Long-time dynamics of a class of nonlocal extensible beams with time delay

Abstract

This work is devoted to the following nonlocal extensible beam equation with time delay:

$$u_{tt}+{\mathrm{\Delta }}^{2}u-M\left({\int }_{\mathrm{\Omega }}|\nabla u{|}^2\mathrm{d}x\right)\mathrm{\Delta }u+a_0{u}_t(x,t)+a_1{u}_t(x,t-\tau )+\phi (u)=f,$$

on a bounded smooth domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ . The main purpose of this paper is to consider the long-time dynamics of the system. Under suitable assumptions, the quasi-stability property of the system is established, based on which the existence and regularity of a finite-dimensional compact global attractor are obtained. Moreover, the existence of exponential attractors is proved.