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.

中文翻译：

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一类具有时滞的非局域可展梁的长时间动力学

这项工作致力于以下具有时滞的非局部可扩展梁方程：

$${你}_{吨吨}+{\mathrm{\Delta }}^2你-米\left({\int }_{\mathrm{\Omega }}|\nabla 你{|}^2\mathrm{d}X\right)\mathrm{\Delta }你+{\mathrm{一种}}_0{你}_{吨}(X,吨)+{\mathrm{一种}}_1{你}_{吨}(X,吨-\tau )+\phi (你)=F,$$

在有界光滑域上 $\mathrm{\Omega }\subset {\mathbb{电阻}}^n$. 本文的主要目的是考虑系统的长期动态。在适当的假设下，建立了系统的拟稳定性质，并以此为基础，得到了有限维致密全局吸引子的存在性和规律性。此外，证明了指数吸引子的存在。