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Long term behavior of stochastic discrete complex Ginzburg-Landau equations with time delays in weighted spaces
1. | School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610031, China |
2. | Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China |
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.
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L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998.
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P. W. Bates, H. Lisei and K. Lu,
Attractors for stochastic lattice dynamical systems, Stoch. Dynam., 6 (2006), 1-21.
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P. W. Bates, K. Lu and B. Wang,
Attractors of non-autonomous stochastic lattice systems in weighted spaces, Physica D, 289 (2014), 32-50.
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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.
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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,
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Y. Lv and J. Sun,
Asymptotic behavior of stochastic discrete complex Ginzburg-Landau equations, Physica D, 221 (2006), 157-169.
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[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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