分数阶非局部反应扩散方程的随机吸引子及不变测度

Abstract

Abstract:

This paper investigates the dynamical behavior of a class of fractional nonlocal reaction-diffusion equation driven by nonlinear colored noise. Firstly, the existence of solutions to the equations with nonlinear colored noise is given by the Galerkin method. Secondly, the existence of the pullback stochastic attractor for this equation is proved in an appropriate Hilbert space. Finally, the generalized Banach limit is used to prove the existence of invariant measures for this equation.

Key words: fractional nonlocal reaction-diffusion equation, random attractor, invariant measure, colored noise.

CLC Number:

O175.2

Yiyao Yang, Jinping Jiang, Nannan Ma. Stochastic Attractors and Invariant Measures for Fractional Nonlocal Reaction-Diffusion Equations[J].Acta mathematica scientia,Series A, 2026, 46(3): 949-962.

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References 24

[1]	Chipot M, Lovat B. On the asymptotic behaviour of some nonlocal problems. Positivity, 1999, 3 (1): 65-81 doi: 10.1023/A:1009706118910
[2]	Caraballo T, Herrera-Cobos M, Marín-Rubio P. Long-time behavior of a non-autonomous parabolic equation with nonlocal diffusion and sublinear terms. Nonlinear Anal: Theory Methods Appl, 2015, 121 : 3-18 doi: 10.1016/j.na.2014.07.011
[3]	Xu J H, Zhang Z C, Caraballo T. Non-autonomous nonlocal partial differential equations with delay and memory. J Differ Equ, 2021, 270 : 505-546 doi: 10.1016/j.jde.2020.07.037
[4]	Anguiano M, Kloeden P E, Lorenz T. Asymptotic behaviour of nonlocal reaction-diffusion equations. Nonlinear Anal: Theory Methods Appl, 2010, 73 (9): 3044-3057 doi: 10.1016/j.na.2010.06.073
[5]	Berna P M, Rossi J D. Nonlocal diffusion equations with dynamical boundary conditions. Nonlinear Anal: Theory Methods Appl, 2020, 195 : Art 111751
[6]	Caraballo T, Herrera-Cobos M, Marín-Rubio P. Time-dependent attractors for non-autonomous non-local reaction-diffusion equations. Proc R Soc Edinb A: Math, 2018, 148 (5): 957-981
[7]	Xu J H, Caraballo T. Long time behavior of stochastic nonlocal partial differential equations and Wong-Zakai approximations. SIAM J Math Anal, 2022, 54 (3): 2792-2844 doi: 10.1137/21M1412645
[8]	Xu J H, Caraballo T. Dynamics of stochastic nonlocal partial differntial equations. Eur Phys J Plus, 2021, 136 (8): Art 849
[9]	Li Y R, Liu G F, Xia H. Mean continuity and local controllability of pullback random attractors for random nonlocal equations. Evol Equ Control Theory, 2024, 13 (3): 950-972
[10]	Di Nezza E, Palatucci G, Valdinoci E. Hitchhiker's guide to the fractional Sobolev spaces. Bull Sci Math, 2012, 136 (5): 521-573 doi: 10.1016/j.bulsci.2011.12.004
[11]	Xu J H, Zhang Z C, Caraballo T. Mild solutions to time fractional stochastic 2D-Stokes equations with bounded and unbounded delay. J Dyn Differ Equ, 2022, 34 (1): 583-603 doi: 10.1007/s10884-019-09809-3
[12]	Chen Z, Wang B X. Invariant measures of fractional stochastic delay reaction-diffusion equations on unbounded domains. Nonlinearity, 2021, 34 (6): 3969-4016 doi: 10.1088/1361-6544/ac0125
[13]	Li F Z, Freitas M M. Asymptotically autonomous dynamics for fractional subcritical nonclassical diffusion equations driven by nonlinear colored noise. Fract Calc Appl Anal, 2023, 26 (1): 414-438 doi: 10.1007/s13540-022-00112-5
[14]	Wang R H, Guo B L, Wang B X. Well-posedness and dynamics of fractional FitzHugh-Nagumo systems on $\mathbb{R}^N$ driven by nonlinear noise. Sci China Math, 2021, 64 (11): 2395-2436 doi: 10.1007/s11425-019-1714-2
[15]	Chen Z, Wang B X. Limit measures and ergodicity of fractional stochastic reaction-diffusion equations on unbounded domains. Stoch Dyn, 2022, 22 (2): Art 2140012
[16]	Li L Y, Chen Z, Caraballo T. Dynamics of a stochastic fractional nonlocal reaction-diffusion model driven by additive noise. Discrete Contin Dyn Syst S, 2022, 16 (10): 2530-2558
[17]	Xu J H, Caraballo T. Dynamics of stochastic nonlocal reaction-diffusion equations driven by multiplicative noise. Anal Appl, 2023, 21 (3): 597-633 doi: 10.1142/S0219530522500075
[18]	Chen Z, Yang D D. Invariant measures and stochastic Liouville type theorem for non-autonomous stochastic reaction-diffusion equations. J Differ Equ, 2023, 353 : 225-267 doi: 10.1016/j.jde.2022.12.030
[19]	Servadei R, Valdinoci E. Variational methods for non-local operators of elliptic type. Discrete Contin Dyn Syst, 2013, 33 (5): 2105-2137 doi: 10.3934/dcds.2013.33.2105
[20]	Wang B X. Asymptotic behavior of non-autonomous fractional stochastic reaction-diffusion equations. Nonlinear Anal: Theory Methods Appl, 2017, 158 : 60-82 doi: 10.1016/j.na.2017.04.006
[21]	Gu A H, Wang B X. Asymptotic behavior of random FitzHugh-Nagumo systems driven by colored noise. Discrete Contin Dyn Syst B, 2018, 23 (4): 1689-1720 doi: 10.3934/dcdsb.2018072
[22]	Wang B X. Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems. J Differ Equ, 2012, 353 (5): 1544-1583
[23]	Foias C, Manley O, Rosa R, et al. Navier-Stokes Equations and Turbulence. Cambridge: Cambridge University Press, 2001
[24]	Medeiros L A, Miranda M M. Weak solutions for a nonlinear dispersive equation. J Math Anal Appl, 1977, 59 (3): 432-441 doi: 10.1016/0022-247X(77)90071-3

Stochastic Attractors and Invariant Measures for Fractional Nonlocal Reaction-Diffusion Equations

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