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# 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

#### 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

Published 16 September 2015