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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

B. Wang, Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245.  doi: 10.1016/j.jde.2005.01.003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

S. Zhou, Attractors for second order lattice dynamical systems, J. Differential Equations, 179 (2002), 605-624.  doi: 10.1006/jdeq.2001.4032.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

B. Wang, Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245.  doi: 10.1016/j.jde.2005.01.003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

S. Zhou, Attractors for second order lattice dynamical systems, J. Differential Equations, 179 (2002), 605-624.  doi: 10.1006/jdeq.2001.4032.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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