Global existence and asymptotic behaviour of solution for a damped nonlinear hyperbolic equation

Abstract

In this paper we study the initial–boundary value problem for a damped nonlinear hyperbolic equation

$$u_{tt}+{\Delta }^{2}u+\alpha {\Delta }^2{u}_t+\Delta f\left(\Delta u\right)=0,x\in \Omega ,t>0,u(x,0)=u_0\left(x\right),u_t(x,0)=u_1\left(x\right),x\in \Omega ,u=0\Delta u=0,x\in \partial \Omega ,t\ge 0,$$

where $\Omega \subset {\mathbb{R}}^n$, $n\ge 1$ and $\alpha$ is a positive constant. Under some assumptions on $f\left(s\right)$ and initial data, we prove the global existence of solution. Furthermore, we prove that the solution decays to zero exponentially.

Introduction

This paper considers the initial boundary value problem for a damped nonlinear hyperbolic equation

$$u_{tt}+{\Delta }^{2}u+\alpha {\Delta }^2{u}_t+\Delta f\left(\Delta u\right)=0,x\in \Omega ,t>0$$$$u(x,0)=u_0\left(x\right),u_t(x,0)=u_1\left(x\right),x\in \Omega ,$$$$u=0,\Delta u=0,x\in \partial \Omega ,t\ge 0,$$

where $\Omega \subset {\mathbb{R}}^n$ is a bounded domain and $\alpha$ is a positive constant.

In the study of the vibration of 3-dimensional nonlinear damped hyperbolic equation, there arises the following equation (see [4, p. 329], an explicit example)

$$u_{tt}+k_1{\nabla }^{4}u+k_2{\nabla }^4{u}_t+{\nabla }^{2}g\left({\nabla }^{2}u\right)=0,(x,t)\in \Omega \times (0,T),$$

where $\Omega$ is a bounded domain in ${\mathbb{R}}^3$ with smooth boundary $\partial \Omega$, $k_1$ and $k_2$ are two positive physical constants. For the 2-dimensional model, Eq. (1.4) denotes the nonlinear damped membrane and for the 1-dimensional model Eq. (1.4) is the nonlinear beam equation (see [2], [3]). For the initial boundary value problem of one-dimensional version of Eq. (1.4) with a monotone function $g$, the existence and uniqueness of global weak solutions were established in [3]. Later, [2] obtained the same results as [3] under the relaxed assumption on the nonlinearity, i.e., $g$ is a locally Lipschitz continuous function. Furthermore, Chen [6] extended the existence result of weak solutions in Ref. [2], [3] to classical solutions for the more smooth $g$, also the blowup of solution are proved in [6]. For the initial boundary value problem of multi-dimensional version of above equation (1.4), the existence and uniqueness of the weak solution was shown in [4] and the energy decay was given in [1] with the boundary condition $u=0$, $\frac{\partial u}{\partial v}=0$, $(x,t)\in \Omega \times (0,T)$, where $v$ denotes the outer unit normal at $\partial \Omega$. And the corresponding well-posedness results for Cauchy problem were given in [5]. It is true that scholars have different points of interest for the same equation with different boundaries, however the existence and long-time behaviour of solutions have always been hot topics in the research of wave equation (see [7], [10]). Most recently, in [8] Chen et al. studied the following initial boundary value problem of multi-dimensional version of Eq. (1.4)

$$u_{tt}+k_1{\Delta }^{2}u+k_2{\Delta }^2{u}_t+\Delta g\left(\Delta u\right)=0,x\in \Omega ,t>0,$$$$u(x,0)=u_0\left(x\right),u_t(x,0)=u_1\left(x\right),x\in \Omega ,$$$$u=0,\Delta u=0,x\in \partial \Omega ,t\ge 0.$$

But its boundary conditions are different from that in [4] and [1]. Under some assumptions on $g\left(s\right)$ and initial data, they proved the local existence and global nonexistence of solution, and the global existence and asymptotic behaviour (as an important description of the long time dynamics as the global attractors [9]) of solution for problem (1.5) are still unsolved, which is the main purpose of the present paper.