Article Contents
Article Contents

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

• * Corresponding author: Shengfan Zhou
The second author is supported by the National Natural Science Foundation of China under Grant Nos. 11871437 and 11971356
• 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 (*)-(**).

Mathematics Subject Classification: Primary: 37L50; Secondary: 35B40, 35Q55.

 Citation:

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