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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 |
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.
References:
[1] |
L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. |
[2] |
P. W. Bates, K. 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. Brzeźniak, E. 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. Brzeźniak, M. 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. Caraballo, X. Y. Han, B. 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. Caraballo, F. 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. Caraballo, M. J. Garrido-Atienza, B. 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. Caraballo, M. 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. Han, W. 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. Li, B. 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. Li, K. N. Lu, B. 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. Li, A. 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. Misiats, O. 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. Wang, K. 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. Wang, K. 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. Wang, Y. 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. Wang, J. 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. Bates, K. 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. Brzeźniak, E. 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. Brzeźniak, M. 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. Caraballo, X. Y. Han, B. 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. Caraballo, F. 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. Caraballo, M. J. Garrido-Atienza, B. 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. Caraballo, M. 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. Han, W. 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. Li, B. 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. Li, K. N. Lu, B. 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. Li, A. 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. Misiats, O. 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. Wang, K. 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. Wang, K. 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. Wang, Y. 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. Wang, J. 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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