Digital Object Identifier (DOI) 10.1007/s00205-004-0345-2Arch. Rational Mech. Anal. 176 (2005) 303–330

Global Existence for Elastic Waveswith Memory

Vladimir Georgiev, Bruno Rubino & Rosella Sampalmieri

Abstract

We treat the Cauchy problem for nonlinear systems of viscoelasticity with amemory term. We study the existence and the time decay of the solution to thisnonlinear problem. The kernel of the memory term includes integrable singularityat zero and polynomial decay at infinity. We prove the existence of a global solutionfor space dimensions n � 3 and arbitrary quadratic nonlinearities.

1. Introduction

The classical models of nonlinear elasticity have been intensively studied andeven the one-dimensional models do not admit, in general, smooth solutions inlarge (see [10] and [12]). To be more precise, let us consider an n-dimensionalbody whose reference configuration is in R

n. The motion of the body in the timeinterval [0, T ] is described by the displacement function u, which is defined in[0, T ] × R

n and is Rn valued. For the classical model of elastic bodies the stress

tensor σ is a function of ∇u = (∂x1u, . . . , ∂xnu

), i.e.,

σ = σ (∇u) .A typical elastic model can be described by the equation

utt − ∇ · σ (∇u) = 0, t ∈ [0, T ]. (1)

The blow-up mechanism can be manifested for the space dimensions n = 3 as isshown in [16] even in the case of small initial data.

The nonlinear viscoelasticity model (see [3] for example) is another dissipativemechanism that allows us to try to find global solutions. This mechanism meansthat

σ = σ (∇u,∇u) ,

304 V. Georgiev, B. Rubino & R. Sampalmieri

where u = ∂tu. The existence of global solutions for the higher-dimensional casen � 3 essentially depends on the dispersive properties of the solution. In [15] aglobal existence result is established on the basis of a suitable Lp −Lq estimate ofthe linearized problem. Since the viscoelasticity model gives an example of a verystrong dissipative effect that dominates the hyperbolic character of the equation, itis natural to try to find alternative models with weaker dissipative terms. The Boltz-mann model of linear viscoelasticity assumes that the instantaneous stress dependson the instantaneous strain and the entire history of the strain rate (see [11]). In thissense we can consider the equation

utt − ∇ · [σ (∇u)− �′ ∗ σ (∇u)] = 0, (2)

where a ∗ b(t) = ∫ t0 a(t − τ)b(τ )dτ and �′(t) is the memory kernel.

Our main goal is to establish the global existence and evaluate the decay rateof the solution for the Cauchy problem associated with (2) by using the dissipativeeffect of the equation. TheLp decay estimates are similar to those obtained for vis-coelasticity [15], heat equations [23] and Navier-Stokes equation [7, 9]. Previousresults in this direction have been obtained only for special types of kernel �′ (seeDafermos [4], Dassios & Zafiropoulos [5], Muñoz Rivera [13] Renardy,Hrusa & Nohel [17]). Moreover, Zeng [24] has proved sharp Lp decay esti-mates for the one-dimensional problem. Some a priori estimates for the linearizedproblem are discussed in Nohel & Shea [14], Staffans [21]. The Lp decay esti-mate (under reasonable assumptions on the memory term) is obtained in Kirova

& Yordanov [8] for the case of linear viscoelasticity.To state our main result, we shall describe the assumptions on σ and �. We

assume the function

σ : Mn×n −→ Mn×nto be smooth and that the following conditions are satisfied:

(H1) σ(∇u) = 12

[∇u+ (∇u)T ] c(∇u)+ d(∇u)(div u)I , with

c : Mn×n −→ Mn×n,d : Mn×n −→ R.

Moreover we assume that c and d satisfy

(H2) c(∇u) = 2µI + c(∇u), µ > 0, c(0) = 0,d(∇u) = λ+ d(∇u), λ > 0, d(0) = 0,where µ and λ are positive numbers called Lamè constants.

Following [8, 14, 21], we impose the condition that the memory kernel�(t) ∈ CN+1

with N sufficiently large and satisfies the assumptions

(H3) ‖�′‖L1(R+) < 1;

(H4) Re(�(iω)) � k0

1 + ω2 for some k0 > 0 and any ω ∈ R,

where �(p) = ∫∞0 e−pt�(t)dt is the Laplace transform of the kernel�(t) =

− ∫∞t�′(τ )dτ ;

(H5) ∂κs (sl�′(s)) ∈ L1(R+) for κ � l, l = 0, 1, . . . N, and N � 2n+ 8.

Global Existence for Elastic Waves with Memory 305

Hypotheses (H3)–(H5) enable us to consider various examples of kernels withexponential and even with polynomial decay as t tends to infinity. Some of theseexamples have important application in polymer dynamics [6]. We start with threeexamples with exponential decay of the kernel.

Example 1. A classical example of a memory kernel in viscoelasticity is the caseof an exponential function: if we choose

�(t) = �a,c0(t) = c0

ae−at ,

where a > 0 and c0 > 0, then we obtain

Re(�(iω)) = c0

a2 + ω2 (3)

so the assumption (H4) is true. Note that

‖�′a,c0(·)‖L1(R+) = c0

a(4)

so the assumption (H3) is valid for 0 < c0 < a and the exponential decay of

�′a,c0(t) = −c0e

−at

shows that (H5) is fulfilled.

Example 2. The following example is the case of a function with exponential decayat infinity but a singularity t−1/2 at the origin:

�′(t) = − 1√2π

e−t√t.

In this case

Re(�(iω)) = − Im(√

1 − iω)

ω

1

1 + ω2 .

Example 3. Another case of exponential decay at infinity with a singularity at theorigin is

�′(t) = 1

2log(t)e−et .

Also in this case the hypotheses are satisfied since

Re(�(iω)) = 1

4 + ω2

(1

2log

(4 + ω2

)+ γ − 2 arctan(ω/2)

w

),

where γ = 0, 577 . . . is the Euler constant.

306 V. Georgiev, B. Rubino & R. Sampalmieri

Next, we consider three examples of type

�′(t) =∞∑

j=1

pje−λj t ,

where λj are suitable positive constants, while pj are constants for simplicity. Thiscan be compared with [24], where a finite sum

�′(t) =m∑

j=1

pj (t)e−λj t ,

with pj (t) polynomials in t is considered. In all these examples we use the basicviscoelastic Example 1.

Example 4. Take

�j (t) = e−tj

jβ, (5)

where β = β(N) is a sufficiently large number and set

�(t) = c0

∞∑

j=1

�j (t).

Then it is easy to see that

|�′(t)| � Cc0e−t

so the assumptions (H3) and (H5) are fulfilled for c0 small enough. By using (3),we can see that the assumption (H4) is satisfied.

Example 5. Take

�j (t) = e−tjγ

jγ, (6)

where γ > 1 and set

�(t) = c0

∞∑

j=1

�j (t), c0 > 0.

Then it can be seen that �′(t) decays exponentially as t tends to ∞, while �′(t) ∼t−1/γ near t = 0. The assumptions (H3) and (H5) are fulfilled if c0 > 0 is smallenough. Using (3), it can be seen that the assumption (H4) is also satisfied. Thisexample is typical for models in polymer dynamics (see [6]).

Global Existence for Elastic Waves with Memory 307

Example 6. Take

�j (t) = e−t/j

jβ, (7)

where β = β(N) is a sufficiently large number and set

�(t) = c0

∞∑

j=1

�j (t).

Then it is easy to see that �′(t) decays polynomially as t tends to ∞. The assump-tions (H3) and (H5) are fulfilled if c0 > 0 is small enough. Using (3), it can be seenthat the assumption (H4) is also satisfied.

Now we can state our main result.

Theorem 1. Suppose n � 3 and (H1)–(H5) are fulfilled. There exists ε0 > 0 sothat the nonlinear Cauchy problem

utt − ∇ · σ(∇u)− �′ ∗ ∇ · σ(∇u) = 0,

u(0, x) = u0(x),

ut (0, x) = u1(x) (8)

for the elastic equation with memory term has a unique global solution

u ∈ C([0,∞);Wk,1(Rn; Rn)) ∩ C1([0,∞);Wk−1,1(Rn; R

n))

provided

u0 ∈ Wk,1(Rn; Rn) ∩Hk(Rn; R

n),

u1 ∈ Wk−1,1(Rn; Rn) ∩Hk−1(Rn; R

n)

and

‖u0‖Wk,1 + ‖u1‖Wk−1,1 + ‖u0‖Hk + ‖u1‖Hk−1 � ε0,

where the number k is greater than 2n+ 8.

Remark 1. The solution u(t, x) of the problem (8) satisfies

‖∇u(t)‖L∞ � c(1 + t)−n2 .

Remark 2. For n = 3 we see that the assumption

c(∇u) = O(|∇u|), d(∇u) = O(|∇u|)is sufficient for the existence of a global solution. Therefore, in this case, we havethe existence of a global solution for arbitrary linear terms in the nonlinearities.Recent results due to Agemi [1] and Sideris [20] show that the global existenceresult for pure elasticity and n = 3 needs an additional algebraic assumption onthe linear parts of c(∇u), d(∇u).

In conclusion, the weak dissipative term of memory type we study is sufficientto assure the existence of global (in time) solutions and prevents the developmentof singularities.

308 V. Georgiev, B. Rubino & R. Sampalmieri

2. Preliminaries

Our main purpose is to establish the existence of global solutions by using thedissipative effect of the problem (8). Since Theorem 1 establishes this result inthe case of singular kernels �′ ∈ L1, we need a suitable L∞ decay estimate forthe linearized problem. The key assumption which enables us to control the L∞norm of the solution is the positivity of the equilibrium stress module and this isexpressed by the hypothesis in (H3) ‖�′‖L1(R+) < 1.

First of all, we substitute (H1) and (H2) in (8) and obtain a new, convenientform for the equation of the problem (8)

utt − L [u] = R [∇u] , (9)

where

L[u] = µ�u+ (λ+ µ)∇(div u)+ µ�′ ∗ �u+ (λ+ µ)�′ ∗ ∇(div u).

The remainder R in (9) is given by

R [∇u] = ∇ · g (∇u)+ �′ ∗ ∇ · g (∇u) = ∇ · (g (∇u)+ �′ ∗ g (∇u)) , (10)

where

g (∇u) = ∇uc(∇u)+ d(∇u)(tr∇u)I. (11)

Now we will state some useful estimates that we shall deduce (see the Appendix 1for the proof) from the result in [8] for the linear Cauchy problem

utt − L [u] = R [∇u] ,

u(0, x) = u0(x),

ut (0, x) = u1(x) (12)

in two special cases. The first one is an L1 − L∞ estimate, of special importancewhen t � 1

‖∇j u(t)‖L∞ � Ct−(j+n)/2‖u0‖L1 + Ct−(j+n−1)/2‖u1‖L1

+C∫ t/2

0(t − s)−(n+j−1)/2‖R(s)‖L1ds

+C∫ t

t/2‖|ξ |j+n−1−δ R(s, ξ)‖L∞

ξds

+C∫ t

t/2‖|ξ |j+n−1+δ R(s, ξ)‖L∞

ξds, (13)

where j � 0 is an integer, δ > 0 and

R(s, ξ) =∫e−ixξR(s, x)dx

Global Existence for Elastic Waves with Memory 309

is the space Fourier transform of R(s, x). For 0 � t � 1, we can use the followingestimate:

‖∇j u(t)‖L∞ � C‖∇j+n−1u0‖L1 + C‖∇j+n+1u0‖L1

+C‖∇j+n−2u1‖L1 + C‖∇j+nu1‖L1

+C∫ t

0‖∇j+n−2R(s)‖L1ds + C

∫ t

0‖∇j+nR(s)‖L1ds, (14)

The second case is an L2 − L2 estimate given by

‖∇j u(t)‖L2 � C(t−j/2‖u0‖L2 + t−(j−1)/2‖u1‖L2)

+C∫ t/2

0(t − s)−j/2‖∇−1R(s)‖L2ds

+C∫ t

t/2(t − s)−1/2‖∇j−2R(s)‖L2ds (15)

when t � 1, while for 0 � t � 1, we have

‖∇j u(t)‖L2 � C(‖∇j u0‖L2 + ‖∇j−1u1‖L2)

+C∫ t

0(t − s)−1/2‖∇j−2R(s)‖L2ds. (16)

It is clear that the decay of the L∞ norm is better than the decay of the lin-

ear wave equation t− n−12 , as j > 0. Moreover, we observe that the L2 estimates

obtained for the space and time derivatives of the solution of (12) are improved,when compared to those obtained in the absence of the dissipative memory term.

The last two inequalities replace the classical energy estimates in order to stopderivative losses in this quasilinear problem.

To establish our main result, we plan to apply a contraction argument for asuitable Banach space. For this purpose, given any integer k2 � 1, we consider theSobolev norm

‖∇u‖Hk2−1;T =k2∑

j=1

sup0�t�T

(1 + t)bj ‖∇j u(t)‖L2 , (17)

where Hi(Rn) = Wi,2(Rn) denotes the standard Sobolev space in Rn. For sim-

plicity, we shall denote these Sobolev spaces simply by Hi. Moreover, we have

|∇j u(t, x)| =∑

|β|=j|∂βx u(t, x)|.

We shall choose bj as follows:

bj = j − 1

2.

310 V. Georgiev, B. Rubino & R. Sampalmieri

The above definition (17) implies in particular that

‖∇u(t)‖H j−1 � C

(1 + t)(j−1)/2‖∇u‖Hk2−1;T

for any integer j, 1 � j � k2. Here and below H s = H s(Rn) is the homogeneousSobolev space with norm

‖f ‖H s = ‖| · |s f ‖L2 .

After interpolation we get

‖∇u(t)‖H s � C

(1 + t)s/2‖∇u‖Hk2−1;T (18)

for any real s, 0 � s � k2 − 1.Further, we define a weighted L∞ norm. Namely, for any integer k1 � 1 we

consider the norm

‖∇u‖Wk1−1,∞;T =k1∑

j=1

sup0�t�T

(1 + t)aj ‖∇j u(t)‖L∞ , (19)

where aj are suitable positive numbers defined as follows:

aj = n− 2 + j − δ

2= bj + n− 1 − δ

2,

where δ > 0 is a sufficiently small number. To obtain the existence of the globalsolution we use a continuation principle for a Banach space with norm

�u k1,k2;T= ‖∇u‖Wk1−1,∞;T + ‖∇u‖Hk2−1;T , (20)

where 1 � k1 < k2 are defined by k1 = n + 6 and k2 = 2n + 8 (see (43)). Ourgoal is to establish a priori estimates

(S1) ‖∇u‖Hk2−1;T � Cε + C �u 2k1,k2;T ,

‖∇u‖Wk1,∞;T � Cε + C �u 2k1,k2;T ,

where 0 � ε � ε0 .Once (S1) is established, we get

(S2) �u k1,k2;T� C1 ε

with some constant C1 independent of T and ε. From this a priori estimate and thecontinuation principle (see [19]) we complete the proof.

In the next section we will apply suitable energy estimates and shall obtaina local existence result. A bootstrap argument of the proof of our main result ispresented in the last section.

Global Existence for Elastic Waves with Memory 311

3. Local existence of solutions

In this section we study the existence of local solutions of the Cauchy problem(8). The main result is

Theorem 2. Suppose n � 1 and (H1)–(H5) are fulfilled. There exists T > 0 andr0 > 0 so that the Cauchy problem (8) has a solution

u ∈ ∩kj=1C([0, T ]; H j (Rn; Rn))

provided

u0 ∈ ∩kj=1Hj (Rn; R

n), u1 ∈ Hk−1(Rn; Rn)

and

k∑

j=1

‖∇j u0‖L2 + ‖u1‖Hk−1 � r0,

where k � n/2 + 2.

Proof. In what follows we will consider the Cauchy problem (12). To show theexistence of a local solution, we equip the space

∩kj=1C([0, T ]; H j (Rn; Rn))

with the natural norm

sup0�t�T

‖∇u(t)‖Hk−1(Rn)

and consider the map

ψ ∈ ∩kj=1C([0, T ]; H j (Rn; Rn)) → u = A(ψ) ∈ ∩kj=1C([0, T ]; H j (Rn; R

n))

defined by the solution of

utt − L [u] = R [∇ψ] ,

u(0, x) = u0(x),

ut (0, x) = u1(x).

Our purpose is to show that it is possible to find T > 0 and r0 > 0 so that A mapsthe ball of radius r0 of

∩kj=1C([0, T ]; H j (Rn; Rn))

into itself and that A is a contraction on this ball.

312 V. Georgiev, B. Rubino & R. Sampalmieri

By taking k � n/2 + 2 and applying the estimate (16), we get

‖∇u(t)‖Hk−1 � C(‖∇u0‖Hk−1 + ‖u1‖Hk−1)

+C∫ t

0(t − s)−1/2‖∇−1R [∇ψ(s)] ‖Hk−1ds.

Note that

∇−1R = ∇−1 (∇ · (g (∇ψ)+ �′ ∗ g (∇ψ)))

holds, so

‖∇−1R(∇ψ)‖Hk−1 � C(‖g (∇ψ) ‖Hk−1 + ∣

∣�′ ∗ ‖g (∇ψ) ‖Hk−1

∣∣)

and by using the inequality

|�′ ∗ h(T )| � ‖�′‖L1(R+) sup0�t�T

|h(t)|,

combined with the Moser-type estimate (see (84) in Appendix 1)

‖g(v)‖Hs � C(‖v‖Hs )‖v‖2Hs , s > n/2, (21)

we get

‖∇u(t)‖Hk−1 � C(‖∇u0‖Hk−1 + ‖u1‖Hk−1)+ C

∫ t

0(t − s)−1/2‖∇ψ(s)‖2

Hk−1ds

so we arrive at the estimate

‖∇u(t)‖Hk−1) � C(‖u0‖∩kj=1H

j + ‖u1‖Hk−1

)+ Ct1/2 sup

0�s�t‖∇ψ(s)‖2

Hk−1 .

Hence, for T > 0 small enough, we obtain

‖∇u‖Hk−1,T � C(‖u0‖∩kj=1H

j + ‖u1‖Hk−1

)+ CT 1/2‖∇ψ‖2

Hk−1,T

and this estimate with suitable r0 > 0 and T > 0 small enough, implies that Amaps the ball of radius r0 into itself.

By taking two elementsψ1 andψ2 in the same ball, we denote by u1 and u2 thecorresponding solutionsA(ψ1) andA(ψ2).Then we can repeat the above argumentbased on the estimate (21) and in this way we obtain

‖∇(u1 − u2)‖Hk−1,T � CT 1/2‖∇(ψ1 − ψ2)‖Hk−1,T

(‖∇ψ1‖Hk−1,T

+ ‖∇ψ2‖Hk−1,T

).

With T small enough we see that A is a contraction and the contraction mappingprinciple completes the proof of the theorem. ��It is not difficult to obtain also the following variant of the local existence result.

Global Existence for Elastic Waves with Memory 313

Corollary 1. Suppose n � 1 and (H1)–(H5) are fulfilled. There exists T > 0 andr0 > 0 so that the Cauchy problem (8) has a solution

u ∈ ( ∩kj=1 C([0, T ]; H j (Rn; Rn))) ∩ C1([0, T ];Hk−1(Rn; R

n))

provided

u0 ∈ ∩kj=1Hj (Rn; R

n), u1 ∈ Hk−1(Rn; Rn))

and

k∑

j=1

‖∇j u0‖L2 + ‖u1‖Hk−1 � r0,

where k � n/2 + 2.

Proof. From the equation we have

utt ∈ C([0, T ];Hk−2).

From this property and

u ∈ ∩kj=1C([0, T ]; H j (Rn; Rn)),

we conclude that

ut ∈ C([0, T ];Hk−1). ��

4. Existence of a global solution

Let u(t, x) be a local solution of the equation

utt − L [u] = R [∇u] ,

u(0, x) = u0(x),

ut (0, x) = u1(x). (22)

In order to establish the estimates (S1), we need to control ‖∇j u(t)‖L∞ and‖∇j u(t)‖L2 . We shall estimate these norms provided the absolute values of thefunctions ∇j u(t, x) behave like

|∇j u(t, x)| � Cε

(1 + t)(n−2+j−δ)/2 , 1 � j � k1. (23)

Here ε is a small positive number.First, we shall consider the norms in the integrals on the right-hand side of (15),

for t � 1. For this purpose we have to estimate ‖∇j−2R(t)‖L2 . In fact we have

∇−1R = ∇−1 (∇ · (g (∇u)+ �′ ∗ g (∇u)))

314 V. Georgiev, B. Rubino & R. Sampalmieri

so we find

‖∇j−2R(t)‖L2 � ‖∇j−1g (∇u) (t)‖L2 +∫ t

0|�′(t − τ)|‖∇j−1g (∇u) (τ )‖L2dτ.

(24)

Now we have to estimate the term ∇j−1g (∇u) . For j = 1 we take advantageof (11), (17) and (19) and derive

‖g (∇u) ‖L2 � ‖∇u‖L2‖∇u‖L∞ � (1 + t)−(n−1−δ)/2 �u 2k1,k2;T . (25)

The above estimate for n � 3 and the explicit form of the memory term

�′ ∗ g (∇u)

suggest that we should evaluate a double convolution term of type

IA(t) =∫ t

t/2(t − s)−1/2

∫ s

0�′(s − τ)(1 + τ)−Adτds,A � 1. (26)

To this end we shall use the following estimates.

Lemma 1. For any N � 0, A � 0 and s � 0, the following estimate holds:

∣∣∣∣

∫ s

0�′(s − τ)(1 + τ)−Adτ

∣∣∣∣ � C

(1 + s)min(A,N)(‖�′‖L1 + ‖| · |N�′‖L1).

Proof. It is sufficient to consider only the case s � 1.Then we can use the inequal-ities

∫ s

s/2|�′(s − τ)|(1 + τ)−Adτ � C(1 + s)−A

∫ s

s/2|�′(s − τ)|dτ

and

∫ s/2

0|�′(s − τ)|(1 + τ)−Adτ � C(1 + s)−N

∫ s/2

0|s − τ |N |�′(s − τ)|dτ

This completes the proof of the Lemma. ��Lemma 2. For any N > 0, A > 1/2 and for

0 < δ < min(N,A− 1/2),

the double integral IA(t), defined in (26), satisfies the estimate

|IA(t)| � C

(1 + t)min(A−1/2−δ,N−δ) (‖�′‖L1 + ‖| · |N�′‖L1).

Global Existence for Elastic Waves with Memory 315

Proof. It is clear that we can take t � 1. Further, the integral

JA(t) ≡∫ t

t−1/2(t − s)−1/2

∫ s

0�′(s − τ)(1 + τ)−Adτds

� C supt−1/2�s�t

∫ s

0|�′(s − τ)|(1 + τ)−Adτds

obviously satisfies a stronger estimate

|JA(t)| � C

(1 + t)min(A,N)(‖�′‖L1 + ‖| · |N�′‖L1)

in view of Lemma 1. For

IA(t)− JA(t) =∫ t−1/2

t/2(t − s)−1/2

∫ s

0�′(s − τ)(1 + τ)−Adτds,

we have

|IA(t)− JA(t)| � C ln1/2(2 + t)

∥∥∥∥

∫ s

0|�′(s − τ)|(1 + τ)−Adτ

∥∥∥∥L2(t/2,t)

.

For any positive number δ ∈ (0, A− 1/2) we can use the inequalities∥∥∥∥

∫ s

s/2|�′(s − τ)|(1 + τ)−Adτ

∥∥∥∥L2(t/2,t)

� C(1 + t)−A+1/2+δ∥∥∥∥

∫ s

s/2|�′(s − τ)|(1 + τ)−1/2−δdτ

∥∥∥∥L2(t/2,t)

and∥∥∥∥

∫ s/2

0|�′(s − τ)|(1 + τ)−Adτ

∥∥∥∥L2(t/2,t)

� C(1 + t)−N∥∥∥∥

∫ s/2

0|s − τ |N |�′(s − τ)|(1 + τ)−Adτ

∥∥∥∥L2(t/2,t)

so another application of the Young inequality for convolutions gives∥∥∥∥

∫ s

0|�′(s − τ)|(1 + τ)−Adτ

∥∥∥∥L2(t/2,t)

� C(1 + t)− min(A−1/2−δ,N) (‖�′‖L1 + ‖| · |N�′‖L1

)

and we arrive at the needed inequality.This completes the proof of the lemma. ��

Lemma 3. For any A � 1, the following estimate holds:∣∣∣∣

∫ t/2

0

∫ s

0�′(s − τ)(1 + τ)−Adτds

∣∣∣∣ � C ln(2 + t)‖�′‖L1 .

316 V. Georgiev, B. Rubino & R. Sampalmieri

Proof. It is clear that we can take t � 1.We have∣∣∣∣

∫ t/2

0

∫ s

0�′(s − τ)(1 + τ)−1dτds

∣∣∣∣ � C ln(2 + t)‖�′‖L1 . (27)

This completes the proof. ��Unifying the above Lemmas (more precisely we takeN = 1, A = 1 in Lemma

2), we get in particular.

Corollary 2. For any δ ∈ (0, 1/2),∣∣∣∣

∫ t

0(t − s)−1/2

∫ s

0�′(s − τ)(1 + τ)−1dτds

∣∣∣∣ � C

(‖�′‖L1 + ‖| · |�′‖L1)

(1 + t)1/2−δ .

For j = 1, the sum of the last two integrals in the right-hand side of (15) is∫ t

0(t − τ)−

12 ‖∇−1R‖L2dτ � c �u 2

k1,k2;T (28)

due to the estimate (25), Corollary 2 and the assumption that n � 3.For j � 2, we have to estimate the term ∇j−1g (∇u) or more generally a term

of type ∇hf (u), where h � 1 is an integer.The estimate for a nonlinear term of the form ∇hf (u) is based on the formula

∇hf (u) =h∑

s=1

∑

β1+···+βs=hcsβ1,··· ,βs f

(s)(u)∂β1x u · · · ∂βsx u (29)

with h �= 0.Now we use the following modification of the relation (29):

∇hf (u) = A1 + A2,

A1 =∑

β=hcβf

′(u)∂βx u,

A2 =h∑

s=2

∑

β1+···+βs=hcsβ1,··· ,βs f

(s)(u)∂β1x u · · · ∂βsx u.

We have to specify that the second term, for the case h = 1, is identically zero. Forh � 2 the sum A2 has a correct sense (the summation is for 2 � s � h).

Note that we can order the indices β1 � · · · � βs in the sum A2 and we haveβs � h− 1 in this sum. Then, considering the term ∇j−1g(∇u), we can write

∇j−1g(∇u) = B1 + B2, (30)

B1 = C(∇u)∇j u, (31)

B2 =j−1∑

s=2

∑

l1+···+ls=j−1

Cβ1,... ,βs (∇u)∇ l1+1u · · · ∇ ls+1u, (32)

Global Existence for Elastic Waves with Memory 317

where the functions C, Cβ1,... ,βs satisfy

C(y) = O(|y|), Cβ1,... ,βs (y) = O(1)

near y = 0. We assume the indices l1, . . . , ls ordered as follows

0 � l1 � l2 � · · · � ls , ls−1 � 1

with ls � j − 2 and lr � j−22 for 1 � r � s − 1 in the sum B2. Therefore, taking

2 � j � k2 and choosing

k1 =[k2

2

]+ 2, (33)

we have

lr + 1 � j

2� k2

2� k1 − 1, 1 � r � s − 1. (34)

From (32), we have the estimate (only for j � 3 since B2 = 0 if j = 2 as before)

|B2| � C

j−1∑

s=2

∑

l1+···+ls=j−1

|∇ l1+1u(t, x)| · · · |∇ ls+1u(t, x)|.

From the definition of the norms (17), (19) we obtain the estimates

‖∇ ls+1u(τ)‖L2 � C

(1 + τ)(ls−δ)/2‖∇u‖Hk2−1;t

and

|∇ lr+1u(τ, x)| � C

(1 + τ)(n+lr−1−δ)/2 ‖∇u‖W [(k1−1)],∞;t

since (34), for 0 � τ � t . As a consequence we get

‖B2(τ )‖L2 � C

j−1∑

s=2

‖∇u‖s−1W [(k1−1)/2],∞;t‖∇u‖Hk2−1;t

(1 + τ)((n−1)(s−1)+j−1−δ)/2

for 0 � τ � t. Since s � 2 we simplify the above estimate as follows:

‖B2(τ )‖L2 � C(1 + τ)−(n+j−2−δ)/2‖∇u‖W [(k1−1)/2],∞;t‖∇u‖Hk2−1;t . (35)

We see from the representation of B1 in (31) that we can proceed with the sameargument for the estimate of the term B2 obtaining

‖B1(τ )‖L2 � C(1 + τ)−(n+j−2−δ)/2‖∇u‖W [(k1−1)/2],∞;t‖∇u‖Hk2−1;t . (36)

Now the representation (30) leads to

‖∇j−2R(∇)(τ )‖L2 � C(�′ ∗ (1 + | · |)−(n+j−2−δ)/2(τ )

+ (1 + τ)−(n+j−2−δ)/2) ‖∇u‖W [(k1−1)/2],∞;t‖∇u‖Hk2−1;t .

318 V. Georgiev, B. Rubino & R. Sampalmieri

In this way, applying Lemma 2, we conclude that the last integral in the right-handside of (15) can be estimated as follows:

∫ t

t/2(t − τ)−1/2 ‖∇j−2R(τ)‖L2dτ

� C(1 + t)− min((n+j−3)/2−δ,N−δ)‖∇u‖W [(k1−1)/2],∞;t‖∇u‖Hk2−1;t .

This estimate for the case when

(n+ j − 3)/2 − δ � (j − 1)/2,

i.e., n > 2 + 2δ, 2 � j � k2 and (33) implies∫ t

t/2(t − τ)−1/2 ‖∇j−2R(τ)‖L2dτ

� C(1 + t)−(j−1)/2‖∇u‖W [(k1−1)/2],∞;t‖∇u‖Hk2−1;t . (37)

To close the L2 estimate, we need to consider the integral∫ t/2

0(t − s)−j/2 ‖∇−1R(s)‖L2ds � C

(1 + t)j/2

∫ t/2

0‖∇−1R(s)‖L2ds

appearing in (15) with t � 1. From (24) (with j = 1) and (25) we have∫ t/2

0‖∇−1R(τ)‖L2dτ � C

∫ t/2

0

(1

(1 + τ)(n−1−δ)/2

+ |�′| ∗ (1 + | · |)−(n−1−δ)/2(τ ))

‖∇u‖W [(k1−1)/2],∞;t‖∇u‖Hk2−1;t dτ.

Applying Lemma 1, we find (for n � 3)∫ t/2

0‖∇−1R(τ)‖L2dτ � C(1 + t)δ/2‖∇u‖W [(k1−1)/2],∞;t‖∇u‖Hk2−1;t .

In this way, taking δ > 0 sufficiently small (say |δ| < 1/2) we get∫ t/2

0(t − s)−j/2 ‖∇−1R(s)‖L2ds � C

‖∇u‖W [(k1−1)/2],∞;t‖∇u‖Hk2−1;t(1 + t)(j−1)/2

.

This estimate and (37) close the proof of the first inequality in (S1).

Turning to the second inequality in (S1), we use (23) and evaluate the weightedL∞ norm defined in (19). From the estimate (13), for 1 � j � k1 we have

‖∇j u(t)‖L∞ � Ct−(j+n)/2‖u0‖L1 + Ct−(j+n−1)/2‖u1‖L1

+C∫ t/2

0(t − s)−(n+j−1)/2‖R(s)‖L1ds

+C∫ t

t/2‖|ξ |j+n−1−δ R(s, ξ)‖L∞

ξds

+C∫ t

t/2‖|ξ |j+n−1+δ R(s, ξ)‖L∞

ξds. (38)

Global Existence for Elastic Waves with Memory 319

Note that the definition of the norm �u k1,k2;T guarantees that

‖R(s)‖L1 � C((1 + s)−1/2 + |�′| ∗ (1 + | · |)−1/2(s)|

)�u 2

k1,k2;t

From the assumption

(1 + | · |)N�′ ∈ L1, N � 2

we easily get

|�′| ∗ (1 + | · |)−1/2(s)| � C(1 + |s|)−1/2

so

‖R(s)‖L1 � C(1 + s)−1/2 �u 2k1,k2;t

and∫ t/2

0(t − s)−(n+j−1)/2‖R(s)‖L1ds � C(1 + t)−(n+j−2)/2 �u 2

k1,k2;t . (39)

The estimate for the nonlinear term

‖|ξ |j+n−1±δR [∇u](s, ξ)‖L∞

is based on the estimates of Appendix 2 and the relation

R [∇u] = ∇ (g(∇u)+ �′ ∗ g(∇u)) .Applying Proposition 1, we find

‖|ξ |j+n±δg [∇u](s, ξ)‖L∞ � C‖∇u‖L2‖∇u‖H j+n±δ .

Then (18) yields

‖|ξ |j+n±δg [∇u](s, ξ)‖L∞ � C

(1 + τ)(j+n±δ)/2�u 2

k1,k2;t

for 1 � j � k1, k1 + n+ 2 � k2.Applying the estimate of Lemma 1, we obtain

‖|ξ |j+n−1±δR [∇u](s, ξ)‖L∞ � C

(1 + τ)(j+n±δ)/2�u 2

k1,k2;t

The application of the above inequality together with the estimate∫ t

t/2(1 + τ)−Bdτ � C

(1 + t)−B+1 , B > 1,

leads to the estimate∫ t

t/2‖|ξ |j+n−1±δR [∇u](τ, ξ)‖L∞

ξdτ � C(1 + t)−(n+j−2±δ)/2 �u 2

k1,k2;t(40)

320 V. Georgiev, B. Rubino & R. Sampalmieri

provided j + n+ 2 � k2 for all j � k1. This requirement can be rewritten as

k1 + n+ 2 � k2. (41)

This inequality and (33) show that one sufficient condition to have the above esti-mates of the solution to the Cauchy problem is the inequality

[k2/2] + n+ 4 � k2. (42)

It is clear that taking k2 � 2(n+ 4) we have (42).By using (39) and (40) in (38), we conclude

‖∇j u(t)‖L∞ � Cε

(1 + t)(n−1+j)/2 + C �u 2k1,k2;t (1 + t)−(j+n−1)/2.

This argument shows that the second estimate in (S1) is valid provided

k2 = 2n+ 8, k1 = n+ 6. (43)

Thus, for this choice of k1 and k2 both the estimates in (S1) are valid and thiscompletes the bootstrap argument.

5. Appendix 1

In this appendix we prove the estimates (13), (14) and (15), (16). First, we shallmake a reduction from the operator

Q = µ� + (λ+ µ)∇div (44)

to the classical Laplace operator. For the purpose we use the standard projections

P1[u](x) = 1

(2π)n

∫eixξ 〈 ξ|ξ | , u(ξ)〉

ξ

|ξ |dξ,P2[u] = u− P1[u].

Here u(ξ) is the Fourier transform. It is easy to verify the properties

∇divP1[u] = �P1[u],divP2[u] = 0.

From these relations we get (see (44))

QP1[u] = (λ+ 2µ)�P1[u],QP2[u] = µ�P2[u].

Hence, the projections P1 and P2 diagonalize Q to multiples of �. For this theestimate of the Cauchy problem associated with

utt − L [u] = utt − Q [u] + �′ ∗ Q [u]

Global Existence for Elastic Waves with Memory 321

can be reduced to the estimate of the linear integro-differential Cauchy problem

�u+ �′ ∗ �u = 0,

u(0, x) = 0,

ut (0, x) = f (x). (45)

More precisely the proof of (13) and (15) is reduced to the proof of

‖∇ku(t)‖L∞ � C

(χ{t�1}‖f ‖L1

t (k+n−1)/2

+ χ{t�1}(‖|ξ |k+n−1−εf ‖L∞ + ‖|ξ |k+n−1+εf ‖L∞

))

, (46)

where ε > 0, and

‖∇ku(t)‖L2 �(χ{t�1}t−

k−12 ‖ (1 − t�)−N f ‖L2

+ χ{t<1}‖ (1 − t�)−N ∇k−1f ‖L2

). (47)

To achieve the above estimates we will employ the following result obtained in [8]:

|E(t, ξ)| � c

|ξ |(

1 + t |ξ |2)−N

, |ξ | > 0, N = N(n, k) � 0, N ∈ N, (48)

|∂t E(t, ξ)| � C(

1 + t |ξ |2)−N

, |ξ | > 0, N = N(n, k) � 0, N ∈ N, (49)

where E ∈ C2(R

+, S1 (Rn))

is the fundamental solution that satisfies

�E + �′ ∗ �E = 0,

E(0, x) = 0,

Et (0, x) = δ. (50)

Here and below η(t, ξ) = ∫e−ix·ξ η(t, x)dx denotes the space Fourier transform

of η(t, x). Making a space Fourier transform in x we find

∂tt E + |ξ |2E − |ξ |2 �′ ∗ E = 0,

E(0, ξ) = 0,

∂t E(0, ξ) = 1. (51)

Further, we use the representation formula

u(t, x) = (2π)−n∫eix·ξ E(t, ξ)f (ξ) dξ, (52)

as well as

u(t, ξ) = E(t, ξ)f (ξ) . (53)

322 V. Georgiev, B. Rubino & R. Sampalmieri

Now we have (taking t � 1)

‖∇ku(t)‖L∞ � C‖|·|ku(t)‖L1 = C‖|·|kE(t)f ‖L1 � C‖|·|kE(t)‖L1‖f ‖L∞ .

The estimate (48) implies

‖|·|kE(t)‖L1 � C‖|·|k−1(1 + t |·|2)−N‖L1 � C

t(n+k−1)/2

for N > k − 1 + n and we arrive at

‖∇ku(t)‖L∞ � Ct−(k+n−1)/2‖f ‖L∞ � Ct−(k+n−1)/2‖f ‖L1 . (54)

For 0 < t � 1, we use the inequalities

‖∇ku(t)‖L∞ � ‖|·|ku(t)‖L1 � ‖|·|k+n−ε(1 + |·|)2εE(t)f ‖L∞

� ‖|·|k+n−1−εf ‖L∞ + ‖|·|k+n−1+εf ‖L∞ . (55)

To evaluate the L2 norm we use the inequalities

‖∇ku(t)‖L2 = (2π)−n/2‖|·|ku(t)‖L2 = (2π)−n/2‖|·|kE(t)f ‖L2

� C‖|·|k−1(

1 + t |·|2)−N

f ‖L2 . (56)

Then we shall use the inequalities (here s � 0, k − 1 � 0 and N � k − 1 + s)

|ξ |k−1(1 + t |ξ |2)−N �{

|ξ |k−1−s t−s/2 if 0 � t � 1;

|ξ |−s t−(k−1+s)/2 if t � 1.(57)

We now combine the estimates (56) and (57) and split the estimates into twocases, t � 1 and t < 1.When t < 1, we simply obtain

‖∇ku(t)‖L2 � C

ts/2‖f ‖H k−1−s , 0 � s � 1. (58)

In the special case s = 0, we get in particular

‖∇ku(t)‖L2 � C‖f ‖H k−1 . (59)

In the case t � 1, we have

‖∇ku(t)‖L2 � C

t(k−1+s)/2 ‖f ‖H−s , s � 0. (60)

For s = 0, we get

‖∇ku(t)‖L2 � C

t(k−1)/2‖f ‖L2 . (61)

Global Existence for Elastic Waves with Memory 323

The above inequalities show that the solution of the general homogeneous Cau-chy problem

�u+ �′ ∗ �u = 0,

u(0, x) = u0(x),

ut (0, x) = u1(x) (62)

for t � 1 satisfies the inequality

‖∇ku(t)‖L∞ � Ct−(k+n)/2‖u0‖L1 + Ct−(k+n−1)/2‖u1‖L1 . (63)

Taking ε = 1 in (55), we get for 0 � t � 1,

‖∇j u(t)‖L∞ � C‖∇j+n−1u0‖L1 + ‖∇j+n+1u0‖L1

+C‖∇j+n−2u1‖L1 + C‖∇j+nu1‖L1 . (64)

For t � 1, the corresponding L2 estimate can be written as

‖∇j u(t)‖L2 � C(t−j/2‖u0‖L2 + t−(j−1)/2‖u1‖L2). (65)

For 0 � t � 1, we have the estimate

‖∇j u(t)‖L2 � C(‖∇j u0‖L2 + ‖∇j−1u1‖L2). (66)

To treat the inhomogeneous problem

�u+ �′ ∗ �u = F

u(0, x) = 0,

ut (0, x) = 0, (67)

we note that the following representation of the solution holds:

u(t, x) = (2π)−n∫ t

0

∫eixξ E(t − s, ξ)F (s, ξ)dξds. (68)

This relation immediately implies that

u(t, ξ) =∫ t

0E(t − s, ξ)F (s, ξ)ds. (69)

To check the identity (68) we denote by v(t, x) the right-hand side of (68), andfrom the fact that E is the solution to (51) we see that v satisfies

∂tt v −�v = F(t, x)

+(2π)−n∫ t

0

∫ t−s

0

∫eixξ�′(t − s − τ)|ξ |2E(τ, ξ)F (s, ξ)dξdτds.

Now by making a change of variables

τ → σ = s + τ,

324 V. Georgiev, B. Rubino & R. Sampalmieri

we obtain the relation∫ t

0

∫ t−s

0

∫eixξ �′(t − s − τ)|ξ |2E(τ, ξ)F (s, ξ)dξdτds

=∫ t

0

∫ σ

0

∫eixξ �′(t − σ)|ξ |2E(σ − s, ξ)F (s, ξ)dξdsdσ

= −(2π)n∫ t

0�′(t − σ)�v(σ, x)dσ.

Due to the definition of v we find

∂tt v −�v = F(t, x)−∫ t

0�′(t − σ)�v(σ, x)dσ.

The verification of the fact that v satisfies the initial conditions in (67) is trivial, sou = v and the relation (68) is established.

With the aid of (68), we can now obtain the estimate (15) for the solution ofthe linear inhomogeneous problem (67).

Indeed, from (69) we have the inequalities

‖∇ku(t)‖L∞ � C

∫ t

0

∥∥∥|ξ |kE(t − τ, ξ)F (τ, ξ)

∥∥∥L1dτ (70)

and

‖∇ku(t)‖L2 � C

∫ t

0

∥∥∥|ξ |kE(t − τ, ξ)F (τ, ξ)

∥∥∥L2dτ. (71)

The above argument of the proof of (46) and (70) leads to

‖∇ku(t)‖L∞ � C

∫ t/2

0(t − s)−(n+k−1)/2‖R(τ)‖L1dτ

+C∫ t

t/2‖|ξ |k+n−1−δ R(τ, ξ)‖L∞

ξdτ

+C∫ t

t/2‖|ξ |k+n−1+δ R(τ, ξ)‖L∞

ξdτ, (72)

when t � 1, while for 0 � t � 1 we use the following variant of (14):

‖∇ku(t)‖L∞ � C

∫ t

0‖∇k+n−2R(τ)‖L1dτ + C

∫ t

0‖∇k+nR(τ)‖L1dτ. (73)

To use (71) we note that the following variant of (57) is fulfilled:

|ξ |k|E(t − τ, ξ)| �{

|ξ |k−2(t − τ)−1/2 if t/2 � τ � t ;

|ξ |−1(t − τ)−k/2 if 0 � τ � t/2.(74)

Global Existence for Elastic Waves with Memory 325

Combining (71) and (74), we see that the following estimate holds (here t � 1):

‖∇ku(t)‖L2 � C

∫ t/2

0(t − τ)−k/2‖R(τ)‖H−1dτ

+C∫ t

t/2(t − τ)−1/2‖R(τ)‖H k−2dτ. (75)

For 0 � t � 1, we have

‖∇j u(t)‖L2 � C

∫ t

0(t − s)−1/2‖∇j−2R(s)‖L2ds. (76)

The general inhomogeneous problem

utt − L [u] = F,

u(0, x) = u0(x),

ut (0, x) = u1(x) (77)

can be treated easily now. In fact, for t � 1 we combine (63) and (72) and obtain(13). For 0 � t � 1, from (64) and (73) we get (14). From (65) and (75) we get(15) for t � 1. For 0 � t � 1, the estimate (16) follows from (66) and (76).

6. Appendix 2

In this Appendix we shall evaluate the term of type

‖| · |s f (u)‖L∞ ,

where s � 0, f (u) is a CN function with N > s and

u ∈ L∞(Rn) ∩Hs(Rn).

Our first step is the case of a bilinear estimate for terms of type

‖| · |s vw‖L∞ ,

where

v,w ∈ L∞(Rn) ∩Hs(Rn).

Lemma 4. The following estimate holds:

‖| · |s vw‖L∞ � C(‖v‖H s‖w‖L2 + ‖v‖L2‖w‖H s

). (78)

326 V. Georgiev, B. Rubino & R. Sampalmieri

Proof. Recall that the Fourier multiplier ϕ(D), defined for ϕ(ξ) ∈ C∞0 (R

n), is anoperator of convolution type

f ∈ C∞0 (R

n) →∈ ϕ(D)(f )(x) =∫

RnK(x − y)f (y)dy,

where the kernel K(x) of this operator is given by

K(x) = (2π)−n∫eixξ ϕ(ξ)dξ.

Any Fourier multiplier ϕ(D) commutes with the operator � + �′ ∗ �.We can usea Paley-Littlewood partition of unity (see [2])

1 =∞∑

j=−∞ϕj (ξ), ξ ∈ R

n

such that ϕj (ξ) ∈ C∞0 (R

n), all functions ϕj (ξ), j ∈ Z are nonnegative and thefollowing property is satisfied:

suppϕj (ξ) ⊆ {2j−1 � |ξ | � 2j+1}, j ∈ Z.

Then we have the following property:

‖g‖2L2(Rn)

∼∞∑

j=−∞‖gj‖2

L2(Rn), gj = ϕj (D)g. (79)

We can represent |ξ |s vw(ξ) as

|ξ |s vw(ξ) = |ξ |s∫v(ξ − η)w(η)dη.

Take k ∈ Z so that

2k−1 � |ξ | � 2k+1.

Then we have the relation

|ξ |s |vw(ξ)| � C|ξ |s

∑

j∈Z

Ij (ξ)

, (80)

where

Ij (ξ) =∫ ∣∣v(ξ − η)w(η)

∣∣ϕj (η)dη.

If k � j+2, then the assumptionη ∈ suppϕj implies |η| � 2j−1 � 2k−3 � 2−4|ξ |,so applying the Cauchy inequality and the property

ϕj−1(η)+ ϕj (η)+ ϕj+1(η) = 1

Global Existence for Elastic Waves with Memory 327

for η ∈ suppϕj , we get

|ξ |sIj (ξ) � C

∫ ∣∣v(ξ − η)∣∣ |η|s ∣∣w(η)∣∣ dη

� C(‖ϕj−1v‖L2 + ‖ϕj v‖L2 + ‖ϕj+1v‖L2

) ‖| · |swϕj‖L2 .

So from (79), the Plancherel identity and the Cauchy inequality we find

∞∑

j=k−2

|ξ |sIj (ξ) � C‖v‖L2‖w‖H s . (81)

If k � j + 3, then the assumption η ∈ suppϕj implies

|ξ − η| � |ξ | − |η| � 2k−1 − 2j+1 � 2k−1 − 2k−2 = 2k−2 � 2−3|ξ |and we can use the inequalities

|ξ |sIj (ξ) � C

∫ ∣∣v(ξ − η)

∣∣ |ξ − η|s ∣∣w(η)∣∣ dη

� C(‖ϕj−1| · |s v‖L2 + ‖ϕj | · |s v‖L2 + ‖ϕj+1| · |s v‖L2

) ‖wϕj‖L2

so we arrive at

j=k−3∑

j=−∞|ξ |sIj (ξ) � C‖v‖H s‖w‖L2 . (82)

From this inequality, (81) and (80) we complete the proof of the Lemma. ��Now we turn to the term

‖| · |s f (u)‖L∞ ,

where s � 0, f (0) = 0, f (u) is a CN function with N > s and

u ∈ L∞(Rn) ∩Hs(Rn).

Let R > 0 be chosen so that

‖u‖L∞(Rn) + ‖u‖Hs(Rn) � R.

By setting

g(u) ={f (u)/u if u �= 0,

f ′(0) if u = 0,(83)

then g(u) ∈ CN−1 and we can use the following estimate obtained essentially inChapter V, [18]

‖g(u)‖H s � ‖u‖H s G (‖u‖L∞) , (84)

where G(r) denotes a bounded measurable function defined for |r| � R.

328 V. Georgiev, B. Rubino & R. Sampalmieri

We shall sketch the proof of this estimate for completeness. We shall need thefollowing equivalent norm the Sobolev spaces H s for fractional s > 0, such thats = k+ θ,where k � 0, k ∈ Z and 0 < θ < 1 (see for instance [22], section 2.5.1)

‖f ‖H s (Rn) =(∫

Rn|h|−(n+2θ)‖�h∇kf (x)‖2

L2(Rn)d h

)1/2

� C < ∞. (85)

Here �hf (x) is the finite difference given by �hf (x) = f (x + h)− f (x).

Lemma 5 (see Chapter V [18]). Suppose that g(u) ∈ CN, where N � 1 is aninteger. Then for any s, 0 < s � N and for any

u ∈ H s(Rn) ∩ L∞(Rn)

such that

‖u‖H s (Rn) + ‖u‖L∞(Rn) � Rfor some R > 0,

the inequality

‖g(u)‖H s � ‖u‖H s G (‖u‖L∞) (86)

holds, where G(r) is a bounded measurable function defined for |r| � R.

Proof. If s > 0 is an integer, then we can use the argument based on the relation(29). For this, there is no loss of generality if we assume 0 < s < 1. Then the normin (85) is

‖f ‖H s (Rn) =(∫

Rn|h|−(n+2s)‖�hf (x)‖2

L2(Rn)d h

)1/2

. (87)

We can use the relation

g(u(x + h))− g(u(x)) = (u(x + h)− u(x))

×∫ 1

0f ′(u(x)+ τ(u(x + h)− u(x)))dτ,

and then, substituting into (87) f (x) with g(u(x)), we get the needed inequality(86), since

∣∣∣∣

∫ 1

0f ′(u(x)+ τ(u(x + h)− u(x)))dτ

∣∣∣∣ � G(‖u‖L∞) . ��

From the estimate of Lemma 4 we obtain

Proposition 1. With the previous hypotheses and notation, the following estimateholds:

‖| · |s f (u)‖L∞ � C‖u‖H s‖u‖L2G(‖u‖L∞) . (88)

Proof. Let us call f (u) = g(u)u and apply Lemma 4. We get the estimate

‖| · |s f (u)‖L∞ � C(‖u‖H s‖g(u)‖L2 + ‖g(u)‖H s‖u‖L2

).

Applying (86), we obtain (88). This completes the proof of the Proposition. ��

Global Existence for Elastic Waves with Memory 329

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330 V. Georgiev, B. Rubino & R. Sampalmieri

Department of Mathematics,University of Pisa,via Buonarroti,2 – 56100 Pisa, Italy

e-mail: georgiev@dm.unipi.it

and

Division of Mathematics for Engineering,Department of Pure and Applied Mathematics – University of L’Aquila,

loc. Monteluco di Roio – 67040 L’Aquila, Italye-mail: rubino@univaq.it, rosella@univaq.it

(Accepted July 12, 2004)Published online April 11, 2005 – © Springer-Verlag (2005)