Communications in Mathematical Sciences

Volume 21 (2023)

Number 3

$G$-mean random attractors for complex Ginzburg–Landau equations with probability-uncertain initial data

Pages: 709 – 730



Tomás Caraballo (Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Spain)

Zhang Chen (School of Mathematics, Shandong University, Jinan, China)

Dandan Yang (School of Mathematics, Shandong University, Jinan, China)


In this paper, a class of complex Ginzburg–Landau equations with random initial data is investigated, where the randomness may be of probability uncertainty. The existence and uniqueness of global solution for such system are proved under the framework of nonlinear expectation. Then, the existence of pullback $\mathrm{G}$-mean random attractors for the $\mathrm{G}$-mean random dynamical system generated by the solution operators of (1.1) is investigated not only in $L^2_G (\Omega, L^2 (\mathbb{R}))$, but also in a weighted space $L^2_G (\Omega, L^2_\sigma (\mathbb{R}))$. Moreover, such attractor is periodic if the nonautonomous deterministic forcing is time periodic.


complex Ginzburg–Landau equation, random initial data, nonlinear expectation, $\mathrm{G}$-mean random dynamical system, $\mathrm{G}$-mean random attractor

2010 Mathematics Subject Classification

35B40, 35Q56, 37H05

This work is partially supported by the NNSF of China (11471190, 11971260), the NSF of Shandong Province (ZR2014AM002), the PSF (2012M511488, 2013T60661, 201202023), and LMNS of Fudan University.

The research of T. Caraballo has been partially supported by Ministerio de Ciencia, Innovación y Universidades (Spain) and FEDER (European Community) under grant PGC2018- 096540-B-I00, and by Junta de Andalucía (Consejería de Economía y Conocimiento) and FEDER under projects US-1254251 and P18-FR-4509.

Received 21 March 2021

Received revised 27 June 2022

Accepted 25 July 2022

Published 27 February 2023