Journal of Function Spaces and ApplicationsVolume 2012 (2012), Article ID 530861, 15 pagesdoi:10.1155/2012/530861

Research Article

Global Nonexistence of Solutions for Viscoelastic Wave Equations of Kirchhoff Type with High Energy

Gang Li, Linghui Hong, and Wenjun Liu

College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

Received 12 February 2012; Accepted 26 March 2012

Academic Editor: Amol Sasane

Copyright © 2012 Gang Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We consider viscoelastic wave equations of the Kirchhoff type with Dirichlet boundary conditions, where denotes the norm in the Lebesgue space . Under some suitable assumptions on and the initial data, we establish a global nonexistence result for certain solutions with arbitrarily high energy, in the sense that for some .

1. Introduction

In this paper we consider the following problem: where is a bounded domain in () with a smooth boundary is a nonnegative function like for ,,,, and represents the kernel of memory term.

Problem (1.1) without the viscoelastic term (i.e., ) has been extensively studied and many results concerning global existence, decay, and blowup have been established. For example, the following equation: has been considered by Matsuyama and Ikehata in [1] for and . The authors proved existence of the global solutions by using Faedo-Galerkin method and the decay of energy based on the method of Nakao [2–4]. Later, Ono [5] investigated (1.2) for and . When , or , the author showed that the solutions blow up in finite time with . For and , this model was considered by the same author in [6]. By applying the potential well method he obtained the blow-up properties with positive initial energy . Recently, Zeng et al. [7] studied (1.2) for the case with the same initial and boundary conditions as that of problem (1.1). By using the concavity argument, they proved that the solutions to (1.2) blow up in finite time with arbitrarily high energy.

In the case of and in the presence of the viscoelastic term (i.e., ), the equation was studied by Messaoudi in [8], where the author proved that any weak solution with negative initial energy blows up in finite time if and

while the solution continues to exist globally for any initial data in the appropriate space if . This blow-up result was improved by the same author in [9] for positive initial energy under suitable conditions on ,, and . More recently, Wang [10] investigated (1.3) and established a blow-up result with arbitrary positive initial energy. In the related work, Cavalcanti et al. [11] studied the following equation: where is a function which may be null on a part of . Under the condition that on , with satisfying some geometric restrictions and , to guarantee that is small enough, they proved an exponential decay rate.

When and is not a constant function, problems related to (1.1) have been treated by several authors. Wu and Tsai [12] considered the global existence, asymptotic behavior, and blow-up properties for the following equation: with the same initial and boundary conditions as that of problem (1.1). They obtained the blow-up properties of local solution with small positive initial energy by using the direct method of [13]. Global existence and decay properties of the solutions were also obtained there. In [14], Wu then extended the decay result of [12] under a weaker condition on .

For other papers related to existence, uniform decay and blowup of solutions of nonlinear wave equations, see [15–33] and references therein.

Motivated by the above research, we consider problem (1.1) for in this paper and establish a global nonexistence result for certain solutions with arbitrarily high energy by using concavity technique. In this way, we can extend the result of [7] to nonzero term and the result of [10] to nonconstant . Throughout the rest of this paper, we always assume that .

The structure of this paper is as follows. In Section 2, we present some assumptions, notations and the main result. In Section 3, we give the proof of the main result. Some further remarks are stated in Section 4.

2. Preliminaries and Main Result

In this section, we will give some assumptions, notations and state the main result. We first give the following assumptions:(A1) is a nonnegative and non-increasing function satisfying(A2) The function is of positive type in the following sense (see [10]):

Remark 2.1. Assumption (A2) is needed to prove Lemma 3.1 below.

In order to prove our result, we make the following assumption on and :(A3) there exists a positive constant such that where .

Remark 2.2. It is clear that when for ,,, , and , condition A3 can be replaced by which is the same as the one in [10, Theorem  1.1] for the case and , where is the constant from the Poincaré inequality . Then, the possible choice of the positive constant in A3 can be easily obtained (see Section 4.1 for details).It is necessary to state the local existence theorem for problem (1.1), whose proof follows the arguments in [12, 34].

Theorem 2.3. Assume that A1 holds, and when , when . For , , and , problem (1.1) has a unique local solution for the maximum existence time .

The energy functional and an auxiliary functional of the solution of problem (1.1) are defined as follows: where

As in [7, 10], we can get Then we have Now we are in a position to state the main result.

Theorem 2.4. Assume that A1 holds and when , when . Let be a solution of problem (1.1) with initial data and , and further assume that Then the solution of problem (1.1) blows up in finite time , which means that where is a constant from the Poincaré inequality and comes from condition .

Remark 2.5. We note that the set of the initial data which satisfy conditions (2.11)–(2.14) is not empty (see Section 4.2 for details).

3. Proof of the Main Result

In this section we prove our main result, Theorem 2.4, whose proof follows the ideas already used in [7, 10] and relies on the following lemmas.

Lemma 3.1 (see [10, Lemma  2.1]). Assume that satisfies assumptions A1-A2, and is a function which is twice continuously differentiable satisfying for every , where is the corresponding solution of problem (1.1) with and . Then the function is strictly increasing on .

Lemma 3.2. Suppose that and satisfy If the solution of problem (1.1) exists on and satisfies then is strictly increasing on .

Proof. Since is the solution of problem (1.1), by a simple computation, we have where the last inequality is derived by (3.3). Then we get Therefore, by using Lemma 3.1, we finish our proof.

Lemma 3.3. If and satisfy the assumptions in Theorem 2.4, then the solution of problem (1.1) satisfies for all .

Proof. We will prove the above lemma by contradiction. First we assume that (3.6) is not true over , it means that there exists a time such that Since on , by Lemma 3.2, we see that is strictly increasing over , which implies And by the continuity of on , we note that On the other hand, by (2.6) and (2.9), we get Combining (3.11) with (3.8) yields By (A3), we get By Poincaré’s inequality, we have Obviously, there is a contradiction between (3.10) and (3.14), thus we prove that for every . By Lemma 3.2, it follows that is strictly increasing on , which implies that for every . This completes the proof of Lemma 3.3.

Proof of Theorem 2.4. We prove our main result by adopting concavity method. We assume by contradiction that the is sufficiently large. Then we consider the auxiliary function where , and are positive constants, which will be chosen later.A straightforward calculation gives consequently, By using Young’s inequality, we obtain Substituting (2.6) and (3.20) for the third and the fifth terms of the right hand side of (3.19), respectively, we have By (A3), we deduce Noting that (2.10), we obtain that Combining (3.22)-(3.23) yields By Poincaré’s inequality, Lemma 3.2, and (2.14), we see that by (3.24)-(3.25), we get which means that for every . Thus, by and , we get and are strictly increasing on .We first choose small enough satisfying consequently, As far as is fixed, we select large enough satisfying From (3.17), (3.18), and (3.29), we now choose such that , which ensures that Letting Since we have assumed that the solution to problem (1.1) exists for every , where is sufficiently large, we have for every . Then it follows that Furthermore, we have for every , which implies that . Thus, we obtain As , letting , we have According to concavity technique, there exists a real number such that and we have that is, which contradicts the assumption that the is sufficiently large.This completes the proof of Theorem 2.4.

4. Some Further Remarks

4.1. The Possible Choice of the Positive Constant in (A3)

When for , ,,, and , by straightforward calculation, we obtain If and , it follows from (2.4) that . Thus, we have where we have used a obvious conclusion: . Therefore, we can choose in condition A3.

If and , then

Taking , applying Lemma 3.2 and Poincaré’s inequality, we can get So, we can choose in condition A3.

4.2. The Set of the Initial Data Satisfying Conditions (2.11)–(2.14) Is Not Empty

For any real value of the initial energy , there exists such initial data which leads to blow up in finite time.

For instance, in the case , then for any with , we may take some , such that satisfies the above conditions (2.11)–(2.14).

Indeed, for , conditions (2.11)–(2.14) become

Now taking such that , and letting for any scaling parameter and , then we have

We suppose that (i.e., for , so we can choose sufficiently large such that And when is fixed, we may choose such that .

Similarly, in the case , we can also take initial data satisfying the above conditions (2.11)–(2.14) (see [7, Remark  1.4]).

Thus the set of the initial data which satisfy conditions (2.11)–(2.14) is not empty.

Acknowledgments

This work was partly supported by the Tianyuan Fund of Mathematics (Grant no. 11026211) and the Natural Science Foundation of the Jiangsu Higher Education Institutions (Grant no. 09KJB110005).

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