DISTRIBUTION-VALUED INITIAL DATA FOR THE COMPLEX GINZBURG-LANDAU EQUATION

The generalized complex Ginzburg-Landau (CGL) equation with a nonlinea rity of order 2 sigma + 1 in d spatial dimensions has a unique local c lassical solution for distributional initial data in the Soholcv space H-q provided that q > d/2 - 1/sigma. This result directly corresponds to a theorem for the nonlinear Schrodinger (NLS) equation which has b een proved by Cazenave and Weissler in 1990. While the proof in the NL S case relies on Besov space techniques, it is shown here that for the CGL equation, the smoothing properties of the linear semigroup can he used to obtain an almost optimal result by elementary means.