Upper semicontinuity of pullback attractors for non-autonomous lattice systems under singular perturbations

Consider the second order nonautonomous lattice systemswith singular perturbations

$ \begin{equation*} \epsilon \ddot{u}_{m}+\dot{u}_{m}+(Au)_{m}+\lambda_{m}u_{m}+f_{m}(u_{j}|j\in I_{mq}) = g_{m}(t),\; \; m\in \mathbb{Z}^{k},\; \; \epsilon>0 \tag{*} \label{0} \end{equation*} $

and the first order nonautonomous lattice systems

$ \begin{equation*} \dot{u}_{m}+(Au)_{m}+\lambda _{m}u_{m}+f_{m}(u_{j}|j∈I_{mq}) = g_{m}(t),\; \; m\in \mathbb{Z}^{k}. \tag{**} \label{00} \end{equation*} $

Under certain conditions, there are pullback attractors $ \{\mathcal{A}_{\epsilon }(t)\subset \ell ^{2}\times \ell ^{2}\}_{t\in \mathbb{R}} $ and $ \{\mathcal{A}(t)\subset \ell ^{2}\}_{t\in \mathbb{R}} $ for systems (*)and (**), respectively. In this paper, we mainly consider the uppersemicontinuity of attractors $ \mathcal{A}_{\epsilon }(t)\subset \ell^{2}\times \ell ^{2} $, $ t\in \mathbb{R} $, with respect to the coefficient $ \epsilon $ of second derivative term under Hausdorff semidistance. First, we studythe relationship between $ \mathcal{A}_{\epsilon }(t) $ and $ \mathcal{A}(t) $ when $ \epsilon \rightarrow 0^{+} $. We construct a family of compact sets $ \mathcal{A}_{0}(t)\subset \ell ^{2}\times \ell ^{2} $, $ t\in \mathbb{R} $ such that $ \mathcal{A}(t) $ is naturally embedded into $ \mathcal{A}_{0}(t) $ as the firstcomponent, and prove that $ \mathcal{A}_{\epsilon }(t) $ can enter anyneighborhood of $ \mathcal{A}_{0}(t) $ when $ \epsilon $ is small enough. Thenfor $ \epsilon _{0}>0 $, we prove that $ \mathcal{A}_{\epsilon }(t) $ can enterany neighborhood of $ \mathcal{A}_{\epsilon _{0}}(t) $ when $ \epsilon\rightarrow \epsilon _{0} $. Finally, we consider the existence andexponentially attraction of the singleton pullback attractors of systems (*)-(**).