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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 |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[9] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Springer, New York, 2013. |
[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. |
[11] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics. AMS Colloquium Publications, 49. AMS, Providence, R. I., 2002. |
[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. |
[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.
|
[14] |
S. N. Chow, Lattice Dynamical Systems, Lecture Notes in Math., 1822 (2003) 1-102. |
[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. |
[16] |
C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, Cambridge, 2001. |
[17] |
I. Fukuda and M. Tsutsumi,
On coupled Klein-Gordon-Schrödinger equations, Math. Japan., 24 (1979), 307-321.
|
[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. |
[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. |
[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. |
[21] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys and Monographs, AMS Colloquium Publications, 25. AMS, Providence, R. I., 1988. |
[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. |
[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. |
[24] |
R. Kapval,
Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113-163.
doi: 10.1007/BF01192578. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[32] |
K. Lu and B. Wang,
Attractor for dissipative Klein-Gordon-Schrödinger equations in $ \textbf{R}^3 $, J. Differential Equations, 170 (2001), 281-316.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[38] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge, Cambridge University Press, 2001. |
[39] |
G. Sell and Y. You, Dynamics of Evolutionary Equations, New York, Springer, 2002. |
[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. |
[41] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, Berlin, 1988. |
[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. |
[43] |
B. Wang,
Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245.
doi: 10.1016/j.jde.2005.01.003. |
[44] |
X. Wang,
Upper-semicontinuity of stationary statistical properties of dissipative systems, Discrete Cont. Dyn. Syst.-A, 23 (2009), 521-540.
|
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[51] |
S. Zhou,
Attractors for second order lattice dynamical systems, J. Differential Equations, 179 (2002), 605-624.
doi: 10.1006/jdeq.2001.4032. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[9] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Springer, New York, 2013. |
[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. |
[11] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics. AMS Colloquium Publications, 49. AMS, Providence, R. I., 2002. |
[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. |
[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.
|
[14] |
S. N. Chow, Lattice Dynamical Systems, Lecture Notes in Math., 1822 (2003) 1-102. |
[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. |
[16] |
C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, Cambridge, 2001. |
[17] |
I. Fukuda and M. Tsutsumi,
On coupled Klein-Gordon-Schrödinger equations, Math. Japan., 24 (1979), 307-321.
|
[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. |
[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. |
[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. |
[21] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys and Monographs, AMS Colloquium Publications, 25. AMS, Providence, R. I., 1988. |
[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. |
[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. |
[24] |
R. Kapval,
Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113-163.
doi: 10.1007/BF01192578. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[32] |
K. Lu and B. Wang,
Attractor for dissipative Klein-Gordon-Schrödinger equations in $ \textbf{R}^3 $, J. Differential Equations, 170 (2001), 281-316.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[38] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge, Cambridge University Press, 2001. |
[39] |
G. Sell and Y. You, Dynamics of Evolutionary Equations, New York, Springer, 2002. |
[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. |
[41] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, Berlin, 1988. |
[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. |
[43] |
B. Wang,
Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245.
doi: 10.1016/j.jde.2005.01.003. |
[44] |
X. Wang,
Upper-semicontinuity of stationary statistical properties of dissipative systems, Discrete Cont. Dyn. Syst.-A, 23 (2009), 521-540.
|
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[51] |
S. Zhou,
Attractors for second order lattice dynamical systems, J. Differential Equations, 179 (2002), 605-624.
doi: 10.1006/jdeq.2001.4032. |
[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. |
[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. |
[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. |
[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. |
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