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

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 Ôtani (1982) to show the local well-posedness of (CGL) and the existence of small global solutions provided that the nonlinearity is the Sobolev subcritical.