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Long-time asymptotics of solutions for the second initial-boundary value problem for the damped Boussinesq equation. (English) Zbl 0933.35165

Summary: For the damped Boussinesq equation \[ u_{tt}-2bu_{txx}=-\alpha u_{xxxx}+ u_{xx}+\beta(u^2)_{xx},\;x\in(0,\pi),\;t>0; \] \(\alpha,b= \text{const}>0\), \(\beta=\text{const} \in\mathbb{R}^1\), the second initial-boundary value problem is considered with small initial data. Its classical solution is constructed as a series in a small parameter present in the initial conditions, and the uniqueness of solutions is proved. The long-time asymptotics is obtained in explicit form and the blow up of the solution is examined in a certain case. The possibility of passing to the limit \(b\to+0\) in the constructed solution is investigated.

MSC:

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