June  2004, 3(2): 319-328. doi: 10.3934/cpaa.2004.3.319

The global solution of an initial boundary value problem for the damped Boussinesq equation

1. 

Department of Applied Mathematics, Southwest Jiaotong University, 610066, Chengdu

2. 

Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA6845, Australia

3. 

Department of Applied Mathematics, Southwest Jiaotong University, Chengdu, China

Received  January 2003 Revised  December 2003 Published  March 2004

This paper deals with an initial-boundary value problem for the damped Boussinesq equation

$u_{t t} - a u_{t t x x} - 2 b u_{t x x} = - c u_{x x x x} + u_{x x} + \beta(u^2)_{x x},$

where $ t > 0,$ $a,$ $b,$ $c$ and $\beta$ are constants. For the case $a \geq 1$ and $a+ c > b^2$, corresponding to an infinite number of damped oscillations, we derived the global solution of the equation in the form of a Fourier series. The coefficients of the series are related to a small parameter present in the initial conditions and are expressed as uniformly convergent series of the parameter. Also we prove that the long time asymptotics of the solution in question decays exponentially in time.

Citation: Shaoyong Lai, Yong Hong Wu, Xu Yang. The global solution of an initial boundary value problem for the damped Boussinesq equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 319-328. doi: 10.3934/cpaa.2004.3.319
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