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

Abstract

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 \({\mathscr A}={\{A_{t}}\}_{t\in{\mathbb R}}\) in ℳ_t. Furthermore, we achieve the regularity of \({\mathscr A}\), that is, A_t is bounded in ℳ ¹_t with a bound independent of t.