# American Institute of Mathematical Sciences

July  2019, 39(7): 3717-3747. doi: 10.3934/dcds.2019151

## Limiting dynamics for non-autonomous stochastic retarded reaction-diffusion equations on thin domains

 1 School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610031, China 2 Department of Mathematics, Brigham Young University, Provo, Utah 84602, USA 3 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China 4 Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA

* Corresponding author: Xiaohu Wang, wangxiaohu@scu.edu.cn

Received  January 2018 Published  April 2019

Fund Project: This work was supported by NSFC (111871049, 11601446 and 1331007), NSF (1413603), Excellent Youth Scholars of Sichuan University (2016SCU04A15) and Fundamental Research Funds for the Central Universities (2682015CX059).

A system of stochastic retarded reaction-diffusion equations with multiplicative noise and deterministic non-autonomous forcing on thin domains is considered. Relations between the asymptotic behavior for the stochastic retarded equations defined on thin domains in ${\mathbb R}^{n+1}$ and an equation on a domain in ${\mathbb R}^{n}$ are investigated. We first show the existence and uniqueness of tempered random attractors for these equations. Then, we analyze convergence properties of the solutions as well as the attractors.

Citation: Dingshi Li, Kening Lu, Bixiang Wang, Xiaohu Wang. Limiting dynamics for non-autonomous stochastic retarded reaction-diffusion equations on thin domains. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3717-3747. doi: 10.3934/dcds.2019151
##### References:
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Wang, Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains, J. Differential Equations, 262 (2017), 1575-1602.  doi: 10.1016/j.jde.2016.10.024.  Google Scholar [35] W. Liu and B. Wang, Poisson-Nernst-Planck systems for narrow tubular-like membrane channels, J. Dynam. Differential Equations, 22 (2010), 413-437.  doi: 10.1007/s10884-010-9186-x.  Google Scholar [36] M. Prizzi and K. P. Rybakowski, The effect of domain squeezing upon the dynamics of reaction-diffusion equations, J. Differential Equations, 173 (2001), 271-320.  doi: 10.1006/jdeq.2000.3917.  Google Scholar [37] Z. Shen, S. Zhou and W. Shen, One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation, J. Differential Equations, 248 (2010), 1432-1457.  doi: 10.1016/j.jde.2009.10.007.  Google Scholar [38] B. Wang, Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains, J. 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Kloeden, Pullback attractors of a multi-valued process generated by parabolic differential equations with unbounded delays, Nonlinear Anal., 90 (2013), 86-95.  doi: 10.1016/j.na.2013.05.026.  Google Scholar [44] Y. Wang and P. E. Kloeden, The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain, Discrete Contin. Dyn. Syst., 34 (2014), 4343-4370.  doi: 10.3934/dcds.2014.34.4343.  Google Scholar [45] X. Wang, S. Li and D. Xu, Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Anal., 72 (2010), 483-494.  doi: 10.1016/j.na.2009.06.094.  Google Scholar [46] X. Wang, K. Lu and B. Wang, Long term behavior of delay parabolic equations with additive noise and deterministic nonautonomous forcing, SIAM J. Appl. Dynam. Syst., 14 (2015), 1018-1047.  doi: 10.1137/140991819.  Google Scholar [47] X. Wang, K. Lu and B. 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Zhao, Random attractors for damped non-autonomous wave equations with memory and white noise, Nonlinear Anal., 120 (2015), 202-226.  doi: 10.1016/j.na.2015.03.009.  Google Scholar

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##### References:
 [1] F. Antoci and M. Prizzi, Reaction-diffusion equations on unbounded thin domains, Topol. Methods Nonlinear Anal., 18 (2001), 283-302.  doi: 10.12775/TMNA.2001.035.  Google Scholar [2] L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar [3] J. M. Arrieta, A. N. Carvalho and G. Lozada-Cruz, Dynamics in dumbbell domains Ⅰ. Continuity of the set of equilibria, J. Differential Equations, 231 (2006), 551-597.  doi: 10.1016/j.jde.2006.06.002.  Google Scholar [4] J. M. Arrieta, A. N. Carvalho and G. Lozada-Cruz, Dynamics in dumbbell domains Ⅱ. The limiting problem, J. Differential Equations, 247 (2009), 174-202.  doi: 10.1016/j.jde.2009.03.014.  Google Scholar [5] J. M. Arrieta, A. N. Carvalho and G. Lozada-Cruz, Dynamics in dumbbell domains Ⅲ. Continuity of attractors, J. Differential Equations, 247 (2009), 225-259.  doi: 10.1016/j.jde.2008.12.014.  Google Scholar [6] P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stochastics Dyn., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.  Google Scholar [7] P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.  doi: 10.1016/j.jde.2008.05.017.  Google Scholar [8] P. W. Bates, K. Lu and B. Wang, Tempered random attractors for parabolic equations in weighted spaces, J. Math. Phy., 54 (2013), 081505, 26pp. doi: 10.1063/1.4817597.  Google Scholar [9] P. W. Bates, K. Lu and B. Wang, Attractors for non-autonomous stochastic lattice systems in weighted space, Physica D, 289 (2014), 32-50.  doi: 10.1016/j.physd.2014.08.004.  Google Scholar [10] D. Cao, C. Sun and M. Yang, Dynamics for a stochastic reaction-diffusion equation with additive noise., J. Differential Equations, 259 (2015), 838-872.  doi: 10.1016/j.jde.2015.02.020.  Google Scholar [11] T. Caraballo, I. D. Chueshov and P. E. Kloeden, Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain, SIAM J. Math. Anal., 38 (2007), 1489-1507.  doi: 10.1137/050647281.  Google Scholar [12] T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443.  doi: 10.3934/dcds.2008.21.415.  Google Scholar [13] T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 439-455.  doi: 10.3934/dcdsb.2010.14.439.  Google Scholar [14] T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero, Pullback attractors for nonautonomous and stochastic multivalued dynamical systems, Set-Valued Anal., 11 (2003), 153-201.  doi: 10.1023/A:1022902802385.  Google Scholar [15] T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297.  doi: 10.1016/j.jde.2004.04.012.  Google Scholar [16] T. Caraballo, J. Real and I. D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 525-539.  doi: 10.3934/dcdsb.2008.9.525.  Google Scholar [17] I. Chueshov and S. Kuksin, Random kick-forced 3D Navier-Stokes equations in a thin domain, Arch. Ration. Mech. Anal., 188 (2008), 117-153.  doi: 10.1007/s00205-007-0068-2.  Google Scholar [18] I. Chueshov and S. Kuksin, Stochastic 3D Navier-Stokes equations in a thin domain and its $\alpha$-approximation., Phys. D, 237 (2008), 1352-1367.  doi: 10.1016/j.physd.2008.03.012.  Google Scholar [19] I. S. Ciuperca, Reaction-diffusion equations on thin domains with varying order of thinness, J. Differential Equations, 126 (1996), 244-291.  doi: 10.1006/jdeq.1996.0051.  Google Scholar [20] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.  Google Scholar [21] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Th. Re. Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar [22] X. Ding and J. Jiang, Random attractors for stochastic retarded reaction-diffusion equations on unbounded domains., Abstr. Appl. Anal., 2013 (2013), Art. ID 981576, 16pp. doi: 10.1155/2013/981576.  Google Scholar [23] J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations., Ann. Probab., 31 (2003), 2109-2135.  doi: 10.1214/aop/1068646380.  Google Scholar [24] J. Duan and B. Schmalfuss, The 3D quasigeostrophic fluid dynamics under random forcing on boundary, Commun. Math. Sci., 1 (2003), 133-151.  doi: 10.4310/CMS.2003.v1.n1.a9.  Google Scholar [25] F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45.  doi: 10.1080/17442509608834083.  Google Scholar [26] J. Hale and G. Raugel, Reaction-diffusion equations on thin domains, J. Math. Pures Appl., 71 (1992), 33-95.   Google Scholar [27] J. Hale and G. Raugel, A reaction-diffusion equation on a thin L-shaped domain, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 283-327.  doi: 10.1017/S0308210500028043.  Google Scholar [28] X. Han, W. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018.  Google Scholar [29] P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. London, Ser. A, 463 (2007), 163-181.  doi: 10.1098/rspa.2006.1753.  Google Scholar [30] Y. Li and B. Guo, Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations, J. Differential Equations, 245 (2008), 1775-1800.  doi: 10.1016/j.jde.2008.06.031.  Google Scholar [31] Y. Li, A. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differential Equations, 258 (2015), 504-534.  doi: 10.1016/j.jde.2014.09.021.  Google Scholar [32] D. Li, K. Lu, B. Wang and X. Wang, Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains, Discrete Contin. Dyn. Syst., 38 (2018), 187-208.  doi: 10.3934/dcds.2018009.  Google Scholar [33] H. Li, Y. You and J. Tu, Random attractors and averaging for non-autonomous stochastic wave equations with nonlinear damping, J. Differential Equations, 258 (2015), 148-190.  doi: 10.1016/j.jde.2014.09.007.  Google Scholar [34] D. Li, B. Wang and X. Wang, Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains, J. Differential Equations, 262 (2017), 1575-1602.  doi: 10.1016/j.jde.2016.10.024.  Google Scholar [35] W. Liu and B. Wang, Poisson-Nernst-Planck systems for narrow tubular-like membrane channels, J. Dynam. Differential Equations, 22 (2010), 413-437.  doi: 10.1007/s10884-010-9186-x.  Google Scholar [36] M. Prizzi and K. P. Rybakowski, The effect of domain squeezing upon the dynamics of reaction-diffusion equations, J. Differential Equations, 173 (2001), 271-320.  doi: 10.1006/jdeq.2000.3917.  Google Scholar [37] Z. Shen, S. Zhou and W. Shen, One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation, J. Differential Equations, 248 (2010), 1432-1457.  doi: 10.1016/j.jde.2009.10.007.  Google Scholar [38] B. Wang, Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains, J. Differential Equations, 246 (2009), 2506-2537.  doi: 10.1016/j.jde.2008.10.012.  Google Scholar [39] B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb R^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663.  doi: 10.1090/S0002-9947-2011-05247-5.  Google Scholar [40] B. Wang, Suffcient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.  Google Scholar [41] B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269.  Google Scholar [42] B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009, 31pp. doi: 10.1142/S0219493714500099.  Google Scholar [43] Y. Wang and P. E. Kloeden, Pullback attractors of a multi-valued process generated by parabolic differential equations with unbounded delays, Nonlinear Anal., 90 (2013), 86-95.  doi: 10.1016/j.na.2013.05.026.  Google Scholar [44] Y. Wang and P. E. Kloeden, The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain, Discrete Contin. Dyn. Syst., 34 (2014), 4343-4370.  doi: 10.3934/dcds.2014.34.4343.  Google Scholar [45] X. Wang, S. Li and D. Xu, Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Anal., 72 (2010), 483-494.  doi: 10.1016/j.na.2009.06.094.  Google Scholar [46] X. Wang, K. Lu and B. Wang, Long term behavior of delay parabolic equations with additive noise and deterministic nonautonomous forcing, SIAM J. Appl. Dynam. Syst., 14 (2015), 1018-1047.  doi: 10.1137/140991819.  Google Scholar [47] X. Wang, K. Lu and B. 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