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An initial-boundary value problem for a generalized Boussinesq water system in a ball. (English) Zbl 1210.35161

Summary: We consider an initial-boundary value problem for the following generalized Boussinesq water waves equation defined in the unit ball B 3 :

u tt -aΔu tt -2bΔu t =αΔ 3 u-βΔ 2 u+Δu+ηΔ(u 2 )·

The existence of mild solutions is established in the space C 0 ([0,),H 0 κ (B)) where κ<5/2, and the solutions are constructed in the form of series in the small parameter present in the initial conditions. For -1/2<κ<5/2, uniqueness is proved. In addition, the long-time asymptotics is obtained in an explicit form.

MSC:
35L25Higher order hyperbolic equations, general
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction