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

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July 2021, 26(7): 3517-3542. doi: 10.3934/dcdsb.2020244

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

Fuzhi Li ^, and Dongmei Xu

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 July 2021 Early access 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.

Keywords: Stochastic Fitzhugh-Nagumo systems, thin domain, bi-spatial random attractor, regularity, upper semi-continuity.

Mathematics Subject Classification: Primary:35B40;Secondary:35B41, 37L30.

Citation: Fuzhi Li, Dongmei Xu. Regular dynamics for stochastic Fitzhugh-Nagumo systems with additive noise on thin domains. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3517-3542. doi: 10.3934/dcdsb.2020244

References:

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[3]	L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.
[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.
[5]	J. M. Arrieta, A. N. Carvalho, R. 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.
[6]	J. M. Arrieta, A. 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.
[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.
[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.
[9]	D. Cao, C. 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.
[10]	I. Chueshov, Monotone Random Systems Theory and Applications, 1779, Springer Science & Business Media, 2002. doi: 10.1007/b83277.
[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.
[12]	H. Cui, Y. 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.
[13]	H. Cui, P. 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.
[14]	A. Gu, D. Li, B. 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.
[15]	A. Gu, K. 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.
[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.
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[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.
[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.
[23]	D. Li, L. 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.
[24]	D. Li, B. 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.
[25]	D. Li, K. Lu, B. 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.
[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.
[27]	F. Li, Y. 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.
[28]	F. Li, Y. 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.
[29]	F. Li, Y. 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.
[30]	Y. Li, H. 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.
[31]	Y. Li, A. 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.
[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.
[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.
[34]	Y. Li, L. 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.
[35]	Y. Li, L. 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.
[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.
[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.
[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.
[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.
[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.
[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.
[42]	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.
[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.
[44]	X. Song, C. 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.
[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.
[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.
[47]	X. Wang, K 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.
[48]	X. Wang, K 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.
[49]	X. Wang, K 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.
[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.
[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.
[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.

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.
[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.
[3]	L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.
[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.
[5]	J. M. Arrieta, A. N. Carvalho, R. 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.
[6]	J. M. Arrieta, A. 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.
[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.
[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.
[9]	D. Cao, C. 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.
[10]	I. Chueshov, Monotone Random Systems Theory and Applications, 1779, Springer Science & Business Media, 2002. doi: 10.1007/b83277.
[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.
[12]	H. Cui, Y. 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.
[13]	H. Cui, P. 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.
[14]	A. Gu, D. Li, B. 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.
[15]	A. Gu, K. 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.
[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.
[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.
[18]	J. K. Hale and G. Raugel, Reaction-diffusion equations on thin domains, J. Math. Pures Appl., 71 (1992), 33-95.
[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.
[20]	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.
[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.
[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.
[23]	D. Li, L. 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.
[24]	D. Li, B. 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.
[25]	D. Li, K. Lu, B. 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.
[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.
[27]	F. Li, Y. 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.
[28]	F. Li, Y. 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.
[29]	F. Li, Y. 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.
[30]	Y. Li, H. 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.
[31]	Y. Li, A. 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.
[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.
[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.
[34]	Y. Li, L. 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.
[35]	Y. Li, L. 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.
[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.
[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.
[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.
[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.
[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.
[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.
[42]	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.
[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.
[44]	X. Song, C. 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.
[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.
[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.
[47]	X. Wang, K 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.
[48]	X. Wang, K 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.
[49]	X. Wang, K 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.
[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.
[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.
[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.

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