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

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 ${\mathcal{H}}_t(\mathrm{\Omega })$. Next, in same phase space, using a priori estimate and energy methods we establish the existence of time-dependent pullback attractors ${\{A_{\xi }(t)\}}_{t\in \mathbb{R}}$ and the upper semicontinuity of ${\{A_{\xi }(t)\}}_{t\in \mathbb{R}}$ and the global attractor $A$ of the equation with $\xi =0$, that is, $$\underset{\xi \to 0^{+}}{\lim }{\mathrm{dist}}_{{\mathcal{H}}_t}(A_{\xi }(t),A)=0.$$