THE INITIAL BOUNDARY VALUE PROBLEM FOR A CLASS OF NONLINEAR WAVE EQUATIONS WITH DAMPING TERM∗

Zhe Zhang,Desheng Li

(School of Math.and System Sciences,Shenyang Normal University, Shenyang 110034,E-mail:zhangzhesuper@163.com(Z.Zhang))

THE INITIAL BOUNDARY VALUE PROBLEM FOR A CLASS OF NONLINEAR WAVE EQUATIONS WITH DAMPING TERM∗

Zhe Zhang,Desheng Li

(School of Math.and System Sciences,Shenyang Normal University, Shenyang 110034,E-mail:zhangzhesuper@163.com(Z.Zhang))

The initial-boundary value problem for the four-order nonlinear wave equation with damping term is derived from diverse physical background such as the study of plate and beams and the study of interaction of water waves.The existence of the global weak solutions to this problem is proved by means of the potential well methods.

potential well;nonlinear wave equation;global solution existence

2000 Mathematics Subject Classification 35B35;65L15;60G40

Ann.of Applied Math.

31:2(2015),246-252

1　Introduction

In this paper,we consider the following problem

where Ω⊂Rnis a bounded domain with smooth boundary,and α,β,γ,σ,τ≥0.

Strongly damped nonlinear wave equation

is put forward from the motion of objects with viscous effect.There have been many results about the existence of global solution to nonlinear partial differential equations,and many effective approaches have been developed[1-4].In 1968,Sattinger[4]introduced the potential well method to show the existence of global solutions to the nonlinear hyperbolic equations that do not possess positive definite energy.Since then,many authors have applied the potential well method to investigate the existence of global solutions to the initial boundary value problems for various nonlinear evolution equations[5-7].

In recent years,many scholars have proposed many fourth order high dimensional nonlinear hyperbolic equation from elastic plastic rod and other mechanical problems of longitudinal motion.The mathematics workers and developments of these equations have widely received attention increasingly.In these studies,the potential well method plays a very important role as an important means about the existence of solutions to the equation. Specifically,in[8],the following fourth-order evolution equation was analyzed

in a bounded domain Ω⊂Rn.The author defined and used a potential well and several invariant and positive invariant sets to study the characteristics of blowup,boundedness, and the asymptotic properties.In[9],an initial boundary value problem for fourth-order wave equations with nonlinear strain and source terms was considered.By introducing a family of potential wells and proving the invariance of some sets,the authors obtained a threshold result of global existence and nonexistence.Various studies on the solution and properties of fourth-order wave equations have also been conducted and can be found in references[10-13].

When α=σ=τ=0,the problem(1)-(3)was studied in[1,3,14];when α=β=1, σ=τ=0,[15,16]gave the research results.[17]considered the situation of α=σ=τ=1, β=γ=0.In this paper,we research the general situation to these problems.

In this paper,we apply the potential well method to establish the conditions under which the initial boundary value problem in question has global weak solutions.The rest of the paper is organized as follows.In Section 2,we define the potential well for the initial boundary value problem in question and investigate its properties.Based on the results obtained in Section 2,the existence of global weak solutions is established in Section 3.

Throughout this paper,we denote ‖·‖p=‖·‖Lp(Ω),‖·‖=‖·‖2,‖u‖2H=β‖∇u‖2+ τ‖u‖2+α‖Δu‖2,(u,v)=

2　The Potential Well and Its Properties

Assume that f(u)∈C,f(u)u≥0 and

For the initial boundary value problem(1)-(3),we define

We also define the following potential well

and a set outside the potential well

where

Lemma 2.1 If d is defined by(12),then

Proof From(7),(8),I(u)=0 for any u∈N,we have‖u‖H≠0,then

Thus,from I(u)=0,

such that

Furthermore,we have

Lemma 2.2 Suppose that J(u)≤d,then I(u)＞0 if and only if

Proof Combining(14)and J(u)≤d,we obtain

If I(u)＞0,we have

Using Lemma 2.1,we obtain

which yields

On the other hand,the inequalityimplies that

then we conclude that I(u)＞0.

Proof Firstly,we can easily obtain thatSecondly,for any u∈W{0}, since I(u)＞0 and J(u)＜d,applying Lemma 2.2,we can obtain

Hence u∈BR,that is W{0}⊂BR.Finally,for any,the following inequality holds

From the definition of J(u),it follows

Therefore,from Lemma 2.2,I(u)＞0.Thus u∈W,that isHence⊂ W⊂BR.

3　Existence of Global Weak Solutions to the Problem

Definition 3.1We call u(x,t)a global weak solution to problem(1)-(3)if u∈,and for any t∈[0,T), the following equality holds

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

Theorem 3.1 Assume that f(u)∈C,f(u)u≥0,and u1(x)∈ L2(Ω),and inequality(6)holds.If u0(x)∈W and E(0)＜d,then there exists a weak solution to problem(1)-(3)such that,and u∈W for t∈[0,∞).

Proof Let{wj}be a complete orthonormal basis in L2(Ω)of the eigenfunctions of Laplacian and satisfy

In order to construct approximate solutions to problem(1)-(3),we define

where m=1,2,···and gjm(t)satisfy for s=1,2,···,m,

Multiplying(21)by g′sm(t)and summing for s,we obtain

Integrating the above equation with respect to t,we obtain

where

From(22)and(23)and letting m→∞we obtain

From the definition of W,u0(x)∈W implies that I(u0)＞0 and J(u0)＜d or u0=0.If I(u0)＞0 and J(u0)＜d,then we have I(um(0))＞0 and J(um(0))＜d for sufficiently large m.Hence um(0)∈W.On the other hand,if u0=0,from Lemma 2.3,⊂W for sufficiently large m.

Now,we shall prove that um(t)∈W for any t＞0 and sufficiently large m.

By the method of contradiction,we assume that for some large m there exists a t0= t0(m)＞0 such that um(t0)∈∂W,that is

It follows from f(u)u≥0 and inequality(6)that

which gives the following inequality

Hence

From the given condition E(0)＜d,for sufficiently large m,we have Em(0)＜d,and then

Therefore J(um)=d is impossible.

On the other hand,if I(um(t0))=0 and um(t0)≠0,we obtain J(um(t0))≥d,which contradicts(28).Therefore,we conclude that um(t)∈W for sufficiently large m and t＞0.

From(28)and(7),(8),we obtain

Noting that I(um)＞0 and t≥0,it follows that

Hence{um}and{umt}are bounded in L∞(0,∞;H2(Ω)∩H10(Ω)∩Lp+1(Ω))and L∞(0,∞; L2(Ω))∩L2(0,∞;H10(Ω)).We note that{f(um)}is bounded in L∞(0,∞;Lq(Ω))whereTherefore,there exist u,χ and uν,which is a subsequence of{um},such that as ν→∞,

Moreover,f(uν)is bounded with respect to ν in Lq(Ω),then we can obtain f(uν)→f(u) weakly in Lq(Ω).Integrating(21)with respect to t yields

Letting m=ν→∞,we further have

For any v∈H2(Ω)∩H10(Ω),it results that

and u∈W for t∈[0,∞).From(22)and(23),we obtain that u(x,0)=u0(x)in H2(Ω)∩H10(Ω)and ut(x,0)=u1(x)in L2(Ω).Therefore u is a weak solution to problem(1)-(3).

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(edited by Liangwei Huang)

∗Manuscript June 24,2014;Revised December 31,2014