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

TL;DR

This paper investigates the long-term behavior of solutions to non-autonomous diffusion equations with delay, proving the existence and regularity of pullback attractors in time-dependent spaces under critical growth conditions.

Contribution

It establishes the existence and regularity of pullback attractors for nonclassical non-autonomous diffusion equations with delay, using energy estimates and Faedo-Galerkin methods.

Findings

Existence of pullback attractors in time-dependent spaces.

Regularity results for solutions in these attractors.

Well-posedness of solutions under critical growth conditions.

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 $h \in L_{l o c}^{2}\left(\mathbb{R} ; L^{2}(\Omega)\right)$. 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 $C_{\mathcal{H}^{1}_{t}(\Omega)}$ respectively.