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August  2003, 3(3): 401-408. doi: 10.3934/dcdsb.2003.3.401

The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation

Shaoyong Lai 1, and  Yong Hong Wu 2,

1. 

Department of Applied Mathematics, Southwest Jiaotong University, 610066, Chengdu, China
2. 

Department of Mathematics and Statistics, Curtin University of Technology, GOP Box U1987, Perth, WA 6845, Australia

Received  April 2002 Revised  February 2003 Published  May 2003

In this paper, we consider the solution of an initial value problem for the generalized 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} - p^2 u + \beta(u^2)_{x x}, $

where $x\in R^1,$ $t > 0,$ $a ,$ $b$ and $c $ are positive constants, $p \ne 0,$ and $\beta \in R^1$. For the case $a + c > b^2$ corresponding to damped oscillations with an infinite number of oscillation cycles, we establish the well-posedness theorem of the global solution to the problem and derive a large time asymptotic solution.

Keywords: well-posedness, Boussinesq equation., Asymptotic solution.
Mathematics Subject Classification: 34C25, 93C1.
Citation: Shaoyong Lai, Yong Hong Wu. The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 401-408. doi: 10.3934/dcdsb.2003.3.401
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