# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2021108
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## Upper semicontinuity of pullback attractors for non-autonomous lattice systems under singular perturbations

 College of Mathematics and Computer Science, Zhejiang Normal University, Jinhua, 321004, China

* Corresponding author: Shengfan Zhou

Received  November 2020 Revised  March 2021 Early access August 2021

Fund Project: 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 (*)-(**).
Citation: Na Lei, Shengfan Zhou. Upper semicontinuity of pullback attractors for non-autonomous lattice systems under singular perturbations. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021108
##### References:

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