Long-Time Dynamics of a Plate Equation with Memory and Time Delay

A plate equation with past history and time-varying delay in the internal feedback is considered. The main result is the long-time dynamics of the system. Under suitable assumptions on real numbers μ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _1$$\end{document} and μ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _2$$\end{document}, we establish the quasi-stability property of the system and obtain the existence of a global attractor which has finite fractal dimension. We also prove the existence of exponential attractors.