doi: 10.3934/dcds.2021108
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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:
[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. BatesK. 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. BatesK. 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. FreitasP. 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. BatesK. 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. BatesK. 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. FreitasP. 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

[1]

Ahmed Y. Abdallah. Upper semicontinuity of the attractor for a second order lattice dynamical system. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 899-916. doi: 10.3934/dcdsb.2005.5.899

[2]

Xuewei Ju, Desheng Li, Jinqiao Duan. Forward attraction of pullback attractors and synchronizing behavior of gradient-like systems with nonautonomous perturbations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1175-1197. doi: 10.3934/dcdsb.2019011

[3]

Yonghai Wang, Chengkui Zhong. Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 3189-3209. doi: 10.3934/dcds.2013.33.3189

[4]

Linfang Liu, Xianlong Fu, Yuncheng You. Pullback attractor in $H^{1}$ for nonautonomous stochastic reaction-diffusion equations on $\mathbb{R}^n$. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3629-3651. doi: 10.3934/dcdsb.2017143

[5]

Yangrong Li, Lianbing She, Jinyan Yin. Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1535-1557. doi: 10.3934/dcdsb.2018058

[6]

Mustapha Yebdri. Existence of $ \mathcal{D}- $pullback attractor for an infinite dimensional dynamical system. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021036

[7]

Yonghai Wang. On the upper semicontinuity of pullback attractors with applications to plate equations. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1653-1673. doi: 10.3934/cpaa.2010.9.1653

[8]

Yejuan Wang, Chengkui Zhong, Shengfan Zhou. Pullback attractors of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems, 2006, 16 (3) : 587-614. doi: 10.3934/dcds.2006.16.587

[9]

José A. Langa, Alain Miranville, José Real. Pullback exponential attractors. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1329-1357. doi: 10.3934/dcds.2010.26.1329

[10]

Jin Zhang, Peter E. Kloeden, Meihua Yang, Chengkui Zhong. Global exponential κ-dissipative semigroups and exponential attraction. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3487-3502. doi: 10.3934/dcds.2017148

[11]

Zhijian Yang, Yanan Li. Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4899-4912. doi: 10.3934/dcdsb.2019036

[12]

Matheus C. Bortolan, José Manuel Uzal. Upper and weak-lower semicontinuity of pullback attractors to impulsive evolution processes. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3667-3692. doi: 10.3934/dcdsb.2020252

[13]

Christian Lax, Sebastian Walcher. Singular perturbations and scaling. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 1-29. doi: 10.3934/dcdsb.2019170

[14]

Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Invariant manifolds as pullback attractors of nonautonomous differential equations. Discrete & Continuous Dynamical Systems, 2006, 15 (2) : 579-596. doi: 10.3934/dcds.2006.15.579

[15]

Xiao-Qiang Zhao, Shengfan Zhou. Kernel sections for processes and nonautonomous lattice systems. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 763-785. doi: 10.3934/dcdsb.2008.9.763

[16]

María Astudillo, Marcelo M. Cavalcanti. On the upper semicontinuity of the global attractor for a porous medium type problem with large diffusion. Evolution Equations & Control Theory, 2017, 6 (1) : 1-13. doi: 10.3934/eect.2017001

[17]

Zvi Artstein. Invariance principle in the singular perturbations limit. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3653-3666. doi: 10.3934/dcdsb.2018309

[18]

Sana Netchaoui, Mohamed Ali Hammami, Tomás Caraballo. Pullback exponential attractors for differential equations with delay. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1345-1358. doi: 10.3934/dcdss.2020367

[19]

Mohamed Ali Hammami, Lassaad Mchiri, Sana Netchaoui, Stefanie Sonner. Pullback exponential attractors for differential equations with variable delays. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 301-319. doi: 10.3934/dcdsb.2019183

[20]

Ahmed Y. Abdallah. Exponential attractors for second order lattice dynamical systems. Communications on Pure & Applied Analysis, 2009, 8 (3) : 803-813. doi: 10.3934/cpaa.2009.8.803

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (66)
  • HTML views (111)
  • Cited by (0)

Other articles
by authors

[Back to Top]