Regularity of random attractors for non-autonomous stochastic discrete complex Ginzburg-Landau equations

Abstract

In this paper, we consider the asymptotic behaviour of non-autonomous stochastic discrete complex Ginzburg-Landau equations with additive noise in weighted space ${\ell }_{\rho }^p$. The existences of tempered random attractors for this equation in spaces ${\ell }_{\rho }^2$ and ${\ell }_{\rho }^p$ are proved respectively by tail estimates, which implies that the obtained ${\ell }_{\rho }^2$ -random attractor is compact and attracting in the topology of ${\ell }_{\rho }^p$ space. The main difficulty here is the lack of compactness on infinite lattices. To deal with this, we introduce a common embedding space of ${\ell }_{\rho }^2$ and ${\ell }_{\rho }^p$ and derive some tail-estimates of solutions.

Keywords:

• Ginzburg-Landau equation
• random attractor
• additive noise
• weighted space
• asymptotic compactness

2010 Mathematics Subject Classifications:

• 60H15
• 37L55

Acknowledgments

The authors would like to thank the reviewers for their helpful comments. Joint research project of Laurent Mathematics Center of Sichuan Normal University and National-Local Joint Engineering Laboratory of System Credibility Automatic Verification.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was partially supported by the National Natural Science Foundation of China (No. 11871138).