] for some

t0 < 0, then u(t) ∈Mc and u solves (1.1) for t0 ≤ t ≤ 0.(c) Assume that v(t) + u∗ ∈Mc for all t ∈ (a, b) and some a < 0 < b. The then

function y = Pcv satisfies the equations

y(t) = −A0Pcy(t) + PcΠH(y(t) + φc(y(t))) + PcG(y(t) + φc(y(t))),

y(0) = Pc(u0 − u∗),(2.20)


on PcX0 for t ∈ (a, b). Moreover, v ∈ C((a, b);X1) and

B∗φc(Pcv0) = B∗v0 = H(v0), (2.21)

Psu(A∗v0 −G(v0)) = φ′c(Pcv0)Pc(A∗v0 −G(v0)). (2.22)

(d) If u ∈ Eloc1 (R) solves (1.1) on R with |u(t)−u∗|1−1/p < ρ for all t ∈ R, then

u(t) ∈Mc for all t ∈ R.(e) In addition, assume that (RR) holds. Then there is a ρ0 > 0 such that the

map φc : PcX0 ∩BX1−1/p(0, ρ0)→ PsuX1 is Lipschitz.

In addition to Theorem 2.3, one constructs a local center–stable manifold Mcs

assuming (2.18) (see Theorem 5.1 of [10]), and a local center–unstable manifoldMcu assuming (2.17) (see Theorem 5.2 of [10]). These manifolds are of class C1 inX1−1/p, and are tangent to PcsX

01−1/p, resp. to PcuX0, at u∗. They are described

in the following two theorems. Recall that P = I −N1−1/pB∗. In Theorem 2.4 themap ϑcs takes care of the compatibility conditions.

Theorem 2.4. Assume Hypothesis 2.1 and (2.18). Take any β ∈ (ωcs, ωu).Then there exists a radius ρ > 0 and maps φcs ∈ C1(PcsX

01−1/p;PuX0), ϑcs ∈

C1(PcsX01−1/p;PcsX1−1/p) and Φcs ∈ C1(PcsX

01−1/p; E1(R+,−β)) having bounded

derivatives such that the following assertions hold.(a) We have φcs(0) = ϑcs(0) = 0 and φ′cs(0) = ϑ′cs(0) = 0. The local center–

stable manifold is given by

Mcs :=u0 = u∗ + z0 + ϑcs(z0) + φcs(z0) : z0 ∈ PcsX

01−1/p

∩BX1−1/p

(u∗, ρ)

=u0 = u∗ + v(0) : v = Φcs(PcsP(u0 − u∗))

∩BX1−1/p

(u∗, ρ). (2.23)

If u0 ∈ Mcs, then the function v from (2.23) solves the equation (2.11) (at least)for t ∈ [0, 4]. Thus, Mcs ⊂M.

(b) Let u0 ∈Mcs and v be given by (2.23). Assume that a forward or a backwardsolution u of (1.1) exists and stays in BXp(u∗, ρ) on [0, t0] or on [−t0, 0] for somet0 > 0. Set v(t) = u(t)− u∗ for −t0 ≤ t ≤ 0 in the second case. We then have1

u(t) = u∗ + v(t) = u∗ + PcsPv(t) + φcs(PcsPv(t)) + ϑcs(PcsPv(t)) ∈Mcs (2.24)

for 0 ≤ t ≤ t0 or −t0 ≤ t ≤ 0, respectively.(c) It holds Mcs ∩Mu = u∗.

Theorem 2.5. Assume Hypothesis 2.1 and (2.17). Let α ∈ (ωcu, ωs). Then thereexists a radius ρ > 0 and maps φcu ∈ C1(PcuX0;PsX1−1/p) and Φcu ∈ C1(PcuX0;E1(R−, α)) with bounded derivatives such that the following assertions hold.

(a) We have φcu(0)=0 and φ′cu(0)=0. The center-unstable manifold is given by

Mcu :=u0 = u∗ + z0 + φcu(z0) : z0 ∈ PcuX0

∩BX1−1/p

(u∗, ρ)

= u0 = u∗ + v(0) : v = Φcu(Pcu(u0 − u∗)) ∩BX1−1/p(u∗, ρ). (2.25)

If u0 ∈ Mcu, then the function v from (2.25) solves the equation (2.11) (at least)for t ∈ [−4, 0]. Thus, Mcu ⊂M. The dimension of Mcu is equal to dimPcuX0.

(b) Let u0 ∈ Mcu and v be given by (2.25). If the function u = u∗ + v stays inBXp

(u∗, ρ) on [t0, 0] for some t0 < 0, then u(t) = u∗+v(t) ∈Mcu and u solves (1.1)for t0 ≤ t ≤ 0. If the forward solution u of (1.1) exists and stays in BXp

(u∗, ρ) on

1We corrected in formula (2.24) a misprint found in [10].


[0, t1] for some t1 > 0, then u(t) ∈Mcu and we set v(t) = u(t)−u∗, for 0 ≤ t ≤ t1.We futher have v(t) = Pcuv(t) + φcu(Pcuv(t)) for t ∈ [t0, 0], resp. t ∈ [0, t1].

(c) It holds Mcu ∩Ms = u∗.(d) Assume, in addition, that (RR) holds. Then there is a ρ0 > 0 such that the

map φcu : PcuX0 ∩BX1−1/p(0, ρ0)→ PsX1 is Lipschitz.

Moreover, the local stable and unstable manifolds Ms and Mu for (1.1) wereconstructed in Theorem 4.1 of [10]. These manifolds have analogous propertiesand are used to prove further properties of the center manifold summarized in thefollowing corollary (see Corollary 5.3 [10]), where we assume throughout that radiiρ > 0 in the above theorems are the same.

Corollary 2.6. Assume that Hypothesis 2.1 and (2.16) hold. We then have Mc ∩Mcs ∩Mcu, Mc ∩Ms = u∗, and Mc ∩Mu = u∗.

Finally, we cite a result from [10] that describes the stability of the steady stateu∗ of (1.1) and the attractivity of Mc. As in Theorem 2.3, one assumes thatHypothesis 2.1 and (2.16) hold. In parabolic problems, the center and center–unstable manifolds are finite dimensional in many cases; e.g., if the spatial domainΩ is bounded. Moreover, there are important applications where Mc consists ofequilibria only, see e.g. [8], [15], [16]. Thus it is quite possible that one can checkthe stability of u∗ with respect to the semiflow on Mc generated by (1.1) withoutknowing a priori that u∗ is stable with respect to the full semiflow of (1.1) on M.

Theorem 2.7 below states, see Theorem 6.1 of [10], that u∗ is stable onM underthe following conditions: s(−A0) ≤ 0, u∗ is stable on Mcu = Mc, Pcu = Pc hasfinite rank, and the additional regularity assumption (RR) holds. Here s(−A0)= supReλ : λ ∈ σ(−A0). In fact, one has a stronger result saying that eachsolution starting sufficiently close to u∗ converges exponentially to a solution onMc. Here one can assume that s(−A0) ≤ 0 without loss of generality since −A0

has no spectrum in the open right halfplane provided u∗ is stable and Pcu has finiterank, due to Theorem 4.1 of [10].

Theorem 2.7. Assume that the spectrum of −A0 admits a splitting σ(−A0) =σs ∪ σc corresponding to the spectral projections Ps and Pc such that Pc has finiterank, σc ⊂ iR, and there is a number α with max Reσs < −α < 0.

Suppose that for each r > 0 there is a ρ > 0 such that for u0 ∈ Mc with|Pc(u0 − u∗)|0 < ρ the solution u of (1.1) exists and u(t) ∈ Mc ∩ BX1−1/p

(u∗, r)for all t ≥ 0. Then there is a ρ > 0 such that for every u0 = u∗ + v0 ∈ M with|v0|1−1/p ≤ ρ the solution u = u∗ + v of (1.1) exists on R+ and there is a solutionu of (1.1) on R+ such that u(t) ∈Mc for all t ≥ 0 and

|u(t)− u(t)|1 ≤ ce−αt |Psv0 − φc(Pcv0)|1−1/p

for all t ≥ 1 and a constant c independent of u0 and t. Moreover, u∗ is stablefor (1.1), i.e.: For each r > 0 there exists a ρ′ > 0 such that for every u0 ∈M∩BX1−1/p

(u∗, ρ′) the solution u of (1.1) exists on R+ and u(t) ∈ BX1−1/p(u∗, r)

for all t ≥ 0.

3. The Palmer Fundamental Lemma and Pliss Reduction Principle

Let u be a solution of (1.1), u∗ be the equilibrium, and let v = u − u∗, v0 =u(0) − u∗. Let |v0|1−1/p ≤ ρ where ρ > 0 is sufficiently small. Throughout, weassume condition (RR) so that φc and φcu are Lipschitz into X1, see Theorem 2.3(e)


and Theorem 2.5(d). We first establish a version of the Palmer Fundamental Lemmain our PDE context.

Lemma 3.1. [The Fundamental Lemma] Assume that Hypothesis 2.1, condition(RR), and (2.17) hold. Then there exist C,C ′, C ′′r,> 0 and α ∈ (0, ωs) such thatif v is a solution of (2.11) satisfying |v(t)|1−1/p ≤ r for all 0 ≤ t ≤ T with someT > 1, then there exists a solution z of (2.11) on [0, T ] such that u∗ + z(t) ∈ Mc

for all t ∈ [0, T ], Pcuz(T ) = Pcuv(T ), and

|v(t)− z(t)|1 ≤ Ce−αt|Psv0 − φcu(Pcuv0)|1−1/p (3.1)

holds for all 1 ≤ t ≤ T . Given T0 > 1, the constants are uniform for T ≥ T0. Forevery t ∈ [0, T ], we further have

|v(t)− z(t)|1−1/p ≤ C ′e−αt|Psv0 − φcu(Pcuv0)|1−1/p ≤ C ′′e−αt|v0|1−1/p. (3.2)

Here, equation (3.1) holds for t ≥ 1 (or for t ≥ a for any a > 0) because one cancontrol the X1-norm of the solution only for strictly positive t, see Proposition 15of [9] and Remark 2.2 above.

Proof. Part 1. We assume that T ≥ 3. For a general T0 > 0 the proof is similar.Let v be a solution of (2.11) on [0, T ] such that |v(t)|1−1/p ≤ r for all t ∈ [0, T ],where a sufficiently small r > 0 is to be chosen below. Due to (2.17) there areconstants δ ∈ (ωcu, ωs) and N ≥ 1 such that ‖e−tA0PcuPcu‖B(X0) ≤ Ne−δt for allt ≤ 0. Using Theorem 2.5, we find a radius ρcu > 0 such that the restrictionφcu : PcuX0 ∩ BX0(0, ρcu) → X1 is Lipschitz with Lipschitz constant `, such thatu∗ + ξ + φcu(ξ) ∈ Mcu for all ξ ∈ PcuX0 ∩ BX0(0, ρcu), and such that there is aconstant c0 > 0 with |Pcuz(t)|0 ≤ c0|Pcuz(1)|0 for all 0 ≤ t ≤ 1 and any solution zof (2.11) with u∗ + z(t) ∈Mc for t ∈ [0, 1]. We set

ε1(R) = maxx∈X1,|x|1≤R

‖G′(x)‖B(X1,X0) , ‖H′(x)‖B(X1,Y1). (3.3)

Because of (2.10), we can fix a (small) number R > 0 such that

d := Nε1(R)(1 + ‖PcuΠ‖B(Y1,X0))(1 + `) < ωs − δ, (3.4)

(1 + c0)R ‖Pcu‖B(X1,X0) ≤ ρcu . (3.5)

To choose r > 0, we note that Remark 2.2 (with T = 1/2) implies the inequality

|v(t)|1 ≤ c|v(t− 1/2)|1−1/p ≤ cr for all t ∈ [1/2, T ].

Here and below the constants do not depend on v, T, t, R or r, and we let r be lessthan the radius indicated by Remark 2.2. We can now take small r > 0 such that

|v(t)|1 ≤ R for all 1/2 ≤ t ≤ T,r(1 + `)‖Pcu‖B(X1−1/p,X1) ≤ R/2 and r ‖Pcu‖B(X1−1/p,X0) ≤ ρcu.

(3.6)

Part 2. We define

w = Psv − φcu(Pcuv) and ϕ = v − w = Pcuv + φcu(Pcuv)

on [0, T ]. The function w belongs to PsX1−1/p and, due to the last inequality in(3.6), the function ϕ belongs to Mcu. Note that Pcuϕ = Pcuv and that

(A∗ + µ)x = (A0 + µ)(x−N1B∗x) = (A−1 + µ)x−ΠB∗x


for every x ∈ X1, see (2.13). One proves the analogues of formulas (2.21) and (2.22)for φcu as in the proof of Theorem 4.2 of [10] for the map φc. ¿From these identitieswe infer

B∗w = B∗v −B∗φcu(Pcuv) = H(v)−H(ϕ) =: h, (3.7)

w = Ps(−A∗v +G(v))− φ′cu(Pcuv)Pcu(G(v)−A∗v)

− φ′cu(Pcuϕ)Pcu(A∗ϕ−G(ϕ)) + Ps(A∗ϕ−G(ϕ))

= −PsA∗w + Ps(G(v)−G(ϕ)) + φ′cu(Pcuv)Pcu(A∗w +G(ϕ)−G(v))

= −A−1Psw + PsΠh+ Ps(G(v)−G(ϕ))− φ′cu(Pcuv)Pcu(Πh+G(v)−G(ϕ))

=: −A−1Psw + PsΠh+ Psg. (3.8)

In the penultimate line we used that PcuA−1w = A−1Pcuw = 0. Applying thevariation of constant formula in X−1, we therefore obtain

w(t) = T (t− τ)Psw(τ) +∫ t

τ

T−1(t− σ)Ps(g(σ) + Πh(σ)) dσ (3.9)

for 0 ≤ τ < t ≤ T . Let α ∈ [0, ωs). Proposition 10(II) in [9] implies that

max‖g‖E0([τ,t],α), ‖h‖F([τ,t],α)

≤ ε(r)‖w‖E1([τ,t],α). (3.10)

Arguing as in the proof of Proposition 8 of [9] (see inequality (43) there), we canthen estimate

e−ατ‖w‖E1([τ,t],α) ≤ c(|w(τ)|1−1/p + e−ατ‖g‖E0([τ,t],α) + e−ατ‖h‖F([τ,t],α)

)≤ c(|w(τ)|1−1/p + e−ατε(r)‖w‖E1([τ,t],α)

).

We note that the constants do not depend on τ if 0 ≤ τ ≤ t − 1/4, say. Makingr > 0 sufficiently small and using (2.7), we arrive at

e−ατ‖w‖E1([τ,t],α) ≤ c|w(τ)|1−1/p,

|w(t)|1−1/p ≤ ce−αt‖w‖E1([τ,t],α) ≤ ce−α(t−τ)|w(τ)|1−1/p

(3.11)

for all 0 ≤ τ < t ≤ T . Again the constants are uniform for τ ∈ [0, t− 1/4].Part 3. Since |Pcuv(T )|0 ≤ ρcu by (3.6), there exists the backward solution z =

Pcuz+φcu(Pcuz) of (2.11) such that u∗+z belongs toMcu and Pcuz(T ) = Pcuv(T ).Due to Theorem 2.5, z(t) exists at least for t ∈ [T − 3, T ] and ‖z‖E1([T−3,T ]) ≤c|Pcuz(T )|0 ≤ cr. Thus, |z(T − 3)|1−1/p ≤ cr by (2.7) and so Proposition 5 of [9]yields |z(t)|1 ≤ cr ≤ R for all t ∈ [T − 2, T ], after decreasing r > 0 if needed. Lett0 ∈ [1/2, T − 2] be the minimal time such that z(t) with u∗+ z onMcu exists andthe inequality |z(t)|1 ≤ R holds for all t0 ≤ t ≤ T . We set

y = Pcu(v − z). (3.12)

Denoting

g1 = G(v)−G(z) and h1 = B∗(v − z) = H(v)−H(z),

we obtain

y′ = Pcu

(−A∗(v − z) + g1

)= −Pcu

((A0 + µ)(v − z −N1h1)− µ(v − z)

)+ Pcug1

= −A0Pcuy + Pcu(g1 + Πh1). (3.13)


This equation yields

y(t) = −∫ T

t

e−(t−τ)A0PcuPcu(g1(τ) + Πh1(τ)) dτ,

|y(t)|0 ≤ N(1 + ‖PcuΠ‖B(Y1,X0))∫ T

t

e−δ(t−τ)ε1(R)|v(τ)− z(τ)|1 dτ (3.14)

for all t ∈ [t0, T ]. We observe that

v − z = w + φcu(Pcuv)− φcu(Pcuz) + y (3.15)

holds on [t0, T ]. Putting d0 = d(1 + `)−1 and recalling (3.4) and (3.5), we furtherestimate

eδt|y(t)|0 ≤ d∫ T

t

eδτ |y(τ)|0 dτ + d0

∫ T

t

eδτ |w(τ)|1 dτ.

We then deduce from a Gronwall-type inequality that

eδt|y(t)|0 ≤ d0

∫ T

t

eδτ |w(τ)|1 dτ + dd0

∫ T

t

ed(τ−t)∫ T

τ

eδσ|w(σ)|1 dσ dτ

= d0

∫ T

t

eδτ |w(τ)|1 dτ + dd0

∫ T

t

eδσ|w(σ)|1∫ σ

t

ed(τ−t) dτ dσ

= d0

∫ T

t

ed(σ−t)eδσ|w(σ)|1 dσ.

Since we have chosen d and δ as in (3.4), we can take α ∈ (d+ δ, ωs). So, Holder’sinequality and (3.11) imply

|y(t)|0 ≤ d0

∫ T

t

e(d+δ)(σ−t)|w(σ)|1 dσ

≤ d0e−αt

(d+ δ − α)1p′‖w‖E1([t,T ],α) ≤ c|w(t)|1−1/p (3.16)

for all t ∈ [t0, T ], where the constant c is uniform for t ≤ T − 1/4, say. If t ∈[T − 1/4, T ], we can estimate in (3.16) the E1 norm on [t, T ] by that on [t− 1/4, T ]and obtain

|y(t)|0 ≤ c|w(t− 1/4)|1−1/p (3.17)with a uniform constant. In view of (3.12), we have z = Pcu(v − y) + φcu(Pcu(v −y)). Thus, |Pcu(v − y)|0 = |Pcuz|0 ≤ ‖Pcu‖B(X1,X0)R ≤ ρcu due (3.5). Using|v(t0)|1−1/p ≤ r, (3.6), (3.16), and (3.11) with τ = 0, we then deduce

|z(t0)|1 ≤ (1 + `)|Pcu(v − y)|1 ≤ (1 + `)‖Pcu‖B(X1−1/p,X1) |v(t0)|1−1/p + c|y(t0)|0≤ R/2 + c|w(t0)|1−1/p ≤ R/2 + c|v0|1−1/p ≤ R/2 + cr < R,

provided r > 0 is small enough. It follows that t0 = 1/2. Since u∗ + z(1/2) ∈Mcu

we can extend u∗ + z on Mcu to the time interval [0, T ] due to Theorem 2.5(a).The estimates (3.16), (3.17) and (3.11) now imply that the inequalities

|Pcu(v(t)− z(t))|0 = |y(t)|0 ≤ ce−αt|w(0)|1−1/p = ce−αt|Psv0 − φcu(Pcuv0)|1− 1p,

|Ps(v(t)− z(t))|1−1/p = |w(t) + φcu(Pcuv(t))− φcu(Pcuz(t))|1−1/p

≤ |w(t)|1−1/p + c|y(t)|0 ≤ ce−αt|Psv0 − φcu(Pcuv0)|1−1/p

hold on [1/2, T ]. These relations and Theorem A.1 of [10] yield (3.1).


Part 4. To establish (3.2), it remains to show that

|v(t)− z(t)|1−1/p ≤ c|Psv0 − φcu(Pcuv0)|1−1/p for all t ∈ [0, 1]. (3.18)

We first note that (3.15) also holds on [0, T ] and that |Pcuz(t)|0 ≤ ρcu for 0 ≤ t ≤ 1due to (3.5) and the text before this inequality. Let g and h be given by (3.8) and(3.7). Theorem 2.1 of [6] gives a ψ ∈ E1([0, 1]) such that

∂tψ(t) +A∗ψ(t) = g(t) on Ω, a.e. t > 0,

B∗ψ(t) = h(t) on ∂Ω, t ≥ 0,

ψ(0) = w(0), on Ω.

Using also (3.10) with α = 0, we further obtain

‖ψ‖E1([0,1]) ≤ c(|w(0)|1− 1p

+‖g‖E0([0,1]) +‖h‖F([0,1])) ≤ c|w(0)|1− 1p

+ε(r)‖w‖E1([0,1]).

In view of (2.15) and (3.9), we have w = Psψ and thus

‖w‖E1([0,1]) ≤ c‖ψ‖E1([0,1]) ≤ c|w(0)|1− 1p

+ ε(r)‖w‖E1([0,1]),

‖w‖E1([0,1]) ≤ c|w(0)|1−1/p = c|Psv0 + φcu(Pcuv0)|p, (3.19)

possibly after decreasing r > 0. Equation (3.15) and (2.7) now yield

|v(t)− z(t)|1−1/p ≤ c(|w(t)|1−1/p + |y(t)|0) ≤ c(‖w‖E1([0,1]) + |y(t)|0)

≤ c(|Psv0 + φcu(Pcuv0)|1−1/p + |y(t)|0) (3.20)

for all t ∈ [0, 1]. To control y(t), we use again (3.13). As in the proof of Proposi-tion 10 of [9] (letting there v = 0 in steps (1), (4) and (5)), one can show that

‖g1‖E0([t,1]) + ‖h1‖Lp([t,1];Y1) ≤ c‖v − z‖Lp([t,1];X1)

where the constant does not depend on t ∈ [0, 1]. Combined with (3.15), (3.19) and(3.1), these facts yield

y(t) = e−(t−1)A0Pcuy(1)−∫ 1

t

e−(t−τ)A0PcuPcu(g1(τ) + Πh1(τ)) dτ,

|y(t)|0 ≤ c(|y(1)|0 + ‖v − z‖Lp([t,1];X1)) ≤ c(|y(1)|0 + ‖w‖Lp([t,1];X1) + ‖y‖E0([t,1]))

≤ c(|Psv0 + φcu(Pcuv0)|1−1/p + ‖y‖E0([t,1]))

where we could employ that φcu is Lipschitz with values in X1 since |Pcuv|0 and|Pcuz|0 are less than ρcu on [0, 1]. It follows

|y(t)|p0 ≤ c|Psv0 + φcu(Pcuv0)|p1−1/p + c

∫ 1

t

|y(τ)|p0 dτ

for all t ∈ [0, 1], so that |y(t)|p0 ≤ c|Psv0+φcu(Pcuv0)|p1−1/p by Gronwall’s inequality.In view of (3.20), we have established (3.18).

Our first theorem says that the center–unstable manifold attracts solutions whichstay in small ball around u∗ for all t ≥ 0 and that there exists a tracking solutionu∗ + z on Mcu.

Theorem 3.2. [Asymptotic Phase] Assume that Hypothesis 2.1, condition (RR),(2.17), and dimPcuX0 < ∞ hold. Then there exists constants r, c > 0 and α ∈(0, ωs) such that : If a solution v of (2.11) exists and satisfies |v(t)|1−1/p ≤ r forall t ≥ 0, then there is a solution z of (2.11) such that u∗ + z(t) ∈Mc and

|v(t)− z(t)|1 ≤ ce−αt|Psv0 − φcu(Pcuv0)|1−1/p for all t ≥ 1, (3.21)


|v(t)− z(t)|1−1/p ≤ ce−αt|Psv0 − φcu(Pcuv0)|1−1/p for all t ≥ 0. (3.22)

Proof. We choose r > 0 so small that Lemma 3.1 can be applied to v. Lemma 3.1gives solutions zn with u∗ + zn on Mcu tracking v on [0, n] for every n ∈ N withn ≥ 2. Lemma 3.1 also yields

|Pcuzn(0)|0 ≤ |Pcuv(0)|0 + |Pcu(zn(0)− v(0))|0 ≤ cr

for all n ∈ N. Hence, there exists a subsequence nj →∞ so that Pcuznj(0)→ ζ ∈

PcuX0 as j →∞. Let z be the solution of (2.11) on [−2, 2] such that Pcuz(0) = ζand u∗+z(t) ∈Mcu for t ∈ [−2, 2], decreasing r > 0 if needed to apply Theorem 2.5and Remark 2.2. We also have |z(t)|1−1/p ≤ c|z(0)|0 ≤ |ζ|0 ≤ c1r for all t ∈[0, 2] and some constant c1 > 0. Let b denote the supremum of t1 > 1 such thatz(t) exists on [0, t1] and stays in the ball B1−1/p(0, (1 + c1 + C ′′)r), where C ′′ isgiven by Lemma 3.1. We thus have b ≥ 2. If we take a sufficiently small r > 0,Theorem 2.5(b) shows that u∗ + z(t) ∈Mcu for t ∈ [0, b).

As in Theorem 4.2 of [10] one can prove that the functions Pcuzn and Pcuzsatisfy an ordinary differential equation analogous to (2.20). Since the initial dataPcuznj

(0) converge to Pcuz(0), we thus obtain Pcuznj(t)→ Pcuz(t) in X0 as j →∞

for t ∈ [0, b) and hence

znj(t) = Pcuznj

(t) + φcu(Pcuznj(t))→ z(t)

in X1 → X1−1/p, using also the Lipschitz property of φcu stated in Theorem 2.5(d)and decreasing r > 0 if necessary. Lemma 3.1 now implies that

|z(t)|1−1/p ≤ lim supj→∞

|znj(t)− v(t)|1−1/p + |v(t)|1−1/p (3.23)

≤ C ′′e−αt|v0|1−1/p + r ≤ (C ′′ + 1)r < (C ′′ + 1 + c1)r.

for all t ∈ [0, b). As a result, b = ∞ and z exists for all t ≥ 0 and stays inB1−1/p(0, (C ′′ + 1 + c1)r). If r > 0 is chosen small enough, then Lemma 3.3 belowshows that u∗ + z(t) ∈ Mcs for all t ≥ 0. Hence, u∗ + z is contained in Mc byCorollary 2.6 since u∗+z(t) ∈Mcu for all t ≥ 0. The convergence properties followfrom Lemma 3.1 as in (3.23).

Lemma 3.3. Assume that Hypothesis 2.1 and (2.18) hold. If u∗ + v0 ∈Mcs, then

v0 = Pcsv0 + φcs(Pcs P v0), (3.24)

where P = I−N1−1/pB∗. Moreover, there exists a ρ > 0 such that if v is a solutionof (2.11) on R+ staying in B(u∗, ρ), then u∗ + v(t) ∈Mcs for all t ≥ 0.

Proof. Theorem 2.4(b) shows that

v0 = Pcs P v0 + ϑcs(Pcs P v0) + φcs(Pcs P v0)

if u∗ + v0 ∈ Mcs. Then Pcsv0 = Pcs P v0 + ϑcs(Pcs P v0), and the first assertionfollows. In the framework of Theorem 2.4, the second assertion can be shown asTheorem 2.3(d) (see the proof of Theorem 4.2(e) in [10]).

Given a solution u∗ + v near u∗, in Lemma 3.1 we have constructed a trackingsolution on Mcu for finite time intervals. In the next lemma, we construct such asolution on the center manifold in the case of trichotomy if also u∗ + v(0) ∈Mcs.


Lemma 3.4. Assume that Hypothesis 2.1, condition (RR), and (2.16) hold. Thenthere exist C, r > 0 such that: If there is a solution v of (2.11) with u∗ + v onMcs staying in B1−1/p(0, r) on [0, T ] for some T > 1, then there is a solution z of(2.11) on [0, T ] such that u∗+ z(t) ∈Mc for all t ∈ [0, T ], Pcuz(T ) = Pcuv(T ), and

|v(t)− z(t)|1 ≤ Ce−αt|Psv0 − φcu(Pcuv0)|1−1/p for all t ≥ 1,

|v(t)− z(t)|1−1/p ≤ Ce−αt|Psv0 − φcu(Pcuv0)|1−1/p for all t ≥ 0,

Given T0 > 1, the constants are uniform for T ≥ T0.

Proof. We assume that T ≥ 3. For a general T0 > 0 the proof is similar. Formula(3.24) says that v = Pcsv + φcs(Pcs P v). We set w = Psv − φcu(Pcuv). Since|Pcv(T )|1−1/p ≤ cr, for sufficiently small r > 0 Theorem 2.5 gives a solution zof (2.11) on [T − 3, T ] such that Pcz(T ) = Pcv(T ) and u∗ + z belongs to Mc.Moreover, |z(T )|1 ≤ c|Pcz(T )|0 ≤ cr. Here and below the constants do not dependon v, T, t, r and the number R > 0 introduced later. We have Mc = Mcs ∩Mcu

due to Corollary 2.6, and thus

z = Pcz + φc(Pcz) = Pcsz + φcs(Pcs P z) = Pcuz + φcu(Pcuz).

As a result, Psz = φcu(Pcuz) and Puz = φcs(Pcs P z). Hence,

v − z = w + φcu(Pcuv)− φcu(Pcuz) + Pc(v − z) (3.25)

+ φcs(Pcs P v)− φcs(Pcs P z).

We set y = Pc(v− z). Given a small R > 0 to be determined later, let t0 ∈ [1/2, T )be the minimal time such that the solution z(t) of (2.11) with u∗+ z onMc existsand the inequality |z(t)|1 ≤ R holds for all t0 ≤ t ≤ T . As in part 3 of the proof ofLemma 3.1, we obtain that 1/2 ≤ t0 ≤ T − 2 exists if r > 0 is chosen small enough.We further note that Remark 2.2 shows that |v(t)|1 ≤ c|v(t− 1/2)|1−1/p ≤ cr ≤ Rwhere we decrease r > 0 if needed.

Theorems 2.4 and 2.5 imply that the maps φcs : PcsX01−1/p → PuX0 and φcu :

PcuX0 → PsX1−1/p are Lipschitz with a constant ε(R) on balls of radius R in therespective domain spaces. Moreover, φcu : PcuX0 → PsX1 is Lipschitz on this balldue to (RR) and Theorem 2.5(d) (after possibly decreasing R). We thus deduce

|v(t)− z(t)|1−1/p ≤ |w(t)|1−1/p + ε(cR)|Pcu(v(t)− z(t))|0+ cε(cR)|Pcs P(v(t)− z(t))|1−1/p + c|y(t)|0

from (3.25) and X1 → X1−1/p → X0. Decreasing R > 0 if needed, we obtain

|v(t)− z(t)|1−1/p ≤ c(|w(t)|1−1/p + |y(t)|0) (3.26)

for t0 ≤ t ≤ T . Proceeding similarly, inequality (3.26) then leads to

|v(t)− z(t)|1 ≤ |w(t)|1 + c|Pcu(v(t)− z(t))|0 + c|Pcs P(v(t)− z(t))|1−1/p + c|y(t)|0≤ |w(t)|1 + c(|w(t)|1−1/p + |y(t)|0) ≤ c(|w(t)|1 + |y(t)|0), (3.27)

for all t0 ≤ t ≤ T .Let δ ∈ (ωc, ωs). As in the proof of (3.13) and (3.14) in Lemma 3.1, we infer

y′(t) = Pc

(−A∗(v(t)− z(t)) +G(v(t))−G(z(t))

)= −A0Pcy(t) + Pc

(Π(H(v(t))−H(z(t))) +G(v(t))−G(z(t))

),

y(t) = −∫ T

t

e−(t−τ)A0PcPc

(Π(H(v(τ))−H(z(τ))) +G(v(τ))−G(z(τ))

)dτ,


|y(t)|0 ≤ cε1(R)∫ T

t

e−δ(t−τ)|v(τ)− z(τ)|1 dτ.

Inequality (3.27) then implies

eδt|y(t)|0 ≤ cε1(R)∫ T

t

eδτ |y(τ)|0 dτ + cε(R)∫ T

t

eδt|w(τ)|1 dτ.

Arguing as in the proof of (3.16) in Lemma 3.1 (with d = d0 = cε(R) being small),we conclude that

|y(t)|0 ≤ d∫ T

t

e(d+δ)(σ−t)|w(σ)|1 dσ.

We now fix a sufficiently small R > 0 such that d+ δ < ωs, where d = cε(R). Letα ∈ (d+ δ, ωs). If we take a sufficiently small r > 0, we can apply estimates (3.11)and (3.19) from Lemma 3.1. Using first Holder’s inequality, we thus derive

|y(t)|0 ≤ ce−αt‖w‖E1([t,T ],α) ≤ c|w(t)|1−1/p (3.28)

for all t0 ≤ t ≤ T − 1/4. As in (3.17), we also obtain

|y(t)|0 ≤ c|w(t− 1/4)|1−1/p (3.29)

for all t ∈ [T − 1/4, T ]. Observe that

z = Pcz + φc(Pcz) = Pc(v − y) + φc(Pc(v − y)). (3.30)

Remark 2.2 further yields

|v(t)|1 ≤ c|v(t− 1/2)|1−1/p ≤ cr ≤ R/2 (3.31)

for all t ∈ [t0, T ] and a sufficiently small r > 0. Using (3.30), (3.31) and (3.28), weinfer

|z(t0)|1 ≤ c(|v(t0)|1 + |y(t0)|0

)≤ R/2 + c|w(t0)|1−1/p .

Since |w(t0)|1−1/p ≤ c|v(t0)|1−1/p, we finally conclude that |z(t0)|1 ≤ R/2+cr < R,decreasing r > 0 again, if needed. Thus, t0 = 1/2. Now the estimates (3.26), (3.28),(3.29) and (3.11) imply

|v(t)− z(t)|1−1/p ≤ c(|w(t)|1−1/p + |y(t)|0

)≤ ce−αt|w(0)|1−1/p = ce−αt|Psv0 − φcu(Pcuv0)|1−1/p

for all t ∈ [1/2, T ]. We can extend this estimate to t ∈ [0, T ] as in part 4) of theproof of Lemma 3.1. Finally, decreasing r > 0 if necessary, one can use the estimate(A.2) in [10] applied to u∗ + v and u∗ + z, obtaining the inequality

|v(t)− z(t)|1 ≤ c|v(t− 1/2)− z(t− 1/2)|1−1/p

and completing the proof.

Our second theorem extends the stability Theorem 2.7 to the case of unsta-ble spectrum. It says that the center manifold locally attracts the center–stablemanifold with a tracking solution if the flow on Mc is stable.

Theorem 3.5. Assume that Hypothesis 2.1, condition (RR), dimPcX0 <∞, andthe trichotomy condition (2.16) hold. Suppose that u∗ is stable for the flow on Mc.Then there exist sufficiently small r > 0 and ρ > 0 such that for each solution v of(2.11) with |v0|1−1/p ≤ ρ either

(a) there exists t > 0 such that |v(t)|1−1/p > r, or(b) u∗ + v0 ∈Mcs.


Moreover, in case (b) the solution v of (2.11) with u∗ + v(t) ∈ Mcs exists for allt ≥ 0, satisfies |v(t)|1−1/p ≤ r for all t ≥ 0, and there exists a solution z of (2.11)on R+ with u∗ + z on Mc such that

|v(t)− z(t)|1 ≤ Ce−αt|Psv0 − φcu(Pcuv0)|1−1/p for all t ≥ 1, (3.32)

|v(t)− z(t)|1−1/p ≤ Ce−αt|Psv0 − φcu(Pcuv0)|1−1/p for all t ≥ 0. (3.33)

Proof. Let r > 0 be the radius determined in Lemma 3.4. We choose a smallρ ∈ (0, r) to be fixed later. Let v0 with u∗+v0 ∈Mcs satisfy |v0|1−1/p ≤ ρ. Denoteby T the supremum of all t > 0 such that v(t) exists and satisfies |v(τ)|1−1/p < r forall 0 ≤ τ ≤ t and |v(T )|1−1/p = r. Remark 2.2 implies that T > 1 for sufficientlysmall ρ > 0. By Lemma 3.4, there exists a solution zT of (2.11) on [0, T ] such thatPczT (T ) = Pcv(T ), u∗ + zT (t) ∈Mc for 0 ≤ t ≤ T , and

|zT (t)− v(t)|1 ≤ ce−αt|Psv0 − φcu(Pcuv0)|1−1/p, t ∈ [1, T ],

|zT (t)− v(t)|1−1/p ≤ ce−αt|Psv0 − φcu(Pcuv0)|1−1/p ≤ cρ, t ∈ [0, T ].(3.34)

Here and below, c does not depend on T and ρ. Using this estimate and Remark 2.2,it follows that

|zT (0)|1−1/p ≤ |zT (0)− v(0)|1−1/p + |v(0)|1−1/p ≤ cρ+ cρ = cρ. (3.35)

Since u∗ is stable for the flow on Mc, we can choose ρ so small that

|zT (T )|1−1/p = distX1−1/p(u∗ + zT (T ), u∗) ≤ r/2.

We then obtain

|v(T )|1−1/p ≤ |v(T )− zT (T )|1−1/p + |zT (T )|1−1/p ≤ cρ+ r/2 < r

provided ρ > 0 is sufficiently small. This strict inequality is a contradiction if T isfinite, and hence T =∞. As a consequence, (3.35) holds for all T > 1. Since PcX0

is finite dimensional, there thus exists a sequence of Tn → ∞ such that PszTn(0)

converge as n → ∞ to some ζ ∈ PcX0 with |ζ|0 ≤ cρ. Using Theorem 2.3(a), wefind a solution z of (2.11) on some time interval [0, t0) such that Pcz(0) = ζ andu∗ + z belongs to Mc. Since |z(0)|1−1/p ≤ c|ζ|0 ≤ cρ, the stability of u∗ on Mc

implies that z(t) exists and |z(t)|1−1/p ≤ r for all t ≥ 0, if ρ > 0 is sufficiently small.As in the proof of Theorem 3.2 we then deduce that zTn(t) converges to z(t) in X1

as n→∞. Thus, the required estimate (3.32) follows from (3.34).

In our last theorem, we extend the stability Theorem 2.7 from the set K = u∗to larger invariant sets K.

Theorem 3.6. [Reduction Principle] Assume the conditions of Theorem 2.7. Thereexists small numbers ρ, ρ0 > 0 such that if K ⊂ BX1−1/p

(u∗, ρ) is a backward andforward globally invariant set for (1.1), then the following assertions hold:

(a) K ⊂Mc and there exists a set K0 ⊂ BPcX0(0, ρ0) such that

K =u∗ + w0 + φc(w0) : w0 ∈ K0 (3.36)

and K0 is forward and backward invariant with respect to the flow induced by theODE (2.20).

(b) If K0 is stable, resp. asymptotically stable, for the flow induced by the ODE(2.20), then K is stable, resp. asymptotically stable, for the flow of (1.1).


Proof. Let r > 0 be the radius determined in Lemma 3.1 and Theorem 3.2. Takeρ > 0 smaller than the radius described in Theorem 2.3 such that

ρ < r/2 and ρ0 := ρ‖Pc‖B(X1−1/p,X0) < r/2 (3.37)

hold. Let K ⊂ BX1−1/p(u∗, ρ) be a backward and forward globally invariant set for

(1.1); that is, for each u∗ + v0 ∈ K the solution v of (2.11) with v(0) = v0 existsfor all t ∈ R and u∗ + v(t) ∈ K for all t ∈ R.

(a) The inclusion K ⊂Mc follows from Theorem 2.3(d) by our choice of ρ sinceK is invariant. We define K0 =

Pc(u0 − u∗) : u0 ∈ K

. Then K0 ⊂ BPcX0(0, ρ0).

For y0 ∈ K0, the function v0 = y0 +φc(y0) satisfies u0 = u∗+ v0 ∈ K. The solutionv of (2.11) with the initial datum v(0) = v0 thus exists for all t ∈ R and satisfiesu∗ + v(t) ∈ K ⊂ BX1−1/p

(u∗, ρ). By Theorem 2.3 (c), the function y = Pcv solvesthe ODE (2.20) for all t ∈ R and thus K0 is invariant for the flow induced by (2.20).

(b) First, we claim that the following assertions hold provided the numbersr0, r > 0 are chosen small enough:

if distPcX0(y,K0) ≤ r0 then |y|1−1/p ≤ r, (3.38)

if distX1−1/p(u∗ + v,K) ≤ r then |v|1−1/p ≤ r, (3.39)

if distX1−1/p(u∗ + v,K) ≤ r then

|Psv − φc(Pcv)|1−1/p ≤ cdistX1−1/p(u∗ + v,K). (3.40)

To show (3.38), we recall that K0 ⊂ BPcX0(0, ρ0) with ρ0 = ρ‖Pc‖B(X1−1/p,X0).Thus, choosing w0 ∈ K0 appropriately and using (3.37), we have

|y|1−1/p ≤ |y − w0|1−1/p + |w0|1−1/p

≤ 2 distX1−1/p(y,K0) + ρ‖Pc‖B(X1−1/p,X0)

≤ 2r0 + ρ‖Pc‖B(X1−1/p,X0) ≤ r/2 + r/2 = r,

provided r0 > 0 is small enough. The proof of (3.39) is analogous. To show (3.40),let us pick a w ∈ K such that

|u∗ + v − w|1−1/p ≤ 2 distX1−1/p(u∗ + v,K).

We note that w = u∗+w0 +φc(w0) for some w0 ∈ K0 due to (3.36) and recall thatw0 ∈ BPcX0(0, ρ0) and v ∈ BX1−1/p

(0, r) by (3.39). Using the Lipschitz property ofφc on small balls stated in Theorem 2.3, we then obtain

|Psv − φc(Pcv)|1−1/p ≤ |Psv − φc(w0)|1−1/p + |φc(w0)− φc(Pcv)|1−1/p

≤ |Psv − Ps(w − u∗)|1−1/p + c|Pc(w − u∗)− Pcv|1−1/p

≤ c|u∗ + v − w|1−1/p ≤ cdistX1−1/p(u∗ + v,K).

Next, let us assume that K0 is stable, that is, that for each r0 > 0 there isa ρ0 > 0 such that if distPcX0(y(0),K0) ≤ ρ0 then distPcX0(y(t),K0) ≤ r0 forall t ≥ 0 for the solution y in PcX0 of the ODE (2.20). Here we choose r0 > 0such that (3.38) holds, but possibly r0 will be further decreased below. To provethat K is stable, let r > 0 be given where we may assume that r < r and thatr is so small that (3.39) and (3.40) hold. We have have to find ρ > 0 such thatif distX1−1/p

(u∗ + v0,K) ≤ ρ then the solution v of (2.11) with the initial datav(0) = v0 exists for all t ≥ 0 and satisfies distX1−1/p

(u∗ + v(t),K) ≤ r for all t ≥ 0.Let us fix a ρ < r to be determined later.


Since distX1−1/p(u∗+ v0,K) ≤ ρ < r, either the solution v(t) of (2.11) is defined

for all t ≥ 0 and satisfies distX1−1/p(u∗ + v(t),K) < r for all t ≥ 0, or there is a

number T such that distX1−1/p(u∗+ v(t),K) < r for all 0 ≤ t < T with equality for

t = T . (We can again assume that T is larger than 1 due to Remark 2.2.) Supposethat the second option holds. By (3.39), we have |v(t)|1−1/p ≤ r for all 0 ≤ t ≤ T .Lemma 3.1 thus yields a solution z of (2.11) on [0, T ] with u∗ + z(t) ∈ Mc. (Werecall that Mc = Mcu due to the setting assumed in the theorem.) Moreover, zsatisfies Pcz(T ) = Pcv(T ) and the estimate

|v(t)− z(t)|1−1/p ≤ ce−αt|Psv0 − φc(Pcv0)|1−1/p (3.41)

for all 0 ≤ t ≤ T .We pause to remark the inequality

distPcX0(Pcv(0),K0) ≤ cdistX1−1/p(u∗ + v(0),K), (3.42)

proved as follows: Pick a w0 ∈ K0 such that w = u∗ + w0 + φc(w0) ∈ K satisfiesthe inequality |u∗ + v(0)−w|1−1/p ≤ 2 distX1−1/p

(u∗ + v(0),K). We then establishthe claim (3.42) by computing

distPcX0(Pcv(0),K0) ≤ |Pcv(0)− w0|PcX0 = |Pc(v(0)− w0 − φc(w0))|PcX0

≤ ‖Pc‖B(X1−1/p,X0)|v(0)− w0 − φc(w0)|1−1/p

= ‖Pc‖B(X1−1/p,X0)|u∗ + v(0)− w|1−1/p ≤ cdistX1−1/p(u∗ + v(0),K).

Using (3.41), (3.42), and (3.40) with v replaced by v0, we obtain

distPcX0(Pcz(0),K0) ≤ |Pcz(0)− Pcv(0)|0 + distPcX0(Pcv(0),K0)

≤ c|Psv0 − φc(Pcv0)|1−1/p + cdistX1−1/p(u∗ + v(0),K)

≤ cdistX1−1/p(u∗ + v0,K) ≤ cρ. (3.43)

By Theorem 2.3(c), y(t) = Pcz(t) satisfies the ODE (2.20). If ρ > 0 is chosensufficiently small, then (3.43) yields

distPcX0(y(0),K0) ≤ cρ ≤ ρ0,

where ρ0 > 0 was chosen above depending on r0. Since K0 is stable, it follows that

distPcX0(y(t),K0) ≤ r0 for all 0 ≤ t ≤ T (3.44)

(and thus |y(t)|1−1/p ≤ r by (3.38)). Using u∗ + z(t) ∈ Mc and (3.36), and alsothe Lipschitz property of φc, we estimate

distX1−1/p(u∗ + z(t),K)

= infw0∈K0

∣∣(u∗ + Pcz(t) + φc(Pcz(t)))−(u∗ + w0 + φc(w0)

)∣∣1−1/p

≤ infw0∈K0

(c|Pcz(t)− w0|0 + c|Pcz(t)− w0|0

)≤ cdistPcX0(Pcz(t),K0) (3.45)

for all 0 ≤ t ≤ T . By means of (3.45), (3.41), (3.40), and (3.44), we deduce

distX1−1/p(u∗ + v(t),K) ≤ distX1−1/p

(u∗ + z(t),K) + |v(t)− z(t)|1−1/p

≤ cdistPcX0(Pcz(t),K0) + ce−αt|Psv0 − φc(Pcv0)|1−1/p

)(3.46)

≤ cdistPcX0(y(t),K0) + cdistX1−1/p(u∗ + v0,K)

≤ cr0 + cρ < r/2 + r/2 = r


for all 0 ≤ t ≤ T , provided that r0 > 0 and ρ > 0 are sufficiently small. Thisstrict inequality is a contradiction that proves T =∞. In particular, the inequalitydistX1−1/p

(u∗ + v(t),K) < r holds for all t ≥ 0 and thus K is stable.To prove the asymptotic stability ofK assuming thatK0 is asymptotically stable,

we apply Theorem 3.2 to the solution v(t) that has been just constructed for allt ≥ 0. That is, we take the solution z with u∗+ z(t) ∈Mc that tracks the solutionv as described in (3.22). Replacing z by z in (3.43) and (3.46), we obtain

distPcX0(Pcz(0),K0) ≤ cdistX1−1/p(u∗ + v0,K), (3.47)

distX1−1/p(u∗ + v(t),K) ≤ cdistPcX0(Pcz(t),K0) (3.48)

+ ce−αt|Psv0 − φc(Pcv0)|1−1/p

),

for all t ≥ 0. Set y(t) = Pcz(t). Due to the asymptotic stability of K0 and (3.47),if distX1−1/p

(u∗ + v0,K) is sufficiently small, then one has distPcX0(y(t),K0) → 0as t→∞. Finally, due to (3.48), we conclude that distX1−1/p

(u∗ + v(t),K)→ 0 ast→∞.

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R. Johnson, Dipartimento di Sistemi e Informatica, Universita di Firenze, Italy 50139E-mail address: [email protected]

Y. latushkin, Department of Mathematics, University of Missouri, Columbia, MO65211, USA

E-mail address: [email protected]

R. Schnaubelt, Department of Mathematics, Karlsruhe Institute of Technology,

76128 Karlsruhe, Germany

E-mail address: [email protected]

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