Blow-Up of Solutions for a Class of Sixth 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], andAl’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:


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   − Δ  − Δ =  () ,  > 0,  ∈ Ω,  > 0, (1) 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.
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 , ,  ∈ R.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 2

Journal of Applied Mathematics
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  = ±1 and  > 0. They proved the local wellposedness in  2 () and  1 () 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  > 1.When ] = 0, 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  > 0, Ω is a bounded domain of   ( ≥ 1) with a smooth boundary Ω, and B  ,  = 0, 1, 2, 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 dik ⃗ (∇) and the initial data  0 ,  1 .

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.
(a) Assume that the Fréchet derivative ⃗   is a symmetric, bounded, linear operator on   and that  → ⃗   is a continuous map from   to L()  .(b) The scalar valued function  :  → R is defined by where () denotes the potential associated with dik ⃗ (∇).The Fréchet derivative of () is   which can be shown to act as follows: for all , V ∈ .(c) Assume that for some  > 0 for all  ∈ .
Proof.By Lemma 1, we see that Using Young's inequality, we have Journal of Applied Mathematics 3 For the operator , using integration of parts, we have where 0 <  ≤ (1/ 1 ).
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.
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.
Next, we are ready to state the blow-up result of this paper.