Dynamics of non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise

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September 2019, 18(5): 2409-2431. doi: 10.3934/cpaa.2019109

Dynamics of non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise

Yun Lan ^1, and Ji Shu ^2,,

1.	School of Mathematical Science, Sichuan Normal University, Chengdu, Sichuan 610068, China
2.	School of Mathematical Science and V.C. & V.R. Key Lab, Sichuan Normal University, Chengdu, Sichuan 610068, China

^* Corresponding author

Received April 2018 Revised July 2018 Published April 2019

This paper is concerned with the asymptotic behavior of solutions for non-autonomous stochastic fractional complex Ginzburg-Landau equations driven by multiplicative noise with $ \alpha\in(0, 1) $. We first apply the Galerkin method and compactness argument to prove the existence and uniqueness of weak solutions, which is slightly different from the deterministic fractional case with $ \alpha\in(\frac{1}{2}, 1) $ and the real fractional case with $ \alpha\in(0, 1) $. Consequently, we establish the existence and uniqueness of tempered pullback random attractors for the equations in a bounded domain. At last, we obtain the upper semicontinuity of random attractors when the intensity of noise approaches zero.

Keywords: Non-autonomous stochastic fractional Ginzburg-Landau equation, random dynamical system, random attractor, multiplicative noise, upper semicontinuity.

Mathematics Subject Classification: Primary: 37L55, 60H15; Secondary: 35Q56.

Citation: Yun Lan, Ji Shu. Dynamics of non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2409-2431. doi: 10.3934/cpaa.2019109

References:

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[4]	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.
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[6]	I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, Heidelbrg, 2002. doi: 10.1007/b83277.
[7]	H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.
[8]	H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225.
[9]	C. Doering, J. Gibbon and C. Levermore, Weak and strong solutions of the complex Ginzburg-Landau equation, Physica D, 71 (1994), 285-318. doi: 10.1016/0167-2789(94)90150-3.
[10]	J. Dong and M. Xu, Space-time fractional Schrödinger equation with time-independent potentials, J. Math. Anal. Appl., 344 (2008), 1005-1017. doi: 10.1016/j.jmaa.2008.03.061.
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[12]	X. Fan and Y. Wang, Attractors for a second order nonautonomous lattice dynamical systems with nonlinear damping, Phys. Lett. A, 365 (2007), 17-27. doi: 10.1016/j.physleta.2006.12.045.
[13]	F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stoch. Stoch. Rep., 59 (1996), 21-45. doi: 10.1080/17442509608834083.
[14]	M. Garrido-Atienza, K. Lu and B. Schmalfuss, Random dynamical systems for stochastic equations driven by a fractional Brownian motion, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 473-493. doi: 10.3934/dcdsb.2010.14.473.
[15]	B. Guo, Y. Han and J. Xin, Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation, Appl. Math. Comput., 204 (2008), 458-477. doi: 10.1016/j.amc.2008.07.003.
[16]	B. Guo and Z. Huo, Global well-posedness for the fractional nonlinear Schrödinger equation, Commun. Partial Differential Equations, 36 (2011), 247-255. doi: 10.1080/03605302.2010.503769.
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[23]	D. Li and B. Guo, Asymptotic behavior of the 2D generalized stochastic Ginzburg-Landau equation with additive noise, Appl. Math. Mech., 30 (2009), 883-894. doi: 10.1007/s10483-009-0801-x.
[24]	J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Linearires, Dunod, Paris, 1969.
[25]	H. Lu and S. Lv, Random attrator for fractional Ginzburg-Laudau equation with multiplicative noise, Taiwanese J. Math., 18 (2014), 435-450. doi: 10.11650/tjm.18.2014.3053.
[26]	H. Lu, P. W. Bates, S. Lu and M. Zhang, Dynamics of 3-D fractional complex Ginzburg-Landau equation, J. Differential Equations, 259 (2015), 5276-5301. doi: 10.1016/j.jde.2015.06.028.
[27]	H. Lu, P. W. Bates, J. Xin and M. Zhang, Asymptotic behavior of stochastic fractional power dissipative equations on Rⁿ, Nonlinear Anal. TMA, 128 (2015), 176-198. doi: 10.1016/j.na.2015.06.033.
[28]	H. Lu, P. W. Bates, S. Lu and M. Zhang, Dynamics of the 3D fractional Ginzburg-Landau equation with multiplicative noise on a unbounded domain, Commmu. Math. Sci., 14 (2016), 273-295. doi: 10.4310/CMS.2016.v14.n1.a11.
[29]	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.
[30]	B. Maslowski and B. Schmalfuss, Random dynamical systems and stationary solutions of differential equations driven by the fractional Brownian motion, Stoch. Anal. Appl., 22 (2004), 1577-1607. doi: 10.1081/SAP-200029498.
[31]	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.
[32]	X. Pu and B. Guo, Global weak Soltuions of the fractional Landau-Lifshitz -Maxwell equation, J. Math. Anal. Appl., 372 (2010), 86-98. doi: 10.1016/j.jmaa.2010.06.035.
[33]	X. Pu and B. Guo, Well-posedness and dynamics for the fractional Ginzburg-Laudau equation, Appl. Anal., 92 (2013), 318-334. doi: 10.1080/00036811.2011.614601.
[34]	J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, Cambridge, UK, 2001. doi: 10.1007/978-94-010-0732-0.
[35]	B. Schmalfuss, Backward cocycle and atttractors of stochastic differential equations, in International Semilar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior (ed. V. Reitmann, T. Riedrich and N. Koksch), Technishe Universität, Dresden, 1992, pp.185–192.
[36]	G. Sell and Y. You, Dynamics of Evolutional Equations, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.
[37]	R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 831-855. doi: 10.1017/S0308210512001783.
[38]	T. Shen and J. Huang, Well-posedness and dynamics of stochastic fractional model for nonlinear optical fiber materials, Nonlinear Anal. TMA, 110 (2014), 33-46. doi: 10.1016/j.na.2014.06.018.
[39]	Z. Shen, S. Zhou and W. Shen, One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation, J. Differential Equations, 248 (2010), 1432-1457. doi: 10.1016/j.jde.2009.10.007.
[40]	J. Shu, Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations driven by fractional Brownian motions, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1587-1599. doi: 10.3934/dcdsb.2017077.
[41]	J. Shu, P. Li, J. Zhang and O. Liao, Random attractors for the stochastic coupled fractional Ginzburg-Landau equation with additive noise, J. Math. Phys., 56 (2015), 102702. doi: 10.1063/1.4934724.
[42]	Vasily E. Tarasov and George M. Zaslavsky, Fractional Ginzburg-Laudau equation for fractal media, Physica A, 354 (2005), 249-261.
[43]	R. Temam, Infinite Dimension Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4684-0313-8.
[44]	P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York-Berlin, 1982.
[45]	B. Wang, Random attractors for the stochastic FitzHugh-Nagumo system on unbounded domains, Nonlinear Anal. TMA, 71 (2009), 2811-2828. doi: 10.1016/j.na.2009.01.131.
[46]	B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electron J. Differential Equations, 139 (2009), 1-18.
[47]	B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on R³, Trans. Amer. Math. Soc., 363 (2011), 3639-3663. doi: 10.1090/S0002-9947-2011-05247-5.
[48]	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.
[49]	B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 269-300. doi: 10.3934/dcds.2014.34.269.
[50]	B. Wang, Existence and upper-semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2012), 1791-1798. doi: 10.1142/S0219493714500099.
[51]	B. Wang, Asymptotic behavior of non-autonomous fractional stochastic reaction-diffusion equations, Nonlinear Anal. TMA, 158 (2017), 60-82. doi: 10.1016/j.na.2017.04.006.
[52]	B. Wang and X. Gao, Random attractors for stochastic wave equations on unbounded domains, Discrete Contin. Dyn. Syst. Suppl., (2009), 800–809. doi: 10.1016/j.nonrwa.2011.06.008.
[53]	X. Wang, S. Li and D. Xu, Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Anal. TMA, 72 (2010), 483-494. doi: 10.1016/j.na.2009.06.094.
[54]	W. Yan, S. Ji and Y. Li, Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations, Phys. Lett. A, 373 (2009), 1268-1275. doi: 10.1016/j.physleta.2009.02.019.
[55]	F. Yin and L. Liu, D-pullback attractor for a non-autonomous wave equation with additive noise on unbounded domains, Comput. Math. Appl., 68 (2014), 424-438. doi: 10.1016/j.camwa.2014.06.018.
[56]	J. Yin, Y. Li and A. Gu, Backwards compact attractors and periodic attractors for non-autonomous damped wave equations on an unbounded domain, Comput. Math. Appl., 74 (2017), 744-758. doi: 10.1016/j.camwa.2017.05.015.
[57]	J. Zhang, C. Huang and J. Shu, Random attractors for the stochastic discrete complex Ginzburg-Landau equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 24 (2017), 303-315.
[58]	C. Zhao and S. Zhou, Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications, J. Math. Anal. Appl., 354 (2009), 78-95. doi: 10.1016/j.jmaa.2008.12.036.
[59]	S. Zhou and M. Zhao, Random attractors for damped non-autonomous wave equations with memory and white noise, Nonlinear Anal. TMA, 120 (2015), 202-226. doi: 10.1016/j.na.2015.03.009.

show all references

References:

[1]	L. Arnold, Random Dynamical Systems, Springer-Verlag, New York, 1998. doi: 10.1007/978-3-662-12878-7.
[2]	M. Bartuccelli, P. Constantin, C. Doering, J. Gibbon and M. Gisselfält, On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation, Physica D, 44 (1990), 421-444. doi: 10.1016/0167-2789(90)90156-J.
[3]	P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical system, Stoch. Dyn., 6 (2006), 1-21. doi: 10.1142/S0219493706001621.
[4]	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.
[5]	Z. Brzezniak and Y. Li, Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains, Trans. Amer. Math. Soc., 358 (2006), 5587-5629. doi: 10.1090/S0002-9947-06-03923-7.
[6]	I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, Heidelbrg, 2002. doi: 10.1007/b83277.
[7]	H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.
[8]	H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225.
[9]	C. Doering, J. Gibbon and C. Levermore, Weak and strong solutions of the complex Ginzburg-Landau equation, Physica D, 71 (1994), 285-318. doi: 10.1016/0167-2789(94)90150-3.
[10]	J. Dong and M. Xu, Space-time fractional Schrödinger equation with time-independent potentials, J. Math. Anal. Appl., 344 (2008), 1005-1017. doi: 10.1016/j.jmaa.2008.03.061.
[11]	J. Duan, P. Holme and E. S. Titi, Global existence theory for a generalized Ginzburg-Landau equation, Nonlinearity, 5 (2009), 1303-1314.
[12]	X. Fan and Y. Wang, Attractors for a second order nonautonomous lattice dynamical systems with nonlinear damping, Phys. Lett. A, 365 (2007), 17-27. doi: 10.1016/j.physleta.2006.12.045.
[13]	F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stoch. Stoch. Rep., 59 (1996), 21-45. doi: 10.1080/17442509608834083.
[14]	M. Garrido-Atienza, K. Lu and B. Schmalfuss, Random dynamical systems for stochastic equations driven by a fractional Brownian motion, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 473-493. doi: 10.3934/dcdsb.2010.14.473.
[15]	B. Guo, Y. Han and J. Xin, Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation, Appl. Math. Comput., 204 (2008), 458-477. doi: 10.1016/j.amc.2008.07.003.
[16]	B. Guo and Z. Huo, Global well-posedness for the fractional nonlinear Schrödinger equation, Commun. Partial Differential Equations, 36 (2011), 247-255. doi: 10.1080/03605302.2010.503769.
[17]	B. Guo, X. Pu and F. Huang, Fractional Partial Differential Equations and their Numerical Solutions, Science Press, Beijing, 2011. doi: 10.1142/9543.
[18]	B. Guo and X. Wang, Finite dimensional behavior for the derivative Ginzburg-Landau equation in two soatial dimensions, Physica D, 89 (1995), 83-99. doi: 10.1016/0167-2789(95)00216-2.
[19]	B. Guo and M. Zeng, Soltuions for the fractional Landau-Lifshitz equation, J. Math. Anal. Appl., 361 (2010), 131-138. doi: 10.1016/j.jmaa.2009.09.009.
[20]	J. K. Hale, Asymptotic Behavior of Dissipative Systems, in Math. Surveys Monogr., vol. 25, AMS, Providence, 1988.
[21]	X. Han, W. Shen and S. Zhou, Random attractors for stochastic lattice dynamical system in weighted space, J. Differential Equations, 250 (2011), 1235-1266. doi: 10.1016/j.jde.2010.10.018.
[22]	D. Li, Z. Dai and X. Liu, Long time behavior for generalized complex Ginzburg-Landau equation, J. Math. Anal. Appl., 330 (2007), 938-948. doi: 10.1016/j.jmaa.2006.07.095.
[23]	D. Li and B. Guo, Asymptotic behavior of the 2D generalized stochastic Ginzburg-Landau equation with additive noise, Appl. Math. Mech., 30 (2009), 883-894. doi: 10.1007/s10483-009-0801-x.
[24]	J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Linearires, Dunod, Paris, 1969.
[25]	H. Lu and S. Lv, Random attrator for fractional Ginzburg-Laudau equation with multiplicative noise, Taiwanese J. Math., 18 (2014), 435-450. doi: 10.11650/tjm.18.2014.3053.
[26]	H. Lu, P. W. Bates, S. Lu and M. Zhang, Dynamics of 3-D fractional complex Ginzburg-Landau equation, J. Differential Equations, 259 (2015), 5276-5301. doi: 10.1016/j.jde.2015.06.028.
[27]	H. Lu, P. W. Bates, J. Xin and M. Zhang, Asymptotic behavior of stochastic fractional power dissipative equations on Rⁿ, Nonlinear Anal. TMA, 128 (2015), 176-198. doi: 10.1016/j.na.2015.06.033.
[28]	H. Lu, P. W. Bates, S. Lu and M. Zhang, Dynamics of the 3D fractional Ginzburg-Landau equation with multiplicative noise on a unbounded domain, Commmu. Math. Sci., 14 (2016), 273-295. doi: 10.4310/CMS.2016.v14.n1.a11.
[29]	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.
[30]	B. Maslowski and B. Schmalfuss, Random dynamical systems and stationary solutions of differential equations driven by the fractional Brownian motion, Stoch. Anal. Appl., 22 (2004), 1577-1607. doi: 10.1081/SAP-200029498.
[31]	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.
[32]	X. Pu and B. Guo, Global weak Soltuions of the fractional Landau-Lifshitz -Maxwell equation, J. Math. Anal. Appl., 372 (2010), 86-98. doi: 10.1016/j.jmaa.2010.06.035.
[33]	X. Pu and B. Guo, Well-posedness and dynamics for the fractional Ginzburg-Laudau equation, Appl. Anal., 92 (2013), 318-334. doi: 10.1080/00036811.2011.614601.
[34]	J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, Cambridge, UK, 2001. doi: 10.1007/978-94-010-0732-0.
[35]	B. Schmalfuss, Backward cocycle and atttractors of stochastic differential equations, in International Semilar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior (ed. V. Reitmann, T. Riedrich and N. Koksch), Technishe Universität, Dresden, 1992, pp.185–192.
[36]	G. Sell and Y. You, Dynamics of Evolutional Equations, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.
[37]	R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 831-855. doi: 10.1017/S0308210512001783.
[38]	T. Shen and J. Huang, Well-posedness and dynamics of stochastic fractional model for nonlinear optical fiber materials, Nonlinear Anal. TMA, 110 (2014), 33-46. doi: 10.1016/j.na.2014.06.018.
[39]	Z. Shen, S. Zhou and W. Shen, One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation, J. Differential Equations, 248 (2010), 1432-1457. doi: 10.1016/j.jde.2009.10.007.
[40]	J. Shu, Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations driven by fractional Brownian motions, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1587-1599. doi: 10.3934/dcdsb.2017077.
[41]	J. Shu, P. Li, J. Zhang and O. Liao, Random attractors for the stochastic coupled fractional Ginzburg-Landau equation with additive noise, J. Math. Phys., 56 (2015), 102702. doi: 10.1063/1.4934724.
[42]	Vasily E. Tarasov and George M. Zaslavsky, Fractional Ginzburg-Laudau equation for fractal media, Physica A, 354 (2005), 249-261.
[43]	R. Temam, Infinite Dimension Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4684-0313-8.
[44]	P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York-Berlin, 1982.
[45]	B. Wang, Random attractors for the stochastic FitzHugh-Nagumo system on unbounded domains, Nonlinear Anal. TMA, 71 (2009), 2811-2828. doi: 10.1016/j.na.2009.01.131.
[46]	B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electron J. Differential Equations, 139 (2009), 1-18.
[47]	B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on R³, Trans. Amer. Math. Soc., 363 (2011), 3639-3663. doi: 10.1090/S0002-9947-2011-05247-5.
[48]	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.
[49]	B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 269-300. doi: 10.3934/dcds.2014.34.269.
[50]	B. Wang, Existence and upper-semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2012), 1791-1798. doi: 10.1142/S0219493714500099.
[51]	B. Wang, Asymptotic behavior of non-autonomous fractional stochastic reaction-diffusion equations, Nonlinear Anal. TMA, 158 (2017), 60-82. doi: 10.1016/j.na.2017.04.006.
[52]	B. Wang and X. Gao, Random attractors for stochastic wave equations on unbounded domains, Discrete Contin. Dyn. Syst. Suppl., (2009), 800–809. doi: 10.1016/j.nonrwa.2011.06.008.
[53]	X. Wang, S. Li and D. Xu, Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Anal. TMA, 72 (2010), 483-494. doi: 10.1016/j.na.2009.06.094.
[54]	W. Yan, S. Ji and Y. Li, Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations, Phys. Lett. A, 373 (2009), 1268-1275. doi: 10.1016/j.physleta.2009.02.019.
[55]	F. Yin and L. Liu, D-pullback attractor for a non-autonomous wave equation with additive noise on unbounded domains, Comput. Math. Appl., 68 (2014), 424-438. doi: 10.1016/j.camwa.2014.06.018.
[56]	J. Yin, Y. Li and A. Gu, Backwards compact attractors and periodic attractors for non-autonomous damped wave equations on an unbounded domain, Comput. Math. Appl., 74 (2017), 744-758. doi: 10.1016/j.camwa.2017.05.015.
[57]	J. Zhang, C. Huang and J. Shu, Random attractors for the stochastic discrete complex Ginzburg-Landau equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 24 (2017), 303-315.
[58]	C. Zhao and S. Zhou, Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications, J. Math. Anal. Appl., 354 (2009), 78-95. doi: 10.1016/j.jmaa.2008.12.036.
[59]	S. Zhou and M. Zhao, Random attractors for damped non-autonomous wave equations with memory and white noise, Nonlinear Anal. TMA, 120 (2015), 202-226. doi: 10.1016/j.na.2015.03.009.

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