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doi: 10.3934/dcdsb.2021303
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Continuity of random attractors on a topological space and fractional delayed FitzHugh-Nagumo equations with WZ-noise

1. 

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

2. 

Department of Mathematics, Nanning Normal University, Nanning 530001, China

* Corresponding author: liyr@swu.edu.cn (Yangrong Li)

Received  July 2021 Revised  November 2021 Early access December 2021

Fund Project: This work was supported by the Natural Science Foundation of China Grant 11571283

We study the continuity of a family of random attractors parameterized in a topological space (perhaps non-metrizable). Under suitable conditions, we prove that there is a residual dense subset $ \Lambda^* $ of the parameterized space such that the binary map $ (\lambda, s)\mapsto A_\lambda(\theta_s \omega) $ is continuous at all points of $ \Lambda^*\times \mathbb{R} $ with respect to the Hausdorff metric. The proofs are based on the generalizations of Baire residual Theorem (by Hoang et al. PAMS, 2015), Baire density Theorem and a convergence theorem of random dynamical systems from a complete metric space to the general topological space, and thus the abstract result, even restricted in the deterministic case, is stronger than those in literature. Finally, we establish the residual dense continuity and full upper semi-continuity of random attractors for the random fractional delayed FitzHugh-Nagumo equation driven by nonlinear Wong-Zakai noise, where the size of noise belongs to the parameterized space $ (0, \infty] $ and the infinity of noise means that the equation is deterministic.

Citation: Yangrong Li, Shuang Yang, Guangqing Long. Continuity of random attractors on a topological space and fractional delayed FitzHugh-Nagumo equations with WZ-noise. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021303
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Z. BrzezniakU. Manna and D. Mukherjee, Wong-Zakai approximation for the stochastic Landau-Lifshitz-Gilbert equations, J. Differential Equations, 267 (2019), 776-825.  doi: 10.1016/j.jde.2019.01.025.  Google Scholar

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[21]

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[22]

Y. Li and S. Yang, Almost continuity of a pullback random attractor for the stochastic g-Navier-Stokes equation, Dyn. Partial Differ. Equ., 18 (2021), 231-256.  doi: 10.4310/DPDE.2021.v18.n3.a4.  Google Scholar

[23]

Y. Li, S. Yang and Q. Zhang, Continuous Wong-Zakai approximations of random attractors for quasi-linear equations with nonlinear noise, Qual. Theory Dyn. Syst., 19 (2020), Paper No: 87, 31 pp. doi: 10.1007/s12346-020-00423-z.  Google Scholar

[24]

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

[25]

H. LuJ. QiB. Wang and M. Zhang, Random attractors for non-autonomous fractional stochastic parabolic equations on unbounded domains, Discrete Contin. Dyn. Syst., 39 (2019), 683-706.  doi: 10.3934/dcds.2019028.  Google Scholar

[26]

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

[27]

U. MannaD. Mukherjee and A. A. Panda, Wong-Zakai approximation for the stochastic Landau-Lifshitz-Gilbert equations with anisotropy energy, J. Math. Anal. Appl., 480 (2019), 1-13.  doi: 10.1016/j.jmaa.2019.123384.  Google Scholar

[28]

J. C. Oxtoby, Measure and Category, 2$^{nd}$ edition, Graduate Texts in Mathematics, 2. Springer-Verlag, New York-Berlin, 1980.  Google Scholar

[29]

J. C. Robinson, Stability of random attractors under perturbation and approximation, J. Differential Equations, 186 (2002), 652-669.  doi: 10.1016/S0022-0396(02)00038-4.  Google Scholar

[30]

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

[31]

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.  Google Scholar

[32]

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.  Google Scholar

[33]

B. Wang, Random attractors for the stochastic FitzHugh-Nagumo system on unbounded domains, Nonlinear Anal., 71 (2009), 2811-2828.  doi: 10.1016/j.na.2009.01.131.  Google Scholar

[34]

F. Wang, J. Li and Y. Li, Random attractors for Ginzburg-Landau equations driven by difference noise of a Wiener-like process, Adv. Difference Equ., (2019), Paper No. 224, 17 pp. doi: 10.1186/s13662-019-2165-6.  Google Scholar

[35]

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

[36]

S. Wang and Y. Li, Probabilistic continuity of a pullback random attractor in time-sample, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 2699-2722.  doi: 10.3934/dcdsb.2020028.  Google Scholar

[37]

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

[38]

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

[39]

X. WangJ. ShenK. Lu and B. Wang, Wong-Zakai approximations and random attractors for non-autonomous stochastic lattice systems, J. Differential Equations, 280 (2021), 477-516.  doi: 10.1016/j.jde.2021.01.026.  Google Scholar

[40]

E. Wong and M. Zakai, On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Statist., 36 (1965), 1560-1564.  doi: 10.1214/aoms/1177699916.  Google Scholar

[41]

W. Zhao, Y. Zhang and S. Chen, Higher-order Wong-Zakai approximations of stochastic reaction-diffusion equations on R-N, Physica D, 401 (2020), Paper No. 132147, 15 pp. doi: 10.1016/j.physd.2019.132147.  Google Scholar

[42]

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

[43]

S. Zhou, Upper-semicontinuity of attractors for random lattice systems perturbed by small white noises, Nonlinear Anal., 75 (2012), 2793-2805.  doi: 10.1016/j.na.2011.11.022.  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]

S. Aida and K. Sasaki, Wong-Zakai approximation of solutions to reflecting stochastic differential equations on domains in Euclidean spaces, Stoch. Proc. Appl., 123 (2013), 3800-3827.  doi: 10.1016/j.spa.2013.05.004.  Google Scholar

[3]

P. W. BatesK. Lu and B. 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.  Google Scholar

[4]

Z. BrzezniakU. Manna and D. Mukherjee, Wong-Zakai approximation for the stochastic Landau-Lifshitz-Gilbert equations, J. Differential Equations, 267 (2019), 776-825.  doi: 10.1016/j.jde.2019.01.025.  Google Scholar

[5]

T. CaraballoJ. A. Langa and J. C. Robinson, Upper semicontinuity of attractors for small random perturbations of dynamical systems, Comm. Partial Differential Equations, 23 (1998), 1557-1581.  doi: 10.1080/03605309808821394.  Google Scholar

[6]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-dimensional Non-Autonomous Dynamical Systems, Appl. Math. Sciences, 182, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[7]

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

[8]

H. CuiJ. 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.  Google Scholar

[9]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[10]

F. FlandoliM. Gubinelli and E. Priola, Well-posedness of the transport equation by stochastic perturbation, Invent. Math., 180 (2010), 1-53.  doi: 10.1007/s00222-009-0224-4.  Google Scholar

[11]

A. Gu, Asymptotic behavior of random lattice dynamical systems and their Wong-Zakai approximations, Discrete Contin. Dyn. Syst. B, 24 (2019), 5737-5767.  doi: 10.3934/dcdsb.2019104.  Google Scholar

[12]

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

[13]

L. T. HoangE. J. Olson and J. C. Robinson, On the continuity of global attractors, Proc. Amer. Math. Soc., 143 (2015), 4389-4395.  doi: 10.1090/proc/12598.  Google Scholar

[14]

L. T. HoangE. J. Olson and J. C. Robinson, Continuity of pullback and uniform attractors, J. Differential Equations, 264 (2018), 4067-4093.  doi: 10.1016/j.jde.2017.12.002.  Google Scholar

[15]

M. Jara, Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps, Comm. Pure Appl. Math., 62 (2009), 198-214.  doi: 10.1002/cpa.20253.  Google Scholar

[16]

P. E. KloedenJ. Simsen and M. S. Simsen, Asymptotically autonomous multivalued Cauchy problems with spatially variable exponents, J. Math. Anal. Appl., 445 (2017), 513-531.  doi: 10.1016/j.jmaa.2016.08.004.  Google Scholar

[17]

D. LiK. LuB. Wang and X. Wang, Limiting dynamics for non-autonomous stochastic retarded reaction-diffusion equations on thin domains, Discrete Contin. Dyn. Syst., 39 (2019), 3717-3747.  doi: 10.3934/dcds.2019151.  Google Scholar

[18]

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

[19]

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

[20]

Y. LiF. Wang and S. Yang, Part-convergent cocycles and semi-convergent attractors of stochastic 2D-Ginzburg-Landau delay equations toward zero-memory, Discrete Contin. Dyn. Syst. B, 26 (2021), 3643-3665.  doi: 10.3934/dcdsb.2020250.  Google Scholar

[21]

Y. Li and S. Yang, Hausdorff sub-norm spaces and continuity of random attractors for bi-stochastic g-Navier-Stokes equations with respect to tempered forces, J. Dyn. Differential Equations, (2021).  doi: 10.1007/s10884-021-10026-0.  Google Scholar

[22]

Y. Li and S. Yang, Almost continuity of a pullback random attractor for the stochastic g-Navier-Stokes equation, Dyn. Partial Differ. Equ., 18 (2021), 231-256.  doi: 10.4310/DPDE.2021.v18.n3.a4.  Google Scholar

[23]

Y. Li, S. Yang and Q. Zhang, Continuous Wong-Zakai approximations of random attractors for quasi-linear equations with nonlinear noise, Qual. Theory Dyn. Syst., 19 (2020), Paper No: 87, 31 pp. doi: 10.1007/s12346-020-00423-z.  Google Scholar

[24]

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

[25]

H. LuJ. QiB. Wang and M. Zhang, Random attractors for non-autonomous fractional stochastic parabolic equations on unbounded domains, Discrete Contin. Dyn. Syst., 39 (2019), 683-706.  doi: 10.3934/dcds.2019028.  Google Scholar

[26]

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

[27]

U. MannaD. Mukherjee and A. A. Panda, Wong-Zakai approximation for the stochastic Landau-Lifshitz-Gilbert equations with anisotropy energy, J. Math. Anal. Appl., 480 (2019), 1-13.  doi: 10.1016/j.jmaa.2019.123384.  Google Scholar

[28]

J. C. Oxtoby, Measure and Category, 2$^{nd}$ edition, Graduate Texts in Mathematics, 2. Springer-Verlag, New York-Berlin, 1980.  Google Scholar

[29]

J. C. Robinson, Stability of random attractors under perturbation and approximation, J. Differential Equations, 186 (2002), 652-669.  doi: 10.1016/S0022-0396(02)00038-4.  Google Scholar

[30]

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

[31]

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.  Google Scholar

[32]

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.  Google Scholar

[33]

B. Wang, Random attractors for the stochastic FitzHugh-Nagumo system on unbounded domains, Nonlinear Anal., 71 (2009), 2811-2828.  doi: 10.1016/j.na.2009.01.131.  Google Scholar

[34]

F. Wang, J. Li and Y. Li, Random attractors for Ginzburg-Landau equations driven by difference noise of a Wiener-like process, Adv. Difference Equ., (2019), Paper No. 224, 17 pp. doi: 10.1186/s13662-019-2165-6.  Google Scholar

[35]

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

[36]

S. Wang and Y. Li, Probabilistic continuity of a pullback random attractor in time-sample, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 2699-2722.  doi: 10.3934/dcdsb.2020028.  Google Scholar

[37]

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

[38]

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

[39]

X. WangJ. ShenK. Lu and B. Wang, Wong-Zakai approximations and random attractors for non-autonomous stochastic lattice systems, J. Differential Equations, 280 (2021), 477-516.  doi: 10.1016/j.jde.2021.01.026.  Google Scholar

[40]

E. Wong and M. Zakai, On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Statist., 36 (1965), 1560-1564.  doi: 10.1214/aoms/1177699916.  Google Scholar

[41]

W. Zhao, Y. Zhang and S. Chen, Higher-order Wong-Zakai approximations of stochastic reaction-diffusion equations on R-N, Physica D, 401 (2020), Paper No. 132147, 15 pp. doi: 10.1016/j.physd.2019.132147.  Google Scholar

[42]

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

[43]

S. Zhou, Upper-semicontinuity of attractors for random lattice systems perturbed by small white noises, Nonlinear Anal., 75 (2012), 2793-2805.  doi: 10.1016/j.na.2011.11.022.  Google Scholar

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