Journal of Applied Mathematics

Volume 2014, Article ID 310297, 6 pages

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

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

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.

#### References

- G. F. Webb, “Existence and asymptotic behavior for a strongly damped nonlinear wave equation,”
*Canadian Journal of Mathematics*, vol. 32, no. 3, pp. 631–643, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. C. Liu and D. C. Liu, “The initial-boundary value problem for the equation ${u}_{tt}-\alpha \Delta {u}_{t}-\Delta u=f(u)$,”
*Journal of Huazhong University of Science and Technology*, vol. 16, no. 6, pp. 169–173, 1988. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. D. Shang, “Blow-up of solutions for two classes of strongly damped nonlinear wave equations,”
*Journal of Engineering Mathematics*, vol. 17, no. 2, pp. 65–70, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. X. Chen, B. L. Guo, and P. Wang, “Long time behavior of strongly damped nonlinear wave equations,”
*Journal of Differential Equations*, vol. 147, no. 2, pp. 231–241, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. F. Zhou, “Attractors for strongly damped wave equation in uniform spaces,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 64, pp. 174–187, 2006. View at Google Scholar - A. B. Al'shin, M. O. Korpusov, and A. G. Sveshnikov,
*Blow-Up in Nonlinear Sobolev Type Equations*, vol. 15 of*De Gruyter Series in Nonlinear Analysis and Applications*, Walter de Gruyter, Berlin, Germany, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - Y. D. Shang, “Initial-boundary value problem for the equation ${u}_{tt}-\Delta u-\Delta {u}_{t}-\Delta {u}_{tt}=f(u)$,”
*Acta Mathematicae Applicatae Sinica*, vol. 23, no. 3, pp. 385–393, 2000. View at Google Scholar · View at MathSciNet - R. Z. Xu, S. Wang, Y. B. Yang, and Y. H. Ding, “Initial boundary value problem for a class of fourth-order wave equation with viscous damping term,”
*Applicable Analysis*, vol. 92, no. 7, pp. 1403–1416, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. C. Liu and R. Z. Xu, “Fourth order wave equations with nonlinear strain and source terms,”
*Journal of Mathematical Analysis and Applications*, vol. 331, no. 1, pp. 585–607, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Schneider and C. E. Wayne, “Kawahara dynamics in dispersive media,”
*Physica D: Nonlinear Phenomena*, vol. 152-153, pp. 384–394, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. J. Boussinesq,
*Essai sur la Théorie des Eaux Courantes*, vol. 3 of*Mémoires Présentés par Divers Savans à l'Académie des Sciences de l'Institut de France (séries 2)*, 1877. - Y. Wang and C. L. Mu, “Blow-up and scattering of solution for a generalized Boussinesq equation,”
*Applied Mathematics and Computation*, vol. 188, no. 2, pp. 1131–1141, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Esfahani, L. G. Farah, and H. Wang, “Global existence and blow-up for the generalized sixth-order Boussinesq equation,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 75, no. 11, pp. 4325–4338, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. W. Wang and S. B. Wang, “Global existence and asymptotic behavior of solution for the Rosenau equation with hydrodynamical damped term,”
*Journal of Mathematical Analysis and Applications*, vol. 401, no. 2, pp. 763–773, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. W. Wang and S. B. Wang, “Decay and scattering of small solutions for Rosenau equations,”
*Applied Mathematics and Computation*, vol. 218, no. 1, pp. 115–123, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. C. Evance,
*Partial Differential Equations*, American Mathematical Society, Providence, RI, USA, 1988. - H. A. Levine, “Instability and nonexistence of global solutions to nonlinear wave equations of the form $P{u}_{tt}=-Au+\mathcal{F}(u)$,”
*Transactions of the American Mathematical Society*, vol. 192, pp. 1–21, 1974. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. L. Lions,
*Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires*, Dunod, Paris, France, 1969. View at MathSciNet - J. C. Robinson,
*Infinite-Dimensional Dynamical Systems, An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors*, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, UK, 2001. View at Publisher · View at Google Scholar · View at MathSciNet