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

Abstract

<p style='text-indent:20px;'>Consider the second order nonautonomous lattice systemswith singular perturbations</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \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*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>and the first order nonautonomous lattice systems</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \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*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>Under certain conditions, there are pullback attractors <inline-formula><tex-math id="M1">\begin{document}$ \{\mathcal{A}_{\epsilon }(t)\subset \ell ^{2}\times \ell ^{2}\}_{t\in \mathbb{R}} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \{\mathcal{A}(t)\subset \ell ^{2}\}_{t\in \mathbb{R}} $\end{document}</tex-math></inline-formula> for systems (*)and (**), respectively. In this paper, we mainly consider the uppersemicontinuity of attractors <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{A}_{\epsilon }(t)\subset \ell^{2}\times \ell ^{2} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ t\in \mathbb{R} $\end{document}</tex-math></inline-formula>, with respect to the coefficient <inline-formula><tex-math id="M5">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> of second derivative term under Hausdorff semidistance. First, we studythe relationship between <inline-formula><tex-math id="M6">\begin{document}$ \mathcal{A}_{\epsilon }(t) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ \mathcal{A}(t) $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M8">\begin{document}$ \epsilon \rightarrow 0^{+} $\end{document}</tex-math></inline-formula>. We construct a family of compact sets <inline-formula><tex-math id="M9">\begin{document}$ \mathcal{A}_{0}(t)\subset \ell ^{2}\times \ell ^{2} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ t\in \mathbb{R} $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M11">\begin{document}$ \mathcal{A}(t) $\end{document}</tex-math></inline-formula> is naturally embedded into <inline-formula><tex-math id="M12">\begin{document}$ \mathcal{A}_{0}(t) $\end{document}</tex-math></inline-formula> as the firstcomponent, and prove that <inline-formula><tex-math id="M13">\begin{document}$ \mathcal{A}_{\epsilon }(t) $\end{document}</tex-math></inline-formula> can enter anyneighborhood of <inline-formula><tex-math id="M14">\begin{document}$ \mathcal{A}_{0}(t) $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M15">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> is small enough. Thenfor <inline-formula><tex-math id="M16">\begin{document}$ \epsilon _{0}>0 $\end{document}</tex-math></inline-formula>, we prove that <inline-formula><tex-math id="M17">\begin{document}$ \mathcal{A}_{\epsilon }(t) $\end{document}</tex-math></inline-formula> can enterany neighborhood of <inline-formula><tex-math id="M18">\begin{document}$ \mathcal{A}_{\epsilon _{0}}(t) $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M19">\begin{document}$ \epsilon\rightarrow \epsilon _{0} $\end{document}</tex-math></inline-formula>. Finally, we consider the existence andexponentially attraction of the singleton pullback attractors of systems (*)-(**).</p>