American Institute of Mathematical Sciences

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September  2019, 24(9): 5121-5148. doi: 10.3934/dcdsb.2019046

Long term behavior of stochastic discrete complex Ginzburg-Landau equations with time delays in weighted spaces

Dingshi Li 1, , Lin Shi 1, and  Xiaohu Wang 2,,

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  September 2018 Published  September 2019 Early access  February 2019

Fund Project: This work was supported by NSFC (11331007, 11601446, 11701475 and 11871049) and Excellent Youth Scholars of Sichuan University (2016SCU04A15).

In this paper, we investigate the long term behavior of the solutions to a class of stochastic discrete complex Ginzburg-Landau equations with time-varying delays and driven by multiplicative white noise. We first prove the existence and uniqueness of random attractor in a weighted space for these equations. Then, we analyze the upper semicontinuity of the random attractors as the time delay approaches to zero.

Keywords: Discrete complex Ginzburg-Landau equation, delay, random attractor, weighted spaces.
Mathematics Subject Classification: Primary: 35B40; Secondary: 35B41, 37L30.
Citation: Dingshi Li, Lin Shi, Xiaohu Wang. Long term behavior of stochastic discrete complex Ginzburg-Landau equations with time delays in weighted spaces. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 5121-5148. doi: 10.3934/dcdsb.2019046
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L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dynam., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.

[3]

P. W. Bates, K. Lu and B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Physica D, 289 (2014), 32-50.  doi: 10.1016/j.physd.2014.08.004.

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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. - B, 14 (2010), 439-455.  doi: 10.3934/dcdsb.2010.14.439.

[5]

T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearity, J. Differential Equations, 253 (2012), 667-693.  doi: 10.1016/j.jde.2012.03.020.

[6]

T. Caraballo, F. Morillas and J. Valero, On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems, Discrete Contin. Dyn. Syst. -A, 34 (2014), 51-77.  doi: 10.3934/dcds.2014.34.51.

[7]

T. Chen, S. Zhou and C. Zhao, Attractors for discrete nonlinear Schrödinger equation with delay, Acta Math. Appl. Sin. Engl. Ser., 26 (2010), 633-642.  doi: 10.1007/s10255-007-7101-y.

[8]

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.

[9]

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.

[10]

A. Gu and Y. Li, Dynamic behavior of stochastic $p$-Laplacian-type lattice equations, Stoch. Dynam., 17 (2017), 1750040, 19pp. doi: 10.1142/S021949371750040X.

[11]

X. Han and P. E. Kloeden, Non-autonomous lattice systems with switching effects and delayed recovery, J. Differential Equations, 261 (2016), 2986-3009.  doi: 10.1016/j.jde.2016.05.015.

[12]

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.

[13]

X. Han, Random attractors for second order stochastic lattice dynamical systems with multiplicative noise in weighted spaces, Stoch. Dynam., 12 (2012), 1150024, 20pp. doi: 10.1142/S0219493711500249.

[14]

X. Han, Asymptotic behaviors for second order stochastic lattice dynamical systems on $\mathbb Z^k$ in weighted spaces, J. Math. Anal. Appl., 397 (2013), 242-254.  doi: 10.1016/j.jmaa.2012.07.015.

[15]

N. Karachalios, H. Nistazakis and A. Yannacopoulos, Asymptotic behavior of solutions of complex discrete evolution equations: the discrete Ginzburg-Landau equation, Discrete Contin. Dyn. Syst -A, 19 (2007), 711-736.  doi: 10.3934/dcds.2007.19.711.

[16]

D. Li and L. Shi, Upper semicontinuity of random attractors of stochastic discrete complex Ginzburg-Landau equations with time-varying delays in the delay, J. Difference Equ. Appl., 24 (2018), 872-897.  doi: 10.1080/10236198.2018.1437913.

[17]

X. Li, K. Wei and H. Zhang, Exponential attractors for lattice dynamical systems in weighted spaces, Acta Appl. Math., 114 (2011), 157-172.  doi: 10.1007/s10440-011-9606-x.

[18]

V. A. Liskevich and M. A. Perelmuter, Analyticity of submarkovian semigroups, Proc. Amer. Math. Soc., 123 (1995), 1097-1104.  doi: 10.2307/2160706.

[19]

Y. Lv and J. Sun, Asymptotic behavior of stochastic discrete complex Ginzburg-Landau equations, Physica D, 221 (2006), 157-169.  doi: 10.1016/j.physd.2006.07.023.

[20]

J. Shu, Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations driven by fractional Brownian motions, Discrete Contin. Dyn. Syst. - B, 22 (2017), 1587-1599.  doi: 10.3934/dcdsb.2017077.

[21]

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.

[22]

B. Wang, Uniform attractors of nonautonomous discrete reaction-diffusion systems in weighted spaces, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), 695-716.  doi: 10.1142/S0218127408020598.

[23]

Y. Wang and K. Bai, Pullback attractors for a class of nonlinear lattices with delays, Discrete Contin. Dyn. Syst. - B, 20 (2015), 1213-1230.  doi: 10.3934/dcdsb.2015.20.1213.

[24]

P. Wang, Y. Huang and X. Wang, Random attractors for stochastic discrete complex non-autonomous Ginzburg-Landau equations with multiplicative noise, Adv. Difference Equ., 2015 (2015), 15pp. doi: 10.1186/s13662-015-0575-7.

[25]

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.

[26]

X. Wang, K. Lu and B. Wang, Exponential stability of non-autonomous stochastic delay lattice systems driven by a multiplicative white noise, J. Dynam. Differential Equations, 28 (2016), 1309-1335.  doi: 10.1007/s10884-015-9448-8.

[27]

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.

[28]

Z. Wang and S. Zhou, Existence and upper semicontinuity of attractors for non-autonomous stochastic lattice systems with random coupled coefficients, Commun. Pure Appl. Anal., 15 (2016), 2221-2245.  doi: 10.3934/cpaa.2016035.

[29]

Z. Wang and S. Zhou, Random attractors for stochastic retarded lattice systems, J. Difference Equations Appl., 19 (2013), 1523-1543.  doi: 10.1080/10236198.2013.765412.

[30]

W. Yan, Y. Li and S. Ji, Random attractors for first order stochastic retarded lattice dynamical systems, J. Math. Phys., 51 (2010), 032702, 17pp. doi: 10.1063/1.3319566.

[31]

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.

[32]

C. Zhang and L. Zhao, The attractors for 2nd-order stochastic delay lattice systems, Discrete Contin. Dyn. Syst. - A, 37 (2017), 575-590.  doi: 10.3934/dcds.2017023.

[33]

M. Zhao and S. Zhou, Random attractor of non-autonomous stochastic Boussinesq lattice system, J. Math. Phys., 56 (2015), 092702, 16pp. doi: 10.1063/1.4930195.

[34]

C. Zhao and S. Zhou, Limit behavior of global attractors for the complex Ginzburg-Landau equation on infinite lattices, Appl. Math. Lett., 21 (2008), 628-635.  doi: 10.1016/j.aml.2007.07.016.

[35]

C. Zhao and S. Zhou, Attractors of retarded first order lattice systems, Nonlinearity, 20 (2007), 1987-2006.  doi: 10.1088/0951-7715/20/8/010.

[36]

S. Zhou and Z. Wang, Random attractors for stochastic retarded lattice systems, J. Difference Equ. Appl., 19 (2013), 1523-1543.  doi: 10.1080/10236198.2013.765412.

show all references

References:
[1]

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

[2]

P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dynam., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.

[3]

P. W. Bates, K. Lu and B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Physica D, 289 (2014), 32-50.  doi: 10.1016/j.physd.2014.08.004.

[4]

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. - B, 14 (2010), 439-455.  doi: 10.3934/dcdsb.2010.14.439.

[5]

T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearity, J. Differential Equations, 253 (2012), 667-693.  doi: 10.1016/j.jde.2012.03.020.

[6]

T. Caraballo, F. Morillas and J. Valero, On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems, Discrete Contin. Dyn. Syst. -A, 34 (2014), 51-77.  doi: 10.3934/dcds.2014.34.51.

[7]

T. Chen, S. Zhou and C. Zhao, Attractors for discrete nonlinear Schrödinger equation with delay, Acta Math. Appl. Sin. Engl. Ser., 26 (2010), 633-642.  doi: 10.1007/s10255-007-7101-y.

[8]

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.

[9]

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.

[10]

A. Gu and Y. Li, Dynamic behavior of stochastic $p$-Laplacian-type lattice equations, Stoch. Dynam., 17 (2017), 1750040, 19pp. doi: 10.1142/S021949371750040X.

[11]

X. Han and P. E. Kloeden, Non-autonomous lattice systems with switching effects and delayed recovery, J. Differential Equations, 261 (2016), 2986-3009.  doi: 10.1016/j.jde.2016.05.015.

[12]

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.

[13]

X. Han, Random attractors for second order stochastic lattice dynamical systems with multiplicative noise in weighted spaces, Stoch. Dynam., 12 (2012), 1150024, 20pp. doi: 10.1142/S0219493711500249.

[14]

X. Han, Asymptotic behaviors for second order stochastic lattice dynamical systems on $\mathbb Z^k$ in weighted spaces, J. Math. Anal. Appl., 397 (2013), 242-254.  doi: 10.1016/j.jmaa.2012.07.015.

[15]

N. Karachalios, H. Nistazakis and A. Yannacopoulos, Asymptotic behavior of solutions of complex discrete evolution equations: the discrete Ginzburg-Landau equation, Discrete Contin. Dyn. Syst -A, 19 (2007), 711-736.  doi: 10.3934/dcds.2007.19.711.

[16]

D. Li and L. Shi, Upper semicontinuity of random attractors of stochastic discrete complex Ginzburg-Landau equations with time-varying delays in the delay, J. Difference Equ. Appl., 24 (2018), 872-897.  doi: 10.1080/10236198.2018.1437913.

[17]

X. Li, K. Wei and H. Zhang, Exponential attractors for lattice dynamical systems in weighted spaces, Acta Appl. Math., 114 (2011), 157-172.  doi: 10.1007/s10440-011-9606-x.

[18]

V. A. Liskevich and M. A. Perelmuter, Analyticity of submarkovian semigroups, Proc. Amer. Math. Soc., 123 (1995), 1097-1104.  doi: 10.2307/2160706.

[19]

Y. Lv and J. Sun, Asymptotic behavior of stochastic discrete complex Ginzburg-Landau equations, Physica D, 221 (2006), 157-169.  doi: 10.1016/j.physd.2006.07.023.

[20]

J. Shu, Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations driven by fractional Brownian motions, Discrete Contin. Dyn. Syst. - B, 22 (2017), 1587-1599.  doi: 10.3934/dcdsb.2017077.

[21]

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.

[22]

B. Wang, Uniform attractors of nonautonomous discrete reaction-diffusion systems in weighted spaces, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), 695-716.  doi: 10.1142/S0218127408020598.

[23]

Y. Wang and K. Bai, Pullback attractors for a class of nonlinear lattices with delays, Discrete Contin. Dyn. Syst. - B, 20 (2015), 1213-1230.  doi: 10.3934/dcdsb.2015.20.1213.

[24]

P. Wang, Y. Huang and X. Wang, Random attractors for stochastic discrete complex non-autonomous Ginzburg-Landau equations with multiplicative noise, Adv. Difference Equ., 2015 (2015), 15pp. doi: 10.1186/s13662-015-0575-7.

[25]

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.

[26]

X. Wang, K. Lu and B. Wang, Exponential stability of non-autonomous stochastic delay lattice systems driven by a multiplicative white noise, J. Dynam. Differential Equations, 28 (2016), 1309-1335.  doi: 10.1007/s10884-015-9448-8.

[27]

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.

[28]

Z. Wang and S. Zhou, Existence and upper semicontinuity of attractors for non-autonomous stochastic lattice systems with random coupled coefficients, Commun. Pure Appl. Anal., 15 (2016), 2221-2245.  doi: 10.3934/cpaa.2016035.

[29]

Z. Wang and S. Zhou, Random attractors for stochastic retarded lattice systems, J. Difference Equations Appl., 19 (2013), 1523-1543.  doi: 10.1080/10236198.2013.765412.

[30]

W. Yan, Y. Li and S. Ji, Random attractors for first order stochastic retarded lattice dynamical systems, J. Math. Phys., 51 (2010), 032702, 17pp. doi: 10.1063/1.3319566.

[31]

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.

[32]

C. Zhang and L. Zhao, The attractors for 2nd-order stochastic delay lattice systems, Discrete Contin. Dyn. Syst. - A, 37 (2017), 575-590.  doi: 10.3934/dcds.2017023.

[33]

M. Zhao and S. Zhou, Random attractor of non-autonomous stochastic Boussinesq lattice system, J. Math. Phys., 56 (2015), 092702, 16pp. doi: 10.1063/1.4930195.

[34]

C. Zhao and S. Zhou, Limit behavior of global attractors for the complex Ginzburg-Landau equation on infinite lattices, Appl. Math. Lett., 21 (2008), 628-635.  doi: 10.1016/j.aml.2007.07.016.

[35]

C. Zhao and S. Zhou, Attractors of retarded first order lattice systems, Nonlinearity, 20 (2007), 1987-2006.  doi: 10.1088/0951-7715/20/8/010.

[36]

S. Zhou and Z. Wang, Random attractors for stochastic retarded lattice systems, J. Difference Equ. Appl., 19 (2013), 1523-1543.  doi: 10.1080/10236198.2013.765412.

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