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

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

$$\begin{array}{rcl}& & |y_t{|}^{\rho }{y}_{tt}-\mathrm{\Delta }{y}_{tt}+{\mathrm{\Delta }}^{2}y-{\int }_0^t{h}_1(t-s){\mathrm{\Delta }}^{2}y(s)ds+f_1(y,z)=0,\\ \\ & & |z_t{|}^{\rho }{z}_{tt}-\mathrm{\Delta }{z}_{tt}+{\mathrm{\Delta }}^{2}z-{\int }_0^t{h}_2(t-s){\mathrm{\Delta }}^{2}z(s)ds+f_2(y,z)=0.\end{array}$$

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.