GlobalExistenceandBlowUpofSolutions inaCauchy Viscoelastic Problem

LIU Gongweiand TIAN Shuying

1College of Science,Henan University of Technology,Zhengzhou 450001,China.

2School of Science,Wuhan University of Technology,Wuhan 430070,China.

Abstract.The Cauchy problem for a viscoelastic equation with nonlinear damping and source terms is considered.We establish the nonexistence result of global solutions with initial energy controlled above by a critical value by modifying the method introduced in a work by Autuori et al.in 2010.Then we establish global existence for arbitrary initial data or the energy in potential well.These improve earlier results in the literatures.

Key Words:Cauchy problem;viscoelastic equation;global existence;blow up.

1 Introduction

In[1],Messaoudi considered the following initial-boundary value problem

where ? is a bounded domain of RN(N≥1)with a smooth boundary??,p＞2,m≥2,and g is a nonincreasing positive function.He showed,under suitable assumptions on g,that the solutions with negative initial energy blow up in finite time if p＞m and continue to exist if m≥p provided thatif N≥3.

In the absence of the viscoelastic term,the problem(1.1)has been extensively studied and results concerning the global existence and nonexistence have been established by many authors.Levine[2,3]showed that solutions with negative initial energy blow up in finite time in the case of linear damping.Georgiev and Todorova[4]extendedLevine’s results to the nonlinear damping via a method different from the one known as the concavity method used by Levine.Vitillaro[5]extended the result in the situation where the solution has positive energy.In the presence of the viscoelastic term(i.e.g/=0),the blow up result[1]was improved by the same author[6]and Wu[7].Recently,Wu et al.[8]considered the problem(1.1)with general damping and source terms,they established the nonexistence result of global solutions with the initial energy controlled by a critical value by modifying the method in a work Autuori et al.[9]in 2010,see[10,11]for more general results.

For the problem in RN,Todorova[12]studied the following problem

He showed the results similar to[4]when q(x)is a decaying function.For more related results,we refer the readers to[13–15].In[15],the author considered the linear damping case and obtained that the solution blows up in finite time when the initial energy is nonpositive.We also mention the work[16],Todorova considered the Cauchy problem(1.2)with q(x)=1 and obtained the solution is global and blows up in finite time under suitable conditions by the ideas of the potential well theory.

In the presence of the viscoelastic term,Ka fini and Messaoudi[17]considered problem(1.1)when m=2 and ?=RN,and obtained the results similar to[15].Later,Lu and Li[18]considered the following Cauchy problem with nonlinear damping term

with compactly supported initial data u(0,x)=u0(x),ut(0,x)=u1(x).They also obtained that the solution blows up in finite time when the initial energy is negative and a global existence result under suitable assumptions.

Motivated by these papers,in this work,we intend to study the following Cauchy problem

where g,f,u0and u1are functions to be specified later.Such problems arise in viscoelasticity and in systems governing the longitudinal motion of a viscoelastic configuration obeying a nonlinear Boltzmann’s model.Our aim is to extend the results of[8,12,17,18]to our problem.

The remainder of this paper is organized as follows.Section 2 is concerned with some notations,statementsof assumptions and local existence result.In Section 3,we state and prove the non-existence result of global solutions of(1.3).In Section 4,we show that the solution of(1.3)is global for arbitrary initial data and in the potential respectively under suitable conditions.

2 Preliminaries and local existence results

In this section,we shall give some notations and preliminaries used throughout this article.Now,we make the following assumptions on g and f.

(F1)f(x,u)∈C(RN×R→R),such that f(x,u)=Fu(x,u)with F(x,0)=0,where F(x,u)=R(x,η)dη is a potential for f in u.There exists a constant d1＞0 such that

for all x∈RN,where the number p satisfies

(F2)There exists ε0＞0 such that for all ε∈(0,ε0],there exists d2＞0 such that

for all x∈RN.

Now,we are in a position to state the local existence result for the Cauchy problem(1.3),which can be established by combining the arguments of[4,18,19].

Theorem 2.1.Assume that(G)and(F1)hold and m≥1.Then for any initial data u0∈H1(RN),u1∈L2(RN)with compact support,problem(1.3)has a unique solution such that

for T small enough.

Now,we define the natural total energy function associated with the problem(1.3)by

for t≥0,where

Now,we give some energy estimates.Assume(G)and(F1)hold,then for all t∈that

where

It is easy to verify that G(λ)has a maximum atand the maximum value is

for all t≥0.In other words,Fu(t)is bounded below inalong any solution associated with the problem(1.3).We put for convenience that

3 Blow up result

In this section,we shall discuss that the solution for the problem(1.3)blows up in finite time under suitable conditions.For this purpose,we assume that g satisfies,in addition to(G),the following assumption:

where d1,d2are constants given in(2.2)and(2.4).

Before we prove the main result,we need the following properties.As in[6,18],we see that

Lemma 3.1.Let u be a solution of problem(1.3)and suppose that(G)and(F1)hold,then,where=[0,∞).Moreover,if,then ω2＞0 and

Proof.We see that ω2is bounded below for all t≥ 0 from Eq.(2.8).Then,using the equation(2.7),(3.2)and the fact,we deduce that

Now,we are in a position to prove λ(t)＞λ0for all t∈R+.In fact,from(2.6)and(2.7),we obtain that

which implies that

This finishes the proof.

Next we state our main result.

Theorem3.1.Assumethat(G),(F1),(F2)and(3.1)hold,and u0∈H1(RN),u1∈L2(RN)with compact support.Then any solution of problem(1.3)with initial energy satisfying E(0)＜blows up in finite time.

Proof.Let u be a solution of problem(1.3),satisfying E(0)＜.We define the function

where E2＞0,E2∈([E(0)]+,).We see that H′(t)=?E′(t)≥0 by(3.2).Thus,we have

Furthermore,by the choice of E2,Eq.(2.7)and the definition of ω2,we have that

Noticing(F2),there exists ε0=ε0(ω2,u)＞0 such that(F2)holds true and,without loss of generality,we take ε0＞0 sufficiently small such that

which is possible since ω2＞0 and E2＜?E1.Note that(3.6)forces ε0≤p?2 being E2＞0.

We then define the weighted functional

where B(t)=RRNuutdx,the constant A＞0 and θ＞0 shall be chosen later.The constant L is such that supp(u0)∪supp(u1)?BL(0).Taking the derivative of B(t)and using(1.3),then exploiting H?lder’s inequality and Young’s inequality,we deduce that

Fix ε∈(0,ε0),by recalling(F2)and(2.6),equation(3.8)becomes

Moreover,since εω2＜ε0ω2≤(p?2)ω2?2E2by(3.6),we deduce that

Now by the finite speed of propagation for problem(1.3),we obtain that

Thuswecan estimatethelast termof(3.9)bycombining(3.2),(3.5)and(3.10)as following

By differentiating(3.7)and using(3.9)and(3.11),we see that

Of course(3.12)remains valid even if δ is time dependentsince the integral is taking over the x variable,therefore by taking δ such thatfor large K to be chosen later,by the definition of H(t)and set,we arrive that

for some positive constant k.We observe from the definition of?E1and ω2,(3.5)becomes that

Hence

where we have used the assumption(3.1).

Now,we are in a position to choose θ＞0 sufficiently small such that

and

Therefore(3.14)takes the form

where γ＞0 is the minimum of the coefficients ofand(g??u)(t).Hence we haveL(t)≥L(0)＞0 for all t≥0.

On other hand,we estimate

for c3=max{c2,1}.Therefore,by Minkowski’s inequality and(3.16),it follows that

for some positive constant c4,where.From(3.15)and(3.17),we get

for c5=θγ/c4.Now,we shall diminish α such that

which is possible since 1?A1＞0.In fact,is equivalent towhich is obvious from the assumption(2.3);is equivalent tojust as the assumption.Setting,an integration of(3.18)over(0,t),we obtain that

which implies that L(t)blows up in a finite time since L(0)＞0 and+∞by virtue of(3.19).The proof is complete.

Utilizing(2.8)and E(t)≤E(0)by(3.2),we have

Therefore,we improve the nonexistence of the result of[12,17,18]to the initial energy E(0)being controlled by a positive numberonly,and the initial boundary value problem in[8]to our Cauchy problem(1.3).

4 Global existence

In this section we show that the solution of(1.3)is global for arbitrary initial data or in the potential respectively.

Theorem 4.1.Assume that(G),(F1)hold and F(x,u)≥d3|u|pfor some constant d3,let p≤m satisfy(2.3).Then for any initial data u0∈H1(RN),u1∈L2?RN?with compact support,problem(1.3)has a unique global solution such that

Proof.The proof is similar to[4]and[18],for the convenience of the reader,we sketch the proof.We now introduce the modified auxiliary functional

Clearly

We estimate the last term in the above equation as following

By virtue of 2＜p≤m and the convexity of uy/y in y for u≥0 and y＞0,we have

where C1and C2are positive constants.Then from(4.1),we have

Choosing δ sufficiently small in(4.3),we obtain that F′(t)≤CF(t).On the other hand,by(4.1)

Then we complete the proof of the global existence result for problem(1.3)by the Gronwall lemma and the continuation principle.

Now,we give the global existence in the potential well for(1.3).

Theorem 4.2.Under the assumptions in Theorem 2.1,if the initial data(u0,u1)satisfies

then the corresponding solution to problem(1.3)globally exists.

Proof.By(3.2)we see that E(t)≤E(0)＜E0for every t∈[0,Tmax).We next prove that

for every t∈[0,Tmax).

We establish(4.4)by contradiction,suppose not,using the continuity of u(t)as a function of t,we then see that there exists a time t0∈[0,Tmax)such that

By virtue of(2.7),we observe that

which contradicts E(t)≤E(0)＜E0,t≥0.Thus we have proved(4.4)for every t∈[0,Tmax).

Using(2.7)and(4.4)we canalso concludethat‖ut‖2∈L∞(0,Tmax).Thus,by a standard continuation argument,we see that Tmax=∞.The proof is complete.

Acknowledgement

The authors would like to thank the referees for the careful reading of this paper and for the valuable suggestions to improve the presentation and the style of the paper.This project is supported by Key Scientific Research Foundation of the Higher Education Institutions of Henan Province,China(No.15A110017)and National Natural Science Foundation of China(No.11526077).