Analyticity of Essentially Bounded Solutions to Semilinear Parabolic Systems and Validity of the GinzburgLandau Equation - Texas A&M University (TAMU) Scholar

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.