doi: 10.3934/dcdsb.2020233

Asymptotic behavior of non-autonomous random Ginzburg-Landau equation driven by colored noise

School of Mathematics, Shandong University, Jinan 250100, China

* Corresponding author: Zhang Chen

Received  November 2019 Revised  May 2020 Published  August 2020

This paper investigates mainly the long term behavior of the non-autonomous random Ginzburg-Landau equation driven by nonlinear colored noise on unbounded domains. Due to the noncompactness of Sobolev embeddings on unbounded domains, pullback asymptotic compactness of random dynamical system associated with such random Ginzburg-Landau equation is proved by the tail-estimates method. Moreover, it is proved that the pullback random attractor of the non-autonomous random Ginzburg-Landau equation driven by a linear multiplicative colored noise converges to that of the corresponding stochastic system driven by a linear multiplicative white noise.

Citation: Lingyu Li, Zhang Chen. Asymptotic behavior of non-autonomous random Ginzburg-Landau equation driven by colored noise. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020233
References:
[1]

P. Acquistapace and B. Terreni, An approach to Ito linear equations in Hilbert spaces by approximation of white noise with coloured noise, Stochastic Anal. Appl., 2 (1984), 131-186.  doi: 10.1080/07362998408809031.  Google Scholar

[2]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[3]

M. BartuccelliP. ConstantinC. R. DoeringJ. D. Gibbon and M. Gisselfält, On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation, Phys. D, 44 (1990), 421-444.  doi: 10.1016/0167-2789(90)90156-J.  Google Scholar

[4]

D. Blömker and Y. Han, Asymptotic compactness of stochastic complex Ginzburg-Landau equation on an unbounded domain, Stoch. Dyn., 10 (2010), 613-636.  doi: 10.1142/S0219493710003121.  Google Scholar

[5]

C. Bu, On the Cauchy problem for the 1+2 complex Ginzburg-Landau equation, J. Austral. Math. Soc. Ser. B, 36 (1995), 313-324.  doi: 10.1017/S0334270000010468.  Google Scholar

[6]

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

[7]

C. R. DoeringJ. D. Gibbon and C. D. Levermore, Weak and strong solutions of the complex Ginzburg-Landau equation, Phys. D, 71 (1994), 285-318.  doi: 10.1016/0167-2789(94)90150-3.  Google Scholar

[8]

J. DuanE. S. Titi and P. Holmes, Regularity, approximation and asymptotic dynamics for a generalized Ginzburg-Landau equation, Nonlinearity, 6 (1993), 915-933.  doi: 10.1088/0951-7715/6/6/005.  Google Scholar

[9]

J. Duan and P. Holmes, On the Cauchy problem of a generalized Ginzburg-Landau equation, Nonlinear Anal., 22 (1994), 1033-1040.  doi: 10.1016/0362-546X(94)90065-5.  Google Scholar

[10]

J. DuanP. Holmes and E. S. Titi, Global existence theory for a generalized Ginzburg-Landau equation, Nonlinearity, 5 (1992), 1303-1314.  doi: 10.1088/0951-7715/5/6/005.  Google Scholar

[11]

H. Gao and C. Bu, A Dirichlet boundary value problem for a generalized Ginzburg-Landau equation, Appl. Math. Lett., 16 (2003), 179-184.  doi: 10.1016/S0893-9659(03)80029-0.  Google Scholar

[12]

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

[13]

A. Gu and B. Wang, Random attractors of FitzHugh-Nagumo systems driven by colored noise on unbounded domains, Stoch. Dyn., 19 (2019), 1950035, 38. doi: 10.1142/S0219493719500357.  Google Scholar

[14]

A. GuK. Lu and B. Wang, Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations, Discrete Contin. Dyn. Syst., 39 (2019), 185-218.  doi: 10.3934/dcds.2019008.  Google Scholar

[15]

B. GuoG. Wang and D. Li, The attractor of the stochastic generalized Ginzburg-Landau equation, Sci. China Ser. A: Math., 51 (2008), 955-964.  doi: 10.1007/s11425-007-0181-6.  Google Scholar

[16]

Y. F. Guo and D. L. Li, Random attractor of stochastic complex Ginzburg-Landau equation with multiplicative noise on unbounded domain, Stoch. Anal. Appl., 35 (2017), 409-422.  doi: 10.1080/07362994.2016.1259075.  Google Scholar

[17]

T. JiangX. Liu and J. Duan, Approximation for random stable manifolds under multiplicative correlated noises, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3163-3174.  doi: 10.3934/dcdsb.2016091.  Google Scholar

[18]

M. M. Kłosek-DygasB. J. Matkowsky and Z. Schuss, Colored noise in dynamical systems, SIAM J. Appl. Math., 48 (1988), 425-441.  doi: 10.1137/0148023.  Google Scholar

[19]

Y. Lan and J. Shu, Fractal dimension of random attractors for non-autonomous fractional stochastic {G}inzburg-{L}andau equations with multiplicative noise, Dyn. Syst., 34 (2019), 274-300.  doi: 10.1080/14689367.2018.1523368.  Google Scholar

[20]

D. LiZ. Dai and X. Liu, Long time behaviour for generalized complex Ginzburg-Landau equation, J. Math. Anal. Appl., 330 (2007), 934-948.  doi: 10.1016/j.jmaa.2006.07.095.  Google Scholar

[21]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires, Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[22]

Z. Liu and Z. Qiao, Wong-Zakai approximation of stochastic Allen-Cahn equation, Int. J. Numer. Anal. Model., 16 (2019), 681-694.   Google Scholar

[23]

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

[24]

H. LuP. W. BatesS. Lü and M. Zhang, Dynamics of the 3D fractional Ginzburg-Landau equation with multiplicative noise on an unbounded domain, Commun. Math. Sci., 14 (2016), 273-295.   Google Scholar

[25]

T. Nakayama and S. Tappe, Wong-Zakai approximations with convergence rate for stochastic partial differential equations, Stoch. Anal. Appl., 36 (2018), 832-857.  doi: 10.1080/07362994.2018.1471402.  Google Scholar

[26]

L. Ridolfi, P. D'Odorico and F. Laio, Noise-Induced Phenomena in the Environmental Sciences, Cambridge University Press, Cambridge, 2011. doi: 10.1017/CBO9780511984730.  Google Scholar

[27]

J. ShenK. Lu and W. Zhang, Heteroclinic chaotic behavior driven by a Brownian motion, J. Differential Equations, 255 (2013), 4185-4225.  doi: 10.1016/j.jde.2013.08.003.  Google Scholar

[28]

J. ShenJ. ZhaoK. Lu and B. Wang, The Wong-Zakai approximations of invariant manifolds and foliations for stochastic evolution equations, J. Differential Equations, 266 (2019), 4568-4623.  doi: 10.1016/j.jde.2018.10.008.  Google Scholar

[29]

R. Temam, Navier-Stokes Equations, Revised Edition, North-Holland Publishing Co., Amsterdam-New York, 1979.  Google Scholar

[30]

B. Wang, Sufficient 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

[31]

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

[32]

G. WangB. 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.  Google Scholar

[33]

M. C. Wang and G. E. Uhlenbeck, On the theory of the Brownian motion. II, Rev. Modern Phys., 17 (1945), 323-342.  doi: 10.1103/RevModPhys.17.323.  Google Scholar

[34]

R. WangY. 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.  Google Scholar

[35]

X. WangK. Lu and B. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 264 (2018), 378-424.  doi: 10.1016/j.jde.2017.09.006.  Google Scholar

[36]

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.  Google Scholar

[37]

J. Zhang and J. Shu, Existence and upper semicontinuity of random attractors for non-autonomous fractional stochastic Ginzburg-Landau equations, J. Math. Phys., 60 (2019), 042702. doi: 10.1063/1.5037480.  Google Scholar

[38]

Q. Zhang, Random attractors for a Ginzburg-Landau equation with additive noise, Chaos Solitons Fractals, 39 (2009), 463-472.  doi: 10.1016/j.chaos.2007.03.001.  Google Scholar

show all references

References:
[1]

P. Acquistapace and B. Terreni, An approach to Ito linear equations in Hilbert spaces by approximation of white noise with coloured noise, Stochastic Anal. Appl., 2 (1984), 131-186.  doi: 10.1080/07362998408809031.  Google Scholar

[2]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[3]

M. BartuccelliP. ConstantinC. R. DoeringJ. D. Gibbon and M. Gisselfält, On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation, Phys. D, 44 (1990), 421-444.  doi: 10.1016/0167-2789(90)90156-J.  Google Scholar

[4]

D. Blömker and Y. Han, Asymptotic compactness of stochastic complex Ginzburg-Landau equation on an unbounded domain, Stoch. Dyn., 10 (2010), 613-636.  doi: 10.1142/S0219493710003121.  Google Scholar

[5]

C. Bu, On the Cauchy problem for the 1+2 complex Ginzburg-Landau equation, J. Austral. Math. Soc. Ser. B, 36 (1995), 313-324.  doi: 10.1017/S0334270000010468.  Google Scholar

[6]

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

[7]

C. R. DoeringJ. D. Gibbon and C. D. Levermore, Weak and strong solutions of the complex Ginzburg-Landau equation, Phys. D, 71 (1994), 285-318.  doi: 10.1016/0167-2789(94)90150-3.  Google Scholar

[8]

J. DuanE. S. Titi and P. Holmes, Regularity, approximation and asymptotic dynamics for a generalized Ginzburg-Landau equation, Nonlinearity, 6 (1993), 915-933.  doi: 10.1088/0951-7715/6/6/005.  Google Scholar

[9]

J. Duan and P. Holmes, On the Cauchy problem of a generalized Ginzburg-Landau equation, Nonlinear Anal., 22 (1994), 1033-1040.  doi: 10.1016/0362-546X(94)90065-5.  Google Scholar

[10]

J. DuanP. Holmes and E. S. Titi, Global existence theory for a generalized Ginzburg-Landau equation, Nonlinearity, 5 (1992), 1303-1314.  doi: 10.1088/0951-7715/5/6/005.  Google Scholar

[11]

H. Gao and C. Bu, A Dirichlet boundary value problem for a generalized Ginzburg-Landau equation, Appl. Math. Lett., 16 (2003), 179-184.  doi: 10.1016/S0893-9659(03)80029-0.  Google Scholar

[12]

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

[13]

A. Gu and B. Wang, Random attractors of FitzHugh-Nagumo systems driven by colored noise on unbounded domains, Stoch. Dyn., 19 (2019), 1950035, 38. doi: 10.1142/S0219493719500357.  Google Scholar

[14]

A. GuK. Lu and B. Wang, Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations, Discrete Contin. Dyn. Syst., 39 (2019), 185-218.  doi: 10.3934/dcds.2019008.  Google Scholar

[15]

B. GuoG. Wang and D. Li, The attractor of the stochastic generalized Ginzburg-Landau equation, Sci. China Ser. A: Math., 51 (2008), 955-964.  doi: 10.1007/s11425-007-0181-6.  Google Scholar

[16]

Y. F. Guo and D. L. Li, Random attractor of stochastic complex Ginzburg-Landau equation with multiplicative noise on unbounded domain, Stoch. Anal. Appl., 35 (2017), 409-422.  doi: 10.1080/07362994.2016.1259075.  Google Scholar

[17]

T. JiangX. Liu and J. Duan, Approximation for random stable manifolds under multiplicative correlated noises, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3163-3174.  doi: 10.3934/dcdsb.2016091.  Google Scholar

[18]

M. M. Kłosek-DygasB. J. Matkowsky and Z. Schuss, Colored noise in dynamical systems, SIAM J. Appl. Math., 48 (1988), 425-441.  doi: 10.1137/0148023.  Google Scholar

[19]

Y. Lan and J. Shu, Fractal dimension of random attractors for non-autonomous fractional stochastic {G}inzburg-{L}andau equations with multiplicative noise, Dyn. Syst., 34 (2019), 274-300.  doi: 10.1080/14689367.2018.1523368.  Google Scholar

[20]

D. LiZ. Dai and X. Liu, Long time behaviour for generalized complex Ginzburg-Landau equation, J. Math. Anal. Appl., 330 (2007), 934-948.  doi: 10.1016/j.jmaa.2006.07.095.  Google Scholar

[21]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires, Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[22]

Z. Liu and Z. Qiao, Wong-Zakai approximation of stochastic Allen-Cahn equation, Int. J. Numer. Anal. Model., 16 (2019), 681-694.   Google Scholar

[23]

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

[24]

H. LuP. W. BatesS. Lü and M. Zhang, Dynamics of the 3D fractional Ginzburg-Landau equation with multiplicative noise on an unbounded domain, Commun. Math. Sci., 14 (2016), 273-295.   Google Scholar

[25]

T. Nakayama and S. Tappe, Wong-Zakai approximations with convergence rate for stochastic partial differential equations, Stoch. Anal. Appl., 36 (2018), 832-857.  doi: 10.1080/07362994.2018.1471402.  Google Scholar

[26]

L. Ridolfi, P. D'Odorico and F. Laio, Noise-Induced Phenomena in the Environmental Sciences, Cambridge University Press, Cambridge, 2011. doi: 10.1017/CBO9780511984730.  Google Scholar

[27]

J. ShenK. Lu and W. Zhang, Heteroclinic chaotic behavior driven by a Brownian motion, J. Differential Equations, 255 (2013), 4185-4225.  doi: 10.1016/j.jde.2013.08.003.  Google Scholar

[28]

J. ShenJ. ZhaoK. Lu and B. Wang, The Wong-Zakai approximations of invariant manifolds and foliations for stochastic evolution equations, J. Differential Equations, 266 (2019), 4568-4623.  doi: 10.1016/j.jde.2018.10.008.  Google Scholar

[29]

R. Temam, Navier-Stokes Equations, Revised Edition, North-Holland Publishing Co., Amsterdam-New York, 1979.  Google Scholar

[30]

B. Wang, Sufficient 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

[31]

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

[32]

G. WangB. 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.  Google Scholar

[33]

M. C. Wang and G. E. Uhlenbeck, On the theory of the Brownian motion. II, Rev. Modern Phys., 17 (1945), 323-342.  doi: 10.1103/RevModPhys.17.323.  Google Scholar

[34]

R. WangY. 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.  Google Scholar

[35]

X. WangK. Lu and B. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 264 (2018), 378-424.  doi: 10.1016/j.jde.2017.09.006.  Google Scholar

[36]

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.  Google Scholar

[37]

J. Zhang and J. Shu, Existence and upper semicontinuity of random attractors for non-autonomous fractional stochastic Ginzburg-Landau equations, J. Math. Phys., 60 (2019), 042702. doi: 10.1063/1.5037480.  Google Scholar

[38]

Q. Zhang, Random attractors for a Ginzburg-Landau equation with additive noise, Chaos Solitons Fractals, 39 (2009), 463-472.  doi: 10.1016/j.chaos.2007.03.001.  Google Scholar

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