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

Shaoyong Lai; Yong Hong Wu; Xu Yang

doi:10.3934/cpaa.2004.3.319

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2004, Volume 3, Issue 2: 319-328. Doi: 10.3934/cpaa.2004.3.319

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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
3.
Department of Applied Mathematics, Southwest Jiaotong University, Chengdu

Received: December 31, 2002

Revised: November 30, 2003

Published: May 2004

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Abstract

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.

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Mathematics Subject Classification: 35Q20, 76B15.

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