DRIDI Hanniand ZENNIR Khaled

1 Laboratory of applied mathematics,Badji Mokhtar university. P.O.Box 12,23000 Annaba,Algeria.

2 Department of Mathematics,College of Sciences and Arts,Qassim University,Ar-Rass,Saudi Arabia.

3 Laboratoire de Math´ematiques Appliqu´ees et de Mod´elisation, Universit´e 8 Mai 1945 Guelma. B.P.401 Guelma 24000 Alg´erie.

Abstract. In this paper, we consider a class of Kirchhoff equation, in the presence of a Kelvin-Voigt type damping and a source term of general nonlinearity forms. Where the studied equation is given as follows

Key Words: Galerkin approximation; variable exponents; Kirchhoff equation; blow-up of solutions;Kelvin-Voigt damping;general nonlinearity.

1 Introduction

1.1 Statement of the problem

The problem with variable exponents occurs in many mathematical models of applied science, for example, viscoelastic fluids, electro rheological fluids, processes of filtration through a porous media, fluids with temperature-dependent viscosity etc. From the mathematical point of view, the question of existence, uniqueness and behavior of solutions remain essential results to describe various phenomena. The blow-up is one of the most important behaviors that have been dealt with in evolution problems. In this article,we consider the following initial boundary value problem

We assume that the nonlinearity F(x,t,u)in(1.1)satisfy the following two assumptions:

(i) F∈C1()and there exist a positive integerland a continuous functionck(x,t)∈QT,for(1≤k≤l)such that

We assume that the Kirchhoff function K defined by(1.3)satisfies the following hypotheses:

We also assume thatp(·),q(·)andr(·)satisfy the log-H¨older’s continuity condition

forL>0 and 0<δ<1.

1.2 Motivations

To motivate our work, we review some related results. In 1883 Kirchhoff [1], proposed the following equation

where the parametersL,h,E,ρandP0represent respectively the length of the string, the area of the cross section,Young modulus of the material,the mass density and the initial tension.Forg(ut)=ut,the global existence and nonexistence results can be found in[2,3];and forg(ut)=|ut|put,p>0,the main results of existence and nonexistence are in paper[4]. In recent years, there has been significant advancement in the study of evolution equations when the exponent of coefficient affecting the source or the damping terms,we refer to interesting works[5–9].

In the Kirchhoff model (1.1), the expression K(Nu(t))∆r(·)ut, involved in the third term, represents the internal material damping of Kelvin-Voigt type of the body structure(see[10,11]for one-dimension and[12,13]for multidimensional spaces).The idea of viscosity of a solid body was first proposed by Kelvin in 1878 such that the concept suggested by Kelvin is according to the Stokes law for fluid viscosity, the imagined porous vibrator is damping as a simple vibrator with resistance directly proportional to the velocity of motion, for more details see[14]. In addition, the corresponding equation that describes the behavior of this model was obtained by Voigt in 1890.For a detailed physical discussion in the case K(Nu(t))≡1 andp(x)≡r(x)≡2,we refer to[15,16],as well as the references therein.

Let us begin by mentioning some relevant physical applications where this kind of problem(1.1)appears.

• If F=0.In spaces 1-dimension and 2-dimension,models the transversal vibrations of a homogeneous string and the longitudinal vibrations of a homogeneous bar(see e.g.[17]),respectively,subject to viscous effects.

• If F=0.In the three-dimensional case,(1.1)describes the variation from the configuration at rest of a homogeneous and isotropic linear viscoelastic solid with short memory.

• If F/=0.In the three-dimensional case,(1.1)occurring in quantum mechanics.

1.3 Known results

Indeed, looking at articles previously published, a few works have been appeared regarding hyperbolic problems of the Kirchhoff type equation with nonlinearity of variable exponent. Here we have a number of detailed articles and reviews, among which we note the work by Jun Zhou[18], where the author considered in a bounded domain the following equation

whereα,β,a,b>0,p>2 are constants. The author proved the blow-up of solutions and established the global existence of solutions through the potential well theory and also discussed the behavior of solutions at infinite time.

In[19],Messaoudi and Talahmeh studied a quasilinear wave equation with variableexponent nonlinearities in

wherea,b>0 are constants and the exponents of nonlinearitym,pandrare given functions. Consequently, they proved a blow up result for solutions with negative initial energy and for certain solutions with positive energy.

Recently,Erhan Piskin[20]studied the following wave equation with variable exponent nonlinearities

the author proved by using modified energy functional method the blow up of solutions in finite time.

We refer to some interesting works regarding hyperbolic problem with nonlinearities of variable-exponent,see[21–25]. Concerning the asymptotic behavior,as well as the rate of decay,of the Kirchhoff wave equation problems,it was the subject of many researchers,see e.g.[26–28].

Motivated by previous studies,we proposed,in this work,the problem(1.1)which is more complicated and interesting and as a result,it has been proved a finite-time blowup result for certain solutions with negative/positive initial energy and also for solutions with negative initial energy. More precisely, we look to find a sufficient conditions forp(·),q(·),r(·),F(·,t,u)and the initial data for which the blow up occurs.

The paper is organized as follows. Firstly, we give a series of notations and preliminary results in Section 2. While, the Section 3 is dedicated to the local existence of solutions result. Finally, in Section 4, we prove our main Theorem 4.1 and Theorem 4.2 concerning the blow up at infinity of solutions to the problem(1.1).

2 Preliminaries

We recall here some definitions and basic properties of the generalized Lebesgue-Sobolev spacesLp(x)(Ω)andW1,p(x)(Ω)where Ω is an open and connected subset ofN. We refer to the books[29,30]and papers[31,32]. Let

Lemma 2.3.Let1

3 Local existence result

Definition 3.1.Let p(x)and r(x)defined on QT,then we define the following spaces

a.e. t∈[0,T],which are equipped with the norms

3.1 Technical Lemmas

We first define some energy functionals.

Let

Lemma 3.4.For u1∈L2(Ω),there exists a sequence φn with φn∈Vn such that φn→u1in L2(Ω)as n→∞.

Proof.The proof is similar to the proof of the previous lemma.

3.2 Proof of Theorem(3.1)

We will need the following three steps:

Step 1: Galerkin approximation.

We shall construct the approximate solutions of(1.1)as follows

Then,(3.10)takes the form

By integration,we obtain

whereλ=max1≤i≤mλi>2.

Now,substituting the estimates(3.16)and(3.14)in(3.12),to get

We can conclude thaty(t)≤C(t).Hence,|z|≤C(t),and therefore,|z′|≤C(t),∀t∈(0,T].

Finally, owing to the Peano’s Theorem we get that the (3.9) admits aC1solution.Hence,we obtain the Galerkin approximation solution

for all(x,t)∈QT,(T

From(3.9),for each 1≤i≤nandt∈(0,T],we have

Step 3: The limit process.

The inequalities(3.39)-(3.43)are sufficient to pass to the limit. Hence,we get a subsequence of(un)n∈∗such that

The proof of local existence is now completed.

4 Blow-up result

LetBkbe the best constant of the Sobolev embedding

Our first result of the blow-up for certain solutions with positive initial energy is as follows

Theorem 4.1.Let(1.2)-(1.10)hold. Assume that

whereµis positif constant andL(t),λ are defined in(4.23),(4.24)respectively.

Our second result of the blow-up for the solutions with initial negative energy is as follows

Theorem 4.2.Let(1.2)-(1.10)hold. Assume that

Then,the solution of(1.1)blows up in finite time(4.6).

To prove our main results,we need the following Lemmas.

4.1 Technical lemmas

Lemma 4.1.Suppose the conditions of Lemma2.1-Lemma2.5hold. Then,there exists a positive constant C,such that

which gives(4.22).

4.2 Proof of Theorems(4.1),(4.2)

Thus,(4.45)and the following Minkowski’s inequality

This completes the proof of Theorem 4.1.

Remark 4.1. From the definition of L(t)we have

According to(4.13),we finally obtain that

that is, the solutions blow up in finite time in theLq1(·)norm. On the other hand, the embedding

are continuous. Then, we can deduce that the solutions blows up in finite time in theLq2(·),Lq3(·),···,Lql(·)norms.

In order to prove Theorem 4.2,let

The following Lemma is needed.

Lemma 4.7. ([19])Let u be the solution of(1.1). Then there exists a constant C>0such that

Proof.Suppose,by contradiction,there exists a sequencetϑsuch that

Then,by using Lemma 2.2 and Lemma 2.5,we get

that contradicts the fact that E(t)<0,∀t≥0.

Proof.(of Theorem 4.2)Using(4.48),(4.49)and by introducing the same procedures used to prove Theorem 4.1,it will end the proof of Theorem 4.2.

Acknowledgments

The authors would like to thank the editor and the referees for their suggestions.