Local well-posedness of the complex Ginzburg-Landau Equation in general domains

TL;DR

This paper establishes local well-posedness for complex Ginzburg-Landau equations with superlinear growth in general domains, extending previous results in bounded domains using advanced parabolic operator theory.

Contribution

It introduces a novel approach based on subdifferential operators and Yosida approximation to prove existence and uniqueness of solutions in broader domains.

Findings

Proves local well-posedness in energy space H1 for general domains.

Establishes global existence for small initial data.

Extends previous bounded domain results to unbounded and complex domains.

Abstract

In this paper, complex Ginzburg-Landau (CGL) equations with superlinear growth terms are studied. We discuss the local well-posedness in the energy space H1 for the initial-boundary value problem of the equations in general domains. The local well-posedness in H1 in bounded domains is already examined by authors (2019). Our approach to CGL equations is based on the theory of parabolic equations governed by subdifferential operators with non-monotone perturbations. By using this method together with the Yosida approximation procedure, we discuss the existence and the uniqueness of local solutions as well as the global existence of solutions with small initial data.