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

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