Existence and stability results of a plate equation with nonlinear damping and source term

The main goal of this work is to investigate the following nonlinear plate equation

$$u_{tt}+{\mathrm{\Delta }}^{2}u+\alpha (t)g(u_t)=u|u{|}^{\beta },$$

which models suspension bridges. Firstly, we prove the local existence using Faedo-Galerkin method and Banach fixed point theorem. Secondly, we prove the global existence by using the well-depth method. Finally, we establish explicit and general decay results for the energy of solutions of the problem. Our decay results depend on the functions $\alpha$ and $g$ and obtained without any restriction growth assumption on $g$ at the origin. The multiplier method, properties of the convex functions, Jensen's inequality and the generalized Young inequality are used to establish the stability results.