Well-posedness and decay property for the generalized damped Boussinesq equation with double rotational inertia

Abstract

In this paper, we investigate the initial value problem (IVP henceforth) associated with the generalized damped Boussinesq equation with double rotational inertia $$\left\{\begin{array}{ll} u_{tt}+\gamma\Delta^2 u_{tt}-a\Delta u_{tt}-2b\Delta u_t-\alpha\Delta^3u+\beta\Delta^2 u-\Delta u=\Delta f(u),\quad x \in\mathbf{R}^n, \; t>0, \\ u(x,0)=u_0(x),\quad u_t(x,0)=u_1(x),\quad x \in\mathbf{R}^n. \end{array}\right.$$ Based on decay estimates of solutions to the corresponding linear equation, we establish the decay estimates and the pointwise estimates by using Fourier transform. Under small condition on the initial data, we obtain the existence and asymptotic behavior of global solutions in the corresponding Sobolev spaces by time weighted norms technique and the contraction mapping principle.