Existence and regularity of pullback attractors for nonclassical non-autonomous diffusion equations with delay

Abstract

In this paper, we consider the asymptotic behavior of weak solutions for non-autonomous diffusion equations with delay in time-dependent spaces when the nonlinear function $f$ is critical growth, the delay term $g(t,u_t)$ contains some hereditary characteristics and the external force $h\in L_{loc}^2\left(\mathbb{R};L^2(\Omega )\right)$. Firstly, we prove the well-posedness of solutions by using the Faedo-Galerkin approximation method. Then after a series of elaborate energy estimates and calculations, we establish the existence and regularity of pullback attractors in time-dependent spaces $C_{{\mathcal{H}}_t(\Omega )}$ and $C_{{\mathcal{H}}_t^1(\Omega )}$ respectively.

30 pages