November  2018, 23(9): 4021-4044. doi: 10.3934/dcdsb.2018122

Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations

1. 

College of Mathematical Physics and Electronic Information Engineering, Wenzhou University, Wenzhou 325035, China

2. 

Institute of Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland

* Corresponding author: Caidi Zhao

Received  June 2017 Revised  December 2017 Published  April 2018

In this article, we first provide a sufficient and necessary condition for the existence of a pullback-$ {\mathcal D} $ attractor for the process defined on a Hilbert space of infinite sequences. As an application, we investigate the non-autonomous discrete Klein-Gordon-Schrödinger system of equations, prove the existence of the pullback-$ {\mathcal D} $ attractor and then the existence of a unique family of invariant Borel probability measures associated with the considered system.

Citation: Caidi Zhao, Gang Xue, Grzegorz Łukaszewicz. Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 4021-4044. doi: 10.3934/dcdsb.2018122
References:
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P. W. BatesK. Lu and B. Wang, Attractors for lattice dynamical systems, Inter. J. Bifur. Chaos, 11 (2001), 143-153. doi: 10.1142/S0218127401002031. Google Scholar

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W.-J. Beyn and S. Yu. Pilyugin, Attractors of reaction diffusion systems on infinite lattices, J. Dyn. Differential Equations, 15 (2003), 485-515. doi: 10.1023/B:JODY.0000009745.41889.30. Google Scholar

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P. Biler, Attractors for the system of Schrödinger and Klein-Gordon equations with Yukawa Coupling, SIAM J. Math. Anal., 21 (1990), 1190-1212. doi: 10.1137/0521065. Google Scholar

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T. CaraballoG. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical system, Nonlinear Anal, 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111. Google Scholar

[6]

T. CaraballoP. E. Kloeden and J. Real, Invariant measures and statitical solutions of the globally modified Navier-Stokes equations, Discrete Cont. Dyn. Syst.-B, 10 (2008), 761-781. doi: 10.3934/dcdsb.2008.10.761. Google Scholar

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T. CaraballoF. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multipliative noise and non-Lipschitz nonlinearities, J. Differential Equations, 253 (2012), 667-693. doi: 10.1016/j.jde.2012.03.020. Google Scholar

[8]

T. CaraballoF. Morillas and J. Valero, On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems, Discrete Cont. Dyn. Syst.-B, 34 (2014), 51-77. Google Scholar

[9]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Springer, New York, 2013. Google Scholar

[10]

M. Chekroun and N. E. Glatt-Holtz, Invariant measures for dissipative dynamical systems: Abstract results and applications, Comm. Math. Phys., 316 (2012), 723-761. doi: 10.1007/s00220-012-1515-y. Google Scholar

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V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics. AMS Colloquium Publications, 49. AMS, Providence, R. I., 2002. Google Scholar

[12]

S. N. Chow and J. M. Paret, Pattern formation and spatial chaos in lattice dynamical systems, IEEE Trans. Circuits Syst., 42 (1995), 752-756. doi: 10.1109/81.473583. Google Scholar

[13]

S. N. ChowJ. M. Paret and E. S. Van Vleck, Pattern formation and spatial chaos in spatially discrete evolution equations, Random Comp. Dyn., 4 (1996), 109-178. Google Scholar

[14]

S. N. Chow, Lattice Dynamical Systems, Lecture Notes in Math., 1822 (2003) 1-102. Google Scholar

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L. FabinyP. Colet and R. Roy, Coherence and phase dynamics of spatially coupled solid-state lasers, Phys. Rev. A, 47 (1993), 4287-4296. doi: 10.1103/PhysRevA.47.4287. Google Scholar

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C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, Cambridge, 2001. Google Scholar

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I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations, Math. Japan., 24 (1979), 307-321. Google Scholar

[18]

J. García-LuengoP. Marín-Rubio and J. Real, Pullback attractors in $ V $ for non-autonomous 2D-Navier-Stokes equations and their tempered behavior, J. Differential Equations, 252 (2012), 4333-4356. doi: 10.1016/j.jde.2012.01.010. Google Scholar

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J. García-LuengoP. Marín-Rubio and J. Real, Pullback attractors for three-dimensional non-autonomous Navier-Stokes-Voigt equations, Nonlinearity, 25 (2012), 905-930. doi: 10.1088/0951-7715/25/4/905. Google Scholar

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B. Guo and Y. Li, Attractor for dissipative Klein-Gordon-Schrödinger equations in $ \textbf{R}^3 $, J. Differential Equations, 136 (1997), 356-377. doi: 10.1006/jdeq.1996.3242. Google Scholar

[21]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys and Monographs, AMS Colloquium Publications, 25. AMS, Providence, R. I., 1988. Google Scholar

[22]

X. HanW. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266. doi: 10.1016/j.jde.2010.10.018. Google Scholar

[23]

X. Han and P. Kloeden, Non-autonomous lattice systems with switching effects and delayed recovery, J. Differential Equations, 261 (2016), 2986-3009. doi: 10.1016/j.jde.2016.05.015. Google Scholar

[24]

R. Kapval, Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113-163. doi: 10.1007/BF01192578. Google Scholar

[25]

N. I. Karachalios and A. N. Yannacopoulos, Global existence and compact attractors for the discrete nonlinear Schrödinger equation, J. Differential Equations, 217 (2005), 88-123. doi: 10.1016/j.jde.2005.06.002. Google Scholar

[26]

J. P. Keener, Propagation and its failure in coupled systems of discret excitable cells, SIAM J. Appl. Math., 47 (1987), 556-572. doi: 10.1137/0147038. Google Scholar

[27]

P. E. KloedenP. Marín-Rubio and J. Real, Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations, Comm. Pure Appl. Anal., 8 (2009), 785-802. doi: 10.3934/cpaa.2009.8.785. Google Scholar

[28]

B. LevantF. Ramos and E. S. Titi, On the statistical properties of the 3D incompressible Navier-Stokes-Voigt model, Comm. Math. Sci., 8 (2010), 277-293. doi: 10.4310/CMS.2010.v8.n1.a14. Google Scholar

[29]

Y. Li and B. Guo, Asymptotic smoothing effect for weakly dissipative Klein-Gordon-Schrödinger equations, J. Math. Anal. Appl., 282 (2003), 256-265. doi: 10.1016/S0022-247X(03)00152-5. Google Scholar

[30]

Y. LiS. Wang and H. Wu, Pullback attractors for non-autonomous reaction-diffusion equations in $ L^p $, Appl. Math. Comp., 207 (2009), 373-379. doi: 10.1016/j.amc.2008.10.065. Google Scholar

[31]

X. LiW. Shen and C. Sun, Invariant measures for complex-valued dissipative dynamical systems and applications, Discrete Cont. Dyn. Syst.-B, 22 (2017), 2427-2466. Google Scholar

[32]

K. Lu and B. Wang, Attractor for dissipative Klein-Gordon-Schrödinger equations in $ \textbf{R}^3 $, J. Differential Equations, 170 (2001), 281-316. Google Scholar

[33]

K. Lu and B. Wang, Upper semicontinuity of attractors for the Klein-Gordon-Schrödinger equations, Inter. J. Bifur. Choas, 15 (2005), 157-168. doi: 10.1142/S0218127405012077. Google Scholar

[34]

G. Łukaszewicz, Pullback attractors and statistical solutions for 2-D Navier-Stokes equations, Discrete Cont. Dyn. Syst.-B, 9 (2008), 643-659. doi: 10.3934/dcdsb.2008.9.643. Google Scholar

[35]

G. ŁukaszewiczJ. Real and J. C. Robinson, Invariant measures for dissipative dynamical systems and generalised Banach limits, J. Dyn. Differential Equations, 23 (2011), 225-250. doi: 10.1007/s10884-011-9213-6. Google Scholar

[36]

G. Łukaszewicz and J. C. Robinson, Invariant measures for non-autonomous dissipative dynamical systems, Discrete Cont. Dyn. Syst.-A, 34 (2014), 4211-4222. doi: 10.3934/dcds.2014.34.4211. Google Scholar

[37]

Q. MaS. Wang and C. Zhong, Necessary and sufficient conditions for the existence of global attractor for semigroup and application, Indiana Univ. Math. J., 51 (2002), 1541-1559. doi: 10.1512/iumj.2002.51.2255. Google Scholar

[38]

J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge, Cambridge University Press, 2001. Google Scholar

[39]

G. Sell and Y. You, Dynamics of Evolutionary Equations, New York, Springer, 2002. Google Scholar

[40]

H. Song, Pullback attractors of non-autonomous reaction-diffusion equations in $ H^1_0 $, J. Differential Equations, 249 (2010), 2357-2376. doi: 10.1016/j.jde.2010.07.034. Google Scholar

[41]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, Berlin, 1988. Google Scholar

[42]

B. Wang and H. Lange, Attractors for the Klein-Gordon-Schrödinger equations, J. Math. Phys., 40 (1999), 2445-2457. doi: 10.1063/1.532875. Google Scholar

[43]

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

[44]

X. Wang, Upper-semicontinuity of stationary statistical properties of dissipative systems, Discrete Cont. Dyn. Syst.-A, 23 (2009), 521-540. Google Scholar

[45]

Y. WangC. Zhong and S. Zhou, Pullback attractors of non-autonomous dynamical systems, Discrete Cont. Dyn. Syst.-A, 16 (2006), 587-614. doi: 10.3934/dcds.2006.16.587. Google Scholar

[46]

Y. Wang and K. Bai, Pullback attractors for a class of nonlinear lattices with delays, Discrete Cont. Dyn. Syst.-A, 20 (2015), 1213-1230. doi: 10.3934/dcdsb.2015.20.1213. Google Scholar

[47]

X. Xiang and S. Zhou, Attractors for second order non-autonomous lattice system with dispersive term, Topological Meth. Nonl. Anal., 46 (2015), 893-914. Google Scholar

[48]

C. Zhao and S. Zhou, Compact kernel sections for nonautonomous Klein-Gordon-Schrödinger equations on infinite lattices, J. Math. Anal. Appl., 332 (2007), 32-56. doi: 10.1016/j.jmaa.2006.10.002. Google Scholar

[49]

C. Zhao and L. Yang, Pullback attractor and invariant measure for the globally modified Navier-Stokes equations, Comm. Math. Sci., 15 (2017), 1565-1580. doi: 10.4310/CMS.2017.v15.n6.a4. Google Scholar

[50]

X. Zhao and S. Zhou, Kernel sections for processes and nonautonomous lattice systems, Discrete Cont. Dyn. Syst.-B, 9 (2008), 763-785. doi: 10.3934/dcdsb.2008.9.763. Google Scholar

[51]

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

[52]

S. Zhou, Attractors for first order dissipative lattice dynamical systems, Physica D, 178 (2003), 51-61. doi: 10.1016/S0167-2789(02)00807-2. Google Scholar

[53]

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

[54]

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

[55]

S. Zhou, Random exponential attractor for cocycle and application to non-autonomous stochastic lattice systems with multiplicative white noise, J. Differential Equations, 263 (2017), 2247-2279. doi: 10.1016/j.jde.2017.03.044. Google Scholar

show all references

References:
[1]

A. Y. Abdallah, Uniform exponential attractor for first order non-autonomous lattice dynamical systems, J. Differential Equations, 251 (2011), 1489-1504. doi: 10.1016/j.jde.2011.05.030. Google Scholar

[2]

P. W. BatesK. Lu and B. Wang, Attractors for lattice dynamical systems, Inter. J. Bifur. Chaos, 11 (2001), 143-153. doi: 10.1142/S0218127401002031. Google Scholar

[3]

W.-J. Beyn and S. Yu. Pilyugin, Attractors of reaction diffusion systems on infinite lattices, J. Dyn. Differential Equations, 15 (2003), 485-515. doi: 10.1023/B:JODY.0000009745.41889.30. Google Scholar

[4]

P. Biler, Attractors for the system of Schrödinger and Klein-Gordon equations with Yukawa Coupling, SIAM J. Math. Anal., 21 (1990), 1190-1212. doi: 10.1137/0521065. Google Scholar

[5]

T. CaraballoG. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical system, Nonlinear Anal, 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111. Google Scholar

[6]

T. CaraballoP. E. Kloeden and J. Real, Invariant measures and statitical solutions of the globally modified Navier-Stokes equations, Discrete Cont. Dyn. Syst.-B, 10 (2008), 761-781. doi: 10.3934/dcdsb.2008.10.761. Google Scholar

[7]

T. CaraballoF. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multipliative noise and non-Lipschitz nonlinearities, J. Differential Equations, 253 (2012), 667-693. doi: 10.1016/j.jde.2012.03.020. Google Scholar

[8]

T. CaraballoF. Morillas and J. Valero, On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems, Discrete Cont. Dyn. Syst.-B, 34 (2014), 51-77. Google Scholar

[9]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Springer, New York, 2013. Google Scholar

[10]

M. Chekroun and N. E. Glatt-Holtz, Invariant measures for dissipative dynamical systems: Abstract results and applications, Comm. Math. Phys., 316 (2012), 723-761. doi: 10.1007/s00220-012-1515-y. Google Scholar

[11]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics. AMS Colloquium Publications, 49. AMS, Providence, R. I., 2002. Google Scholar

[12]

S. N. Chow and J. M. Paret, Pattern formation and spatial chaos in lattice dynamical systems, IEEE Trans. Circuits Syst., 42 (1995), 752-756. doi: 10.1109/81.473583. Google Scholar

[13]

S. N. ChowJ. M. Paret and E. S. Van Vleck, Pattern formation and spatial chaos in spatially discrete evolution equations, Random Comp. Dyn., 4 (1996), 109-178. Google Scholar

[14]

S. N. Chow, Lattice Dynamical Systems, Lecture Notes in Math., 1822 (2003) 1-102. Google Scholar

[15]

L. FabinyP. Colet and R. Roy, Coherence and phase dynamics of spatially coupled solid-state lasers, Phys. Rev. A, 47 (1993), 4287-4296. doi: 10.1103/PhysRevA.47.4287. Google Scholar

[16]

C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, Cambridge, 2001. Google Scholar

[17]

I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations, Math. Japan., 24 (1979), 307-321. Google Scholar

[18]

J. García-LuengoP. Marín-Rubio and J. Real, Pullback attractors in $ V $ for non-autonomous 2D-Navier-Stokes equations and their tempered behavior, J. Differential Equations, 252 (2012), 4333-4356. doi: 10.1016/j.jde.2012.01.010. Google Scholar

[19]

J. García-LuengoP. Marín-Rubio and J. Real, Pullback attractors for three-dimensional non-autonomous Navier-Stokes-Voigt equations, Nonlinearity, 25 (2012), 905-930. doi: 10.1088/0951-7715/25/4/905. Google Scholar

[20]

B. Guo and Y. Li, Attractor for dissipative Klein-Gordon-Schrödinger equations in $ \textbf{R}^3 $, J. Differential Equations, 136 (1997), 356-377. doi: 10.1006/jdeq.1996.3242. Google Scholar

[21]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys and Monographs, AMS Colloquium Publications, 25. AMS, Providence, R. I., 1988. Google Scholar

[22]

X. HanW. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266. doi: 10.1016/j.jde.2010.10.018. Google Scholar

[23]

X. Han and P. Kloeden, Non-autonomous lattice systems with switching effects and delayed recovery, J. Differential Equations, 261 (2016), 2986-3009. doi: 10.1016/j.jde.2016.05.015. Google Scholar

[24]

R. Kapval, Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113-163. doi: 10.1007/BF01192578. Google Scholar

[25]

N. I. Karachalios and A. N. Yannacopoulos, Global existence and compact attractors for the discrete nonlinear Schrödinger equation, J. Differential Equations, 217 (2005), 88-123. doi: 10.1016/j.jde.2005.06.002. Google Scholar

[26]

J. P. Keener, Propagation and its failure in coupled systems of discret excitable cells, SIAM J. Appl. Math., 47 (1987), 556-572. doi: 10.1137/0147038. Google Scholar

[27]

P. E. KloedenP. Marín-Rubio and J. Real, Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations, Comm. Pure Appl. Anal., 8 (2009), 785-802. doi: 10.3934/cpaa.2009.8.785. Google Scholar

[28]

B. LevantF. Ramos and E. S. Titi, On the statistical properties of the 3D incompressible Navier-Stokes-Voigt model, Comm. Math. Sci., 8 (2010), 277-293. doi: 10.4310/CMS.2010.v8.n1.a14. Google Scholar

[29]

Y. Li and B. Guo, Asymptotic smoothing effect for weakly dissipative Klein-Gordon-Schrödinger equations, J. Math. Anal. Appl., 282 (2003), 256-265. doi: 10.1016/S0022-247X(03)00152-5. Google Scholar

[30]

Y. LiS. Wang and H. Wu, Pullback attractors for non-autonomous reaction-diffusion equations in $ L^p $, Appl. Math. Comp., 207 (2009), 373-379. doi: 10.1016/j.amc.2008.10.065. Google Scholar

[31]

X. LiW. Shen and C. Sun, Invariant measures for complex-valued dissipative dynamical systems and applications, Discrete Cont. Dyn. Syst.-B, 22 (2017), 2427-2466. Google Scholar

[32]

K. Lu and B. Wang, Attractor for dissipative Klein-Gordon-Schrödinger equations in $ \textbf{R}^3 $, J. Differential Equations, 170 (2001), 281-316. Google Scholar

[33]

K. Lu and B. Wang, Upper semicontinuity of attractors for the Klein-Gordon-Schrödinger equations, Inter. J. Bifur. Choas, 15 (2005), 157-168. doi: 10.1142/S0218127405012077. Google Scholar

[34]

G. Łukaszewicz, Pullback attractors and statistical solutions for 2-D Navier-Stokes equations, Discrete Cont. Dyn. Syst.-B, 9 (2008), 643-659. doi: 10.3934/dcdsb.2008.9.643. Google Scholar

[35]

G. ŁukaszewiczJ. Real and J. C. Robinson, Invariant measures for dissipative dynamical systems and generalised Banach limits, J. Dyn. Differential Equations, 23 (2011), 225-250. doi: 10.1007/s10884-011-9213-6. Google Scholar

[36]

G. Łukaszewicz and J. C. Robinson, Invariant measures for non-autonomous dissipative dynamical systems, Discrete Cont. Dyn. Syst.-A, 34 (2014), 4211-4222. doi: 10.3934/dcds.2014.34.4211. Google Scholar

[37]

Q. MaS. Wang and C. Zhong, Necessary and sufficient conditions for the existence of global attractor for semigroup and application, Indiana Univ. Math. J., 51 (2002), 1541-1559. doi: 10.1512/iumj.2002.51.2255. Google Scholar

[38]

J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge, Cambridge University Press, 2001. Google Scholar

[39]

G. Sell and Y. You, Dynamics of Evolutionary Equations, New York, Springer, 2002. Google Scholar

[40]

H. Song, Pullback attractors of non-autonomous reaction-diffusion equations in $ H^1_0 $, J. Differential Equations, 249 (2010), 2357-2376. doi: 10.1016/j.jde.2010.07.034. Google Scholar

[41]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, Berlin, 1988. Google Scholar

[42]

B. Wang and H. Lange, Attractors for the Klein-Gordon-Schrödinger equations, J. Math. Phys., 40 (1999), 2445-2457. doi: 10.1063/1.532875. Google Scholar

[43]

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

[44]

X. Wang, Upper-semicontinuity of stationary statistical properties of dissipative systems, Discrete Cont. Dyn. Syst.-A, 23 (2009), 521-540. Google Scholar

[45]

Y. WangC. Zhong and S. Zhou, Pullback attractors of non-autonomous dynamical systems, Discrete Cont. Dyn. Syst.-A, 16 (2006), 587-614. doi: 10.3934/dcds.2006.16.587. Google Scholar

[46]

Y. Wang and K. Bai, Pullback attractors for a class of nonlinear lattices with delays, Discrete Cont. Dyn. Syst.-A, 20 (2015), 1213-1230. doi: 10.3934/dcdsb.2015.20.1213. Google Scholar

[47]

X. Xiang and S. Zhou, Attractors for second order non-autonomous lattice system with dispersive term, Topological Meth. Nonl. Anal., 46 (2015), 893-914. Google Scholar

[48]

C. Zhao and S. Zhou, Compact kernel sections for nonautonomous Klein-Gordon-Schrödinger equations on infinite lattices, J. Math. Anal. Appl., 332 (2007), 32-56. doi: 10.1016/j.jmaa.2006.10.002. Google Scholar

[49]

C. Zhao and L. Yang, Pullback attractor and invariant measure for the globally modified Navier-Stokes equations, Comm. Math. Sci., 15 (2017), 1565-1580. doi: 10.4310/CMS.2017.v15.n6.a4. Google Scholar

[50]

X. Zhao and S. Zhou, Kernel sections for processes and nonautonomous lattice systems, Discrete Cont. Dyn. Syst.-B, 9 (2008), 763-785. doi: 10.3934/dcdsb.2008.9.763. Google Scholar

[51]

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

[52]

S. Zhou, Attractors for first order dissipative lattice dynamical systems, Physica D, 178 (2003), 51-61. doi: 10.1016/S0167-2789(02)00807-2. Google Scholar

[53]

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

[54]

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

[55]

S. Zhou, Random exponential attractor for cocycle and application to non-autonomous stochastic lattice systems with multiplicative white noise, J. Differential Equations, 263 (2017), 2247-2279. doi: 10.1016/j.jde.2017.03.044. Google Scholar

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