# American Institute of Mathematical Sciences

February  2019, 24(2): 449-465. doi: 10.3934/dcdsb.2018181

## Asymptotic behavior of stochastic complex Ginzburg-Landau equations with deterministic non-autonomous forcing on thin domains

 1 School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610031, China 2 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

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

Received  November 2017 Revised  January 2018 Published  February 2019 Early access  June 2018

Fund Project: This work was supported by NSFC (11271270, 11601446 and 11331007) and Excellent Youth Scholars of Sichuan University (2016SCU04A15).

In this paper, we investigate the asymptotic behavior for non-autonomous stochastic complex Ginzburg-Landau equations with multiplicative noise on thin domains. For this aim, we first show that the existence and uniqueness of random attractors for the considered equations and the limit equations. Then, we establish the upper semicontinuity of these attractors when the thin domains collapse onto an interval.

Citation: Dingshi Li, Xiaohu Wang. Asymptotic behavior of stochastic complex Ginzburg-Landau equations with deterministic non-autonomous forcing on thin domains. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 449-465. doi: 10.3934/dcdsb.2018181
##### 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. [2] L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998. doi: 10.1007/978-3-662-12878-7. [3] J. Arrieta, A. Carvalho, M. Pereira and R. P. Da Silva, Semilinear parabolic problems in thin domains with a highly oscillatory boundary, Nonlinear Anal., 74 (2011), 5111-5132.  doi: 10.1016/j.na.2011.05.006. [4] P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621. [5] 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. [6] 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. [7] I. D. 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. [8] I. D. Chueshov and S. Kuksin, Stochastic 3D Navier-Stokes equations in a thin domain and its α-approximation, Physica D, 237 (2008), 1352-1367.  doi: 10.1016/j.physd.2008.03.012. [9] I. 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. [10] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225. [11] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Relat. Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705. [12] H. Cui, Y. Li and J. Yin, Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles, Nonlinear Anal., 128 (2015), 303-324.  doi: 10.1016/j.na.2015.08.009. [13] 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. [14] 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. [15] J. K. Hale and G. Raugel, Reaction-diffusion equations on thin domains, J. Math. Pures Appl., 71 (1992), 33-95. [16] J. K. 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. [17] R. Johnson, M. Kamenskii and P. Nistri, Existence of periodic solutions of an autonomous damped wave equation in thin domains, J. Dynam. Differential Equations, 10 (1998), 409-424.  doi: 10.1023/A:1022601213052. [18] P. E. Kloeden and J. 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. [19] S. Lü, H. Lu and Z. Feng, Stochastic dynamics of 2D fractional Ginzburg-Landau equation with multiplicative noise, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 575-590.  doi: 10.3934/dcdsb.2016.21.575. [20] 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. [21] 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. [22] 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. [23] Y. Morita, Stable solutions to the Ginzburg-Landau equation with magnetic effect in a thin domain, Japan J. Indust. Appl. Math., 21 (2004), 129-147.  doi: 10.1007/BF03167468. [24] M. Prizzi and K. P. Rybakowski, Recent results on thin domain problems, Ⅱ, Topol. Methods Nonlinear Anal., 19 (2002), 199-219.  doi: 10.12775/TMNA.2002.010. [25] M. Prizzi and K. P. Rybakowski, The effect of domain squeezing upon the dynamics of reaction-diffusion equations, J. Differential Equations, 237 (2001), 271-320.  doi: 10.1006/jdeq.2000.3917. [26] G. Raugel, Dynamics of partial differential equations on thin domains, Dynamical Systems (Montecatini Terme, 1994), 208-315, Lecture Notes in Math., 1609, Springer, Berlin, 1995. doi: 10.1007/BFb0095241. [27] G. Raugel and G. Sell, Navier-Stokes equations on thin 3D domains. Ⅰ. Global attractors and global regularity of solutions, J. Amer. Math. Soc., 6 (1993), 503-568.  doi: 10.2307/2152776. [28] A. Rodriguez-Bernal, B. Wang and R. Willie, Asymptotic behaviour of time-dependent Ginzburg-Landau equations of superconductivity, Math. Meth. Appl. Sci., 22 (1999), 1647-1669.  doi: 10.1002/(SICI)1099-1476(199912)22:18<1647::AID-MMA97>3.0.CO;2-W. [29] B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, 1992,185-192. [30] 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. [31] B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269. [32] G. Wang, B. Guo and Y. Li, The asymptotic behavior of the stochastic Ginzburg-Landau equation with additive noise, Appl. Math. Comput., 198 (2008), 849-857.  doi: 10.1016/j.amc.2007.09.029. [33] X. Wang, K. Lu and B. Wang, Long term behavior of delay parabolic equations with additive noise and deterministic time dependent forcing, SIAM J. Appl. Dynam. Syst., 14 (2015), 1018-1047.  doi: 10.1137/140991819. [34] Z. Wang and S. Zhou, Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise, Discrete Contin. Dyn. Syst., 37 (2017), 2787-2812.  doi: 10.3934/dcds.2017120. [35] D. Yang, The asymptotic behavior of the stochastic Ginzburg-Landau equation with multiplicative noise, J. Math. Phys., 45 (2004), 4064-4076.  doi: 10.1063/1.1794365.

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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. [2] L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998. doi: 10.1007/978-3-662-12878-7. [3] J. Arrieta, A. Carvalho, M. Pereira and R. P. Da Silva, Semilinear parabolic problems in thin domains with a highly oscillatory boundary, Nonlinear Anal., 74 (2011), 5111-5132.  doi: 10.1016/j.na.2011.05.006. [4] P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621. [5] 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. [6] 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. [7] I. D. 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. [8] I. D. Chueshov and S. Kuksin, Stochastic 3D Navier-Stokes equations in a thin domain and its α-approximation, Physica D, 237 (2008), 1352-1367.  doi: 10.1016/j.physd.2008.03.012. [9] I. 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. [10] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225. [11] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Relat. Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705. [12] H. Cui, Y. Li and J. Yin, Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles, Nonlinear Anal., 128 (2015), 303-324.  doi: 10.1016/j.na.2015.08.009. [13] 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. [14] 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. [15] J. K. Hale and G. Raugel, Reaction-diffusion equations on thin domains, J. Math. Pures Appl., 71 (1992), 33-95. [16] J. K. 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. [17] R. Johnson, M. Kamenskii and P. Nistri, Existence of periodic solutions of an autonomous damped wave equation in thin domains, J. Dynam. Differential Equations, 10 (1998), 409-424.  doi: 10.1023/A:1022601213052. [18] P. E. Kloeden and J. 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. [19] S. Lü, H. Lu and Z. Feng, Stochastic dynamics of 2D fractional Ginzburg-Landau equation with multiplicative noise, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 575-590.  doi: 10.3934/dcdsb.2016.21.575. [20] 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. [21] 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. [22] 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. [23] Y. Morita, Stable solutions to the Ginzburg-Landau equation with magnetic effect in a thin domain, Japan J. Indust. Appl. Math., 21 (2004), 129-147.  doi: 10.1007/BF03167468. [24] M. Prizzi and K. P. Rybakowski, Recent results on thin domain problems, Ⅱ, Topol. Methods Nonlinear Anal., 19 (2002), 199-219.  doi: 10.12775/TMNA.2002.010. [25] M. Prizzi and K. P. Rybakowski, The effect of domain squeezing upon the dynamics of reaction-diffusion equations, J. Differential Equations, 237 (2001), 271-320.  doi: 10.1006/jdeq.2000.3917. [26] G. Raugel, Dynamics of partial differential equations on thin domains, Dynamical Systems (Montecatini Terme, 1994), 208-315, Lecture Notes in Math., 1609, Springer, Berlin, 1995. doi: 10.1007/BFb0095241. [27] G. Raugel and G. Sell, Navier-Stokes equations on thin 3D domains. Ⅰ. Global attractors and global regularity of solutions, J. Amer. Math. Soc., 6 (1993), 503-568.  doi: 10.2307/2152776. [28] A. Rodriguez-Bernal, B. Wang and R. Willie, Asymptotic behaviour of time-dependent Ginzburg-Landau equations of superconductivity, Math. Meth. Appl. Sci., 22 (1999), 1647-1669.  doi: 10.1002/(SICI)1099-1476(199912)22:18<1647::AID-MMA97>3.0.CO;2-W. [29] B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, 1992,185-192. [30] 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. [31] B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269. [32] G. Wang, B. Guo and Y. Li, The asymptotic behavior of the stochastic Ginzburg-Landau equation with additive noise, Appl. Math. Comput., 198 (2008), 849-857.  doi: 10.1016/j.amc.2007.09.029. [33] X. Wang, K. Lu and B. Wang, Long term behavior of delay parabolic equations with additive noise and deterministic time dependent forcing, SIAM J. Appl. Dynam. Syst., 14 (2015), 1018-1047.  doi: 10.1137/140991819. [34] Z. Wang and S. Zhou, Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise, Discrete Contin. Dyn. Syst., 37 (2017), 2787-2812.  doi: 10.3934/dcds.2017120. [35] D. Yang, The asymptotic behavior of the stochastic Ginzburg-Landau equation with multiplicative noise, J. Math. Phys., 45 (2004), 4064-4076.  doi: 10.1063/1.1794365.
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