Blow-up of Solutions for a Class of Fourth-order EquationInvolvingDissipativeBoundaryCondition and Positive Initial Energy

SHAHROUZI Mohammad

Department of Mathematics,Jahrom University,Jahrom 74137-66171,Iran.

Received 18 October 2014;Accepted 25 December 2014

Blow-up of Solutions for a Class of Fourth-order EquationInvolvingDissipativeBoundaryCondition and Positive Initial Energy

SHAHROUZI Mohammad?

Department of Mathematics,Jahrom University,Jahrom 74137-66171,Iran.

Received 18 October 2014;Accepted 25 December 2014

.In this paper we consider a forth order nonlinear wave equation with dissipative boundary condition.We show that there are solutions under some conditions on initial data which blow up in finite time with positive initial energy.

Blow up;fourth order;boundary dissipation.

1 Introduction

In this article,we are concerned with the problem

where ??Rnis a bounded domain with smooth boundary?? in order that the divergence theorem can be applied.ν is the unit normal vector pointing toward the exterior of ? and p＞m+1＞3.Moreover,the constants a0,a,b,c0are positive numbers and g(x,t,u,Δu)is a real function that satisfies specific condition that will be enunciated later.

The one-dimension case of the fourth order wave equation is written aswhich was first introduced in[1]to describe the elasto-plastic-microstructure models for the longitudinal motion of an elasto-plastic bar.Chen and Yang[2]studied the Cauchy problem for the more general Eq.(1.4).

Young Zhou,in[3]studiedthe following nonlinear wave equation with damping and source term on the whole space:

where a,b＞0,m≥1 are constants and φ(x):RN→R,n≥2.He has obtained the criteria to guarantee blow up of solutions with positive initial energy,both for linear and nonlinear damping cases.

In[4],the same author has been studied the following Cauchy problem

where a,b＞0.He proved that the solution blows up in finite time even for vanishing initial energy if the initial datum(u0,u1)satisfiesRRNu0u1dx≥0.(See also[5])

Recently,in[6]Bilgin and Kalantarov investigated blow up of solutions for the following initial-boundary value problem

They obtained sufficient conditions on initial functions for which there exists a finite time that some solutions blow up at this time.

Tahamtani and Shahrouzi studied the following fourth order viscoelastic equation

in a bounded domain and proved the existence of weak solutions in[7].Furthermore, they showed that there are solutions under some conditions on initial data which blow up in finite time with non-positive initial energy as well as positive initial energy.Later, the same authors investigatedglobal behavior of solutionsto some class of inverse source problems.In[8],the same authors investigated the global in time behavior of solutions for an inverse problem of determining a pair of functions{u,f}satisfying the equation

the initial conditions

the boundary conditions

and the over-determination condition

Also,the asymptoticstability resulthas beenestablished withthe oppositesignofpowertype nonlinearities.

In[9],Tahamtani and Shahrouzi considered

They showedthat the solutions of this problem undersome suitable conditions are stable if α1,α2being large enough,α3≥0 and φ(t)tends to zero as time goes to infinity.Also, established a blow-up result,if α3＜0 and φ(t)=k be a constant.Their approaches are based on the Lyapunov function and perturbed energy method for stability result and concavity argument for blow-up result.The interested reader is referred to the papers [10–14].

Motivated by the aforementioned works,we take b,λ and c0in the appropriately domain and prove that somesolutions of(1.1)-(1.4)blow up in a finite time.Our approaches are based on the modified concavity argument method.

2 Preliminaries and main results

In this section,we present some material needed in the proof of our main results.We shall assume that the function g(x,t,u,Δu)and the functions appearing in the data satisfy the following conditions

with some positive M＞0.

We sometimes use the Poincar′e inequality

and Young’s inequality

We will use the trace inequality

where B is the optimal constant.

The following lemma was introduced in[15];it will be used in Section 3 in order to prove the blow-up result.(see also[16,17])

Lemma 2.1.Letμ＞0,c1,c2≥0 and c1+c2＞0.Assume that ψ(t)is a twice differentiable positive function such that

for all t≥0.If

then

Here

We consider the following problem that is obtained from(1.1)-(1.3)by substituting

The energy associated with problem(2.7)-(2.9)is given by

where

Now we are in a position to state our blow-up result as follows.

Theorem 2.1.Let the conditions(A1)and(A2)are satisfied.Assume that Eλ(0)＞0 and

then there exists a finite time t1such that the solution of the problem(1.1)-(1.3)blows up in a finite time,that is

3 Blow up

In this section we are going to prove that for sufficiently large initial data some of the solutions blow up in a finite time.To prove the blow-up result for certain solutions with positive initial energy,we need the following Lemma.

Lemma 3.1.Under the conditions of Theorem 2.2,the energy functional Eλ(t),defined by(2.10), satisfies

Proof.A multiplication of Eq.(2.7)by vtand integrating over ? gives

where

It is easy to verify that

Employing the last inequality,we obtain from(2.10)the following inequality

At this point,let us recall the Poincar′e and trace inequalities to estimate the terms on the right side of(3.5),we obtain

Now,if we choose ε0=aλ/m and ε1=λ(p-m-1)/p then we obtain from(3.6)the inequality

Since Eλ(0)＞0,we obtain from(3.8)that Eλ(t)≥eλ(m-1)tEλ(0)≥0.Therefore integrating (3.8)yields

and proof of Lemma is competed.

Proof of Theorem 2.2.For obtain the blow-up result,we consider the following functional

where C is a positive constant that will be chosen appropriately.

and consequently

A multiplication of Eq.(2.7)by v and integrating over ? gives

By using boundary conditions in terms of(3.12)and combining with(3.11)we obtain

Due to the condition(A2)we have

and so by inserting(3.14)into(3.13),we get

by using definition of Eλ(t)in(3.15),we have

where Poincar′e inequality(2.1)has been used.Since p+m≥4 and by choosing δ0= a(p-m)/m and δ1=(p-m)/p,we get

Finally,thanks to the assumptions of Theorem 2.2 about λ and b,we deduce

Combining estimation of Lemma 3.1 with(3.18)yields

where the inequality 2(p+m)/(p+m+4)≥1 has been used.

Taking into account the estimation(3.19),it is easy to verify

Hence we see that the hypotheses of Lemma 2.1 are fulfilled with

and the conclusion of Lemma 2.1 gives us that some solutions of problem(2.7)-(2.9)blow up in a finite time t1.Since this system is equivalent to(1.1)-(1.3),the proof is complete.

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?Corresponding author.Email addresses:mshahrouzi@shirazu.ac.ir(M.Shahrouzi)

10.4208/jpde.v27.n4.5 December 2014

AMS Subject Classifications:35B30;35B44;35G31

Chinese Library Classifications:O175.27