BLOW-UP OF SOLUTION FOR A VISCOELASTIC WAVE EQUATION WITH DELAY?

Shun-Tang WU

General Education Center,National Taipei University of Technology,Taipei,Taiwan 106,China

E-mail:stwu@ntut.edu.tw

Abstract In this paper,we consider the following viscoelastic wave equation with delay

Key words blow up;nonlinear source;wave equation;delay;viscoelastic

1 Introduction

In this paper,we study a nonlinear viscoelastic equation with delay of the form

where ρ >0,b,μ1are positive constants,μ2is a real number,τ>0 represents the time delay,g is the kernel of the memory term,the initial data(u0,u1,f0)are given functions belonging to suitable spaces,and ? ?RN(N ≥1)is a bounded domain with a smooth boundary??.

It is well known that time delay e ff ects arise in many applications and practical problems such as physical,chemical,biological,thermal and economic phenomena.These hereditary e ff ects might induce some instabilities.Hence,questions related to the behavior of solutions for the PDEs with time delay e ff ects became an active area of research in recent years.Many authors focused on this problem and several results concerning existence,decay and instability were obtained,see[2–4,7,17–21,23–25]and reference therein.In this regard,Datko et al.[3]showed that a small delay in a boundary control is a source of instability.Nicaise et al.[17]studied a system of wave equation with linear damping and delay as follows

Under the condition μ2< μ1,they established a stabilization result.Conversely,ifμ2≥ μ1,they showed that there exists a sequence of delays for which the corresponding solution of(1.5)is unstable.Also,they obtained the same results if both the damping and the delay act on the boundary.Later,Pignotti[22]considered the wave equation with internal distributed time delay and local damping in a bounded and smooth domain.She/He showed that an exponential stability result holds if the coefficient of the delay term is sufficiently small.

In the absence of the delay term(i.e.,μ2=0),problem(1.1)is extensively studied and there are numerous results related to existence,asymptotic behavior and blow-up of solutions.For example,Cavalcanti et al.[1]considered the following problem

with the same initial and boundary conditions(1.3)–(1.4),where a global existence result for γ ≥ 0 and an exponential decay result for γ >0 were established under the assumptions 0<ρ≤if N ≥ 3 or ρ >0 if N=1,2 and g(t)decays exponentially.Later,these decay results were extended by Messaoudi and Tatar[14]to a situation where a source term is present.Liu[9]considered the nonlinear viscoelastic problem

He obtained the general decay result for the global solution and the fi nite time blow-up of solution.Song[23]studied a problem similar to(1.6),and She/He showed the nonexistence of global solutions with positive initial energy.For the more works on viscoelastic wave equations,we refer the reader to Refs.[8,12,13,16].

Recently,Ka fi ni et al.[5]investigated the following nonlinear wave equation with delay

They established the new result of global non-existence for nonlinear wave equation under suitable conditions on the initial data,the weights of the damping,the delay term and the nonlinear source.Later,Ka fi ni et al.[6]generalized this result to a second-order abstract evolution system with delay.

Motivated by previous works,in this paper,we investigate problem(1.1)–(1.4)and prove the blow-up result with nonpositive and positive initial energy by modifying the method in[5,6].In this way,we can extend the results of[5]where the authors considered(1.1)with ρ=0 and in the absence of the memory term.The paper is organized as follows.In Section 2,we provide some assumptions and lemmas needed for our work.In Section 3,we state and prove our main result that is given in Theorem 3.4.

2 Preliminary Results

In this section,we shall give some lemmas and assumptions which will be used throughout this work.We use the standard Lebesgue space Lp(?)and Sobolev spacewith their usual products and norms.

Lemma 2.1Let 2≤p≤the inequality

holds with some positive constant cs.

Assume that ρ and p satisfy

and

Regarding the kernel function g(t),we assume that it veri fi es

(A1) g:R+→R+is a bounded C1function satisfying

We also need the following lemma in the course of the investigation.

Lemma 2.2(see[5]) Suppose that(2.2)holds,then there exists a positive constant C such that

Similarly to[17],we introduced the new variable z,

which implies that

Therefore,problem(1.1)–(1.4)can be transformed as follows

We now state,without a proof,the local existence result,which can be established by combining the arguments of[7,26].

Theorem 2.3Suppose thatμ2< μ1,(A1),and(2.1)–(2.2)hold.Assume thet u0,u1∈and f0∈ L2(? ×(0,1)).Then there exists a unique solution(u,z)of(2.4)satisfying

for T>0.

3Blow-Up

In this section,we shall investigate the blow-up result for certain solutions with nonpositive initial energy as well as positive initial energy.First,we de fi ne the energy function of problem(2.4)as

Therefore,based on above arguments,(3.16)becomes

which together with(3.20)implies that

On the other hand,by H?lder inequality and Young’s inequality,we have