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On the initial-boundary value problem for the damped Boussinesq equation. (English) Zbl 0952.35103

Summary: The first initial-boundary value problem for the damped Boussinesq equation \[ u_{tt}- 2bu_{txx}=-\alpha u_{xxxx}+ u_{xx}+ \beta(u^2)_{xx},\quad x\in (0,\pi),\quad t>0, \] with \(\alpha, b=\text{const}> 0\), \(\beta= \text{const}\in \mathbb{R}^1\), 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 \([0,T]\), \(T<+\infty\), is obtained by means of passing to the limit \(b\to 0\). Long-time asymptotics of the solution in question is calculated which shows the presence of the damped oscillations decaying exponentialy in time. This is in contrast with the long time behavior of the solution of the periodic problem which major term increases linearly with time.

MSC:

35Q35 PDEs in connection with fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35B40 Asymptotic behavior of solutions to PDEs
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