Time-dependent Global Attractors for the Nonclassical Diffusion Equations with Fading Memory

In this paper, we discuss the long-time behavior of solutions to the nonclassical diffusion equation with fading memory when the nonlinear term f satisfies critical exponential growth and the external force g(x) ∈ L2(Ω). In the framework of time-dependent spaces, we verify the existence of absorbing sets and the asymptotic compactness of the process, then we obtain the existence of the time-dependent global attractor A={At}t∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr A}={\{A_{t}}\}_{t\in{\mathbb R}}$$\end{document} in ℳt. Furthermore, we achieve the regularity of A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr A}$$\end{document}, that is, At is bounded in ℳt1 with a bound independent of t.