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
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T. CaraballoI. 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

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[21]

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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

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[32]

D. LiK. LuB. 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

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H. LiY. 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

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D. LiB. 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. ShenS. 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. WangS. 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. WangK. 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. WangK. Lu and B. Wang, Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise, J. Dynam. Differential Equations, 28 (2016), 1309-1335.  doi: 10.1007/s10884-015-9448-8.  Google Scholar

[48]

J. Wang and Y. Wang, Pullback attractors for reaction-diffusion delay equations on unbounded domains with non-autonomous deterministic and stochastic forcing terms, J. Math. Phys., 54 (2013), 082703, 25 pp. doi: 10.1063/1.4817862.  Google Scholar

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Y. Wang and J. Wang, Pullback attractors for multi-valued non-compact random dynamical systems generated by reaction-diffusion equations on an unbounded domain, J. Differential Equations, 259 (2015), 728-776.  doi: 10.1016/j.jde.2015.02.026.  Google Scholar

[50]

C. Zhao and J. Duan, Random attractor for the Ladyzhenskaya model with additive noise, J. Math. Anal. Appl., 362 (2010), 241-251.  doi: 10.1016/j.jmaa.2009.08.050.  Google Scholar

[51]

S. Zhou and M. 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

show all references

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. ArrietaA. 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. ArrietaA. 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. ArrietaA. 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. BatesH. 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. BatesK. 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. BatesK. 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. CaoC. 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. CaraballoI. 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. CaraballoM. J. Garrido-AtienzaB. 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. CaraballoM. J. Garrido-AtienzaB. 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. CaraballoJ. A. LangaV. 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. CaraballoJ. 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. CrauelA. 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. DuanK. 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. HanW. 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. LiA. 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. LiK. LuB. 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. LiY. 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. LiB. 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. ShenS. 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. WangS. 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. WangK. 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. WangK. Lu and B. Wang, Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise, J. Dynam. Differential Equations, 28 (2016), 1309-1335.  doi: 10.1007/s10884-015-9448-8.  Google Scholar

[48]

J. Wang and Y. Wang, Pullback attractors for reaction-diffusion delay equations on unbounded domains with non-autonomous deterministic and stochastic forcing terms, J. Math. Phys., 54 (2013), 082703, 25 pp. doi: 10.1063/1.4817862.  Google Scholar

[49]

Y. Wang and J. Wang, Pullback attractors for multi-valued non-compact random dynamical systems generated by reaction-diffusion equations on an unbounded domain, J. Differential Equations, 259 (2015), 728-776.  doi: 10.1016/j.jde.2015.02.026.  Google Scholar

[50]

C. Zhao and J. Duan, Random attractor for the Ladyzhenskaya model with additive noise, J. Math. Anal. Appl., 362 (2010), 241-251.  doi: 10.1016/j.jmaa.2009.08.050.  Google Scholar

[51]

S. Zhou and M. 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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