Advanced Search
Article Contents
Article Contents

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

  • * Corresponding author

    * Corresponding author
Abstract Full Text(HTML) Related Papers Cited by
  • 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.

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


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] L. Arnold, Random Dynamical Systems, Springer-Verlag, New York, 1998. doi: 10.1007/978-3-662-12878-7.
    [2] M. BartuccelliP. ConstantinC. DoeringJ. 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. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical system, Stoch. Dyn., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.
    [4] P. W. BatesK. 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. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.
    [9] C. DoeringJ. 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. DuanP. 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-AtienzaK. 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. GuoY. 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. HanW. 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. LiZ. 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. LuP. W. BatesS. 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. LuP. W. BatesJ. 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. LuP. W. BatesS. 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 NezzaG. 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. ShenS. 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. WangS. 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. YanS. 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. YinY. 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. ZhangC. 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.
  • 加载中

Article Metrics

HTML views(649) PDF downloads(311) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint