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Existence and asymptotic behavior of solutions to the generalized damped Boussinesq equation. (English) Zbl 1254.35196
Summary: We consider the Cauchy problem for the $n$-dimensional generalized damped Boussinesq equation. Based on decay estimates of solutions to the corresponding linear equation, we define a solution space with time weighted norms. Under a smallness condition on the initial value, the existence and asymptotic behavior of global solutions in the corresponding Sobolev spaces are established by the contraction mapping principle.
##### MSC:
 35Q35 PDEs in connection with fluid mechanics 35B40 Asymptotic behavior of solutions of PDE 35L30 Higher order hyperbolic equations, initial value problems 35L75 Nonlinear hyperbolic PDE of higher $\left(>2\right)$ order