PENG Xiaoming and SHANG Yadong

1 School of Statistics and Mathematics,Guangdong University of Finance and Economics,Guangzhou 510320,China.

2 School of Mathematics and Information Science,Guangzhou University,Guangzhou 510006,China.

Abstract. This paper is concerned with the asymptotic behavior of a quasilinear viscoelastic equation with nonlinear damping and memory.Assuming that the kernelμ(s)satisfieswe establish the exponential stability result for m=1 and the polynomial stability result for 1＜m＜.

Key Words:Exponential stability;polynomial stability;quasilinear;nonlinear damping;memory.

1 Introduction

In this paper,we investigate the long-time dynamics of solutions for the quasilinear viscoelastic equation with nonlinear damping and memory

in the unknownu=u(x,t):Ω×R+R,complemented with the Dirichlet boundary condition

and the initial conditions

where Ω is a bounded domain of RN(N≥1) with smooth boundary?Ω,u0is the prescribed past history ofu.

Eq.(1.1)provides a generalization,accounting for memory effects in the material,of equations of the form

Whenf(ut) is a constant,Eq.(1.4) was introduced to model extensional vibrations of thin rods[1,Chapter 20]and ion-sound waves[2,Section 6].There have been extensive researches on the well-posedness and the longtime dynamics for Eq.(1.4)with a different kind of damping term and source term,(see[3,4]and the references therein).Whenf(ut)is not a constant,Eq.(1.4) can model materials whose density depends on the velocityut[5].We refer the reader to Fabrizio and Morro[6]for several other related models.

Let us recall some results concerning quasilinear viscoelastic wave equations with finite memory.In[7],the authors studied the following equation with Dirichlet boundary conditions

By assuming

they proved a global existence result forγ≥0 and an exponential decay result forγ＞0.In the absence of the strong damping (γ=0),Messaoudi and Tatar [8] established the exponential and polynomial decay rates of energy.Messaoudi and Mustafa[9]improved the results in [8] and proved an explicit and general energy decay formula that allows a larger class of functionsg(s).Recently,based on integral inequalities and multiplier techniques,Li and Hu[10]proved a general decay rate from which the exponential decay and the polynomial decay are only special cases.

In[11],Messaoudi and Tatar studied the following problem

with boundary and initial conditions (1.2) and (1.3).By introducing a new functional and using a potential well method,they obtained the global existence of solutions and the uniform decay of the energy if the initial data are in some stable set.When the only dissipation effect is given by the memory (i.e.b=0 in (1.7)),Messaoudi and Tatar [12]proved the exponential decay of global solutions to (1.7),without smallness of initial data.Liu [13] considered a system of two coupled quasilinear viscoelastic equations in canonical form with Dirichlet boundary condition.Using the perturbed energy method,the author proved that the dissipations given by the viscoelastic terms are strong enough to ensure uniform decay(with exponential and polynomial rates)of the solutions energy.In [14],the same author proved that the solution decays exponentially when the initial data belongs to the stable set,and the solution blows up in finite time when the initial data belongs to the unstable set.Replacing polynomial nonlinearityb|u|p-2uin(1.7)with logarithmic nonlinearityu|u|γ-2ln|u|k,Boulaaras et al.[15]proved a general decay result of the energy of solutions for the reltaed problem.

In[16],Han and Wang investigated the global existence and exponential decay rate of the energy for solutions for the nonlinear viscoelastic problem with linear weak damping

By introducing two auxiliary functionals,the same author[17]proved the energy decay for the viscoelastic equation with nonlinear damping

wherem＞0 is a constant.Later,Park and Park[18]established the general decay for the viscoelastic problem with nonlinear damping

wherehis a continuous function.

Now,we focus on the quasilinear viscoelastic wave equations with infinite memory.Recently,Araújo et al.[19]studied the following equation

and proved the global existence,uniqueness and exponential stability of solutions and existence of the global attractor.Lately,the authors[20]established an existence,uniqueness and continuous dependence result for the weak solutions to the semigroup generated for the system (1.11) in a three-dimensional space whenρ∈[0,4] andfhas polynomial growth of (at most) critical order 5.Recently,the authors [21] proved the uniform exponential decay of energy.Besides,they also showed that the sole weak dissipation (γ=0) given by the memory term is enough to ensure existence and optimal regularity of the global attractor.In the recent work [22],Li and Jia proved the existence of a global solution by means of the Galerkin method when the kernelμ(s)satisfiesμ′(s)≤-k1μq(s),1≤q＜3/2.Moreover,they established the exponential stability result forq=1 and the polynomial stability result for 1＜q＜.However,the parameterρmust satisfy the condition (1.6) and the nonlinearityf(u) grows at most asin [22].In particular,the parameterρbelongs to the interval (0,2] and the nonlinearityfis only allowed to reach the polynomial order 3 forN=3.

On the other hand,after [20] we know the fact that the well-posedness result holds in more general assumptions,and in particular,for allρ∈[0,4] in case ofN=3.This opens a new scenario which is worth to be investigated.Motivated by the works above mentioned,our aim is to present some results about energy decay rates for the problem(1.1)-(1.3).Under the conditionμ′(s)≤-k1μm(s),1≤m＜3/2,we establish the exponential stability result form=1 and the polynomial stability result for 1＜m＜.Especially,the parameterρbelongs to the interval (0,4] and the nonlinearityfhas polynomial critical growth of(at most)critical order 5 in case ofN=3.

Following the framework proposed in [23-25],we shall introduce a new variableηtto the system which corresponds to the relative displacement history.Let us define

Note that

Thus,the original memory term can be rewritten as

and Eq.(1.1)becomes

with boundary condition

and initial conditions

where

2 Assumptions and the main result

We begin with precise hypotheses on problem(1.12)-(1.15).

(A1)Assume that

(A2)Concerning the source termf:RR,we assume that

wherec0＞0 and

In addition,we assume that

whereF(u)=f(s)ds.

(A3)The damping functiong∈C1(R) is a non-decreasing function withg(0)=0 and satisfies the polynomial condition

wherec1＞0 and

(A4)With respect to the memory component,we assume that

and there existk0,k1＞0 such that

and

Remark 2.1.The restriction onmis to ensure thatμ2-m(s)ds＜∞.

Ifm=1,indeed,it follows from(2.9)that

This immediately implies that(s)ds＜∞.

If 1＜m＜,by assumption(2.9)onμ(s),we have

whereC′=α1-m,α=min{k1(m-1),μ1-m(0)}.Since＞1,this yields

As usual,‖·‖pdenotes theLp-norms as well as(·,·)denotes theL2-inner product.Letλ1＞0 be the first eigenvalue of-Δ in(Ω).

In order to consider the relative displacementηtas a new variable,one introduces the weightedL2-space

which is a Hilbert space endowed with inner product and norm

respectively.Next let us introduce the phase space

endowed with the norm

The energy of problem(1.12)-(1.15)is given by

According to the arguments[22]with slightly modified,we state the following existence result of solution without proof.

Theorem 2.1.Assume that(2.1)-(2.7)hold.If initial data(u0,u1,η0)∈H and h∈L2(Ω),then the problem(1.12)-(1.15)has a global weak solution

satisfying

Now,we state our main stability result.

Theorem 2.2.Assume that(2.1)-(2.7)hold.

(1)If m=1,then

where K1,ν are positive constants.

(2)If1＜m＜,then

where K2is a positive constant.

3 Stability

In this section,we shall prove the exponential decay of the solutions energy.For this purpose,we introduce the following two functionals

Then we define a Lyapunov functional

whereMandεare positive constants to be determined later.

In what follows,the generic positive constants will be denoted asC.Next,we give some a priori estimates used later.

Lemma 3.1.Assume that(2.1)-(2.7)hold.Let u(t)be a solution of problem(1.12)-(1.15),then E(t)is nonincreasing,that is E′(t)≤0.Moreover,the following energy inequality holds

And

Proof.Multiplying(1.12)byutand(1.13)byηt,we obtain

According toηt(x,0)=0 and the limit lims∞μ(s)=0,one can easily see that

Inserting(3.6)into(3.5)yields

Sinceμ′≤0 andg′≥0,we have

which implies that

Then making use of(2.4)we obtain

This means that

The proof is finished.

Lemma 3.2.For M＞0sufficiently large,there exists two positive constants β1and β2such that

Proof.Using H¨older inequality,Young inequality and Sobolev inequality,it yields

whereCsis the best embedding constant from(Ω)+2(Ω).It follows from Young inequality that

Therefore,

On the other hand,we can get from(3.2)that

Applying H¨older inequality,Young inequality and Sobolev inequality,the two terms in the right-hand side of(3.10)can be estimated as follows

Thus(3.10)can be written as

It follows from(3.9)and(3.11)that

for someC2＞0.Then we chooseMlarge enough such thatM＞C2.If we takeβ1=M-C2andβ2=M+C2,we obtain(3.8).The proof of Lemma 3.2 is complete.

Lemma 3.3.There exists a positive constant C1,depending on E(0),such that

Proof.Using the definition of Φ(t)and Eq.(1.12),we see that

Using H¨older inequality and Cauchy inequality,we have

Using H¨older inequality,Young inequality and Sobolev inequality,taking into account(2.5)and(2.6),we arrive at

Combining the last two estimates,we have

This lemma is complete.

Lemma 3.4.There exists C2,C3＞0,dependent on E(0),such that

Proof.From the definition of Ψ(t)and Eq.(1.12),we have

Next,we will estimate two integrals in the right-hand side of the above identity.Integrating by parts with respect toxand using Young inequality,we obtain

and

Applying(2.5),H¨older inequality,Sobolev embedding inequality,Young inequality and Lemma 3.1,we have

and

On the other hand,since

we find

Applying H¨older inequality,Young inequality and Sobolev embedding inequality,we obtain

and

Collecting all the above inequalities,we end up with the differential inequality

This lemma is complete.

Proof of Theorem2.2(2).By Lemmas 3.3 and 3.4,we have

Using(2.9),we have

This means that

Let us takeε＞0 so small that.Then,for fxiedε,we chooseδ1,δ2small enough andMso large that

Recalling(2.4),the inequality(3.15)can be rewritten

(1)Casem=1.Using(3.16),we can deduce that

which together(3.8)imply that

An application of the Gronwall inequality yields

Using(3.8)again we get

which implies(2.11)withν=C/β2andK1=β2E(0)/β1.

(2)Case 1＜m＜.Exploiting the uniform bound(s)ds＜∞for anyθ＜2-m,we have

with positive constantL＞1.Using H¨older inequality and(3.18),we obtain

In light of Lemma 3.1,for anyσ＞1,we get

Combining(3.16)and(3.20),we end up with

Hence from(3.8)we obtain

Integrating(3.21)from 0 totand using(3.8)again,we have

whereK2is a positive constant.The proof is complete.

Acknowledgement

The authors are grateful to the anonymous referees for the constructive comments and useful suggestions.This work is supported by the Basic Research Project of Guangzhou Science and Technology Plan(No.202201011341).