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

Na Lei; Shengfan Zhou

doi:10.3934/dcds.2021108

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January 2022, 42(1): 73-108. doi: 10.3934/dcds.2021108

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

Na Lei and Shengfan Zhou ^,

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

^* Corresponding author: Shengfan Zhou

Received November 2020 Revised March 2021 Published January 2022 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 (*)-(**).

Keywords: Nonautonomous lattice system, singular perturbations, pullback attractor, semicontinuity, exponential attraction.

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

Citation: Na Lei, Shengfan Zhou. Upper semicontinuity of pullback attractors for non-autonomous lattice systems under singular perturbations. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 73-108. doi: 10.3934/dcds.2021108