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

Previous Article
Asymptotic decay for the classical solution of the chemotaxis system with fractional Laplacian in high dimensions
DCDS-B Home
This Issue
Next Article
Analysis of a stage-structured dengue model

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

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

Caidi Zhao ^1,, , Gang Xue ^1, and Grzegorz Łukaszewicz ^2,

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.

Keywords: Non-autonomous lattice system, pullback attractor, invariant measures, discrete Klein-Gordon-Schrödinger equations.

Mathematics Subject Classification: 35B41, 35D99, 76F20.

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:

[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. Bates, K. 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. Caraballo, G. 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. Caraballo, P. 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. Caraballo, F. 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. Caraballo, F. 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. Chow, J. 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. Fabiny, P. 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-Luengo, P. 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-Luengo, P. 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. Han, W. 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. Kloeden, P. 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. Levant, F. 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. Li, S. 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. Li, W. 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. Łukaszewicz, J. 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. Ma, S. 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. Wang, C. 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. Bates, K. 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. Caraballo, G. 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. Caraballo, P. 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. Caraballo, F. 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. Caraballo, F. 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. Chow, J. 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. Fabiny, P. 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-Luengo, P. 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-Luengo, P. 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. Han, W. 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. Kloeden, P. 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. Levant, F. 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. Li, S. 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. Li, W. 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. Łukaszewicz, J. 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. Ma, S. 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. Wang, C. 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

[1]	Ahmed Y. Abdallah. Asymptotic behavior of the Klein-Gordon-Schrödinger lattice dynamical systems. Communications on Pure & Applied Analysis, 2006, 5 (1) : 55-69. doi: 10.3934/cpaa.2006.5.55
[2]	Salah Missaoui, Ezzeddine Zahrouni. Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system with cubic nonlinearities in $\mathbb{R}^2$. Communications on Pure & Applied Analysis, 2015, 14 (2) : 695-716. doi: 10.3934/cpaa.2015.14.695
[3]	Ji Shu. Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations driven by fractional Brownian motions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1587-1599. doi: 10.3934/dcdsb.2017077
[4]	Fábio Natali, Ademir Pastor. Orbital stability of periodic waves for the Klein-Gordon-Schrödinger system. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 221-238. doi: 10.3934/dcds.2011.31.221
[5]	Marilena N. Poulou, Nikolaos M. Stavrakakis. Finite dimensionality of a Klein-Gordon-Schrödinger type system. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 149-161. doi: 10.3934/dcdss.2009.2.149
[6]	Fábio Natali, Ademir Pastor. Stability properties of periodic standing waves for the Klein-Gordon-Schrödinger system. Communications on Pure & Applied Analysis, 2010, 9 (2) : 413-430. doi: 10.3934/cpaa.2010.9.413
[7]	Weizhu Bao, Chunmei Su. Uniform error estimates of a finite difference method for the Klein-Gordon-Schrödinger system in the nonrelativistic and massless limit regimes. Kinetic & Related Models, 2018, 11 (4) : 1037-1062. doi: 10.3934/krm.2018040
[8]	Olivier Goubet, Wided Kechiche. Uniform attractor for non-autonomous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2011, 10 (2) : 639-651. doi: 10.3934/cpaa.2011.10.639
[9]	A. F. Almeida, M. M. Cavalcanti, J. P. Zanchetta. Exponential decay for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2039-2061. doi: 10.3934/cpaa.2018097
[10]	Adriana Flores de Almeida, Marcelo Moreira Cavalcanti, Janaina Pedroso Zanchetta. Exponential stability for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping. Evolution Equations & Control Theory, 2019, 8 (4) : 847-865. doi: 10.3934/eect.2019041
[11]	Pavlos Xanthopoulos, Georgios E. Zouraris. A linearly implicit finite difference method for a Klein-Gordon-Schrödinger system modeling electron-ion plasma waves. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 239-263. doi: 10.3934/dcdsb.2008.10.239
[12]	E. Compaan, N. Tzirakis. Low-regularity global well-posedness for the Klein-Gordon-Schrödinger system on $ \mathbb{R}^+ $. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3867-3895. doi: 10.3934/dcds.2019156
[13]	V. V. Chepyzhov, M. I. Vishik, W. L. Wendland. On non-autonomous sine-Gordon type equations with a simple global attractor and some averaging. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 27-38. doi: 10.3934/dcds.2005.12.27
[14]	Rodrigo Samprogna, Tomás Caraballo. Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 509-523. doi: 10.3934/dcdsb.2017195
[15]	T. Caraballo, J. A. Langa, J. Valero. Structure of the pullback attractor for a non-autonomous scalar differential inclusion. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 979-994. doi: 10.3934/dcdss.2016037
[16]	Olivier Goubet, Marilena N. Poulou. Semi discrete weakly damped nonlinear Klein-Gordon Schrödinger system. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1525-1539. doi: 10.3934/cpaa.2014.13.1525
[17]	Zeqi Zhu, Caidi Zhao. Pullback attractor and invariant measures for the three-dimensional regularized MHD equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1461-1477. doi: 10.3934/dcds.2018060
[18]	Grzegorz Łukaszewicz, James C. Robinson. Invariant measures for non-autonomous dissipative dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4211-4222. doi: 10.3934/dcds.2014.34.4211
[19]	Yang Han. On the cauchy problem for the coupled Klein Gordon Schrödinger system with rough data. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 233-242. doi: 10.3934/dcds.2005.12.233
[20]	Xiaolin Jia, Caidi Zhao, Juan Cao. Uniform attractor of the non-autonomous discrete Selkov model. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 229-248. doi: 10.3934/dcds.2014.34.229

American Institute of Mathematical Sciences