Weak mean attractors of fractional stochastic lattice systems driven by nonlinear delay noise

Ailin Bai; Pengyu Chen

doi:10.61383/ejam.20242485

Authors

Ailin Bai Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
Pengyu Chen Department of Mathematics, Northwest Normal University, Lanzhou 730070, China Corresponding Author https://orcid.org/0000-0003-2891-4763

DOI:

https://doi.org/10.61383/ejam.20242485

Keywords:

Fractional stochastic lattice system, nonlinear delay noise, weak mean random attractor

Abstract

This paper deals with the existence and uniqueness of weak pullback mean random attractors for fractional stochastic lattice systems driven by nonlinear delay noise defined on the entire integer set \(\mathbb{Z}\). We first establish the global well-posedness to stochastic lattice system in \(C([\tau,\infty), L^2(\Omega, \ell^2))\) when the nonlinear diffusion terms and drift terms are locally Lipschitz continuous functions. Then we define a mean random dynamical system through the solution operators and prove the existence and uniqueness of weak pullback mean random attractors in \(L^2(\Omega,\mathcal{F};\ell^2)\times L^2\big(\Omega,\mathcal{F};L^2((-\rho,0), \ell^2)\big)\) under certain conditions.

References

[1] L. Arnold, Stochastic Differential Equations: Theory and Applications, John Wiley and Sons Inc, New York, 1974.

[2] P.W. Bates, K. Lu, B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Phys. D289(2014), 32-50. DOI: https://doi.org/10.1016/j.physd.2014.08.004

[3] ´ O. 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., 330 (2018), 688-738. DOI: https://doi.org/10.1016/j.aim.2018.03.023

[4] ´ O. Ciaurri, L. Roncal, Hardy’s inequality for the fractional powers of a discrete Laplacian, J. Anal., 26 (2018), 211-225. DOI: https://doi.org/10.1007/s41478-018-0141-2

[5] T. Caraballo, F. Morillas, J. Valerom, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, J. Differential Equations 253 (2012), 667-693. DOI: https://doi.org/10.1016/j.jde.2012.03.020

[6] 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. DOI: https://doi.org/10.3934/dcdsb.2021271

[7] Y. Chen, X. Wang, K. Wu, Wong-Zakai approximations and pathwise dynamics stochastic fractional lattice systems, Commun. Pure Appl. Anal., 21 (2022), 2529-2560. DOI: https://doi.org/10.3934/cpaa.2022059

[8] Z. Chen, B. Wang, Weak mean attractors and invariant measures for stochastic Schr¨odinger delay lattice systems, J. Dynam. Differential Equations, 35 (2023), 3201-3240. DOI: https://doi.org/10.1007/s10884-021-10085-3

[9] Z. Chen, B. Wang, Existence, exponential mixing and convergence of periodic measures of fractional stochastic delay reaction-diffusion equations on Rn, J. Differential Equations, 336 (2022), 505-564. DOI: https://doi.org/10.1016/j.jde.2022.07.026

[10] Z. Chen, X. Li, B. Wang, Invariant measures of stochastic delay lattice systems, Discrete Contin. Dyn. Syst. Ser. B 26 (2021), 3235-3269. DOI: https://doi.org/10.3934/dcdsb.2020226

[11] S.N. Chow, J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems, I, II, IEEE Trans. Circuits Syst. 42 (1995), 746-751. DOI: https://doi.org/10.1109/81.473583

[12] S.N. Chow, Lattice dynamical systems, Dynamical Systems, Lecture Notes in Math., 1822, Springer, Berlin, 1-102 (2003). DOI: https://doi.org/10.1007/978-3-540-45204-1_1

[13] M. Kwasn´ ıcki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20 (2017), 7-51. DOI: https://doi.org/10.1515/fca-2017-0002

[14] P. E. Kloeden, T. Lorenz, Mean-square random dynamical systems, J. Differential Equations, 253 (2012), 1422-1438. DOI: https://doi.org/10.1016/j.jde.2012.05.016

[15] C. Lizama, L. Roncal, H¨older-Lebesgue regularity and almost periodicity for semidiscrete equations with a fractional Laplacian, Discrete Contin. Dyn. Syst., 38 (2018), 1365-1403. DOI: https://doi.org/10.3934/dcds.2018056

[16] D. Li, B. Wang, X. Wang, Periodic measures of stochastic delay lattice systems, J. Differential Equations, 272 (2021), 74-104. DOI: https://doi.org/10.1016/j.jde.2020.09.034

[17] D. Li, B. Wang, X. Wang, Limiting behavior of invariant measures of stochastic delay lattice systems, J. Dynam. Differential Equations, 34 (2022), 1453-1487. DOI: https://doi.org/10.1007/s10884-021-10011-7

[18] Y. Li, A. Gu, 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: https://doi.org/10.1016/j.jde.2014.09.021

[19] B. Wang, Dynamics of systems on infinite lattices, J. Differential Equations 221 (2006), 224-245. DOI: https://doi.org/10.1016/j.jde.2005.01.003

[20] B. Wang, Weak pullback attractors for mean random dynamical systems in bochner spaces, J. Dynam. Differential Equations 31 (2019), 2177-2204. DOI: https://doi.org/10.1007/s10884-018-9696-5

[21] B. Wang, Dynamics of stochastic reaction-diffusion lattice system driven by nonlinear noise, J. Math. Anal. Appl. 477 (2019), 104-132. DOI: https://doi.org/10.1016/j.jmaa.2019.04.015

[22] X. Wang, K. Lu, B. Wang, Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise, J. Dynam. Differ. Equ. 28 (2016), 1309-1335. DOI: https://doi.org/10.1007/s10884-015-9448-8

[23] X. Zhang, P. Chen, Weak mean attractors of stochastic p-Laplacian delay lattice systems driven by nonlinear noise, Bull. Sci. math. 182 (2023), 103230 DOI: https://doi.org/10.1016/j.bulsci.2023.103230