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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 and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021303
References:
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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]

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.

[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.

[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.

[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.

[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.

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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.

[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.

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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.

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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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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]

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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