On the Gevrey wellposedness of the Cauchy problem for some non-Kowalewskian equations
type: Item
instance of: scholarly article
title: On the Gevrey wellposedness of the Cauchy problem for some non-Kowalewskian equations
zbMATH Open document ID: 0962.35115
DOI: 10.1016/S0021-7824(00)00153-7
author: N. Delaunay
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2000-06-22T00:00:00Z
review text: This paper is concerned with the wellposedness of the Cauchy problem for the equation in \(\mathbb{R}^d\): \[ u_{tt}+a(t)\Delta^2 u+p(t,\partial_t, \partial_x)u=0, \] where \(a(t)\) is a real-valued nonnegative function. In the case of \(a(t)\equiv 1\) the global wellposedness is well known in the usual Sobolev space. This paper studies the case that \(a(t)\) is not constant and sufficiently smooth. So, the wellposedness has to be considered in the Gevrey space. Using Fourier analysis, energy estimates and known mathematical results for weakly hyperbolic equations, the authors prove the \(\gamma^s\)-wellposedness for \(p(t,\partial_t,\partial_x)=0\), or \(=-b(t)\Delta\) or \(=-c(t)\Delta\partial_t\) provided some assumptions on \(s\) are satisfied (see also \textit{P. D'Ancona} [Publ. Res. Inst. Math. Sci. 24, No. 3, 433-449 (1988; Zbl 0706.35077)] on the \(\gamma^s\)-wellposedness).
Mathematics Subject Classification ID: 35L30
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35L80
Mathematics Subject Classification ID: 35L15
Mathematics Subject Classification ID: 35Q72
zbMATH DE Number: 1463873
zbMATH Keywords: weakly hyperbolic
zbMATH Keywords: Fourier analysis
zbMATH Keywords: Gevrey space
reviewed by: Song Jiang
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1463873