July  2020, 25(7): 2461-2493. doi: 10.3934/dcdsb.2020019

Random dynamics of lattice wave equations driven by infinite-dimensional nonlinear noise

1. 

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

2. 

Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA

* Corresponding author: rwang-math@outlook.com (Renhai Wang)

Received  June 2019 Revised  August 2019 Published  July 2020 Early access  April 2020

Fund Project: The first author is supported by the Innovation Project of Chongqing grant CYB18115 and the China Scholarship Council(CSC No.201806990064)

This paper is concerned with the global existence and random dynamics of non-autonomous stochastic second-order lattice systems driven by infinite-dimensional nonlinear noise defined on higher-dimensional integer sets. We first show the existence and uniqueness of mean square solutions to the equations when the nonlinear drift term has a polynomial growth of arbitrary order and the diffusion term is locally Lipschitz continuous. We then prove that the mean random dynamical system associated with the solution operator possesses a unique tempered weak pullback mean random attractor in a Bochner space under certain conditions. We finally establish the existence of invariant measures for the stochastic systems in $ \ell^2\times\ell^2 $ by showing the tightness of a family of distribution laws of solutions via the idea of uniform tail-estimates on the solutions.

Citation: Renhai Wang, Bixiang Wang. Random dynamics of lattice wave equations driven by infinite-dimensional nonlinear noise. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2461-2493. doi: 10.3934/dcdsb.2020019
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P. W. BatesK. N. Lu and B. X. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.  doi: 10.1016/j.jde.2008.05.017.

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Z. BrzeźniakE. Motyl and M. Ondrejat, Invariant measure for the stochastic Navier-Stokes equations in unbounded 2D domains, Ann. Probab., 45 (2017), 3145-3201.  doi: 10.1214/16-AOP1133.

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Z. BrzeźniakM. Ondreját and J. Seidler, Invariant measures for stochastic nonlinear beam and wave equations, J. Differential Equations, 260 (2016), 4157-4179.  doi: 10.1016/j.jde.2015.11.007.

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T. CaraballoX. Y. HanB. Schmalfuss and J. Valero, Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise, Nonlinear Anal., 130 (2016), 255-278.  doi: 10.1016/j.na.2015.09.025.

[7]

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

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T. CaraballoM. J. Garrido-AtienzaB. Schmalfuss and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 439-455.  doi: 10.3934/dcdsb.2010.14.439.

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T. CaraballoM. J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal., 74 (2011), 3671-3684.  doi: 10.1016/j.na.2011.02.047.

[10]

S.-N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems, Ⅰ, Ⅱ, IEEE Trans. Circuits Systems I Fund. Theory Appl., 42 (1995), 746-751.  doi: 10.1109/81.473583.

[11]

H. Y. Cui and P. E. Kloeden, Invariant forward attractors of non-autonomous random dynamical systems, J. Differential Equations, 265 (2018), 6166-6186.  doi: 10.1016/j.jde.2018.07.028.

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A. H. Gu and Y. R. Li, Dynamic behavior of stochastic $p$-Laplacian-type lattice equations, Stoch. Dyn., 17 (2017), 1750040, 19 pp. doi: 10.1142/S021949371750040X.

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X. Y. Han, Random attractors for second order stochastic lattice dynamical systems with multiplicative noise in weighted spaces, Stoch. Dyn., 12 (2012), 1150024, 20 pp. doi: 10.1142/S0219493711500249.

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X. Y. Han and P. E. Kloeden, Asymptotic behavior of a neural field lattice model with a Heaviside operator, Physica D, 389 (2019), 1-12.  doi: 10.1016/j.physd.2018.09.004.

[19]

X. Y. Han and P. E. 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.

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J. U. Kim, Periodic and invariant measures for stochastic wave equations, Electronic journal of differential equations, 2004 (2004), 30 pp.

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J. U. Kim, On the stochastic Burgers equation with polynomial nonlinearity in the real line, Discrete and Continuous Dynamical Systems Series B, 6 (2006), 835-866.  doi: 10.3934/dcdsb.2006.6.835.

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J. U. Kim, On the stochastic Benjamin-Ono equation, Journal of Differential Equations, 228 (2006), 737-768.  doi: 10.1016/j.jde.2005.11.005.

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P. E. Kloeden and T. Lorenz, Mean-square random dynamical systems, J. Differential Equations, 253 (2012), 1422-1438.  doi: 10.1016/j.jde.2012.05.016.

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P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 163-181.  doi: 10.1098/rspa.2006.1753.

[26]

D. S. LiB. X. Wang and X. H. Wang, Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains, J. Differential Equations, 262 (2017), 1575-1602.  doi: 10.1016/j.jde.2016.10.024.

[27]

D. S. LiK. N. LuB. X. Wang and X. H. Wang, Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains, Discrete Contin. Dyn. Syst., 38 (2018), 187-208.  doi: 10.1016/j.jde.2016.10.024.

[28]

Y. R. LiA. H. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differential Equations, 258 (2015), 504-534.  doi: 10.1016/j.jde.2014.09.021.

[29]

C. B. Li and J. C. Sprott, An infinite 3-D quasiperiodic lattice of chaotic attractors, Phys. Lett. A, 382 (2018), 581-587.  doi: 10.1016/j.physleta.2017.12.022.

[30]

W.-W. Lin and Y.-Q. Wang, Proof of synchronized chaotic behaviors in coupled map lattices, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 21 (2011), 1493-1500.  doi: 10.1142/S0218127411029069.

[31]

K. N. Lu and B. X. Wang, Wong-Zakai approximations and long term behavior of stochastic partial differential equations, J. Dynam. Differ. Equ., 31 (2019), 1341-1371.  doi: 10.1007/s10884-017-9626-y.

[32]

O. MisiatsO. Stanzhytskyi and N. K. Yip, Existence and uniqueness of invariant measures for stochastic reaction-diffusion equations in unbounded domains, J. Theoret. Probab., 29 (2016), 996-1026.  doi: 10.1007/s10959-015-0606-z.

[33]

X. H. WangK. N. Lu and B. X. Wang, Random attractors for delay parabolic equations with additive noise and deterministic nonautonomous forcing, SIAM J. Appl. Dyn. Syst., 14 (2015), 1018-1047.  doi: 10.1137/140991819.

[34]

X. H. WangK. N. Lu and B. X. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2018), 378-424.  doi: 10.1016/j.jde.2017.09.006.

[35]

B. X. Wang, Attractors for reaction-diffusion equations in unbounded domains, Phys. D, 128 (1999), 41-52.  doi: 10.1016/S0167-2789(98)00304-2.

[36]

B. X. Wang, Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245.  doi: 10.1016/j.jde.2005.01.003.

[37]

B. X. Wang, Weak pullback attractors for mean random dynamical systems in Bochner spaces, J. Dynam. Differ. Equ., 31 (2019), 2177–2204, https://doi.org/10.1007/s10884-018-9696-5. doi: 10.1007/s10884-018-9696-5.

[38]

B. X. Wang, Dynamics of stochastic reaction-diffusion lattice system driven by nonlinear noise, J. Math. Anal. Appl., 477 (2019), 104-132.  doi: 10.1016/j.jmaa.2019.04.015.

[39]

B. X. Wang, Dynamics of fractional stochastic reaction-diffusion equations on unbounded domains driven by nonlinear noise, J. Differential Equations, 268 (2019), 1–59, https://doi.org/10.1016/j.jde.2019.08.007. doi: 10.1016/j.jde.2019.08.007.

[40]

B. X. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.

[41]

B. X. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^{3}$, Tran. Amer. Math. Soc., 363 (2011), 3639-3663.  doi: 10.1090/S0002-9947-2011-05247-5.

[42]

B. X. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269.

[43]

R. H. WangY. R. Li and B. X. Wang, Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dyn. Syst., 39 (2019), 4091-4126.  doi: 10.3934/dcds.2019165.

[44]

R. H. Wang and Y. R. Li, Regularity and backward compactness of attractors for non-autonomous lattice systems with random coefficients, Appl. Math. Comput., 354 (2019), 86-102.  doi: 10.1016/j.amc.2019.02.036.

[45]

Y. J. WangJ. H. Xu and P. E. Kloeden, Asymptotic behavior of stochastic lattice systems with a Caputo fractional time derivative, Nonlinear Anal., 135 (2016), 205-222.  doi: 10.1016/j.na.2016.01.020.

[46]

W. Q. Zhao, Random dynamics of stochastic $p$-Laplacian equations on $\mathbb{R}^N$ with an unbounded additive noise, J. Math. Anal. Appl., 455 (2017), 1178-1203.  doi: 10.1016/j.jmaa.2017.06.025.

[47]

W. Q. Zhao and Y. J. Zhang, Compactness and attracting of random attractors for non-autonomous stochastic lattice dynamical systems in weighted space $\ell^p_\rho$, Appl. Math. Comput., 291 (2016), 226-243.  doi: 10.1016/j.amc.2016.06.045.

[48]

S. F. Zhou and L. Wei, A random attractor for a stochastic second order lattice system with random coupled coefficients, J. Math. Anal. Appl., 395 (2012), 42-55.  doi: 10.1016/j.jmaa.2012.04.080.

[49]

S. F. Zhou and X. Y. Han, Pullback exponential attractors for non-autonomous lattice systems, J. Dyn. Differ. Equ., 24 (2012), 601-631.  doi: 10.1007/s10884-012-9260-7.

[50]

S. F. Zhou and Z. J. Wang, Finite fractal dimensions of random attractors for stochastic FitzHugh-Nagumo system with multiplicative white noise, J. Math. Anal. Appl., 441 (2016), 648-667.  doi: 10.1016/j.jmaa.2016.04.038.

[51]

S. F. Zhou and M. Zhao, Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise, Discrete Contin. Dyn. Syst., 36 (2017), 2887-2914.  doi: 10.3934/dcds.2016.36.2887.

show all references

References:
[1]

L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.

[2]

P. W. BatesK. N. Lu and B. X. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.  doi: 10.1016/j.jde.2008.05.017.

[3]

P. W. Bates, K. N. Lu and B. X. Wang, Tempered random attractors for parabolic equations in weighted spaces, J. Math. Phys., 54 (2013), 081505, 26 pp. doi: 10.1063/1.4817597.

[4]

Z. BrzeźniakE. Motyl and M. Ondrejat, Invariant measure for the stochastic Navier-Stokes equations in unbounded 2D domains, Ann. Probab., 45 (2017), 3145-3201.  doi: 10.1214/16-AOP1133.

[5]

Z. BrzeźniakM. Ondreját and J. Seidler, Invariant measures for stochastic nonlinear beam and wave equations, J. Differential Equations, 260 (2016), 4157-4179.  doi: 10.1016/j.jde.2015.11.007.

[6]

T. CaraballoX. Y. HanB. Schmalfuss and J. Valero, Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise, Nonlinear Anal., 130 (2016), 255-278.  doi: 10.1016/j.na.2015.09.025.

[7]

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

[8]

T. CaraballoM. J. Garrido-AtienzaB. Schmalfuss and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 439-455.  doi: 10.3934/dcdsb.2010.14.439.

[9]

T. CaraballoM. J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal., 74 (2011), 3671-3684.  doi: 10.1016/j.na.2011.02.047.

[10]

S.-N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems, Ⅰ, Ⅱ, IEEE Trans. Circuits Systems I Fund. Theory Appl., 42 (1995), 746-751.  doi: 10.1109/81.473583.

[11]

H. Y. Cui and P. E. Kloeden, Invariant forward attractors of non-autonomous random dynamical systems, J. Differential Equations, 265 (2018), 6166-6186.  doi: 10.1016/j.jde.2018.07.028.

[12]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Second edition. Encyclopedia of Mathematics and its Applications, 152. Cambridge University Press, Cambridge, 2014. doi: 10.1017/CBO9781107295513.

[13]

J.-P. Eckmann and M. Hairer, Invariant measures for stochastic partial differential equations in unbounded domains, Nonlinearity, 14 (2001), 133-151.  doi: 10.1088/0951-7715/14/1/308.

[14]

T. Erneux and G. Nicolis, Propagating waves in discrete bistable reaction systems, Physica D, 67 (1993), 237-244.  doi: 10.1016/0167-2789(93)90208-I.

[15]

A. H. Gu and Y. R. Li, Dynamic behavior of stochastic $p$-Laplacian-type lattice equations, Stoch. Dyn., 17 (2017), 1750040, 19 pp. doi: 10.1142/S021949371750040X.

[16]

J.-S. Guo and C.-C. Wu, Uniqueness and stability of traveling waves for periodic monostable lattice dynamical system, J. Differential Equations, 246 (2009), 3818-3833.  doi: 10.1016/j.jde.2009.03.010.

[17]

X. Y. Han, Random attractors for second order stochastic lattice dynamical systems with multiplicative noise in weighted spaces, Stoch. Dyn., 12 (2012), 1150024, 20 pp. doi: 10.1142/S0219493711500249.

[18]

X. Y. Han and P. E. Kloeden, Asymptotic behavior of a neural field lattice model with a Heaviside operator, Physica D, 389 (2019), 1-12.  doi: 10.1016/j.physd.2018.09.004.

[19]

X. Y. Han and P. E. 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.

[20]

X. Y. HanW. X. Shen and S. F. 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.

[21]

J. U. Kim, Periodic and invariant measures for stochastic wave equations, Electronic journal of differential equations, 2004 (2004), 30 pp.

[22]

J. U. Kim, On the stochastic Burgers equation with polynomial nonlinearity in the real line, Discrete and Continuous Dynamical Systems Series B, 6 (2006), 835-866.  doi: 10.3934/dcdsb.2006.6.835.

[23]

J. U. Kim, On the stochastic Benjamin-Ono equation, Journal of Differential Equations, 228 (2006), 737-768.  doi: 10.1016/j.jde.2005.11.005.

[24]

P. E. Kloeden and T. Lorenz, Mean-square random dynamical systems, J. Differential Equations, 253 (2012), 1422-1438.  doi: 10.1016/j.jde.2012.05.016.

[25]

P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 163-181.  doi: 10.1098/rspa.2006.1753.

[26]

D. S. LiB. X. Wang and X. H. Wang, Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains, J. Differential Equations, 262 (2017), 1575-1602.  doi: 10.1016/j.jde.2016.10.024.

[27]

D. S. LiK. N. LuB. X. Wang and X. H. Wang, Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains, Discrete Contin. Dyn. Syst., 38 (2018), 187-208.  doi: 10.1016/j.jde.2016.10.024.

[28]

Y. R. LiA. H. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differential Equations, 258 (2015), 504-534.  doi: 10.1016/j.jde.2014.09.021.

[29]

C. B. Li and J. C. Sprott, An infinite 3-D quasiperiodic lattice of chaotic attractors, Phys. Lett. A, 382 (2018), 581-587.  doi: 10.1016/j.physleta.2017.12.022.

[30]

W.-W. Lin and Y.-Q. Wang, Proof of synchronized chaotic behaviors in coupled map lattices, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 21 (2011), 1493-1500.  doi: 10.1142/S0218127411029069.

[31]

K. N. Lu and B. X. Wang, Wong-Zakai approximations and long term behavior of stochastic partial differential equations, J. Dynam. Differ. Equ., 31 (2019), 1341-1371.  doi: 10.1007/s10884-017-9626-y.

[32]

O. MisiatsO. Stanzhytskyi and N. K. Yip, Existence and uniqueness of invariant measures for stochastic reaction-diffusion equations in unbounded domains, J. Theoret. Probab., 29 (2016), 996-1026.  doi: 10.1007/s10959-015-0606-z.

[33]

X. H. WangK. N. Lu and B. X. Wang, Random attractors for delay parabolic equations with additive noise and deterministic nonautonomous forcing, SIAM J. Appl. Dyn. Syst., 14 (2015), 1018-1047.  doi: 10.1137/140991819.

[34]

X. H. WangK. N. Lu and B. X. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2018), 378-424.  doi: 10.1016/j.jde.2017.09.006.

[35]

B. X. Wang, Attractors for reaction-diffusion equations in unbounded domains, Phys. D, 128 (1999), 41-52.  doi: 10.1016/S0167-2789(98)00304-2.

[36]

B. X. Wang, Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245.  doi: 10.1016/j.jde.2005.01.003.

[37]

B. X. Wang, Weak pullback attractors for mean random dynamical systems in Bochner spaces, J. Dynam. Differ. Equ., 31 (2019), 2177–2204, https://doi.org/10.1007/s10884-018-9696-5. doi: 10.1007/s10884-018-9696-5.

[38]

B. X. Wang, Dynamics of stochastic reaction-diffusion lattice system driven by nonlinear noise, J. Math. Anal. Appl., 477 (2019), 104-132.  doi: 10.1016/j.jmaa.2019.04.015.

[39]

B. X. Wang, Dynamics of fractional stochastic reaction-diffusion equations on unbounded domains driven by nonlinear noise, J. Differential Equations, 268 (2019), 1–59, https://doi.org/10.1016/j.jde.2019.08.007. doi: 10.1016/j.jde.2019.08.007.

[40]

B. X. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.

[41]

B. X. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^{3}$, Tran. Amer. Math. Soc., 363 (2011), 3639-3663.  doi: 10.1090/S0002-9947-2011-05247-5.

[42]

B. X. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269.

[43]

R. H. WangY. R. Li and B. X. Wang, Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dyn. Syst., 39 (2019), 4091-4126.  doi: 10.3934/dcds.2019165.

[44]

R. H. Wang and Y. R. Li, Regularity and backward compactness of attractors for non-autonomous lattice systems with random coefficients, Appl. Math. Comput., 354 (2019), 86-102.  doi: 10.1016/j.amc.2019.02.036.

[45]

Y. J. WangJ. H. Xu and P. E. Kloeden, Asymptotic behavior of stochastic lattice systems with a Caputo fractional time derivative, Nonlinear Anal., 135 (2016), 205-222.  doi: 10.1016/j.na.2016.01.020.

[46]

W. Q. Zhao, Random dynamics of stochastic $p$-Laplacian equations on $\mathbb{R}^N$ with an unbounded additive noise, J. Math. Anal. Appl., 455 (2017), 1178-1203.  doi: 10.1016/j.jmaa.2017.06.025.

[47]

W. Q. Zhao and Y. J. Zhang, Compactness and attracting of random attractors for non-autonomous stochastic lattice dynamical systems in weighted space $\ell^p_\rho$, Appl. Math. Comput., 291 (2016), 226-243.  doi: 10.1016/j.amc.2016.06.045.

[48]

S. F. Zhou and L. Wei, A random attractor for a stochastic second order lattice system with random coupled coefficients, J. Math. Anal. Appl., 395 (2012), 42-55.  doi: 10.1016/j.jmaa.2012.04.080.

[49]

S. F. Zhou and X. Y. Han, Pullback exponential attractors for non-autonomous lattice systems, J. Dyn. Differ. Equ., 24 (2012), 601-631.  doi: 10.1007/s10884-012-9260-7.

[50]

S. F. Zhou and Z. J. Wang, Finite fractal dimensions of random attractors for stochastic FitzHugh-Nagumo system with multiplicative white noise, J. Math. Anal. Appl., 441 (2016), 648-667.  doi: 10.1016/j.jmaa.2016.04.038.

[51]

S. F. Zhou and M. Zhao, Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise, Discrete Contin. Dyn. Syst., 36 (2017), 2887-2914.  doi: 10.3934/dcds.2016.36.2887.

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