The Cauchy problem in local spaces for the complex Ginzburg-Landau equation I: compactness methods

We continue the study of the initial value problem for the complex Ginzburg—Landau equation $$$$ (with a > 0, b > 0, g≥ 0) in \(\) initiated in a previous paper [I]. We treat the case where the initial data and the solutions belong to local uniform spaces, more precisely to spaces of functions satisfying local regularity conditions and uniform bounds in local norms, but no decay conditions (or arbitrarily weak decay conditions) at infinity in \(\). In [I] we used compactness methods and an extended version of recent local estimates [3] and proved in particular the existence of solutions globally defined in time with local regularity of the initial data corresponding to the spaces L r for r≥ 2 or H 1. Here we treat the same problem by contraction methods. This allows us in particular to prove that the solutions obtained in [I] are unique under suitable subcriticality conditions, and to obtain for them additional regularity properties and uniform bounds. The method extends some of those previously applied to the nonlinear heat equation in global spaces to the framework of local uniform spaces.