### Electronic Research Archive

2021, Issue 2: 1969-1990. doi: 10.3934/era.2020100
Special Issues

# Regular random attractors for non-autonomous stochastic reaction-diffusion equations on thin domains

• Received: 01 February 2020 Revised: 01 July 2020 Published: 23 September 2020
• Primary: 60H15, 35B40, 58F11, 58F36

• This paper deals with the limiting dynamical behavior of non-autonomous stochastic reaction-diffusion equations on thin domains. Firstly, we prove the existence and uniqueness of the regular random attractor. Then we prove the upper semicontinuity of the regular random attractors for the equations on a family of $(n+1)$-dimensional thin domains collapses onto an $n$-dimensional domain.

Citation: Dingshi Li, Xuemin Wang. Regular random attractors for non-autonomous stochastic reaction-diffusion equations on thin domains[J]. Electronic Research Archive, 2021, 29(2): 1969-1990. doi: 10.3934/era.2020100

### Related Papers:

• This paper deals with the limiting dynamical behavior of non-autonomous stochastic reaction-diffusion equations on thin domains. Firstly, we prove the existence and uniqueness of the regular random attractor. Then we prove the upper semicontinuity of the regular random attractors for the equations on a family of $(n+1)$-dimensional thin domains collapses onto an $n$-dimensional domain.

 [1] L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998. doi: 10.1007/978-3-662-12878-7 [2] Random attractors for stochastic reaction-diffusion equations on unbounded domains. J. Differential Equations (2009) 246: 845-869. [3] P. W. Bates, K. Lu and B. Wang, Tempered random attractors for parabolic equations in weighted spaces, J. Math. Phy., 54 (2013), 081505, 26 pp. doi: 10.1063/1.4817597 [4] Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain. SIAM J. Math. Anal. (2006/07) 38: 1489-1507. [5] Random attractors. J. Dynam. Differential Equations (1997) 9: 307-341. [6] Invariant manifolds for stochastic partial differential equations. Ann. Probab. (2003) 31: 2109-2135. [7] Regularity of random attractors for fractional stochastic reaction-diffusion equations on $\mathbb{R}^n$. J. Differential Equations (2018) 264: 7094-7137. [8] A reaction-diffusion equation on a thin L-shaped domain. Proc. Roy. Soc. Edinburgh Sect. A (1995) 125: 283-327. [9] Long term behavior of a random Hopfield neural lattice model. Commun. Pure Appl. Anal. (2019) 18: 809-824. [10] Random attractors for stochastic lattice dynamical systems in weighted spaces. J. Differential Equations (2011) 250: 1235-1266. [11] Strong convergence of bi-spatial random attractors for parabolic equations on thin domains with rough noise. Topol. Methods Nonlinear Anal. (2019) 53: 659-682. [12] Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise. Discrete Contin. Dyn. Syst. (2018) 38: 3663-3685. [13] Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains. Discrete Contin. Dyn. Syst. (2018) 38: 187-208. [14] Limiting dynamics for non-autonomous stochastic retarded reaction-diffusion equations on thin domains. Discrete Contin. Dyn. Syst. (2019) 39: 3717-3747. [15] D. Li and L. Shi, Upper semicontinuity of attractors of stochastic delay reaction-diffusion equations in the delay, J. Math. Phys., 59 (2018), 032703, 35 pp. doi: 10.1063/1.4994869 [16] Asymptotic behavior of stochastic complex Ginzburg-Landau equations with deterministic non-autonomous forcing on thin domains. Discrete Contin. Dyn. Syst. B (2019) 24: 449-465. [17] Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains. J. Differential Equations (2017) 262: 1575-1602. [18] Poisson-Nernst-Planck systems for narrow tubular-like membrane channels. J. Dynam. Differential Equations (2010) 22: 413-437. [19] One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation. J. Differential Equations (2010) 248: 1432-1457. [20] Asymptotic behavior of stochastic FitzHugh-Nagumo systems on unbounded thin domains. J. Differential Equations (2019) 267: 4373-4409. [21] Suffcient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems. J. Differential Equations (2012) 253: 1544-1583. [22] Random attractors for non-autonomous stochastic wave equations with multiplicative noise. Discrete Contin. Dyn. Syst. (2014) 34: 269-300. [23] Pullback attractors for multi-valued non-compact random dynamical systems generated by reaction-diffusion equations on an unbounded domain. J. Differential Equations (2015) 259: 728-776.
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