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.

In this manuscript, we study a thin and narrow plate equation that models the deck of a suspension bridge that is subject to a Balakrishnan-Taylor damping and a strong damping. First, by using the Faedo Galerkin method, we prove the existence of both global weak and regular solutions. Second, we prove the exponential stability of the energy for regular solutions by combining the multiplier method and a well-known result of Komornik.

In this manuscript, we examine a nonlinear Cauchy problem aimed at describing the deformation of the deck of either a footbridge or a suspension bridge in a rectangular domain $\mathrm{\Omega }=(0,\pi )\times (-d,d)$, with $d<<\pi$, incorporating hinged boundary conditions along its short edges, as well as free boundary conditions along its remaining free edges. We establish the existence of solutions and the exponential decay of energy.