EXPONENTIAL STABILITY OF ATTRACTOR FOR RANDOM REACTION-DIFFUSION EQUATION DRIVEN BY NONLINEAR COLORED NOISE ON ℝN

  • Huijuan ZHU ,
  • Xiaojun LI ,
  • Yanjiao LI
Expand
  • 1. School of Mathematics, Hohai University, Nanjing 210098, China;
    2. School of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China
Huijuan ZHU, E-mail: zhuhuijuan2022@hhu.edu.cn; Yanjiao LI, E-mail: yanjiaoli2013@163.com

Received date: 2024-01-17

  Online published: 2025-10-10

Supported by

NSFC (12271141), and Zhu's research was supported by the Fundamental Research Funds for the Central Universities (B240205026), and the Postgraduate Research and Practice Innovation Program of Jiangsu Province (KYCX24_0821).

Abstract

In this paper, we consider the existence of pullback random exponential attractor for non-autonomous random reaction-diffusion equation driven by nonlinear colored noise defined on $\mathbb{R}^n$. The key steps of the proof are the tails estimate and to demonstrate the Lipschitz continuity and random squeezing property of the solution for the equation defined on $\mathbb{R}^n$.

Cite this article

Huijuan ZHU , Xiaojun LI , Yanjiao LI . EXPONENTIAL STABILITY OF ATTRACTOR FOR RANDOM REACTION-DIFFUSION EQUATION DRIVEN BY NONLINEAR COLORED NOISE ON ℝN[J]. Acta mathematica scientia, Series B, 2025 , 45(4) : 1567 -1596 . DOI: 10.1007/s10473-025-0418-x

References

[1] Arnold L, Random Dynamical Systems. Berlin: Springer, 1998
[2] Arrieta J M, Cholewa J W, Dlotko T, Rodríguez-Bernal A. Asymptotic behavior and attractors for reaction diffusion equations in unbounded domains. Nonlinear Anal, 2004, 56(4): 515—554
[3] Arrieta J M, Santamaría E. Distance of attractors of reaction-diffusion equations in thin domains. J Differential Equations, 2017, 263(9): 5459-5506
[4] Bates P W, Lu K, Wang B. Random attractors for stochastic reaction-diffusion equations on unbounded domains. J Differential Equations, 2009, 246(2): 845-869
[5] Beyn W J, Pilyugin S Y. Attractors of reaction diffusion systems on infinite lattices. J Dynam Differential Equations, 2003, 15(2/3): 485-515
[6] Britton N F.Reaction-Diffusion Equations and Their Applications to Biology. London: Academic Press, 1986
[7] Caraballo T, Chen Z, Yang D. Random dynamics and limiting behaviors for 3D globally modified Navier-Stokes equations driven by colored noise. Stud Appl Math, 2023, 151(1): 247-284
[8] Caraballo T, Kloeden P E, Schmalfuß B. Exponentially stable stationary solutions for stochastic evolution equations and their perturbation. Appl Math Optim, 2004, 50(3): 183-207
[9] Chen P, Wang B, Wang R, Zhang X. Multivalued random dynamics of Benjamin-Bona-Mahony equations driven by nonlinear colored noise on unbounded domains. Math Ann, 2023, 386(1/2): 343-373
[10] Chen P, Zhang X. Random dynamics of stochastic BBM equations driven by nonlinear colored noise on unbounded channel. J Evol Equ, 2022, 22(4): Art 87
[11] Crauel H, Debussche A, Flandoli F. Random attractors. J Dynam Differential Equations, 1997, 9(2): 307-341
[12] Crauel H, Flandoli F. Attractors for random dynamical systems. Probab Theory Relat Fields, 1994, 100(3): 365-393
[13] Dafallah A A, Mohamed A, Ma Q. Random attractors for a stochastic wave equations with nonlinear damping and multiplicative noise. J Math, 2016, 12: 39-55
[14] Eden A, Foias C, Nicolaenko B, Temam R.Exponential Attractors for Dissipative Evolution Equations. Pairs: Masson, 1994
[15] Fan X. Attractors for a damped stochastic wave equation of Sine-Gordon type with sublinear multiplicative noise. Stoch Anal Appl, 2006, 24(4): 767-793
[16] Fan X. Random attractors for damped stochastic wave equations with multiplicative noise. Internat J Math, 2008, 19(4): 421-437
[17] Gu A, Guo B, Wang B. Long term behavior of random Navier-Stokes equations driven by colored noise. Discrete Contin Dyn Syst Ser B, 2020, 25(7): 2495-2532
[18] Han Z, Zhou S. Random exponential attractor for the 3D non-autonomous stochastic damped Navier-Stokes equation. J Dynam Differential Equations, 2023, 35(2): 1133-1149
[19] Li H, You Y, Tu J. Random attractors and averaging for non-autonomous stochastic wave equations with nonlinear damping. J Differential Equations, 2015, 258(1): 148-190
[20] Li L, Chen Z. Asymptotic behavior of non-autonomous random Ginzburg-Landau equation driven by colored noise. Discrete Contin Dyn Syst Ser B, 2021, 26(6): 3303-3333
[21] Li Y, Li B, Li X. Uniform random attractors for a non-autonomous stochastic strongly damped wave equation on $\mathbb {R}^{\mathbb {N}}$. Z Angew Math Phys, 2022, 73(3): Art 106
[22] Li Y, Li X, Zuo J. Random attractors for non-autonomous stochastic wave equations with strong damping and additive noise on $\mathbb {R}^N$. J Appl Anal Comput, 2023, 13(4): 1739-1765
[23] Marion M. Attractors for reaction-diffusion equations: existence and estimate of their dimension. Appl Anal, 1987, 25(1/2): 101-147
[24] Ridolfi L, D'Odorico P, Laio F. Noise-Induced Phenomena in the Environmental Sciences. Cambridge: Cambridge University Press, 2011
[25] Shirikyan A, Zelik S. Exponential attractors for random dynamical systems and applications. Stoch Partial Differ Equ Anal Comput, 2013, 1(2): 241-281
[26] Slavík J. Attractors for stochastic reaction-diffusion equation with additive homogeneous noise. Czechoslovak Math J, 2021, 71(1): 21-43
[27] Uhlenbeck G, Ornstein L. On the theory of Brownian motion. Phys Rev, 1930, 36(5): 823-841
[28] Wang B. Attractors for reaction-diffusion equations in unbounded domains. Phys D, 1999, 128(1): 41-52
[29] Wang B. Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms. Stoch Dyn, 2014, 14(4): Art 1450009
[30] Wang B. Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains. J Differential Equations, 2009, 246(6): 2506-2537
[31] Wang B. Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems. J Differential Equations, 2012, 253(5): 1544-1583
[32] Wang B. Well-posedness and long term behavior of supercritical wave equations driven by nonlinear colored noise on $\mathbb{R}^n$. J Funct Anal, 2022, 283(2): Art 109498
[33] Wang G, Tang Y. Random attractors for stochastic reaction-diffusion equations with multiplicative noise in $H^1_0$. Math Nachr, 2014, 287(14/15): 1774-1791
[34] Wang M, Uhlenbeck G. On the theory of Brownian motion II. Rev Modern Phys, 1945, 17(2/3): 323-342
[35] Wang X, Lu K, Wang B. Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains. J Differential Equations, 2018, 264(1): 378-424
[36] Wang Z, Zhou S. Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise. Discrete Contin Dyn Syst, 2017, 37(5): 2787-2812
[37] Wang Z, Zhou S. Random attractor for stochastic reaction-diffusion equation with multiplicative noise on unbounded domains. J Math Anal Appl, 2011, 384(1): 160-172
[38] Zelik S V. Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity. Comm Pure Appl Math, 2003, 56(5): 584-637
[39] Zhang X, Yuan R. Pullback attractor for random chemostat model driven by colored noise. Appl Math Lett, 2021, 112(1): Art 106833
[40] Zhao W Q, Zhang Y J. Upper semi-continuity of random attractors for a non-autonomous dynamical system with a weak convergence condition. Acta Math Sci, 2020, 40(4): 921-933
[41] Zhou S. Random exponential attractor for stochastic reaction-diffusion equation with multiplicative noise in $\mathbb{R}^3$. J Differential Equations, 2017, 263(10): 6347-6383
[42] Zhou S, Zhao M. Fractal dimension of random invariant sets for nonautonomous random dynamical systems and random attractor for stochastic damped wave equation. Nonlinear Anal, 2016, 133: 292-318
Options
Outlines

/