Communications in Mathematical Sciences

Volume 14 (2016)

Number 1

Dynamics of the 3D fractional Ginzburg–Landau equation with multiplicative noise on an unbounded domain

Pages: 273 – 295

DOI: http://dx.doi.org/10.4310/CMS.2016.v14.n1.a11

Authors

Hong Lu (College of Science, China University of Mining and Technology, Jiangsu, China)

Peter W. Bates (Department of Mathematics, Michigan State University, East Lansing, Mich., U.S.A.)

Shujuan Lü (School of Mathematics and Systems Science & LMIB, Beihang University, Beijing, China)

Mingji Zhang (Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, New Mexico, U.S.A)

Abstract

We study a stochastic fractional complex Ginzburg–Landau equation with multiplicative noise in three spatial dimensions with particular interest in the asymptotic behavior of its solutions. We first transform our equation into a random equation whose solutions generate a random dynamical system. A priori estimates are derived when the nonlinearity satisfies certain growth conditions. Applying the estimates for far-field values of solutions and a cut-off technique, asymptotic compactness is proved. Furthermore, the existence of a random attractor in $H^1 (\mathbb{R}^3)$ of the random dynamical system is established.

Keywords

stochastic fractional Ginzburg–Landau equation, asymptotic compactness, random attractor, pullback attractor

2010 Mathematics Subject Classification

35Q99, 37L55, 60H15

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