doi: 10.3934/dcdsb.2020244

Regular dynamics for stochastic Fitzhugh-Nagumo systems with additive noise on thin domains

School of Mathematics and Computer Science, Shangrao Normal University, Shangrao 334001 China

* Corresponding author: Fuzhi Li, lifuzhisru@163.com

Received  December 2019 Revised  June 2020 Published  August 2020

This paper is devoted to bi-spatial random attractors of the stochastic Fitzhugh-Nagumo equations with additive noise on thin domains when the terminate space is the Sobolev space. We first established the existence of random attractor on regular space and then show that the upper semi-continuity of these attractors under the Sobolev norm when a family of $ (n+1) $-dimensional thin domains degenerates onto an $ n $-dimensional domain.

Citation: Fuzhi Li, Dongmei Xu. Regular dynamics for stochastic Fitzhugh-Nagumo systems with additive noise on thin domains. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020244
References:
[1]

A. Adili and B. Wang, Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 643-666.  doi: 10.3934/dcdsb.2013.18.643.  Google Scholar

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F. Antoci and M. Prizzi, Reaction-diffusion equations on unbounded thin domains, Topol. Methods Nonlinear Anal., 18 (2001), 283-302.  doi: 10.12775/TMNA.2001.035.  Google Scholar

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J. M. Arrieta and A. N. Carvalho, Spectral convergence and nonlinear dynamics of reaction-diffusion equations under perturbations of the domain, J. Differential Equations, 199 (2004), 143-178.  doi: 10.1016/j.jde.2003.09.004.  Google Scholar

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J. M. ArrietaA. N. CarvalhoR. P. Silva and M. C. Pereira, Semilinear parabolic problems in thin domains with a highly oscillatory boundary, Nonlinear Anal., 74 (2011), 5111-5132.  doi: 10.1016/j.na.2011.05.006.  Google Scholar

[6]

J. M. ArrietaA. Nogueira and M. C. Pereira, Semilinear elliptic equations in thin regions with terms concentrating on oscillatory boundaries, Comput. Math. Appl., 77 (2019), 536-554.  doi: 10.1016/j.camwa.2018.09.056.  Google Scholar

[7]

J. M. Arrieta and M. Villanueva-Pesqueira, Elliptic and parabolic problems in thin domains with doubly oscillatory boundary, Commun. Pure Appl. Anal., 19 (2020), 1891-1914.  doi: 10.3934/cpaa.2020083.  Google Scholar

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T. Q. Bao, Regularity of pullback random attractors for stochastic FitzHugh-Nagumo system on unbounded domains, Disrete Contin. Dyn. Syst., 35 (2015), 441-466.  doi: 10.3934/dcds.2015.35.441.  Google Scholar

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D. CaoC. Sun and M. Yang, Dynamics for a stochastic reaction-diffusion equation with additive noise, J. Differential Equations, 259 (2015), 838-872.  doi: 10.1016/j.jde.2015.02.020.  Google Scholar

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I. Chueshov, Monotone Random Systems Theory and Applications, 1779, Springer Science & Business Media, 2002. doi: 10.1007/b83277.  Google Scholar

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I. Chueshov and S. Kuksin, Random kick-forced 3D Navier-Stokes equations in a thin domain, Arch. Ration. Mech. Anal., 188 (2008), 117-153.  doi: 10.1007/s00205-007-0068-2.  Google Scholar

[12]

H. CuiY. Li and J. Yin, Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles, Nonlinear Anal., 38 (2018), 303-324.  doi: 10.1016/j.na.2015.08.009.  Google Scholar

[13]

H. CuiP. E. Kloeden and F. Wu, Pathwise upper semi-continuity of random pullback attractors along the time axis, Phys. D, 374/375 (2018), 21-34.  doi: 10.1016/j.physd.2018.03.002.  Google Scholar

[14]

A. GuD. LiB. Wang and H. Yang, Regularity of random attractors for fractional stochastic reaction-diffusion equations on $\mathbb R^n$, J. Differential Equations, 264 (2018), 7094-7137.  doi: 10.1016/j.jde.2018.02.011.  Google Scholar

[15]

A. GuK. Lu and B. Wang, Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations, Discrete Contin. Dyn. Syst., 39 (2019), 185-218.  doi: 10.3934/dcds.2019008.  Google Scholar

[16]

A. Gu and B. Wang, Asymptotic behavior of random Fitzhugh-Nagumo systems driven by colored noise, Discrete Contin. Dyn. Syst. B, 23 (2018), 1689-1720.  doi: 10.3934/dcdsb.2018072.  Google Scholar

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[18]

J. K. Hale and G. Raugel, Reaction-diffusion equations on thin domains, J. Math. Pures Appl., 71 (1992), 33-95.   Google Scholar

[19]

J. K. Hale and G. Raugel, A reaction-diffusion equation on a thin L-shaped domain, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 283-327.  doi: 10.1017/S0308210500028043.  Google Scholar

[20]

P. E. KloedenJ. 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.  Google Scholar

[21]

P. E. Kloeden and J. Simsen, Attractors of asymptotically autonomous quasi-linear parabolic equation with spatially variable exponents, J. Math. Anal. Appl., 425 (2015), 911-918.  doi: 10.1016/j.jmaa.2014.12.069.  Google Scholar

[22]

D. Li and L. Shi, Upper semicontinuity of attractors of stochastic delay reaction-diffusion equations in the delay, J. Math. Phys., 59 (2018), 032703. doi: 10.1063/1.5031770.  Google Scholar

[23]

D. LiL. Shi and X. Wang, Long term behavior of stochastic discrete complex Ginzburg-Landau equations with time delays in weighted spaces, Discrete Contin. Dyn. Syst. B, 24 (2019), 5121-5148.  doi: 10.3934/dcdsb.2019046.  Google Scholar

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D. LiB. Wang and X. 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.  Google Scholar

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D. LiK. LuB. Wang and X. Wang, Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains, Discrete Contin. Dyn. Syst., 38 (2018), 187-208.  doi: 10.3934/dcds.2018009.  Google Scholar

[26]

F. Li and Y. Li, Asymptotic behavior of stochastic $g$-Navier-Stokes equations on a sequence of expanding domains, J. Math. Phys., 60 (2019), 061505. doi: 10.1063/1.5083695.  Google Scholar

[27]

F. LiY. Li and R. Wang, Limiting dynamics for stochastic reaction diffusion equations on the Sobolev space with thin domains, Comput. Math. Appl., 79 (2020), 457-475.  doi: 10.1016/j.camwa.2019.07.009.  Google Scholar

[28]

F. LiY. Li and R. Wang, Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise, Discrete Contin. Dyn. Syst., 38 (2018), 3663-3685.  doi: 10.3934/dcds.2018158.  Google Scholar

[29]

F. LiY. Li and R. Wang, Strong convergence of bi-spatial random attractors for parabolic on thin domains with rough noise, Topol. Methods Nonlinear Anal., 53 (2019), 659-682.  doi: 10.12775/TMNA.2019.015.  Google Scholar

[30]

Y. LiH. Cui and J. Li, Upper semi-continuity and regularity of random attractors on p-times integrable spaces and applications, Nonlinear Anal., 109 (2014), 33-44.  doi: 10.1016/j.na.2014.06.013.  Google Scholar

[31]

Y. LiA. 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.  Google Scholar

[32]

Y. Li and B. Guo, Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations, J. Differential Equations, 245 (2008), 1775-1800.  doi: 10.1016/j.jde.2008.06.031.  Google Scholar

[33]

Y. Li and F. Li, Limiting dynamics for stochastic FitzHugh-Nagumo equations on large domains, Stoch. Dyn., 19 (2019), 1950037. doi: 10.1142/S0219493719500370.  Google Scholar

[34]

Y. LiL. She and R. Wang, Asymptotically autonomous dynamics for parabolic equation, J. Math. Anal. Appl., 459 (2018), 1106-1123.  doi: 10.1016/j.jmaa.2017.11.033.  Google Scholar

[35]

Y. LiL. She and J. Yin, Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1535-1557.  doi: 10.3934/dcdsb.2018058.  Google Scholar

[36]

Y. Li and J. Yin, A modified proof of pullback attractors in a Sobolev space for stochastic Fitzhugh-Nagumo equations, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1203-1223.  doi: 10.3934/dcdsb.2016.21.1203.  Google Scholar

[37]

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

[38]

I. Pažanin and M. C. Pereira, On the nonlinear convection-diffusion-reaction problem in a thin domain with a weak boundary absorption, Commun. Pure Appl. Anal., 17 (2018), 579-592.  doi: 10.3934/cpaa.2018031.  Google Scholar

[39]

M. Prizzi and K. P. Rybakowski, Recent results on thin domain problems â…¡, Topol. Methods Nonlinear Anal., 19 (2002), 199-219.  doi: 10.12775/TMNA.2002.010.  Google Scholar

[40]

G. Raugel and G. R. Sell, Navier-Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions, J. Amer. Math. Soc., 6 (1993), 503-568.  doi: 10.2307/2152776.  Google Scholar

[41]

L. Shi, D. Li, X, Li and X. Wang, Dynamics of stochastic FitzHugh-Nagumo systems with additive noise on unbounded thin domains, Stoch. Dyn., 20 (2020), 2050018. doi: 10.1142/S0219493720500185.  Google Scholar

[42]

L. ShiR. WangK 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.  Google Scholar

[43]

M. Sui and Y. Wang, Upper semicontinuity of pullback attractors for lattice nonclassical diffusion delay equations under singular perturbations, Appl. Math. Comput., 242 (2014), 315-327.  doi: 10.1016/j.amc.2014.05.045.  Google Scholar

[44]

X. SongC. Sun and L. Yang, Pullback attractors for 2D Navier-Stokes equations on time-varying domains, Nonlinear Anal. Real World Appl., 45 (2019), 437-460.  doi: 10.1016/j.nonrwa.2018.07.013.  Google Scholar

[45]

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.  Google Scholar

[46]

S. Wang and Y. Li, Longtime robustness of pullback random attractors for stochastic magneto-hydrodynamics equations, Phys. D, 382/383 (2018), 46-57.  doi: 10.1016/j.physd.2018.07.003.  Google Scholar

[47]

X. WangK Lu and B. 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.  Google Scholar

[48]

X. WangK Lu and B. Wang, Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise, J. Dynam. Differential Equations, 28 (2016), 1309-1335.  doi: 10.1007/s10884-015-9448-8.  Google Scholar

[49]

X. WangK Lu and B. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 264 (2018), 378-424.  doi: 10.1016/j.jde.2017.09.006.  Google Scholar

[50]

W. Zhao, $H^1$-random attractors for stochastic reaction-diffusion equations with additive noise, Nonlinear Anal., 84 (2013), 61-72.  doi: 10.1016/j.na.2013.01.014.  Google Scholar

[51]

W. Zhao, Smoothing dynamics of the non-autonomous stochastic Fitzhugh-Nagumo system on $\mathbb{R}^N$ driven by multiplicative noises, Discrete Contin. Dyn. Syst. B, 24 (2019), 3453-3474.  doi: 10.3934/dcdsb.2018251.  Google Scholar

[52]

W. Zhao, Continuity and random dynamics of the non-autonomous stochastic FitzHugh-Nagumo system on $\mathbb{R}^N$, Comput. Math. Appl., 75 (2018), 3801-3824.  doi: 10.1016/j.camwa.2018.02.031.  Google Scholar

show all references

References:
[1]

A. Adili and B. Wang, Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 643-666.  doi: 10.3934/dcdsb.2013.18.643.  Google Scholar

[2]

F. Antoci and M. Prizzi, Reaction-diffusion equations on unbounded thin domains, Topol. Methods Nonlinear Anal., 18 (2001), 283-302.  doi: 10.12775/TMNA.2001.035.  Google Scholar

[3]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[4]

J. M. Arrieta and A. N. Carvalho, Spectral convergence and nonlinear dynamics of reaction-diffusion equations under perturbations of the domain, J. Differential Equations, 199 (2004), 143-178.  doi: 10.1016/j.jde.2003.09.004.  Google Scholar

[5]

J. M. ArrietaA. N. CarvalhoR. P. Silva and M. C. Pereira, Semilinear parabolic problems in thin domains with a highly oscillatory boundary, Nonlinear Anal., 74 (2011), 5111-5132.  doi: 10.1016/j.na.2011.05.006.  Google Scholar

[6]

J. M. ArrietaA. Nogueira and M. C. Pereira, Semilinear elliptic equations in thin regions with terms concentrating on oscillatory boundaries, Comput. Math. Appl., 77 (2019), 536-554.  doi: 10.1016/j.camwa.2018.09.056.  Google Scholar

[7]

J. M. Arrieta and M. Villanueva-Pesqueira, Elliptic and parabolic problems in thin domains with doubly oscillatory boundary, Commun. Pure Appl. Anal., 19 (2020), 1891-1914.  doi: 10.3934/cpaa.2020083.  Google Scholar

[8]

T. Q. Bao, Regularity of pullback random attractors for stochastic FitzHugh-Nagumo system on unbounded domains, Disrete Contin. Dyn. Syst., 35 (2015), 441-466.  doi: 10.3934/dcds.2015.35.441.  Google Scholar

[9]

D. CaoC. Sun and M. Yang, Dynamics for a stochastic reaction-diffusion equation with additive noise, J. Differential Equations, 259 (2015), 838-872.  doi: 10.1016/j.jde.2015.02.020.  Google Scholar

[10]

I. Chueshov, Monotone Random Systems Theory and Applications, 1779, Springer Science & Business Media, 2002. doi: 10.1007/b83277.  Google Scholar

[11]

I. Chueshov and S. Kuksin, Random kick-forced 3D Navier-Stokes equations in a thin domain, Arch. Ration. Mech. Anal., 188 (2008), 117-153.  doi: 10.1007/s00205-007-0068-2.  Google Scholar

[12]

H. CuiY. Li and J. Yin, Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles, Nonlinear Anal., 38 (2018), 303-324.  doi: 10.1016/j.na.2015.08.009.  Google Scholar

[13]

H. CuiP. E. Kloeden and F. Wu, Pathwise upper semi-continuity of random pullback attractors along the time axis, Phys. D, 374/375 (2018), 21-34.  doi: 10.1016/j.physd.2018.03.002.  Google Scholar

[14]

A. GuD. LiB. Wang and H. Yang, Regularity of random attractors for fractional stochastic reaction-diffusion equations on $\mathbb R^n$, J. Differential Equations, 264 (2018), 7094-7137.  doi: 10.1016/j.jde.2018.02.011.  Google Scholar

[15]

A. GuK. Lu and B. Wang, Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations, Discrete Contin. Dyn. Syst., 39 (2019), 185-218.  doi: 10.3934/dcds.2019008.  Google Scholar

[16]

A. Gu and B. Wang, Asymptotic behavior of random Fitzhugh-Nagumo systems driven by colored noise, Discrete Contin. Dyn. Syst. B, 23 (2018), 1689-1720.  doi: 10.3934/dcdsb.2018072.  Google Scholar

[17]

J. K. Hale and G. Raugel, A damped hyperbolic equation on thin domains, Trans. Amer. Math. Soc., 329 (1992), 185-219.  doi: 10.1090/S0002-9947-1992-1040261-1.  Google Scholar

[18]

J. K. Hale and G. Raugel, Reaction-diffusion equations on thin domains, J. Math. Pures Appl., 71 (1992), 33-95.   Google Scholar

[19]

J. K. Hale and G. Raugel, A reaction-diffusion equation on a thin L-shaped domain, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 283-327.  doi: 10.1017/S0308210500028043.  Google Scholar

[20]

P. E. KloedenJ. 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.  Google Scholar

[21]

P. E. Kloeden and J. Simsen, Attractors of asymptotically autonomous quasi-linear parabolic equation with spatially variable exponents, J. Math. Anal. Appl., 425 (2015), 911-918.  doi: 10.1016/j.jmaa.2014.12.069.  Google Scholar

[22]

D. Li and L. Shi, Upper semicontinuity of attractors of stochastic delay reaction-diffusion equations in the delay, J. Math. Phys., 59 (2018), 032703. doi: 10.1063/1.5031770.  Google Scholar

[23]

D. LiL. Shi and X. Wang, Long term behavior of stochastic discrete complex Ginzburg-Landau equations with time delays in weighted spaces, Discrete Contin. Dyn. Syst. B, 24 (2019), 5121-5148.  doi: 10.3934/dcdsb.2019046.  Google Scholar

[24]

D. LiB. Wang and X. 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.  Google Scholar

[25]

D. LiK. LuB. Wang and X. Wang, Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains, Discrete Contin. Dyn. Syst., 38 (2018), 187-208.  doi: 10.3934/dcds.2018009.  Google Scholar

[26]

F. Li and Y. Li, Asymptotic behavior of stochastic $g$-Navier-Stokes equations on a sequence of expanding domains, J. Math. Phys., 60 (2019), 061505. doi: 10.1063/1.5083695.  Google Scholar

[27]

F. LiY. Li and R. Wang, Limiting dynamics for stochastic reaction diffusion equations on the Sobolev space with thin domains, Comput. Math. Appl., 79 (2020), 457-475.  doi: 10.1016/j.camwa.2019.07.009.  Google Scholar

[28]

F. LiY. Li and R. Wang, Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise, Discrete Contin. Dyn. Syst., 38 (2018), 3663-3685.  doi: 10.3934/dcds.2018158.  Google Scholar

[29]

F. LiY. Li and R. Wang, Strong convergence of bi-spatial random attractors for parabolic on thin domains with rough noise, Topol. Methods Nonlinear Anal., 53 (2019), 659-682.  doi: 10.12775/TMNA.2019.015.  Google Scholar

[30]

Y. LiH. Cui and J. Li, Upper semi-continuity and regularity of random attractors on p-times integrable spaces and applications, Nonlinear Anal., 109 (2014), 33-44.  doi: 10.1016/j.na.2014.06.013.  Google Scholar

[31]

Y. LiA. 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.  Google Scholar

[32]

Y. Li and B. Guo, Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations, J. Differential Equations, 245 (2008), 1775-1800.  doi: 10.1016/j.jde.2008.06.031.  Google Scholar

[33]

Y. Li and F. Li, Limiting dynamics for stochastic FitzHugh-Nagumo equations on large domains, Stoch. Dyn., 19 (2019), 1950037. doi: 10.1142/S0219493719500370.  Google Scholar

[34]

Y. LiL. She and R. Wang, Asymptotically autonomous dynamics for parabolic equation, J. Math. Anal. Appl., 459 (2018), 1106-1123.  doi: 10.1016/j.jmaa.2017.11.033.  Google Scholar

[35]

Y. LiL. She and J. Yin, Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1535-1557.  doi: 10.3934/dcdsb.2018058.  Google Scholar

[36]

Y. Li and J. Yin, A modified proof of pullback attractors in a Sobolev space for stochastic Fitzhugh-Nagumo equations, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1203-1223.  doi: 10.3934/dcdsb.2016.21.1203.  Google Scholar

[37]

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

[38]

I. Pažanin and M. C. Pereira, On the nonlinear convection-diffusion-reaction problem in a thin domain with a weak boundary absorption, Commun. Pure Appl. Anal., 17 (2018), 579-592.  doi: 10.3934/cpaa.2018031.  Google Scholar

[39]

M. Prizzi and K. P. Rybakowski, Recent results on thin domain problems â…¡, Topol. Methods Nonlinear Anal., 19 (2002), 199-219.  doi: 10.12775/TMNA.2002.010.  Google Scholar

[40]

G. Raugel and G. R. Sell, Navier-Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions, J. Amer. Math. Soc., 6 (1993), 503-568.  doi: 10.2307/2152776.  Google Scholar

[41]

L. Shi, D. Li, X, Li and X. Wang, Dynamics of stochastic FitzHugh-Nagumo systems with additive noise on unbounded thin domains, Stoch. Dyn., 20 (2020), 2050018. doi: 10.1142/S0219493720500185.  Google Scholar

[42]

L. ShiR. WangK 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.  Google Scholar

[43]

M. Sui and Y. Wang, Upper semicontinuity of pullback attractors for lattice nonclassical diffusion delay equations under singular perturbations, Appl. Math. Comput., 242 (2014), 315-327.  doi: 10.1016/j.amc.2014.05.045.  Google Scholar

[44]

X. SongC. Sun and L. Yang, Pullback attractors for 2D Navier-Stokes equations on time-varying domains, Nonlinear Anal. Real World Appl., 45 (2019), 437-460.  doi: 10.1016/j.nonrwa.2018.07.013.  Google Scholar

[45]

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.  Google Scholar

[46]

S. Wang and Y. Li, Longtime robustness of pullback random attractors for stochastic magneto-hydrodynamics equations, Phys. D, 382/383 (2018), 46-57.  doi: 10.1016/j.physd.2018.07.003.  Google Scholar

[47]

X. WangK Lu and B. 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.  Google Scholar

[48]

X. WangK Lu and B. Wang, Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise, J. Dynam. Differential Equations, 28 (2016), 1309-1335.  doi: 10.1007/s10884-015-9448-8.  Google Scholar

[49]

X. WangK Lu and B. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 264 (2018), 378-424.  doi: 10.1016/j.jde.2017.09.006.  Google Scholar

[50]

W. Zhao, $H^1$-random attractors for stochastic reaction-diffusion equations with additive noise, Nonlinear Anal., 84 (2013), 61-72.  doi: 10.1016/j.na.2013.01.014.  Google Scholar

[51]

W. Zhao, Smoothing dynamics of the non-autonomous stochastic Fitzhugh-Nagumo system on $\mathbb{R}^N$ driven by multiplicative noises, Discrete Contin. Dyn. Syst. B, 24 (2019), 3453-3474.  doi: 10.3934/dcdsb.2018251.  Google Scholar

[52]

W. Zhao, Continuity and random dynamics of the non-autonomous stochastic FitzHugh-Nagumo system on $\mathbb{R}^N$, Comput. Math. Appl., 75 (2018), 3801-3824.  doi: 10.1016/j.camwa.2018.02.031.  Google Scholar

[1]

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