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On the damped Boussinesq equation in a circle. (English) Zbl 0938.35146

The author studies the long-time behaviour of solutions to a radially symmetric initial-boundary problem for the damped Boussinesq equation \(u_{tt}- 2bu_{txx}= -\alpha u_{xxx} +u_{xx}+ \beta(u^2)_{xx}\) in a circle, which describes the propagation of long waves on the surface of shallow water. The main tool is the use of Fourier-Bessel series, and the main difficulty consists in the comparatively poor convergence of these series. The author proves existence and uniqueness of the solution and gives explicit long-time asymptotics together with an interesting discussion of damped and overdamped oscillations.
Reviewer: O.Titow (Berlin)

MSC:

35Q35 PDEs in connection with fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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