Asymptotic stability for solutions of a coupled system of quasi-linear viscoelastic Kirchhoff plate equations

Abstract

In this manuscript, we study the asymptotic stability of solutions of two coupled quasi-linear viscoelastic Kirchhoff plate equations involving free boundary conditions, and accounting for rotational forces $ \begin{eqnarray*} &&\vert y_t\vert^{\rho}y_{tt}-\Delta y_{tt}+\Delta^{2}y- \int_0^t h_1(t-s)\Delta^2 y(s)\;ds+f_1(y, z) = 0,\\\\ &&\vert z_t\vert^{\rho}z_{tt}-\Delta z_{tt}+\Delta^{2}z- \int_0^t h_2(t-s)\Delta^2 z(s)\;ds+f_2(y, z) = 0. \end{eqnarray*} $ The system under study in this contribution could be seen as a model for two stacked plates. This work is motivated by previous works about coupled quasi-linear wave equations or concerning single quasi-linear Kirchhoff plate. The existence of local weak solutions is established by the Faedo-Galerkin approach. By using the perturbed energy method, we prove a general decay rate of the energy for a wide class of relaxation functions.

Keywords

• kirchhoff plate equation
• free boundary conditions
• viscoelastic
• faedo-galerkin method
• asymptotic stability