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July  1998, 4(3): 431-444. doi: 10.3934/dcds.1998.4.431

On the initial boundary value problem for the damped Boussinesq equation

Vladimir V. Varlamov 1,

1. 

Departamento de Matematicas y Estadistica, Facultad de Ciencias, Universidad Nacional de Colombia, Bogota, Colombia

Received  March 1996 Revised  April 1997 Published  April 1998

The first initial-boundary value problem for the damped Boussinesq equation $ u_{t t}-2bu_{t x x}=-\alpha u_{x x x x}+u_{x x}+\beta (u^2)_{x x}, x\in (0,\pi ),\quad t>0,$ with $\alpha, b=const>0,\quad \beta =const\in R^1,$ is considered with small initial data. For the most interesting case $\alpha >b^2$ corresponding to an infinite number of damped oscillations its solution is constructed in the form of a Fourier series which coefficients in their own turn are represented as series in small parameter present in the initial conditions. The solution of the corresponding problem for the classical Boussinesq equation on $[0,T],\quad T<+\infty,$ is obtained by means of passing to the limit $b\rightarrow +0.$ Long-time asymptotics of the solution in question is calculated which shows the presence of the damped oscillations decaying exponentially in time. This is in contrast with the long time behavior of the solution of the periodic problem studied in [30] which major term increases linearly with time.
Keywords: first initial-boundary value problem, long-time asymptotics., Damped Boussinesq equation.
Mathematics Subject Classification: 35Q20, 76B1.
Citation: Vladimir V. Varlamov. On the initial boundary value problem for the damped Boussinesq equation. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 431-444. doi: 10.3934/dcds.1998.4.431
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