Pullback attractors for nonclassical diffusion equations with a delay operator

TL;DR

This paper studies the long-term behavior of solutions to nonclassical diffusion equations with delays, proving the existence of pullback attractors in time-dependent spaces under certain conditions.

Contribution

It establishes well-posedness and the existence of pullback attractors for nonclassical diffusion equations with delay operators in time-dependent spaces.

Findings

Proved well-posedness of solutions using Faedo-Galerkin method.

Established existence and regularity of pullback attractors.

Analyzed asymptotic behavior in time-dependent function spaces.

Abstract

In this paper, we consider the asymptotic behavior of weak solutions for nonclassical non-autonomous diffusion equations with a delay operator in time-dependent spaces when the nonlinear function $g$ satisfies subcritical exponent growth conditions, the delay operator $\varphi(t, u_t)$ contains some hereditary characteristics and the external force $k \in L_{l o c}^{2}\left(\mathbb{R} ; L^{2}(\Omega)\right)$. First, 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.