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

Shaoyong Lai; Yong Hong Wu

doi:10.3934/dcdsb.2003.3.401

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2003, Volume 3, Issue 3: 401-408. Doi: 10.3934/dcdsb.2003.3.401

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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
2.
Department of Mathematics and Statistics, Curtin University of Technology, GOP Box U1987, Perth, WA 6845

Received: April 2002

Revised: February 2003

Published: August 2003

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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.

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Mathematics Subject Classification: 34C25, 93C15.

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