Dynamical behaviors of non-autonomous fractional FitzHugh-Nagumo system driven by additive noise in unbounded domains

Abstract

The regularity of random attractors is considered for the nonautonomous fractional stochastic FitzHugh-Nagumo system. We prove that the system has a pullback random attractor that is compact in H^s({\mathrm{ℝ}}^n)\times L^{\mathrm{2}}({\mathrm{ℝ}}^n) and attracts all tempered random sets of L^s({\mathrm{ℝ}}^n)\times L^{\mathrm{2}}({\mathrm{ℝ}}^n) in the topology of H^s({\mathrm{ℝ}}^n)\times L^{\mathrm{2}}({\mathrm{ℝ}}^n) with s\in (\mathrm{0},\mathrm{1}). By the idea of positive and negative truncations, spectral decomposition in bounded domains, and tail estimates, we achieved the desired results.

Keywords

Fractional stochastic FitzHugh-Nagumo system / random attractor / asymptotic compactness

References

[1] Adili A,Wang B. Random attractors for non-autonomous stochastic FitzHugh-Nagumo systems with multiplicative noise. Discrete Contin Dyn Syst Ser S, 2013, 2013(Special): 1–10

[2] Adili A, Wang B. Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing. Discrete Contin Dyn Syst Ser B, 2013, 18: 643–666

[3] Arnold L. Random Dynamical Systems. New York: Springer-Verlag, 1998

[4] Bates P W, Lu K, Wang B. Random attractors for stochastic reaction-diffusion equations on unbounded domains. J Differential Equations, 2009, 246: 845–869

[5] Crauel H, Debussche A, Flandoli F. Random attractors. J Dynam Differential Equations, 1997, 9: 307–341

[6] Di Nezza E, Palatucci G,Valdinoci E. Hitchhiker's guide to the fractional Sobolev spaces. Bull Sci Math, 2012, 136: 521–573

[7] FitzHugh R. Impulses and physiological states in theoretical models of nerve membrane. Biophys J, 1961, 1: 445–466

[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: 7094–7137

[9] Gu A, Li Y. Singleton sets random attractor for stochastic FitzHugh-Nagumo lattice equations driven by fractional Brownian motions. Commun Nonlinear Sci Numer Simul, 2014, 19: 3929–3937

[10] Guo B, Huo Z. Global well-posedness for the fractional nonlinear Schrödinger equation. Comm Partial Differential Equations, 2011, 36: 247–255

[11] Guo B, Pu X, Huang F. Fractional Partial Differential Equations and their Numerical Solutions. Beijing: Science Press, 2011(in Chinese)

[12] Huang J, Shen W.Global attractors for partly dissipative random stochastic reaction diffusion systems. Int J Evol Equ, 2010, 4: 383–411

[13] Li Y, Yin J. A modified proof of pullback attractors in a Sobolev space for stochastic FitzHugh-Nagumo equations. Discrete Contin Dyn Syst Ser B, 2016, 21: 1203–1223

[14] Liu F, Turner I, Anh V, Yang Q, Burrage K. A numerical method for the fractional FitzHugh-Nagumo monodomain model. ANZIAM J, 2013, 54: C608–C629

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

[16] Lu H, Bates P W, Xin J, Zhang M. Asymptotic behavior of stochastic fractional power dissipative equations on Rn: Nonlinear Anal, 2015, 128: 176–198

[17] Marion M. Finite-dimensional attractors associated with partly dissipative reactiondi ffusion systems. SIAM J Math Anal, 1989, 20: 816–844

[18] Marion M. Inertial manifolds associated to partly dissipative reaction-diffusion systems. J Math Anal Appl, 1989, 143: 295–326

[19] Morillas F, Valero J. Attractors for reaction-diffusion equations in Rn with continuous nonlinearity. Asymptot Anal, 2005, 44: 111–130

[20] Nagumo J, Arimoto S, Yosimzawa S. An active pulse transmission line simulating nerve axon. Proc Inst Radio Eng, 1964, 50: 2061–2070

[21] Pu X, Guo B. Global weak soltuions of the fractional Landau-Lifshitz-Maxwell equation. J Math Anal Appl, 2010, 372: 86–98

[22] Ruelle D. Characteristic exponents for a viscous fluid subjected to time dependent forces. Comm Math Phys, 1984, 93: 285–300

[23] Shao Z. Existence of inertial manifolds for partly dissipative reaction diffusion systems in higher space dimensions. J Differential Equations, 1998, 144: 1–43

[24] Shu J, Li P, Zhang J, Liao O. Random attractors for the stochastic coupled fractional Ginzburg-Landau equation with additive noise. J Math Phys, 2015, 56: 102702

[25] Wang B. Random attractors for the stochastic FitzHugh-Nagumo system on unbounded domains. Nonlinear Anal, 2009, 71: 2811–2828

[26] Wang B. Upper semicontinuity of random for non-compact random systems. J Differential Equations, 2009, 139: 1–18

[27] Wang B. Sufficient and necessary criteria for existence of pullback attractors for noncompact random dynamical systems. J Differential Equations, 2012, 253: 1544–1583

[28] Wang B. Asymptotic behavior of non-autonomous fractional stochastic reactiondi ffusion equations. Nonlinear Anal, 2017, 158: 60{82

[29] Zhou S,Wang Z. Finite fractal dimensions of random attractors for stochastic FitzHugh-Nagumo system with multiplicative white noise. J Math Anal Appl, 2016, 441: 648–667

RIGHTS & PERMISSIONS

2021 Higher Education Press