Taiwanese Journal of Mathematics


Hong Lu and Shujuan Lü

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In this paper, we consider the asymptotic behavior of solutions to the stochastic fractional complex Ginzburg-Landau equation with multiplicative noise in one spatial dimensions. We first transfer stochastic fractional Ginzburg-Landau equation into random equation which solutions generate a random dynamical system. Then, we consider the existence of a random attractor for the random dynamical system. At last we estimate the Hausdorff dimension of the random attractor by using linearization and Lyapunov exponents.

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Taiwanese J. Math., Volume 18, Number 2 (2014), 435-450.

First available in Project Euclid: 10 July 2017

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Primary: 37L55: Infinite-dimensional random dynamical systems; stochastic equations [See also 35R60, 60H10, 60H15] 60H15: Stochastic partial differential equations [See also 35R60] 35Q99: None of the above, but in this section

stochastic fractional Ginzburg-Landau equation asymptotic compactness random attractor Hausdorff dimension


Lu, Hong; Lü, Shujuan. RANDOM ATTRACTOR FOR FRACTIONAL GINZBURG-LANDAU EQUATION WITH MULTIPLICATIVE NOISE. Taiwanese J. Math. 18 (2014), no. 2, 435--450. doi:10.11650/tjm.18.2014.3053. https://projecteuclid.org/euclid.twjm/1499706395

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  • G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Physics Reports, 371 (2002), 461-580.
  • A. I. Saichev and G. M. Zaslavsky, Fractional kinetic equations: solutions and applications, Chaos, 7 (1997), 753-764.
  • G. M. Zaslavsky and M. Edelman, Weak mixing and anomalous kinetics along filamented surfaces, Chaos, 11 (2001), 295-305.
  • R. R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Stat. Solidi. B, 133 (1986), 425-430.
  • E. W. Montroll and M. F. Shlesinger, On the wonderful world of random walks, in: Nonequilibrium Phenomena II: from Stochastics to Hydrodynamics, J. Leibowitz and E. W. Montroll (eds.), North-Holland, Amsterdam, 1984, pp. 1-121.
  • M. F. Shlesinger, G. M. Zaslavsky and J. Klafter, Strange Kinetics, Nature, 363 (1993), 31-37.
  • G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, 2005.
  • Boling Guo, Yongqian Han and Jie Xin, Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation, Appliled Mathematics and Computation, 204 (2008), 468-477.
  • Boling Guo and Z. Huo, Global well-posedness for the fractional nonlinear Schrödinger equation, Commun. Partial Differential Equations, 36(2) (2011), 247-255.
  • Jianping Dong and Mingyu Xu, Space-time fractional Schrödinger equation with time-independent potentials, J. Math. Anal. Appl., 344 (2008), 1005-1017.
  • Boling Guo and Ming Zeng, Solutions for the fractional Landau-Lifshitz equation, J. Math. Anal. Appl., 361 (2010), 131-138.
  • Xueke Pu and Boling Guo, Global weak solutions of the fractional Landau-Lifshitz-Maxwell equation, J. Math. Anal. Appl., 372 (2010), 86-98.
  • Vasily E. Tarasov and George M. Zaslavsky, Fractional Ginzburg-Landau equation for fractal media, Physica A, 354 (2005), 249-261.
  • C. W. Gardiner, Handbooks of Stochastic Methods for Physics, Chemistry and Natural Sciences, Springer-Verlag, Berlin, 1983.
  • H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probability Theory and Related Fields, 100 (1994), 365-393.
  • H. Crauel, A. Debussche, F. Flandoli and Random Attractors, J. Dynamics and Differential Equations, 9 (1997), 307-341.
  • R. Temam, Infinite Dimension Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1995.
  • A. Debussche, Hausdorff dimension of a random invariant set, J. Math. Pures Appl., 77 (1998), 967-988.
  • Xueke Pu and Boling Guo, Well-posedness and dynamics for the fractional Ginzburg-Landau equation, Applicable Analysis, iFirst (2011), 1-17.
  • L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.
  • C. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46(4) (1993), 453-620.