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

Pages: 319 - 328, Volume 3, Issue 2, June 2004

doi:10.3934/cpaa.2004.3.319       Abstract        Full Text (200.7K)       Related Articles

Shaoyong Lai - Department of Applied Mathematics, Southwest Jiaotong University, 610066, Chengdu, China (email)
Yong Hong Wu - Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA6845, Australia (email)
Xu Yang - Department of Applied Mathematics, Southwest Jiaotong University, Chengdu, China (email)

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.

Keywords:  Global solution, initial-boundary value problem, Boussinesq equations.
Mathematics Subject Classification:  35Q20, 76B15.

Received: January 2003;      Revised: December 2003;      Published: March 2004.