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December  2020, 19(12): 5367-5386. doi: 10.3934/cpaa.2020242

Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains

School of Mathematics, Southwest Jiaotong University, Chengdu 610031, China, National Engineering Laboratory of Integrated Transportation, Big Data Application Technology, Chengdu 61756, China

* Corresponding author

Received  February 2020 Revised  July 2020 Published  December 2020 Early access  September 2020

Fund Project: The first author is supported by the National Natural Science Foundation of China (grant No. 11701475, 1197130 and 11971394)

This article is concerned with the limiting behavior of dynamics of a class of non-autonomous stochastic partial differential equations driven by colored noise on unbounded thin domains. We first prove the existence of tempered pullback random attractors for the equations defined on $ (n+1) $-dimensional unbounded thin domains. Then, we show the upper semicontinuity of these attractors when the $ (n+1) $-dimensional unbounded thin domains collapse onto the $ n $-dimensional space $ \mathbb{R}^n $. Here, the tail estimates are utilized to deal with the non-compactness of Sobolev embeddings on unbounded domains.

Citation: Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242
References:
[1]

L. Arnold, Random Dynamical Systems, Springer, New York, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

P. W. BatesK. Lu and B. Wang, Random attractors for stochastic reaction diffusion equations on unbounded domains, J. Differ. Equ., 246 (2009), 845-869.  doi: 10.1016/j.jde.2008.05.017.

[3]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.

[4]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dyn. Differ. Equ., 9 (1997), 307-341.  doi: 10.1007/BF02219225.

[5]

F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes eqution with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45.  doi: 10.1080/17442509608834083.

[6]

A. Gu and B. Wang, Asymptotic behavior of random FitzHugh-Nagumo systems driven by colored noise, Discrete Contin. Dyn. Syst. B, 23 (2018), 1689-1720.  doi: 10.3934/dcdsb.2018072.

[7]

J. K. Hale and G. Raugel, Reaction-diffusion equations on thin domains, J. Math. Pures Appl., 71 (1992), 33-95. 

[8]

K. Lu and B. Wang, Wong-Zakai approximations and long term behavior of stochastic partial differential equations, J. Dyn. Differ. Equ., 31 (2019), 1341-1371. doi: 10.1007/s10884-017-9626-y.

[9]

K. Lu and Q. Wang, Chaotic behavior in differential equations driven by a Brownian motion, J. Differ. Equ., 251 (2011), 2853-2895.  doi: 10.1016/j.jde.2011.05.032.

[10]

D. LiB. Wang and X. Wang, Limiting behavior of non-autonomous stochastic reaction diffusion equations on thin domains, J. Differ. Equ., 262 (2017), 1575-1602.  doi: 10.1016/j.jde.2016.10.024.

[11]

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.

[12]

D. Ruelle, Characteristic exponents for a viscous fluid sujectied to time dependent forces, Commun. Math. Phys., 93 (1884), 285-300.

[13]

B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, Dresden, (1992), 185–192.

[14]

L. ShiR. WangK. Lu and B. Wang, Asymptotic behavior of stochastic FitzHugh-Nagumo systems on unbounded thin domains, J. Differ. Equ., 267 (2019), 4373-4409.  doi: 10.1016/j.jde.2019.05.002.

[15]

L. Shi and X. Li, Limiting behavior of non-autonomous stochastic reaction-diffusion equations on unbounded thin domains, J. Math. Phycs., 60 (2019), 082702. doi: 10.1063/1.5093890.

[16]

J. ShenK. Lu and B. Wang, Convergence and center manifolds for differential equations driven by colored noise, Discrete Contin. Dynam. Systems, 39 (2019), 4797-4840.  doi: 10.3934/dcds.2019196.

[17]

S. M. Ulam and J. von Neumann, Random egodic theorems, Bull. Amer. Math. Soc., 51 (1945), 660. doi: 10.1090/S0002-9904-1958-10189-5.

[18]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differ. Equ., 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.

[19]

X. WangK. Lu and B. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differ. Equ., 246 (2018), 378-424.  doi: 10.1016/j.jde.2017.09.006.

[20]

R. WangR. Li and B. Wang, Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dyn. Syst., 39 (2019), 4091-4126.  doi: 10.3934/dcds.2019165.

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer, New York, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

P. W. BatesK. Lu and B. Wang, Random attractors for stochastic reaction diffusion equations on unbounded domains, J. Differ. Equ., 246 (2009), 845-869.  doi: 10.1016/j.jde.2008.05.017.

[3]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.

[4]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dyn. Differ. Equ., 9 (1997), 307-341.  doi: 10.1007/BF02219225.

[5]

F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes eqution with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45.  doi: 10.1080/17442509608834083.

[6]

A. Gu and B. Wang, Asymptotic behavior of random FitzHugh-Nagumo systems driven by colored noise, Discrete Contin. Dyn. Syst. B, 23 (2018), 1689-1720.  doi: 10.3934/dcdsb.2018072.

[7]

J. K. Hale and G. Raugel, Reaction-diffusion equations on thin domains, J. Math. Pures Appl., 71 (1992), 33-95. 

[8]

K. Lu and B. Wang, Wong-Zakai approximations and long term behavior of stochastic partial differential equations, J. Dyn. Differ. Equ., 31 (2019), 1341-1371. doi: 10.1007/s10884-017-9626-y.

[9]

K. Lu and Q. Wang, Chaotic behavior in differential equations driven by a Brownian motion, J. Differ. Equ., 251 (2011), 2853-2895.  doi: 10.1016/j.jde.2011.05.032.

[10]

D. LiB. Wang and X. Wang, Limiting behavior of non-autonomous stochastic reaction diffusion equations on thin domains, J. Differ. Equ., 262 (2017), 1575-1602.  doi: 10.1016/j.jde.2016.10.024.

[11]

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.

[12]

D. Ruelle, Characteristic exponents for a viscous fluid sujectied to time dependent forces, Commun. Math. Phys., 93 (1884), 285-300.

[13]

B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, Dresden, (1992), 185–192.

[14]

L. ShiR. WangK. Lu and B. Wang, Asymptotic behavior of stochastic FitzHugh-Nagumo systems on unbounded thin domains, J. Differ. Equ., 267 (2019), 4373-4409.  doi: 10.1016/j.jde.2019.05.002.

[15]

L. Shi and X. Li, Limiting behavior of non-autonomous stochastic reaction-diffusion equations on unbounded thin domains, J. Math. Phycs., 60 (2019), 082702. doi: 10.1063/1.5093890.

[16]

J. ShenK. Lu and B. Wang, Convergence and center manifolds for differential equations driven by colored noise, Discrete Contin. Dynam. Systems, 39 (2019), 4797-4840.  doi: 10.3934/dcds.2019196.

[17]

S. M. Ulam and J. von Neumann, Random egodic theorems, Bull. Amer. Math. Soc., 51 (1945), 660. doi: 10.1090/S0002-9904-1958-10189-5.

[18]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differ. Equ., 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.

[19]

X. WangK. Lu and B. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differ. Equ., 246 (2018), 378-424.  doi: 10.1016/j.jde.2017.09.006.

[20]

R. WangR. Li and B. Wang, Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dyn. Syst., 39 (2019), 4091-4126.  doi: 10.3934/dcds.2019165.

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