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

Abstract

<abstract><p>The main goal of this work is to investigate the following nonlinear plate equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_{tt}+\Delta ^2 u +\alpha(t) g(u_t) = u \vert u\vert ^{\beta}, $\end{document} </tex-math></disp-formula></p> <p>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.</p></abstract>