Periodic measures of reaction-diffusion lattice systems driven by superlinear noise

Yusen Lin; Yusen Lin

doi:10.3934/era.2022002

Electronic Research Archive

2022, Volume 30, Issue 1: 35-51. doi: 10.3934/era.2022002

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Research article

Periodic measures of reaction-diffusion lattice systems driven by superlinear noise

Yusen Lin ^,

School of Mathematics, Southwest Jiaotong University, Chengdu, 610031, China; National Engineering Laboratory of Integrated Transportation Big Data Application Technology, Chengdu, 610031, China

Received: 18 May 2021 Revised: 09 October 2021 Accepted: 09 October 2021 Published: 09 December 2021

The periodic measures are investigated for a class of reaction-diffusion lattice systems driven by superlinear noise defined on $ \mathbb Z^k $. The existence of periodic measures for the lattice systems is established in $ l^2 $ by Krylov-Bogolyubov's method. The idea of uniform estimates on the tails of solutions is employed to establish the tightness of a family of distribution laws of the solutions.
- superlinear noise,
- periodic measure,
- reaction-diffusion lattice system
Citation: Yusen Lin. Periodic measures of reaction-diffusion lattice systems driven by superlinear noise[J]. Electronic Research Archive, 2022, 30(1): 35-51. doi: 10.3934/era.2022002

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Abstract

The periodic measures are investigated for a class of reaction-diffusion lattice systems driven by superlinear noise defined on $ \mathbb Z^k $. The existence of periodic measures for the lattice systems is established in $ l^2 $ by Krylov-Bogolyubov's method. The idea of uniform estimates on the tails of solutions is employed to establish the tightness of a family of distribution laws of the solutions.

References

[1]	J. Bell, C. Cosner, Threshold behaviour and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons, Quart. Appl. Math., 42 (1984), 1–14. https://doi.org/10.1090/qam/736501 doi: 10.1090/qam/736501
[2]	L. O. Chua, T. Roska, The CNN paradigm, IEEE Trans. Circuits Syst., 40 (1993), 147–156. https://doi.org/10.1109/81.222795 doi: 10.1109/81.222795
[3]	L. O. Chua, Y. Yang, Cellular neural networks: theory, IEEE Trans. Circuits Syst., 35 (1988), 1257–1272. https://doi.org/10.1109/31.7600 doi: 10.1109/31.7600
[4]	R. Kapval, Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113–163. https://doi.org/10.1007/BF01192578 doi: 10.1007/BF01192578
[5]	J.P. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47 (1987), 556–572. https://doi.org/10.1137/0147038 doi: 10.1137/0147038
[6]	J.P. Keener, The effects of discrete gap junction coupling on propagation in myocardium, J. Theor. Biol., 148 (1991), 49–82. https://doi.org/10.1016/S0022-5193(05)80465-5 doi: 10.1016/S0022-5193(05)80465-5
[7]	X. Han, P.E. Kloeden, B. Usman, Upper semi-continuous convergence of attractors for a Hopfield-type lattice model, Nonlinearity, 33 (2020), 1881–1906. https://doi.org/10.1088/1361-6544/ab6813 doi: 10.1088/1361-6544/ab6813
[8]	S. Zhou, Attractors and approximations for lattice dynamical systems, J. Differ. Equ., 200 (2004), 342–368. https://doi.org/10.1016/j.jde.2004.02.005 doi: 10.1016/j.jde.2004.02.005
[9]	P. W. Bates, K. Lu, B. Wang, Attractors for lattice dynamical systems, Int. J. Bifurcat. Chaos, 11 (2001), 143–153. https://doi.org/10.1142/S0218127401002031 doi: 10.1142/S0218127401002031
[10]	A. Gu, P.E. Kloeden, Asymptotic behavior of a nonautonomous p-Laplacian lattice system, Int. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1650174. https://doi.org/10.1142/S0218127416501741 doi: 10.1142/S0218127416501741
[11]	Z. Chen, X. Li, B. Wang, Invariant measures of stochastic delay lattice systems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017).
[12]	X. Han, P.E. Kloeden, B. Usman, Long term behavior of a random Hopfield neural lattice model, Commun. Pure Appl. Anal., 18 (2019), 809–824. https://doi.org/10.3934/cpaa.2019039 doi: 10.3934/cpaa.2019039
[13]	Y. Lin, D. Li, Periodic measures of impulsive stochastic Hopfield-type lattice systems, Stoch. Anal. Appl., (2021), 1–17. https://doi.org/10.1080/07362994.2021.1970582 doi: 10.1080/07362994.2021.1970582
[14]	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., 24 (2018), 872–897. https://doi.org/10.1080/10236198.2018.1437913 doi: 10.1080/10236198.2018.1437913
[15]	D. Li, B. Wang, X. Wang, Periodic measures of stochastic delay lattice systems, J. Differ. Equ., 272 (2021), 74–104. https://doi.org/10.1016/j.jde.2020.09.034 doi: 10.1016/j.jde.2020.09.034
[16]	D. Li, B. Wang, X. Wang, Limiting Behavior of Invariant Measures of Stochastic Delay Lattice Systems, J. Dyn. Differ. Equ., (2021), 1–35.
[17]	B. Wang, R. Wang, Asymptotic behavior of stochastic Schrodinger lattice systems driven by nonlinear noise, Stoch. Anal. Appl., 38 (2020), 213–237. https://doi.org/10.1080/07362994.2019.1679646 doi: 10.1080/07362994.2019.1679646
[18]	R. Wang, B. Wang, Random dynamics of p-Laplacian lattice systems driven by infinite-dimensional nonlinear noise, Stoch. Process. Appl., 130 (2020), 7431–7462. https://doi.org/10.1016/j.spa.2020.08.002 doi: 10.1016/j.spa.2020.08.002
[19]	B. Wang, Dynamics of stochastic reaction-diffusion lattice systems driven by nonlinear noise, J. Math. Anal. Appl., 477 (2019), 104–132. https://doi.org/10.1016/j.jmaa.2019.04.015 doi: 10.1016/j.jmaa.2019.04.015
[20]	R. Wang, B. Wang, Global well-posedness and long-term behavior of discrete reaction-diffusion equations driven by superlinear noise, Stoch. Anal. Appl., (2020).
[21]	X. Mao, Stochastic Differential Equations and Applications, Second Edition, Woodhead Publishing Limited, Cambridge, 2011.
[22]	B. Wang, Dynamics of systems on infinite lattices, J. Differ. Equ., 221 (2006), 224–245. https://doi.org/10.1016/j.jde.2005.01.003 doi: 10.1016/j.jde.2005.01.003
[23]	B. Wang, Attractors for reaction-diffusion equations in unbounded domains, Phys. D, 128 (1999), 41–52. https://doi.org/10.1016/S0167-2789(98)00304-2 doi: 10.1016/S0167-2789(98)00304-2
[24]	L. Arnold, Stochastic Differential Equations: Theory and Applications, New York: John Wiley and Sons Inc, 1974.
[25]	P. W. Bates, K. Lu, B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differ. Equ., 246 (2009), 845–869. https://doi.org/10.1016/j.jde.2008.05.017 doi: 10.1016/j.jde.2008.05.017
[26]	L. Chen, Z. Dong, J. Jiang, J. Zhai, On limiting behavior of stationary measures for stochastic evolution systems with small noise intensity, Sci. China Math., (2019), 1–42.
[27]	R. Z. Khasminskii, Stochastic Stability of Differential Equations, Springer, New York, 2012. https://doi.org/10.1007/978-3-642-23280-0
[28]	D. Li, Y. Lin, Periodic measures of impulsive stochastic differential equations, Chaos Soliton. Fract., 148 (2021), 111035. https://doi.org/10.1016/j.chaos.2021.111035 doi: 10.1016/j.chaos.2021.111035

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