Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback

Abstract

In the present paper, we consider a suspension bridge problem with a nonlinear delay term in the internal feedback. Namely, we investigate the following equation: \begin{document}$ \begin{equation*} u_{tt}+ \Delta^2 u + \delta_1 g_1 (u_t (x,y,t))+ \delta_2 g_2 (u_t (x,y, t-\tau))+ h(u(x,y,t)) = f(x,y), \end{equation*} $\end{document} together with some suitable initial data and boundary conditions. We prove the global existence of solutions by means of the energy method combined with the Faedo-Galerkin procedure under a certain relation between the weight of the delay term in the feedback and the weight of the nonlinear frictional damping term without delay. Moreover, we establish the existence of a global attractor for the above-mentioned system by proving the existence of an absorbing set and the asymptotic smoothness of the semigroup \begin{document}$ S(t) $\end{document} .