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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

 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 (*)-(**).
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
##### References:
 [1] A. Y. Abdallah and R. T. Wannan, Second order non-autonomous lattice systems and their uniform attractors, Commun. Pure Appl. Anal., 18 (2019), 1827-1846.  doi: 10.3934/cpaa.2019085. [2] P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 143-153.  doi: 10.1142/S0218127401002031. [3] P. W. Bates, K. Lu and B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Phys. D, 289 (2014), 32-50.  doi: 10.1016/j.physd.2014.08.004. [4] A. Carvalho, J. Langa and J. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4. [5] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, Vol. 49, American Mathematical Society, Providence, RI, 2002. doi: 10.1090/coll/049. [6] M. M. Freitas, P. Kalita and J. A. Langa, Continuity of non-autonomous attractors for hyperbolic perturbation of parabolic equations, J. Differential Equations, 264 (2018), 1886-1945.  doi: 10.1016/j.jde.2017.10.007. [7] J. K. Hale and G. Raugel, Upper semicontinuity of the attractors for a singularly perturbed hyperbolic equation, J. Differential Equations, 73 (1988), 197-214.  doi: 10.1016/0022-0396(88)90104-0. [8] B. Wang, Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245.  doi: 10.1016/j.jde.2005.01.003. [9] B. Wang, Asymptotic behavior of non-autonomous lattice systems, J. Math. Anal. Appl., 331 (2007), 121-136.  doi: 10.1016/j.jmaa.2006.08.070. [10] B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with determinstic non-autonomous, Stoch. Dyn., 14 (2014), 1450009. doi: 10.1142/s0219493714500099. [11] J. Wang and A. Gu, Existence of backwards-compact pullback attractors for non-autonomous lattice dynamical systems, J. Difference Equ. Appl., 22 (2016), 1906–1911. doi: 10.1080/10236198.2016.1254205. [12] C. Zhao and S. Zhou, Upper semicontinuity of attractors for lattice systems under singularly perturbations, Nonlinear Anal., 72 (2010), 2149-2158.  doi: 10.1016/j.na.2009.09.001. [13] X. Zhao and S. Zhou, Kernel sections for processes and nonautonomous lattice systems, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 763-785.  doi: 10.3934/dcdsb.2008.9.763. [14] S. Zhou, Attractors for second order lattice dynamical systems, J. Differential Equations, 179 (2002), 605-624.  doi: 10.1006/jdeq.2001.4032. [15] S. Zhou, Attractors for first order disspative lattice dynamical systems, Phys. D, 178 (2003), 51-61.  doi: 10.1016/S0167-2789(02)00807-2. [16] S. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations, 200 (2004), 342-368.  doi: 10.1016/j.jde.2004.02.005. [17] S. Zhou, Upper-semicontinuity of attractors for random lattice systems perturbed by small white noises, Nonlinear Anal., 75 (2012), 2793-2805.  doi: 10.1016/j.na.2011.11.022. [18] S. Zhou and W. Shi, Attractors and dimension of dissipative lattice systems, J. Differential Equations, 224 (2006), 172-204.  doi: 10.1016/j.jde.2005.06.024.

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##### References:
 [1] A. Y. Abdallah and R. T. Wannan, Second order non-autonomous lattice systems and their uniform attractors, Commun. Pure Appl. Anal., 18 (2019), 1827-1846.  doi: 10.3934/cpaa.2019085. [2] P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 143-153.  doi: 10.1142/S0218127401002031. [3] P. W. Bates, K. Lu and B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Phys. D, 289 (2014), 32-50.  doi: 10.1016/j.physd.2014.08.004. [4] A. Carvalho, J. Langa and J. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4. [5] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, Vol. 49, American Mathematical Society, Providence, RI, 2002. doi: 10.1090/coll/049. [6] M. M. Freitas, P. Kalita and J. A. Langa, Continuity of non-autonomous attractors for hyperbolic perturbation of parabolic equations, J. Differential Equations, 264 (2018), 1886-1945.  doi: 10.1016/j.jde.2017.10.007. [7] J. K. Hale and G. Raugel, Upper semicontinuity of the attractors for a singularly perturbed hyperbolic equation, J. Differential Equations, 73 (1988), 197-214.  doi: 10.1016/0022-0396(88)90104-0. [8] B. Wang, Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245.  doi: 10.1016/j.jde.2005.01.003. [9] B. Wang, Asymptotic behavior of non-autonomous lattice systems, J. Math. Anal. Appl., 331 (2007), 121-136.  doi: 10.1016/j.jmaa.2006.08.070. [10] B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with determinstic non-autonomous, Stoch. Dyn., 14 (2014), 1450009. doi: 10.1142/s0219493714500099. [11] J. Wang and A. Gu, Existence of backwards-compact pullback attractors for non-autonomous lattice dynamical systems, J. Difference Equ. Appl., 22 (2016), 1906–1911. doi: 10.1080/10236198.2016.1254205. [12] C. Zhao and S. Zhou, Upper semicontinuity of attractors for lattice systems under singularly perturbations, Nonlinear Anal., 72 (2010), 2149-2158.  doi: 10.1016/j.na.2009.09.001. [13] X. Zhao and S. Zhou, Kernel sections for processes and nonautonomous lattice systems, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 763-785.  doi: 10.3934/dcdsb.2008.9.763. [14] S. Zhou, Attractors for second order lattice dynamical systems, J. Differential Equations, 179 (2002), 605-624.  doi: 10.1006/jdeq.2001.4032. [15] S. Zhou, Attractors for first order disspative lattice dynamical systems, Phys. D, 178 (2003), 51-61.  doi: 10.1016/S0167-2789(02)00807-2. [16] S. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations, 200 (2004), 342-368.  doi: 10.1016/j.jde.2004.02.005. [17] S. Zhou, Upper-semicontinuity of attractors for random lattice systems perturbed by small white noises, Nonlinear Anal., 75 (2012), 2793-2805.  doi: 10.1016/j.na.2011.11.022. [18] S. Zhou and W. Shi, Attractors and dimension of dissipative lattice systems, J. Differential Equations, 224 (2006), 172-204.  doi: 10.1016/j.jde.2005.06.024.
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