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, strongly parabolic semilinear system of order $2m$ in $realnos^D times (0,T)$ with analytic entries are investigated. These properties are expressed in terms of holomorphic continuation in space and time of essentially bounded global solutions to the system. Given $0 < T' < T le infty$, it is proved that any weak, essentially bounded solution ${bold u} = (u_1,dots,u_N)$ in $realnos^Dtimes (0,T)$ possesses a bounded holomorphic continuation $bold u (x+iy,sigma + itau )$ into a region in $complexnos^Dtimescomplexnos$ defined by $(x,sigma )in realnos^Dtimes (T',T)$, $|y| < A'$ and $|tau | < B'$, where $A'$ and $B'$ are some positive constants depending upon $T'$. The proof is based on analytic smoothing properties of a parabolic Green function combined with a contraction mapping argument in a Hardy space $H^infty$. Applications include weakly coupled semilinear systems of complex reaction-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 instability phenomena in hydrodynamics