Characterization of algebraic curves that satisfy the Phragmén-Lindelöf principle for global evolution
In some previous papers, \textit{C. Boiti} and \textit{M. Nacinovich} [Ann. Inst. Fourier 47, No. 1, 155--199 (1997; Zbl 0865.35091), Ann. Mat. Pura Appl., IV. Ser. 180, No. 1, 81--126 (2001; Zbl 1030.35128)] investigated the Cauchy problem, with initial data on an affine subvariety, for general (overdetermined) systems of partial differential operators with constant coefficients. In particular they characterized evolution in spaces of ultradifferentiable functions in terms of the validity of a Phragmén-Lindelöf maximum principle for plurisubharmonic functions on some associated complex algebraic variety. In this paper, using also the results obtained by \textit{R. Meise, B. A. Taylor} and \textit{D. Vogt} [J. Am. Math. Soc. 11, No. 1, 1--39 (1998; Zbl 0896.32008)] and by \textit{R. W. Braun, R. Meise} and \textit{B. A. Taylor} [Trans. Am. Math. Soc. 356, No. 4, 1315--1383 (2004; Zbl 1039.32010)], quite explicit conditions for evolution in spaces of ultradifferentiable functions are found for the case where the associated complex algebraic variety is a plane curve \(V\). First it is shown that in the homogeneous case all points at infinity must be real; next necessary and sufficient conditions for the general curve are found in terms of its branching at infinity, described in terms of Puiseux series expansions. In the last section these results are illustrated by examples and also to study evolution for some overdetermined systems (the curve \(V\) is no longer assumed to be planar).
scientific article; zbMATH DE number 2119241
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