Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

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

Pages: 401 - 408, Volume 3, Issue 3, August 2003      doi:10.3934/dcdsb.2003.3.401

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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, GOP Box U1987, Perth, WA 6845, Australia (email)

Abstract: 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:  Asymptotic solution, well-posedness, Boussinesq equation.
Mathematics Subject Classification:  34C25, 93C15.

Received: April 2002;      Revised: February 2003;      Available Online: May 2003.