ANALYTICITY OF ESSENTIALLY BOUNDED SOLUTIONS TO SEMILINEAR PARABOLIC-SYSTEMS AND VALIDITY OF THE GINZBURG-LANDAU EQUATION

Some analytic smoothing properties of a general strongly coupled, stro ngly parabolic semilinear system of order 2m in R(D) x (0, T) with ana lytic entries are investigated. These properties are expressed in term s of holomorphic continuation in space and time of essentially bounded global solutions to the system. Given 0 < T' < T less than or equal t o infinity, it is proved that any weak, essentially bounded solution u = (u(1),...,u(N)) in R(D) x (0, T) possesses a bounded holomorphic co ntinuation u(z + iy, sigma + i tau) into a region in C-D x C defined b y (x, sigma) epsilon R(D) x (T',T), \y\ < A' and \tau\ < B', where A' and B' are some positive constants depending upon T'. The proof is bas ed on analytic smoothing properties of a parabolic Green function comb ined with a contraction mapping argument in a Hardy space H-infinity. Applications include weakly coupled semilinear systems of complex reac tion-diffusion equations such as the complex Ginzburg-Landau equations . Special attention is given to the problem concerning the validity of the derivation of amplitude equations which describe various instabil ity phenomena in hydrodynamics.