Existence and upper semicontinuity of time-dependent attractors for the non-autonomous nonlocal diffusion equations

TL;DR

This paper proves the existence of time-dependent attractors for non-autonomous nonlocal diffusion equations and demonstrates their upper semicontinuity as a parameter approaches zero, enhancing understanding of long-term dynamics.

Contribution

It establishes the existence of minimal time-dependent pullback attractors and their upper semicontinuity for nonlocal diffusion equations in time-dependent spaces.

Findings

Existence of minimal time-dependent pullback attractors.

Upper semicontinuity of attractors as a parameter tends to zero.

Convergence of time-dependent attractors to the global attractor.

Abstract

In this paper, under some appropriate assumptions, we prove the existence of the minimal time-dependent pullback $\mathcal D_{\sigma}^{\mathcal{H}_{t}}$-attractors ${\mathcal{A}}_{\mathcal D_{\sigma}^{\mathcal{H}_{t}}}$ for the non-autonomous nonlocal diffusion equations in time-dependent space $\mathcal{H}_{t}(\Omega)$. Next, in same phase space, using a priori estimate and energy methods we establish the existence of time-dependent pullback attractors $\left\{A_{\xi}(t)\right\}_{t \in \mathbb R}$ and the upper semicontinuity of $\left\{A_{\xi}(t)\right\}_{t \in \mathbb R}$ and the global attractor $A$ of the equation with $\xi = 0$, that is, $$ \lim_{\xi \rightarrow 0^{+}} \operatorname{dist}_{\mathcal H_{t}}\left(A_{\xi}(t), A \right)=0. $$