Well-posedness in \(C^\infty\) for some weakly hyperbolic equations
type: Item
instance of: scholarly article
title: Well-posedness in \(C^\infty\) for some weakly hyperbolic equations
zbMATH Open document ID: 1007.35045
author: Ferruccio Colombini
author: Nicola Orrù
published in: Journal of Mathematics of Kyoto University
publication date: 2000-11-15T00:00:00Z
review text: The authors study the well-posedness of the Cauchy problem for the third-order homogeneous equation \[ D^3_tu+ a(t)D^2_t D_xu+b(t) D^2_xD_t u+c(t)D^3_x u=f. \] The coefficients are assumed to vanish at \(t=0\), but \(b(t)\) and \(c(t)\) do not vanish simultaneously to infinite order; moreover, the equation is supposed to be weakly hyperbolic. A necessary and sufficient condition is then given for well-posedness in \(H^\infty\). As an introduction to their result, the authors give a detailed survey on weakly hyperbolic equations with variable multiplicity. An extension of the result to higher-order equations is also proposed at the end of the paper.
Mathematics Subject Classification ID: 35L30
Mathematics Subject Classification ID: 35A05
zbMATH DE Number: 1530988
zbMATH Keywords: necessary and sufficient condition for well-posedness
zbMATH Keywords: third-order homogeneous equation
reviewed by: Luigi Rodino
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250517861
OpenAlex ID: W1561975952
DOI: 10.1215/KJM/1250517861
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description: scientific article; zbMATH DE number 1530988