Random uniform attractors for fractional stochastic FitzHugh-Nagumo lattice systems

Xintao Li; Yunlong Gao; Xintao Li; Yunlong Gao

doi:10.3934/math.20241083

AIMS Mathematics

2024, Volume 9, Issue 8: 22251-22270. doi: 10.3934/math.20241083

Previous Article Next Article

Research article

Random uniform attractors for fractional stochastic FitzHugh-Nagumo lattice systems

Xintao Li ^,,
Yunlong Gao

School of Mathematics and Statistics, Liupanshui Normal University, Liupanshui 553004, China

Received: 30 May 2024 Revised: 01 July 2024 Accepted: 09 July 2024 Published: 16 July 2024
MSC : 35B40, 35B41, 37L30

The present study focuses on the asymptotic behavior of fractional stochastic FitzHugh-Nagumo lattice systems with multiplicative noise. First, we investigate the well-posedness of solutions for these stochastic systems and subsequently establish the existence and uniqueness of tempered random uniform attractors.
- fractional stochastic FitzHugh-Nagumo lattice systems,
- multiplicative noise,
- random uniform attractor,
- almost periodic symbol
Citation: Xintao Li, Yunlong Gao. Random uniform attractors for fractional stochastic FitzHugh-Nagumo lattice systems[J]. AIMS Mathematics, 2024, 9(8): 22251-22270. doi: 10.3934/math.20241083

Related Papers:

Abstract

The present study focuses on the asymptotic behavior of fractional stochastic FitzHugh-Nagumo lattice systems with multiplicative noise. First, we investigate the well-posedness of solutions for these stochastic systems and subsequently establish the existence and uniqueness of tempered random uniform attractors.

References

[1]	S. N. Chow, J. Mallet-Paret, W. Shen, Traveling waves in lattice dynamical systems, J. Differ. Equations, 149 (1998), 248–291. https://doi.org/10.1006/jdeq.1998.3478 doi: 10.1006/jdeq.1998.3478
[2]	C. E. Elmer, E. S. Van Vleck, Traveling waves solutions for bistable differential-difference equations with periodic diffusion, SIAM J. Appl. Math., 61 (2001), 1648–1679. https://doi.org/10.1137/S0036139999357113 doi: 10.1137/S0036139999357113
[3]	S. N. Chow, J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems Ⅰ, IEEE Trans. Circuits Systems, 42 (1995), 746–751. https://doi.org/10.1109/81.473583 doi: 10.1109/81.473583
[4]	S. N. Chow, W. Shen, Dynamics in a discrete Nagumo equation: Spatial topological chaos, SIAM J. Appl. Math., 55 (1995), 1764–1781. https://doi.org/10.1137/S0036139994261757 doi: 10.1137/S0036139994261757
[5]	A. Y. Abdallah, Attractors for first order lattice systems with almost periodic nonlinear part, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 1241–1255. https://doi.org/10.3934/dcdsb.2019218 doi: 10.3934/dcdsb.2019218
[6]	Y. Chen, X. Wang, Random attractors for stochastic discrete complex Ginzburg-Landau equations with long-range interactions, J. Math. Phys., 63 (2022), 032701. https://doi.org/10.1063/5.0077971 doi: 10.1063/5.0077971
[7]	Z. Chen, L. Li, D. Yang, Asymptotic behavior of random coupled Ginzburg-Landau equation driven by colored noise on unbounded domains, Adv. Differ. Equat., 2021 (2021), 291. https://doi.org/10.1186/s13662-020-03127-5 doi: 10.1186/s13662-020-03127-5
[8]	Z. Chen, X. Li, B. Wang, Invariant measures of stochastic delay lattice systems, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 3235–3269. https://doi.org/10.3934/dcdsb.2020226 doi: 10.3934/dcdsb.2020226
[9]	D. Li, L. Shi, Upper semicontinuity of random attractors of stochastic discrete complex Ginzburg-Landau equations with time-varying delays in the delay, J. Differ. Equ. Appl., 4 (2018), 872–897. https://doi.org/10.1080/10236198.2018.1437913 doi: 10.1080/10236198.2018.1437913
[10]	D. Li, B. Wang, X. Wang, Limiting behavior of invariant measures of stochastic delay lattice systems, J. Dyn. Differ. Equ., 34 (2022), 1453–1487. https://doi.org/10.1007/s10884-021-10011-7 doi: 10.1007/s10884-021-10011-7
[11]	R. Wang, Long-time dynamics of stochastic lattice plate equations with nonlinear noise and damping, J. Dynam. Differ. Equ., 33 (2021), 767–803. https://doi.org/ 10.1007/s10884-020-09830-x doi: 10.1007/s10884-020-09830-x
[12]	R. Wang, B. Wang, Random dynamics of p-Laplacian lattice systems driven by infinite-dimensional nonlinear noise, Stoch. Proc. Appl., 130 (2020), 7431–7462. https://doi.org/10.1016/j.spa.2020.08.002 doi: 10.1016/j.spa.2020.08.002
[13]	R. Wang, B. Wang, Random dynamics of lattice wave equations driven by infinite-dimensional nonlinear noise, Discrete Contin. Dynam. Syst. Ser. B, 25 (2020), 2461–2493. https://doi.org/10.3934/dcdsb.2020019 doi: 10.3934/dcdsb.2020019
[14]	R. Wang, B. Wang, Global well-posedness and long-term behavior of discrete reaction-diffusion equations driven by superlinear noise, Stoch. Anal. Appl., 39 (2021), 667–696. https://doi.org/10.1080/07362994.2020.1828917 doi: 10.1080/07362994.2020.1828917
[15]	X. Wang, P. E. Kloeden, X. Han, Stochastic dynamics of a neural field lattice model with state dependent nonlinear noise, Nodea Nonlinear Differ., 28 (2021), 43. https://doi.org/10.1007/s00030-021-00705-8 doi: 10.1007/s00030-021-00705-8
[16]	X. Wang, K. Lu, B. Wang, Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise, J. Dyn. Differ. Equ., 28 (2016), 1309–1335. https://doi.org/10.1007/s10884-015-9448-8 doi: 10.1007/s10884-015-9448-8
[17]	S. Yang, Y. Li, Dynamics and invariant measures of multi-stochastic sine-Gordon lattices with random viscosity and nonlinear noise, J. Math. Phys., 62 (2021), 051510. https://doi.org/10.1063/5.0037929 doi: 10.1063/5.0037929
[18]	B. Wang, Sufficient and necessary criteria for existence of pullback attractors for noncompact random dynamical systems, J. Differ. Equations, 253 (2012), 1544–1583. https://doi.org/10.1016/j.jde.2012.05.015 doi: 10.1016/j.jde.2012.05.015
[19]	H. Cui, J. A. Langa, Uniform attractors for non-autonomous random dynamical systems, J. Differ. Equations, 263 (2017), 1225–1268. https://doi.org/10.1016/j.jde.2017.03.018 doi: 10.1016/j.jde.2017.03.018
[20]	H. Cui, A. C. Cunha, J. A. Langa, Finite-dimensionality of tempered random uniform attractors, J. Nonlinear Sci., 32 (2022), 13. https://doi.org/10.1007/s00332-021-09764-8 doi: 10.1007/s00332-021-09764-8
[21]	A. Y. Abdallah, Random uniform attractors for first order stochastic non-autonomous lattice systems, Qual. Theor. Dyn. Syst., 22 (2023), 60. https://doi.org/10.1007/s12346-023-00758-3 doi: 10.1007/s12346-023-00758-3
[22]	Ó. Ciaurri, L. Roncal, Hardy's inequality for the fractional powers of a discrete Laplacian, J. Anal., 26 (2018), 211–225. https://doi.org/10.1007/s41478-018-0141-2 doi: 10.1007/s41478-018-0141-2
[23]	Ó. Ciaurri, L. Roncal, P. R. Stinga, J. L. Torrea, J. L. Varona, Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications, Adv. Math., 30 (2018), 688–738. https://doi.org/10.1016/j.aim.2018.03.023 doi: 10.1016/j.aim.2018.03.023
[24]	C. Lizama, L. Roncal, Hölder-Lebesgue regularity and almost periodicity for semidiscrete equations with a fractional Laplacian, Discrete Contin. Dyn. Syst., 38 (2018), 1365–1403. https://dx.doi.org/10.3934/dcds.2018056 doi: 10.3934/dcds.2018056
[25]	Y. Chen, X. Wang, Asymptotic behavior of non-autonomous fractional stochastic lattice systems with multiplicative noise, Discrete Contin. Dyn. Syst. Ser. B, 27 (2022), 5205–5224. https://doi.org/10.3934/dcdsb.2021271 doi: 10.3934/dcdsb.2021271
[26]	Y. Chen, X. Wang, K. Wu, Wong-Zakai approximations and pathwise dynamics of stochastic fractional lattice systems, Commun. Pur. Appl. Anal., 21 (2022), 2529–2560. https://doi.org/10.3934/cpaa.2022059 doi: 10.3934/cpaa.2022059
[27]	C. K. R. T. Jones, Stability of the traveling wave solution of the FitzHugh-Nagumo system, Trans. Amer. Math. Soc., 286 (1984), 431–469. https://doi.org/10.1090/S0002-9947-1984-0760971-6 doi: 10.1090/S0002-9947-1984-0760971-6
[28]	B. Wang, Pullback attractors for the non-autonomous FitzHugh-Nagumo system on unbounded domains, Nonlinear Anal.-Theor., 70 (2009), 3799–3815. https://doi.org/10.1016/j.na.2008.07.011 doi: 10.1016/j.na.2008.07.011
[29]	E. Van Vleck, B. Wang, Attractors for lattice FitzHugh-Nagumo systems, Phys. D, 212 (2005), 317–336. https://doi.org/10.1016/j.physd.2005.10.006 doi: 10.1016/j.physd.2005.10.006
[30]	A. M. Boughoufala, A. Y. Abdallah, Attractors for FitzHugh-Nagumo lattice systems with almost periodic nonlinear parts, Discrete. Contin. Dyn. Syst. Ser. B, 26 (2021), 1549–1563. https://doi.org/10.3934/dcdsb.2020172 doi: 10.3934/dcdsb.2020172
[31]	A. Adili, B. Wang, Random attractors for non-autonomous stochasitic FitzHugh-Nagumo systems with multiplicative noise, Discrete Contin. Dyn. Syst., 2013 (2013), 1–10. https://doi.org/10.3934/proc.2013.2013.1 doi: 10.3934/proc.2013.2013.1
[32]	A. Gu, Y. Li, Singleton sets random attractor for stochastic FitzHugh-Nagumo lattice equations driven by fractional Brownian motions, Commun. Nonlinear Sci., 19 (2014), 3929–3937. https://doi.org/10.1016/j.cnsns.2014.04.005 doi: 10.1016/j.cnsns.2014.04.005
[33]	A. Gu, Y. Li, J. Li, Random attractors on lattice of stochastic FitzHugh-Nagumo systems driven by $\alpha$-stable Lévy noises, Int. J. Bifurcat. Chaos, 24 (2014), 1450123. https://doi.org/10.1142/S0218127414501235 doi: 10.1142/S0218127414501235
[34]	Z. Wang, S. Zhou, Random attractors for non-autonomous stochastic lattice FitzHugh-Nagumo systems with random coupled coefficients, Taiwan. J. Math., 20 (2016), 589–616. https://doi.org/10.11650/tjm.20.2016.6699 doi: 10.11650/tjm.20.2016.6699
[35]	Z. Chen, D. Yang, S. Zhong, Limiting dynamics for stochastic FitzHugh-Nagumo lattice systems in weighted spaces, J. Dyn. Diff. Equ., 36 (2024), 321–352. https://doi.org/10.1007/s10884-022-10145-2 doi: 10.1007/s10884-022-10145-2
[36]	P. R. Stinga, J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Commun. Part. Diff. Eq., 35 (2010), 2092–2122. https://doi.org/10.1080/03605301003735680 doi: 10.1080/03605301003735680
[37]	L. Arnold, Random dynamical systems, Springer-Verlag, Berlin, 1998.
[38]	B. M. Levitan, V. V. Zhikov, Almost periodic functions and differential equations, Cambridge Univ. Press, Cambridge, 1982.

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)