Local well-posedness of the complex Ginzburg-Landau equation in bounded domains

TL;DR

This paper establishes local well-posedness results for the complex Ginzburg-Landau equation in bounded domains, focusing on the non-dissipative case and employing a perturbation approach based on Sobolev subcritical nonlinearities.

Contribution

It extends the analysis of the complex Ginzburg-Landau equation to the non-dissipative case in bounded domains, using a perturbation method inspired by Otani (1982).

Findings

Proves local well-posedness for non-dissipative CGL in bounded domains.

Shows existence of small global solutions under Sobolev subcritical conditions.

Adapts perturbation techniques to handle non-monotone nonlinearities.

Abstract

In this paper, we are concerned with the local well-posedness of the initial-boundary value problem for complex Ginzburg-Landau (CGL) equations in bounded domains. There are many studies for the case where the real part of its nonlinear term plays as dissipation. This dissipative case is intensively studied and it is shown that (CGL) admits a global solution when parameters appearing in (CGL) belong to the so-called CGL-region. This paper deals with the non-dissipative case. We regard (CGL) as a parabolic equation perturbed by monotone and non-monotone perturbations and follows the basic strategy developed in \^Otani (1982) to show the local well-posedness of (CGL) and the existence of small global solutions provided that the nonlinearity is Sobolev subcritical.