-
Previous Article
Global existence of solutions to reaction diffusion systems with mass transport type boundary conditions on an evolving domain
- DCDS Home
- This Issue
-
Next Article
On the number of positive solutions to an indefinite parameter-dependent Neumann problem
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 |
$ \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*} $ |
$ \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*} $ |
$ \{\mathcal{A}_{\epsilon }(t)\subset \ell ^{2}\times \ell ^{2}\}_{t\in \mathbb{R}} $ |
$ \{\mathcal{A}(t)\subset \ell ^{2}\}_{t\in \mathbb{R}} $ |
$ \mathcal{A}_{\epsilon }(t)\subset \ell^{2}\times \ell ^{2} $ |
$ t\in \mathbb{R} $ |
$ \epsilon $ |
$ \mathcal{A}_{\epsilon }(t) $ |
$ \mathcal{A}(t) $ |
$ \epsilon \rightarrow 0^{+} $ |
$ \mathcal{A}_{0}(t)\subset \ell ^{2}\times \ell ^{2} $ |
$ t\in \mathbb{R} $ |
$ \mathcal{A}(t) $ |
$ \mathcal{A}_{0}(t) $ |
$ \mathcal{A}_{\epsilon }(t) $ |
$ \mathcal{A}_{0}(t) $ |
$ \epsilon $ |
$ \epsilon _{0}>0 $ |
$ \mathcal{A}_{\epsilon }(t) $ |
$ \mathcal{A}_{\epsilon _{0}}(t) $ |
$ \epsilon\rightarrow \epsilon _{0} $ |
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. |
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. |
[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. |
[1] |
Ahmed Y. Abdallah. Upper semicontinuity of the attractor for a second order lattice dynamical system. Discrete and 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 and 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 and Continuous Dynamical Systems, 2013, 33 (7) : 3189-3209. doi: 10.3934/dcds.2013.33.3189 |
[4] |
Xiaoying Han, Peter E. Kloeden. Pullback and forward dynamics of nonautonomous Laplacian lattice systems on weighted spaces. Discrete and Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021143 |
[5] |
Linfang Liu, Xianlong Fu, Yuncheng You. Pullback attractor in $H^{1}$ for nonautonomous stochastic reaction-diffusion equations on $\mathbb{R}^n$. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3629-3651. doi: 10.3934/dcdsb.2017143 |
[6] |
Yangrong Li, Lianbing She, Jinyan Yin. Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1535-1557. doi: 10.3934/dcdsb.2018058 |
[7] |
Mustapha Yebdri. Existence of $ \mathcal{D}- $pullback attractor for an infinite dimensional dynamical system. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 167-198. doi: 10.3934/dcdsb.2021036 |
[8] |
Yonghai Wang. On the upper semicontinuity of pullback attractors with applications to plate equations. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1653-1673. doi: 10.3934/cpaa.2010.9.1653 |
[9] |
Yejuan Wang, Chengkui Zhong, Shengfan Zhou. Pullback attractors of nonautonomous dynamical systems. Discrete and Continuous Dynamical Systems, 2006, 16 (3) : 587-614. doi: 10.3934/dcds.2006.16.587 |
[10] |
José A. Langa, Alain Miranville, José Real. Pullback exponential attractors. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1329-1357. doi: 10.3934/dcds.2010.26.1329 |
[11] |
Jin Zhang, Peter E. Kloeden, Meihua Yang, Chengkui Zhong. Global exponential κ-dissipative semigroups and exponential attraction. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3487-3502. doi: 10.3934/dcds.2017148 |
[12] |
Zhijian Yang, Yanan Li. Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4899-4912. doi: 10.3934/dcdsb.2019036 |
[13] |
Matheus C. Bortolan, José Manuel Uzal. Upper and weak-lower semicontinuity of pullback attractors to impulsive evolution processes. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3667-3692. doi: 10.3934/dcdsb.2020252 |
[14] |
Christian Lax, Sebastian Walcher. Singular perturbations and scaling. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 1-29. doi: 10.3934/dcdsb.2019170 |
[15] |
Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Invariant manifolds as pullback attractors of nonautonomous differential equations. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 579-596. doi: 10.3934/dcds.2006.15.579 |
[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 and Control Theory, 2017, 6 (1) : 1-13. doi: 10.3934/eect.2017001 |
[17] |
Xiao-Qiang Zhao, Shengfan Zhou. Kernel sections for processes and nonautonomous lattice systems. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 763-785. doi: 10.3934/dcdsb.2008.9.763 |
[18] |
Zvi Artstein. Invariance principle in the singular perturbations limit. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3653-3666. doi: 10.3934/dcdsb.2018309 |
[19] |
Sana Netchaoui, Mohamed Ali Hammami, Tomás Caraballo. Pullback exponential attractors for differential equations with delay. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1345-1358. doi: 10.3934/dcdss.2020367 |
[20] |
Mohamed Ali Hammami, Lassaad Mchiri, Sana Netchaoui, Stefanie Sonner. Pullback exponential attractors for differential equations with variable delays. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 301-319. doi: 10.3934/dcdsb.2019183 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]