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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 & Applied Analysis, 2019, 18 (5) : 2409-2431. doi: 10.3934/cpaa.2019109
References:
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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.  Google Scholar

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

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H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, Heidelbrg, 2002. doi: 10.1007/b83277.  Google Scholar

[7]

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

[8]

H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. Duan, P. Holme and E. S. Titi, Global existence theory for a generalized Ginzburg-Landau equation, Nonlinearity, 5 (2009), 1303-1314.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

B. Guo, X. Pu and F. Huang, Fractional Partial Differential Equations and their Numerical Solutions, Science Press, Beijing, 2011. doi: 10.1142/9543.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, in Math. Surveys Monogr., vol. 25, AMS, Providence, 1988.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Linearires, Dunod, Paris, 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, Cambridge, UK, 2001. doi: 10.1007/978-94-010-0732-0.  Google Scholar

[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. Google Scholar

[36]

G. Sell and Y. You, Dynamics of Evolutional Equations, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[42]

Vasily E. Tarasov and George M. Zaslavsky, Fractional Ginzburg-Laudau equation for fractal media, Physica A, 354 (2005), 249-261.   Google Scholar

[43]

R. Temam, Infinite Dimension Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[44]

P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[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.  Google Scholar

[46]

B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electron J. Differential Equations, 139 (2009), 1-18.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[50]

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