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Asymptotic behavior of non-autonomous fractional stochastic lattice systems with multiplicative noise
Existence and approximation of attractors for nonlinear coupled lattice wave equations
1. | School of Mathematics and Compute Science, Liupanshui Normal University, Liupanshui, Guizhou 553004, China |
2. | Faculty of Mathematics, Federal University of Pará, Raimundo Santana Cruz Street, S/N, 68721-000, Salinópolis, Pará, Brazil |
3. | Ph.D Program in Mathematics, Federal University of Pará, Augusto Corrêa Street, 01, 66075-110, Belém, Pará, Brazil |
4. | Institute of Applied Physics and Computational Mathematics, PO Box 8009, Beijing 100088, China |
This paper is concerned with the asymptotic behavior of solutions to a class of nonlinear coupled discrete wave equations defined on the whole integer set. We first establish the well-posedness of the systems in $ E: = \ell^2\times\ell^2\times\ell^2\times\ell^2 $. We then prove that the solution semigroup has a unique global attractor in $ E $. We finally prove that this attractor can be approximated in terms of upper semicontinuity of $ E $ by a finite-dimensional global attractor of a $ 2(2n+1) $-dimensional truncation system as $ n $ goes to infinity. The idea of uniform tail-estimates developed by Wang (Phys. D, 128 (1999) 41-52) is employed to prove the asymptotic compactness of the solution semigroups in order to overcome the lack of compactness in infinite lattices.
References:
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A. Y. Abdallah,
Uniform exponential attractors for first order non-autonomous lattice dynamical systems, J. Differ. Equ., 251 (2011), 1489-1504.
doi: 10.1016/j.jde.2011.05.030. |
[2] |
P. W. Bates, H. Lisei and K. Lu,
Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.
doi: 10.1142/S0219493706001621. |
[3] |
P. W. Bates, K. Lu and B. Wang,
Attractors for lattice dynamical systems, J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 143-153.
doi: 10.1142/S0218127401002031. |
[4] |
H. 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. |
[5] |
H. Cui, J. A. Langa and Y. Li,
Measurability of random attractors for quasi strong-to-weak continuous random dynamical systems, J. Dynam. Differential Equations, 30 (2018), 1873-1898.
doi: 10.1007/s10884-017-9617-z. |
[6] |
S. N. Chow, J. M. Paret and W. Shen,
Traveling waves in lattice dynamical systems, J. Differ. Equ., 149 (1998), 248-291.
doi: 10.1006/jdeq.1998.3478. |
[7] |
T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero,
Existence of pullback attractors for pullback asymptotically compact processes, Nonlinear Anal., 72 (2010), 1967-1976.
doi: 10.1016/j.na.2009.09.037. |
[8] |
T. Caraballo, I. D. Chueshov and P. E. Kloeden,
Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain, SIAM J. Math. Anal., 38 (2006/07), 1489-1507.
doi: 10.1137/050647281. |
[9] |
T. Caraballo, B. Guo, N. H. Tuan and R. Wang, Asymptotically autonomous robustness of random attractors for a class of weakly dissipative stochastic wave equations on unbounded domains, Proc. Roy. Soc. Edinburgh Sect. A, (2020), 1–31.
doi: 10.1017/prm.2020.77. |
[10] |
T. Caraballo, G. Lukaszewicz and J. Real,
Pullback attractors for non-autonomous 2D Navier-Stokes equations in unbounded domains, C. R. Math. Acad. Sci. Paris, 342 (2006), 263-268.
doi: 10.1016/j.crma.2005.12.015. |
[11] |
T. Caraballo, A. M. Mérquez-Durén and J. Real,
Pullback and forward attractors for a 3D LANS-$\alpha$ model with delay, Discrete Contin Dyn Syst., 15 (2006), 559-578.
doi: 10.3934/dcds.2006.15.559. |
[12] |
T. Caraballo, P. Marín-Rubio and J. Valero,
Autonomous and non-autonomous attractors for differential equations with delays, J. Differential Equations, 208 (2005), 9-41.
doi: 10.1016/j.jde.2003.09.008. |
[13] |
T. L. Carrol and L. M. Pecora,
Synchronization in chaotic systems, Phys. Rev. Lett., 64 (1990), 821-824.
doi: 10.1103/PhysRevLett.64.821. |
[14] |
T. Caraballo and J. Real,
Navier-Stokes equations with delays, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441-2453.
doi: 10.1098/rspa.2001.0807. |
[15] |
T. Erneux and G. Nicolis,
Propagating waves in discrete bistable reaction diffusion systems, Physica D, 67 (1993), 237-244.
doi: 10.1016/0167-2789(93)90208-I. |
[16] |
J. Huang, X. Han and S. Zhou,
Uniform attractors for non-autonomous Klein-Gordon Schrödinger lattice systems, Appl. Math. Mech., 30 (2009), 1597-1607.
doi: 10.1007/s10483-009-1211-z. |
[17] |
X. Han,
Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise, J. Math. Anal. Appl., 376 (2011), 481-493.
doi: 10.1016/j.jmaa.2010.11.032. |
[18] |
X. Han,
Exponential attractors for lattice dynamical systems in weighted spaces, Discrete Contin. Dyn. Syst., 31 (2011), 445-467.
doi: 10.3934/dcds.2011.31.445. |
[19] |
X. Han,
Asymptotic dynamics of stochastic lattice differential equations: A review, Continuous and Distributed Systems II. Stud. Syst. Decis. Control, 30 (2015), 121-136.
doi: 10.1007/978-3-319-19075-4_7. |
[20] |
X. Han,
Random attractors for second order stochastic lattice dynamical systems with multiplicative noise in weighted spaces, Stoch. Dyn., 12 (2012), 1150024.
doi: 10.1142/S0219493711500249. |
[21] |
X. Han,
Asymptotic behaviors for second order stochastic lattice dynamical systems on Zk in weighted spaces, J. Math. Anal. Appl., 397 (2013), 242-254.
doi: 10.1016/j.jmaa.2012.07.015. |
[22] |
X. Han and P. E. Kloeden, Attractors Under Discretisation, SpringerBriefs in Mathematics. BCAM SpringerBriefs. Springer, Cham; BCAM Basque Center for Applied Mathematics, Bilbao, 2017.
doi: 10.1007/978-3-319-61934-7. |
[23] |
X. Han, P. E. Kloeden and S. Sonner,
Discretisation of global attractors for lattice dynamical systems, J. Dynam. Differential Equations, 32 (2020), 1457-1474.
doi: 10.1007/s10884-019-09770-1. |
[24] |
X. Han, W. Shen and S. Zhou,
Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differ. Equ., 250 (2011), 1235-1266.
doi: 10.1016/j.jde.2010.10.018. |
[25] |
J. P. Keener,
Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47 (1987), 556-572.
doi: 10.1137/0147038. |
[26] |
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. |
[27] |
P. E. Kloeden, P. Marín-Rubio and J. Real,
Pullback attractors for a semilinear heat equation in a non-cylindrical domain, J. Differential Equations, 244 (2008), 2062-2090.
doi: 10.1016/j.jde.2007.10.031. |
[28] |
P. E. Kloeden and T. Lorenz,
Construction of nonautonomous forward attractors, Proc. Amer. Math. Soc., 144 (2016), 259-268.
doi: 10.1090/proc/12735. |
[29] |
P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, vol. 176 of Mathematical Surveys and Monographs, Americal Mathematical Society, 2011.
doi: 10.1090/surv/176. |
[30] |
P. E. Kloeden, J. Real and C. Sun,
Pullback attractors for a semilinear heat equation on time-varying domains, J. Differential Equations, 246 (2009), 4702-4730.
doi: 10.1016/j.jde.2008.11.017. |
[31] |
J. C. Robinson, Dimensions, Embeddings and Attractors, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2011.
doi: 10.1017/CBO9780511933912. |
[32] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge, 2001.
doi: 10.1007/978-94-010-0732-0.![]() ![]() ![]() |
[33] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Texts in Applied Mathematics, 2001.
doi: 10.1007/978-94-010-0732-0. |
[34] |
J. C. Robinson,
Global attractors: Topology and finite-dimensional dynamics, J. Dynam. Differential Equations, 11 (1999), 557-581.
doi: 10.1023/A:1021918004832. |
[35] |
L. Shi, R. Wang, K. Lu and B. Wang,
Asymptotic behavior of stochastic FitzHugh-Nagumo systems on unbounded thin domains, J. Differential Equations, 267 (2019), 4373-4409.
doi: 10.1016/j.jde.2019.05.002. |
[36] |
B. Wang,
Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245.
doi: 10.1016/j.jde.2005.01.003. |
[37] |
B. Wang,
Attractors for reaction-diffusion equations in unbounded domains, Phys. D, 128 (1999), 41-52.
doi: 10.1016/S0167-2789(98)00304-2. |
[38] |
B. Wang,
Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains, J. Differential Equations, 246 (2009), 2506-2537.
doi: 10.1016/j.jde.2008.10.012. |
[39] |
B. Wang,
Asymptotic behavior of non-autonomous fractional stochastic reaction-diffusion equations, Nonlinear Anal., 158 (2017), 60-82.
doi: 10.1016/j.na.2017.04.006. |
[40] |
B. Wang,
Weak pullback attractors for stochastic Navier-Stokes equations with nonlinear diffusion terms, Proc. Amer. Math. Soc., 147 (2019), 1627-1638.
doi: 10.1090/proc/14356. |
[41] |
B. 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. |
[42] |
B. 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] |
B. Wang,
Weak pullback attractors for mean random dynamical systems in Bochner spaces, J. Dynam. Differential Equations, 31 (2019), 2177-2204.
doi: 10.1007/s10884-018-9696-5. |
[44] |
B. Wang,
Dynamics of fractional stochastic reaction-diffusion equations on unbounded domains driven by nonlinear noise, J. Differential Equations, 268 (2019), 1-59.
doi: 10.1016/j.jde.2019.08.007. |
[45] |
B. 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. |
[46] |
C. Wang, G. Xue and C. Zhao,
Invariant Borel probability measures for discrete long-wave-short-wave resonance equations, Appl. Math. Comp., 339 (2018), 853-865.
doi: 10.1016/j.amc.2018.06.059. |
[47] |
R. Wang,
Long-time dynamics of stochastic lattice plate equations with nonlinear noise and damping, J. Dynam. Differential Equations, 33 (2021), 767-803.
doi: 10.1007/s10884-020-09830-x. |
[48] |
R. Wang, B. Guo and B. Wang,
Well-posedness and dynamics of fractional FitzHugh-Nagumo systems on $\mathbb{R}^N$ driven by nonlinear noise, Sci. China Math., 64 (2021), 2395-2436.
doi: 10.1007/s11425-019-1714-2. |
[49] |
R. Wang, Y. Li and B. Wang,
Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dyn. Syst., 39 (2019), 4091-4126.
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R. Wang, Y. Li and B. Wang,
Bi-spatial pullback attractors of fractional non-classical diffusion equations on unbounded domains with $(p, q)$-growth nonlinearities, Appl. Math. Optim., 84 (2021), 425-461.
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R. Wang, L. Shi and B. Wang,
Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear colored noise on $\mathbb{R}^N$, Nonlinearity, 32 (2019), 4524-4556.
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R. Wang and B. Wang,
Random dynamics of $p$-laplacian lattice systems driven by infinite-dimensional nonlinear noise, Stochastic Process. Appl., 130 (2020), 7431-7462.
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Statistical solutions and piecewise Liouville theorem for the impulsive reaction-diffusion equations on infinite lattices, Appl. Math. Comput., 404 (2021), 126103.
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C. Zhao, G. Xue and G. Lukaszewicz,
Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations, Discrete Contin. Dyn. Syst.B, 23 (2018), 4021-4044.
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S. Zhou, C. Zhao and Y. Wang,
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show all references
References:
[1] |
A. Y. Abdallah,
Uniform exponential attractors for first order non-autonomous lattice dynamical systems, J. Differ. Equ., 251 (2011), 1489-1504.
doi: 10.1016/j.jde.2011.05.030. |
[2] |
P. W. Bates, H. Lisei and K. Lu,
Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.
doi: 10.1142/S0219493706001621. |
[3] |
P. W. Bates, K. Lu and B. Wang,
Attractors for lattice dynamical systems, J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 143-153.
doi: 10.1142/S0218127401002031. |
[4] |
H. 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. |
[5] |
H. Cui, J. A. Langa and Y. Li,
Measurability of random attractors for quasi strong-to-weak continuous random dynamical systems, J. Dynam. Differential Equations, 30 (2018), 1873-1898.
doi: 10.1007/s10884-017-9617-z. |
[6] |
S. N. Chow, J. M. Paret and W. Shen,
Traveling waves in lattice dynamical systems, J. Differ. Equ., 149 (1998), 248-291.
doi: 10.1006/jdeq.1998.3478. |
[7] |
T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero,
Existence of pullback attractors for pullback asymptotically compact processes, Nonlinear Anal., 72 (2010), 1967-1976.
doi: 10.1016/j.na.2009.09.037. |
[8] |
T. Caraballo, I. D. Chueshov and P. E. Kloeden,
Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain, SIAM J. Math. Anal., 38 (2006/07), 1489-1507.
doi: 10.1137/050647281. |
[9] |
T. Caraballo, B. Guo, N. H. Tuan and R. Wang, Asymptotically autonomous robustness of random attractors for a class of weakly dissipative stochastic wave equations on unbounded domains, Proc. Roy. Soc. Edinburgh Sect. A, (2020), 1–31.
doi: 10.1017/prm.2020.77. |
[10] |
T. Caraballo, G. Lukaszewicz and J. Real,
Pullback attractors for non-autonomous 2D Navier-Stokes equations in unbounded domains, C. R. Math. Acad. Sci. Paris, 342 (2006), 263-268.
doi: 10.1016/j.crma.2005.12.015. |
[11] |
T. Caraballo, A. M. Mérquez-Durén and J. Real,
Pullback and forward attractors for a 3D LANS-$\alpha$ model with delay, Discrete Contin Dyn Syst., 15 (2006), 559-578.
doi: 10.3934/dcds.2006.15.559. |
[12] |
T. Caraballo, P. Marín-Rubio and J. Valero,
Autonomous and non-autonomous attractors for differential equations with delays, J. Differential Equations, 208 (2005), 9-41.
doi: 10.1016/j.jde.2003.09.008. |
[13] |
T. L. Carrol and L. M. Pecora,
Synchronization in chaotic systems, Phys. Rev. Lett., 64 (1990), 821-824.
doi: 10.1103/PhysRevLett.64.821. |
[14] |
T. Caraballo and J. Real,
Navier-Stokes equations with delays, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441-2453.
doi: 10.1098/rspa.2001.0807. |
[15] |
T. Erneux and G. Nicolis,
Propagating waves in discrete bistable reaction diffusion systems, Physica D, 67 (1993), 237-244.
doi: 10.1016/0167-2789(93)90208-I. |
[16] |
J. Huang, X. Han and S. Zhou,
Uniform attractors for non-autonomous Klein-Gordon Schrödinger lattice systems, Appl. Math. Mech., 30 (2009), 1597-1607.
doi: 10.1007/s10483-009-1211-z. |
[17] |
X. Han,
Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise, J. Math. Anal. Appl., 376 (2011), 481-493.
doi: 10.1016/j.jmaa.2010.11.032. |
[18] |
X. Han,
Exponential attractors for lattice dynamical systems in weighted spaces, Discrete Contin. Dyn. Syst., 31 (2011), 445-467.
doi: 10.3934/dcds.2011.31.445. |
[19] |
X. Han,
Asymptotic dynamics of stochastic lattice differential equations: A review, Continuous and Distributed Systems II. Stud. Syst. Decis. Control, 30 (2015), 121-136.
doi: 10.1007/978-3-319-19075-4_7. |
[20] |
X. Han,
Random attractors for second order stochastic lattice dynamical systems with multiplicative noise in weighted spaces, Stoch. Dyn., 12 (2012), 1150024.
doi: 10.1142/S0219493711500249. |
[21] |
X. Han,
Asymptotic behaviors for second order stochastic lattice dynamical systems on Zk in weighted spaces, J. Math. Anal. Appl., 397 (2013), 242-254.
doi: 10.1016/j.jmaa.2012.07.015. |
[22] |
X. Han and P. E. Kloeden, Attractors Under Discretisation, SpringerBriefs in Mathematics. BCAM SpringerBriefs. Springer, Cham; BCAM Basque Center for Applied Mathematics, Bilbao, 2017.
doi: 10.1007/978-3-319-61934-7. |
[23] |
X. Han, P. E. Kloeden and S. Sonner,
Discretisation of global attractors for lattice dynamical systems, J. Dynam. Differential Equations, 32 (2020), 1457-1474.
doi: 10.1007/s10884-019-09770-1. |
[24] |
X. Han, W. Shen and S. Zhou,
Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differ. Equ., 250 (2011), 1235-1266.
doi: 10.1016/j.jde.2010.10.018. |
[25] |
J. P. Keener,
Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47 (1987), 556-572.
doi: 10.1137/0147038. |
[26] |
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. |
[27] |
P. E. Kloeden, P. Marín-Rubio and J. Real,
Pullback attractors for a semilinear heat equation in a non-cylindrical domain, J. Differential Equations, 244 (2008), 2062-2090.
doi: 10.1016/j.jde.2007.10.031. |
[28] |
P. E. Kloeden and T. Lorenz,
Construction of nonautonomous forward attractors, Proc. Amer. Math. Soc., 144 (2016), 259-268.
doi: 10.1090/proc/12735. |
[29] |
P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, vol. 176 of Mathematical Surveys and Monographs, Americal Mathematical Society, 2011.
doi: 10.1090/surv/176. |
[30] |
P. E. Kloeden, J. Real and C. Sun,
Pullback attractors for a semilinear heat equation on time-varying domains, J. Differential Equations, 246 (2009), 4702-4730.
doi: 10.1016/j.jde.2008.11.017. |
[31] |
J. C. Robinson, Dimensions, Embeddings and Attractors, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2011.
doi: 10.1017/CBO9780511933912. |
[32] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge, 2001.
doi: 10.1007/978-94-010-0732-0.![]() ![]() ![]() |
[33] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Texts in Applied Mathematics, 2001.
doi: 10.1007/978-94-010-0732-0. |
[34] |
J. C. Robinson,
Global attractors: Topology and finite-dimensional dynamics, J. Dynam. Differential Equations, 11 (1999), 557-581.
doi: 10.1023/A:1021918004832. |
[35] |
L. Shi, R. Wang, K. Lu and B. Wang,
Asymptotic behavior of stochastic FitzHugh-Nagumo systems on unbounded thin domains, J. Differential Equations, 267 (2019), 4373-4409.
doi: 10.1016/j.jde.2019.05.002. |
[36] |
B. Wang,
Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245.
doi: 10.1016/j.jde.2005.01.003. |
[37] |
B. Wang,
Attractors for reaction-diffusion equations in unbounded domains, Phys. D, 128 (1999), 41-52.
doi: 10.1016/S0167-2789(98)00304-2. |
[38] |
B. Wang,
Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains, J. Differential Equations, 246 (2009), 2506-2537.
doi: 10.1016/j.jde.2008.10.012. |
[39] |
B. Wang,
Asymptotic behavior of non-autonomous fractional stochastic reaction-diffusion equations, Nonlinear Anal., 158 (2017), 60-82.
doi: 10.1016/j.na.2017.04.006. |
[40] |
B. Wang,
Weak pullback attractors for stochastic Navier-Stokes equations with nonlinear diffusion terms, Proc. Amer. Math. Soc., 147 (2019), 1627-1638.
doi: 10.1090/proc/14356. |
[41] |
B. 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. |
[42] |
B. 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] |
B. Wang,
Weak pullback attractors for mean random dynamical systems in Bochner spaces, J. Dynam. Differential Equations, 31 (2019), 2177-2204.
doi: 10.1007/s10884-018-9696-5. |
[44] |
B. Wang,
Dynamics of fractional stochastic reaction-diffusion equations on unbounded domains driven by nonlinear noise, J. Differential Equations, 268 (2019), 1-59.
doi: 10.1016/j.jde.2019.08.007. |
[45] |
B. 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. |
[46] |
C. Wang, G. Xue and C. Zhao,
Invariant Borel probability measures for discrete long-wave-short-wave resonance equations, Appl. Math. Comp., 339 (2018), 853-865.
doi: 10.1016/j.amc.2018.06.059. |
[47] |
R. Wang,
Long-time dynamics of stochastic lattice plate equations with nonlinear noise and damping, J. Dynam. Differential Equations, 33 (2021), 767-803.
doi: 10.1007/s10884-020-09830-x. |
[48] |
R. Wang, B. Guo and B. Wang,
Well-posedness and dynamics of fractional FitzHugh-Nagumo systems on $\mathbb{R}^N$ driven by nonlinear noise, Sci. China Math., 64 (2021), 2395-2436.
doi: 10.1007/s11425-019-1714-2. |
[49] |
R. Wang, Y. Li and B. Wang,
Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dyn. Syst., 39 (2019), 4091-4126.
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