Longtime dynamics for a type of suspension bridge equation with past history and time delay

Abstract

In this paper, we investigate a suspension bridge equation with past history and time delay effects, defined in a bounded domain \begin{document}$ \Omega $\end{document} of \begin{document}$ \mathbb{R}^N $\end{document} . Many researchers have considered the well-posedness, energy decay of solution and existence of global attractors for suspension bridge equation without memory or delay. But as far as we know, there are no results on the suspension bridge equation with both memory and time delay. The purpose of this paper is to show the existence of a global attractor which has finite fractal dimension by using the methods developed by Chueshov and Lasiecka. Result on exponential attractors is also proved. We also establish the exponential stability under some conditions. These results are extension and improvement of earlier results.