Asymptotic Behavior of Non-autonomous Random Ginzburg–Landau Equations with Colored Noise on Unbounded Thin Domains

Abstract

This paper is mainly concerned with non-autonomous random complex Ginzburg–Landau equations with nonlinear multiplicative colored noise on unbounded thin domains. Together with the properties of the colored noise, uniform estimates of solutions for such random equations are derived. Moreover, the pullback asymptotic compactness is proved by the tail-estimates method, and the existence and uniqueness of pullback attractors are further obtained. In addition, when the unbounded thin domain collapses onto the real space ℝ, the upper semi-continuity of pullback attractors is also investigated.

Keywords

Non-autonomous random Ginzburg–Landau equation / colored noise / unbounded thin domain / pullback attractor / upper semi-continuity

References

[1] Acquistapace P, Terreni B An approach to It ö linear equations in Hilbert spaces by approximation of white noise with coloured noise. Stochastic Anal. Appl., 1984, 2(2): 131-186.

[2] Arnold L. Random Dynamical Systems, 1998, Berlin: Springer-Verlag.

[3] Bates PW, Lu K, Wang B. Random attractors for stochastic reaction-diffusion equations on unbounded domains. J. Differential Equations, 2009, 246(2): 845-869.

[4] Blömker D, Han Y. Asymptotic compactness of stochastic complex Ginzburg–Landau equation on an unbounded domain. Stoch. Dyn., 2010, 10(4): 613-636.

[5] Caraballo T, Chueshov I D, Kloeden P E. Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain. SIAM J. Math. Anal., 2006, 38(5): 1489-1507.

[6] Chen Z., Li L., Asymptotic behavior of non-autonomous stochastic complex Ginzburg–Landau equations on unbounded thin domains. J. Math. Phys., 2021, 62 (2): Paper No. 022704, 18 pp.

[7] Crauel H, Flandoli F. Attractors for random dynamical systems. Probab. Theory Related Fields, 1994, 100(3): 365-393.

[8] Gu A, Li D, Wang B, Yang H. Regularity of random attractors for fractional stochastic reaction-diffusion equations on ℝⁿ. J. Differential Equations, 2018, 264(12): 7094-7137.

[9] Gu A, Lu K, Wang B. Asymptotic behavior of random Navier–Stokes equations driven by Wong–Zakai approximations. Discrete Contin. Dyn. Syst., 2019, 39(1): 185-218.

[10] Gu A, Wang B. Asymptotic behavior of random Fitzhugh–Nagumo systems driven by colored noise. Discrete Contin. Dyn. Syst. Ser. B, 2018, 23(4): 1689-1720.

[11] Gu A., Wang B., Random attractors of FitzHugh–Nagumo systems driven by colored noise on unbounded domains. Stoch. Dyn., 2019, 19 (5): Paper No. 1950035, 38 pp.

[12] Guo B, Wang G, Li D. The attractor of the stochastic generalized Ginzburg–Landau equation. Sci. China Ser. A, 2008, 51(5): 955-964.

[13] Guo Y, Li D. Random attractor of stochastic complex Ginzburg–Landau equation with multiplicative noise on unbounded domain. Stoch. Anal. Appl., 2017, 35(3): 409-422.

[14] Hale JK, Raugel G. A damped hyperbolic equation on thin domains. Trans. Amer. Math. Soc., 1992, 329(1): 185-219.

[15] Hale JK, Raugel G. Reaction-diffusion equation on thin domains. J. Math. Pures Appl., 1992, 71(1): 33-95.

[16] Jiang T, Liu X, Duan J. Approximation for random stable manifolds under multiplicative correlated noises. Discrete Contin. Dyn. Syst. Ser. B, 2016, 21(9): 3163-3174.

[17] Kłosek-Dygas MM, Matkowsky BJ, Schuss Z. Colored noise in dynamical systems. SIAM J. Appl. Math., 1988, 48(2): 425-441.

[18] Lan Y, Shu J. Fractal dimension of random attractors for non-autonomous fractional stochastic Ginzburg–Landau equations with multiplicative noise. Dyn. Syst., 2019, 34(2): 274-300.

[19] Li D, Lu K, Wang B, Wang X. Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains. Discrete Contin. Dyn. Syst., 2018, 38(1): 187-208.

[20] Li D, Wang B, Wang X. Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains. J. Differential Equations, 2017, 262(3): 1575-1602.

[21] Li D, Wang X. Asymptotic behavior of stochastic complex Ginzburg–Landau equations with deterministic non-autonomous forcing on thin domains. Discrete Contin. Dyn. Syst. Ser. B, 2019, 24(2): 449-465.

[22] Li F, Li Y, Wang R. Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise. Discrete Contin. Dyn. Syst., 2018, 38(7): 3663-3685.

[23] Li L, Chen Z. Asymptotic behavior of non-autonomous random Ginzburg–Landau equation driven by colored noise. Discrete Contin. Dyn. Syst. B, 2021, 26(6): 3303-3333.

[24] Liu W, Wang B. Poisson–Nernst–Planck systems for narrow tubular-like membrane channels. J. Dynam. Differential Equations, 2010, 22(3): 413-437.

[25] Lu H, Bates PW, Lu S, Zhang M. Dynamics of the 3D fractional Ginzburg–Landau equation with multiplicative noise on an unbounded domain. Commun. Math. Sci., 2016, 14(1): 273-295.

[26] Lu H, Zhang M. Dynamics of non-autonomous fractional Ginzburg–Landau equations driven by colored noise. Discrete Contin. Dyn. Syst. Ser. B, 2020, 25(9): 3553-3576.

[27] Nirenberg L. On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 1959, 13: 115-162.

[28] Raugel G. Dynamics of Partial Differential Equations on Thin Domains, 1995, Berlin: Springer.

[29] Ridolfi L, D’Odorico P, Laio F. Noise-induced Phenomena in the Environmental Sciences, 2011, Cambridge: Cambridge University Press.

[30] Shi L., Li X., Limiting behavior of non-autonomous stochastic reaction-diffusion equations on unbounded thin domains. J. Math. Phys., 2019, 60 (8): Paper No. 082702, 17 pp.

[31] Shi L, Wang R, Lu K, Wang B. Asymptotic behavior of stochastic FitzHugh–Nagumo systems on unbounded thin domains. J. Differential Equations, 2019, 267(7): 4373-4409.

[32] Shi L, Wang X, Li D. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Commun. Pure Appl. Anal., 2020, 19(12): 5367-5386.

[33] Shu J., Huang X., Zhang J., Asymptotic behavior for non-autonomous fractional stochastic Ginzburg–Landau equations on unbounded domains. J. Math. Phys., 2020, 61 (7): Paper No. 072704, 18 pp.

[34] Uhlenbeck GE, Ornstein LS. On the Theory of the Brownian Motion. Phys. Rev., 1930, 36: 823-841.

[35] Wang B. Attractors for reaction-diffusion equations in unbounded domains. Phys. D, 1999, 128(1): 41-52.

[36] 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.

[37] Wang G, Guo B, Li Y. The asymptotic behavior of the stochastic Ginzburg–Landau equation with additive noise. Appl. Math. Comput., 2008, 198(2): 849-857.

[38] Wang MC, Uhlenbeck GE. On the theory of the Brownian motion, II. Rev. Modern Phys., 1945, 17: 323-342.

[39] Wang R, Li Y, Wang B. Random dynamics of fractional nonclassical diffusion equations driven by colored noise. Discrete Contin. Dyn. Syst., 2019, 39(7): 4091-4126.

[40] 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.

[41] Yang D. The asymptotic behavior of the stochastic Ginzburg–Landau equation with multiplicative noise. J. Math. Phys., 2004, 45(11): 4064-4076.

[42] Zhang J, Shu J. Existence and upper semicontinuity of random attractors for non-autonomous fractional stochastic Ginzburg–Landau equations. J. Math. Phys., 2019, 60(4): 042702. 16 pp.

[43] Zhang Q. Random attractors for a Ginzburg–Landau equation with additive noise. Chaos Solitons Fractals, 2009, 39(1): 463-472.

[44] Zhou S. Random exponential attractor for cocycle and application to non-autonomous stochastic lattice systems with multiplicative white noise. J. Differential Equations, 2017, 263(4): 2247-2279.