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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 $B\subset \Bbb R^3$: $$ u_{tt}-a\Delta u_{tt}-2b\Delta u_t=\alpha\Delta^3u-\beta\Delta^2u+\Delta u+\eta\Delta(u^2). $$ The existence of mild solutions is established in the space $C^0([0,\infty), H^\kappa_0(B))$ where $\kappa<5/2$, and the solutions are constructed in the form of series in the small parameter present in the initial conditions. For $-1/2<\kappa<5/2$, uniqueness is proved. In addition, the long-time asymptotics is obtained in an explicit form.

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