Global Existence and Asymptotic Behavior for a CoupledSystemofViscoelasticWaveEquationswith a Delay Term

FERHAT Mohamedand HAKEM Ali

Department of Mathematics,Oran University USTO 31000,AlGERIA.

Computer Science Department,Sidi Bel Abbes University 22000,AlGERIA.

Received 19 May 2014;Accepted 24 November 2014

Global Existence and Asymptotic Behavior for a CoupledSystemofViscoelasticWaveEquationswith a Delay Term

FERHAT Mohamed1,?and HAKEM Ali2

Department of Mathematics,Oran University USTO 31000,AlGERIA.

Computer Science Department,Sidi Bel Abbes University 22000,AlGERIA.

Received 19 May 2014;Accepted 24 November 2014

.Weareinterestedinthe studyof acoupledsystem of viscoelasticwaveequations with a delay term.Firstly global existence of the solutions is proved.The asymptotic behavior is obtained by using multiplier technique proved by A.Guessmia[1], however in the unstable set for certain initial data bolstered with some conditions,we obtain the blow up of the solutions for various sign of initial energy negative,positive or null.We use the same technique in[2]with a necessary modification imposed by our problem.This improves the earlier results in the literature

Global existence;energy decay;nonlinear damping.

1 Introduction

In this paper,we consider the following problem:

where ? is a bounded domain in IRn,n∈IN?,with a smooth boundary??,and g1,g2: IR+→IR+,fi(·,·):IR2→IR,i=1,2,are given functions which will be specified later,τ2＞0 is a time delay,whereμ1,α1,α2,μ2are positive real numbers and the initial data(u0,u1,f0), (υ0,υ1,f1)belonging to a suitable space.Problems of this type arise in material science and physics.

Recently,the authors of[3]considered the following coupled system of quasilinear viscoelastic equation in canonical form without delay terms

where ? is a bounded domain in IRn(n≥1)with a smooth boundary??,γ1,γ2≥0 are constants and ρ is a real number such that 0＜ρ＜2/(n-2)if n≥3 or ρ＞0 if n=1,2. The functions u0,u1,υ0and υ1are given initial data.The relaxations functions g1and g2are continuous functions and f1(u,υ),f2(u,υ)represent the nonlinear terms.The authors proved the energy decay result using the perturbed energy method.

Many authors considered the initial boundary value problem as follows

when the viscoelastic terms gi(i=1,2)are not taken into account in(1.8),Agre and Rammaha[4]obtained several results related to local and global existence of a weak solution.By using the same technique as in[5],they showed that any weak solution blow-up in finite time with negative initial energy.Later Said-Houari[6]extended this blow up result to positive initial energy.Conversly,in the presence of the memory term (gi6=0(i=1,2.),there are numerous results related to the asymptotic behavior and blow up of solutions of viscoelastic systems.For example,Liang and Gao[7]studied problem (1.8)with h1(ut)=-Δut,h2(υt)=-Δut.They obtained that,under suitable conditions on the functions gi,fi,i=1,2,and certain initial data in the stable set,the decay rate of the energy functions is exponential.On the contrary,for certain initial data in the unstable set,there are solutions with positive initial energy that blow-up in finite time.For h1(ut)=|ut|m-1utand h2(υt)=|υt|r-1υt.Hun and Wang[8]established several results related to local existence,global existence and finite time blow-up(the initial energy E(0)＜0).

This latter has been improved by Messaoudi and Said-Houari[9]by considering a larger class of initial data for which the initial energy can take positive values,on the other hand,Messaoudi and Tatar[9]considered the following problem

where the functions f1and f2satisfy the following assumptions

for some constant d＞0 and 0≤βi≤n/(n-2),i=1,2,3,4.They obtained that the solution goes to zero with an exponential or polynomial rate,depending on the decay rate of the relaxation functions gi,i=1,2.

Muhammad[10]considered the following problem

and provedthewell-posednessandenergydecayresultforwiderclass ofrelaxation functions.

In the present paper,we investigate the influence of the viscoelastic terms,damping terms and delay terms on the solutions to(1.1)-(1.6).Under suitable assumptions on the functions gi(·),fi(·,·),(i=1,2),the initial data and the parameters in the equations,we establish several results concerning global existence,boundedness,asymptotic behavior of solutions to(1.1)-(1.6).

Our work is organized as follows.In Section 2,we present the preliminaries and some lemma.In Section 3,a global existence result is obtained.Finally in Section 4 decay property is derived.

2 Preliminary results

In this section,we present some material for the proof of our result.For the relaxation function giwe assume

(A1):The relaxations functions g1and g2are of class C1and satisfy,for s≥0

with a,b＞0.Further,one can easily verify that

where

(A2):There exists c0,c1＞0,such that

and

(A3):

Lemma 2.1.(Sobolev-Poincar′e inequality)Let 2≤m≤2n/(n-2).The inequality

holds with some positive constant cs.

Lemma 2.2.([11])For any g∈C1and φ∈H1(0,T),we have

where

Lemma 2.4.([1])E:IR+→IR+be a differentiable function a1,a2∈IR+?and a3,λ∈IR+such that

and for all 0≤S≤T＜+∞,

then there exist two positive constants and c such that,for all t≥0,

Remark 2.1.Avoiding the complexity of the matter,we take a=b=1 in(2.1)-(2.2).

3 Global existence

In order to prove the existence of solutions of problem(1.1)-(1.6),we introduce the new variables z1,z2as in[12]

which implies that

therefore,problem(1.1)-(1.6)is equivalent to

In the following,we will give sufficient conditions for the well-posedness of problem (3.1)by using the Fadeo-Galerkin’s method.

Theorem 3.1.Suppose thatμ2＜μ1,α2＜α1,(A1)-(A3)hold.Assume that((u0,u1),(υ0,υ1))∈(H10(?))2and(φ0,φ1)∈(L2(?×(0,1))2.Then there exists a unique solution((u,z1),(υ,z2)) of(3.1)satisfying

for j=1.....n.More specifically

where

Let ξ1,ξ2be positive constants such that

Step one:Energy estimates.

Using Lemma 2.2 and integrating(3.10)over(0,t),we get

we multiply the Eq.(3.5)by ξ1/τ2zk1,j(t)and the Eq.(3.6)by ξ2/τ2zk2,j(t),summing with respect to j and integrating the result over ?×(0,1)to obtain

then

in the same manner

Summing(3.10),(3.13)and(3.14),we get

Using Young and Cauchy-Schwartz inequalities,we obtain

where E(t)is the energy of the solution defined by the following formula

We will prove that the problem(3.2)-(3.7)admits a local solution in[0,tm),0＜tm＜T,for an arbitrary T＞0.The extension of the solution to the whole interval[0,T]is a consequence of the estimates below.

Step two:First estimate.

and summing with respect to n from 1 to n,respectively,using Lemma 2.2,we get

and

Using Young’s inequalities,summing(3.18)-(3.19),and integrating over(0,t),we obtain

where

is a positive constant.We just need to estimate the right hand terms of(3.20).Applying H¨older’s inequality,Sobolev embedding theorem inequality,we get

Likewise,we obtain

Then

Let

Then, we infer from (3.21)-(3.24) that

Hence from (3.16) and (3.27), we obtain

where L1is a positive constant depending on the parameter E(0).

Step three:Second estimate.

Summing(3.29)-(3.30)we obtain

Exploiting H¨older,Young’s inequalities,and Lemma 2.2,for ?＞0,c＞0 from the first estimate we have

and

Likewise,we obtain

Substituting these estimates(3.32)-(3.38)into(3.31),then integrating the obtained inequality over(0,t)and using(3.16),we deduce that

where L2is a positive constant independent of n∈N and t∈[0,T).

Further,by Aubin’s Lemma[11],it follows from(3.28)and(3.41)that there exists a subsequence(un,υn)still represented by the same notation,such that

then

and

Analysis of the nonlinear term

In the same way for f2(ui,υi)

From the(3.42)and(3.43)we deduce that

Consequently, are bounded in

using Aubin-Lions theorem[13],we can extract a subsequence(uξ)of(un)and(υξ)of (υn)such that

therefore

similarly

Now,we will pass to the limit in(3.2)-(3.7).Taking n=ξ,?wj∈Wn,??j∈Vnin(3.2)-(3.7) and fixed j＜ξ,

by using the property of continuous of the operator in the distributions space and due to (3.42)-(3.51)we have

as ξ→∞the convergence(3.67)-(3.80)permits us to deduce that

we deduce also

As ξ→+∞.Hence,this completes our proof of existence result of system(3.2)-(3.7)

Remark 3.1.By virtue of the theory of ordinary differential equations,the system(3.2)-(3.7)has local solution which is extendedto a maximal interval[0,Tk[with(0＜Tk≤+∞).

Now we will prove that the solution obtained above is global and bounded in time, for this purpose,we define we observe that

Lemma 3.1.Let((u,z1),(υ,z2)),be the solution of problem(3.2)-(3.7).Assume further that I(0)＞0 and

Then I(t)＞0,?t.Moreover the solution of problem(3.1)is global and bounded,where ρ is a positive constant appeared in Lemma 2.3.

Proof.Since I(0)＞0,then there exists(by continuity of u(t))T?＜T such that

for all t∈[0,T?].From(3.85)(3.86)gives that

Thus by(3.86),(3.87)we deduce

Employing Lemma 2.3,(A1),we obtain

Hence,we conclude from(3.92)that I(t)＞0 on[0,t0]which contradicts thus I(t)＞0 on [0,T],which completes the proof.

Lemma 3.2.Let((u,z1),(υ,z2))be a solution of the problem(3.1).Then the energy functional defined by(3.87)satisfies

Proof.After deriving the Eq.(3.16)we get the desired result.

Remark 3.2.Due to the conditions(3.8),(3.9)we have

4 Asymptotic behavior

In this section we use the multiplier method introduced by A.Guessmia[1],we get the following result

Theorem 4.1.Suppose thatμ2＜μ1,α2＜α1,(A1),(3.88)hold and I(0)＞0.Assume that ((u0,u1),(υ0,υ1))∈(H10(?))2and(φ0,φ1)∈(L2(?×(0,1)))2.Then the solution of problem (3.1)is global and bounded,Furthermore,we have the following decay property

where c,ω,are positive constants,independent of the initial data.

Remark 4.1.We denote by civarious constants which may be different at different occurrences.

We multiply the first equation of(3.1)by Equ,the second equation of(3.1)by Eqυ,integrating over[S,T]×? we get

Similarly,we multiply the third equation in(3.1)by Eqe-2τk1z1and the forth equation by Eqe-2τk2z2we get

in the same manner for the forth equation in(3.1)

Taking their sum,we obtain

where A=2min{1,2τe-2τ/ζ1,2τe-2τ/ζ2}.Using the Cauchy-Schwartz and Poincar′e inequalities and the energy identity,we get using Lemma 2.3,to get

using Lemma 2.3, to get

and

This yields

Similarly

As in[11],employing Young’s inequality for convolutionk??φk≤k?kkφk,and using the potential well method’s,we easily find

in the same manner

exploiting(4.20)to obtain

Likewise we obtain

and

then

for the second and third equation we use the same technique.Combining all above inequalities,and choosing ?2small enough,we deduce from(4.6)-(4.27)that

where c is a positive constant independent of E(0).From the last inequality,and the conditions of Lemma 2.4 are satisfied,then

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?Corresponding author.Email addresses:ferhat22@hotmail.fr(M.Ferhat),hakemali@yahoo.com(A.Hakem)

10.4208/jpde.v27.n4.2 December 2014

AMS Subject Classifications:35L05;35L15;35L70;93D15

Chinese Library Classifications:O175.27