Analyticity of Essentially Bounded Solutions to Semilinear Parabolic Systems and Validity of the GinzburgLandau Equation
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Some analytic smoothing properties of a general strongly coupled, strongly parabolic semilinear system of order 2m in D (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 , it is proved that any weak, essentially bounded solution u = (u1, . . . ,uN) in D (0,T) possesses a bounded holomorphic continuation u(x + iy, + i) into a region in D defined by (x, ) D (T, T), |y| < A and || < 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. 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.
SIAM Journal on Mathematical Analysis
author list (cited authors)
Tak, P., Bollerman, P., Doelman, A., van Harten, A., & Titi, E. S.
complete list of authors
Takáč, P||Bollerman, P||Doelman, A||van Harten, A||Titi, ES