Journal of Applied Mathematics

Volume 2014 (2014), Article ID 310297, 6 pages

http://dx.doi.org/10.1155/2014/310297

Research Article

Blow-Up of Solutions for a Class of Sixth Order Nonlinear Strongly Damped Wave Equation

Huafei Di and Yadong Shang

School of Mathematics and Information Science, Guangzhou University, Guangzhou, Guangdong 510006, China

Received 28 February 2014; Revised 1 May 2014; Accepted 4 May 2014; Published 15 May 2014

Academic Editor: Zhijun Liu

Copyright © 2014 Huafei Di and Yadong Shang. 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 the blow-up phenomenon of sixth order nonlinear strongly damped wave equation. By using the concavity method, we prove a finite time blow-up result under assumptions on the nonlinear term and the initial data.

1. Introduction

It is well known that nonlinear strongly damped wave equation is proposed to describe all kinds of viscous vibration system. The global well-posedness of third order nonlinear strongly damped wave equation

was studied by Webb [1] firstly. He gave the existence and asymptotic behavior of strong solutions for the problem (1). Then this result was improved by Y. C. Liu and D. C. Liu [2]. The existence and uniqueness of strong solutions were proved under the hypothesis of the weaker conditions. For two classes of strongly damped nonlinear wave equation, the finite time blow-up of solutions was proved by Shang [3]. A number of authors (Chen et al. [4], Zhou [5], and Al’shin et al. [6]) have shown the existence of the global weak solutions and the global attractors for third order nonlinear strongly damped wave equation.

For the fourth order nonlinear strongly damped wave equation, there are also some results about initial boundary value problem or Cauchy problem [7–9]. In [7], Shang studied the initial boundary value problem of the following equation:

Under some assumptions on and , he investigated the existence, uniqueness, asymptotic behavior, and blow-up phenomenon of the solutions.

In [8], Xu et al. considered the initial boundary value problem of fourth order wave equation with viscous damping term

They proved the global existence and nonexistence of the solution by argument related to the potential well-convexity method.

In order to investigate the water wave problem with surface tension, Schneider and Wayne [10] studied a class of Boussinesq equation as follows:

where . This type of equations can be formally derived from the 2D water wave problem and models the water wave problem with surface tension. They proved that the long wave limit can be described approximately by two decoupled Kawahara equations. A more natural model seems to be an extension from the classical Boussinesq equation as follows (see [11]):

Wang and Mu [12] studied the Cauchy problem of the equation

They obtained the existence and uniqueness of the local solutions and proved the blow-up of solutions to the problem (6). Esfahani et al. [13] studied the solutions of

where and . They proved the local well-posedness in and and gave finite time blow-up results to the problem (7).

For the sixth order nonlinear wave equation with strong damping term

H. W. Wang and S. B. Wang [14] established a global existence result of small amplitude solutions of the Cauchy problem (8) for all space dimensions . When , H. W. Wang and S. B. Wang [15] studied the long-time behavior of small solutions of the Cauchy problem for a Rosenau equation. The decay and scattering for small amplitude solution are established.

In this paper, we study a class of sixth order nonlinear strongly damped wave equation:

where , is a bounded domain of with a smooth boundary , and , , are homogeneous boundary condition:

By using the ideas of the concavity theory introduced by Levine [17], we prove the finite time blow-up results under assumption on the nonlinear term and the initial data .

2. Preliminaries and Main Results

In this section, we introduce some notations, basic ideas, and important lemmas which will be needed in the course of the paper.

Let be a Hilbert space which is equipped with the scalar product .

Now, we define

where is a symmetric linear operator and satisfies for all .

For the nonlinear term of the problem (9), is a vector function which satisfies the following conditions.

(a)Assume that the Fréchet derivative is a symmetric, bounded, linear operator on and that is a continuous map from to .(b)The scalar valued function is defined by

where denotes the potential associated with . The Fréchet derivative of is which can be shown to act as follows:

for all , .(c)Assume that for some

for all .

To obtain the finite time blow-up result, we need the following interpolation inequality of Evance [16] for function in .

Lemma 1. For all , if , and are integers, and , , then

where , with the constant depending only on , , , , , and .

In particular, if , , and , one has

Lemma 2. Assume that and (with depending on the constant of Sobolev’s interpolation inequality); then .

Proof. By Lemma 1, we see that

Using Young’s inequality, we have

For the operator , using integration of parts, we have

where .

The verification of the action of can be proved from the definition. The details, not being germane to this paper, are omitted here. But a formula will be useful in the sequel as follows.

Lemma 3. Let ; then one has

for with a strongly continuous derivative .

Proof. By the chain rule and the action of , we have

where we have used the symmetry of in the fourth line.

The following lemma contributing to the result of this paper is analogous to Corollary 1.1 of [17] with slight modification.

Lemma 4. Assume that is homogenous of degree for some (i.e., for all and for all ). Let for some . Then there are infinitely many vectors such that

Proof. Let , where is large enough so that

Then for all

we have

The local existence of solution for the problem (9) can be obtained by the standard Faedo-Galerkin approximation methods. The interested reader is referred to Lions [18] or Robinson [19] for details.

Next, we are ready to state the blow-up result of this paper.

Theorem 5. Let be a strongly continuously differentiable solution of (9) in the norm. Suppose that with ( depending on the constant of Sobolev’s interpolation inequality) , where . Finally let satisfy

Then the solution u can only exist on a bounded interval , and in fact

while also

and consequently

Proof. For arbitrary , , and , let

A direct computation yields

Suppressing the argument , we see that

Hence, from (30), (31), and (32), we find after some algebra that

where

Using Schwarz’s inequality, we have

By (35), we have . Let

Thus

Using the positive semidefiniteness of , Lemma 3, and (14), we have

Thus, from what has been discussed above, we have

Therefore, for any such that

and . We see that , for all and , if is sufficiently large. Since a concave function must always lie below any tangent line, so we have

or

we may choose such that . Thus, we see that the interval of existence of must be contained in and that the finite time blow-up of solution of (9) is proved. Let

Since , we have . Even if we take , we have

thus we must choose so large such that . As a function, has a minimum at

and this minimum is

Since is restricted to , we see that attains its minimum at . Thus cannot exceed .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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