Well-posedness of the Cauchy problem for some weakly hyperbolic operators in Gevrey classes
type: Item
instance of: scholarly article
title: Well-posedness of the Cauchy problem for some weakly hyperbolic operators in Gevrey classes
zbMATH Open document ID: 0584.35065
author: Shigeharu Itoh
published in: Proceedings of the Japan Academy. Series A
publication date: 1985-01-01T00:00:00Z
review text: The problem studied in this paper is concerned with the determination of a function space in which the Cauchy problem for a given weakly hyperbolic operator is well-posed. The operator is supposed to be of the form \[ L=L_ 0(t,x,D_ t,D_ x)+L_ 1(t,x,D_ t,D_ x),\quad where \] \[ L_ 0(t,x,D_ t,D_ x)=D^ m_ t +\sum^{m}_{k=1}\sigma (x)^{k\mu} (\sum_{| \alpha | =k}a_{k\alpha}(t,x)D_ x^{\alpha})D_ t^{m-k},\quad and \] \[ L_ 1(t,x,D_ t,D_ x)=\sum^{m}_{k=1}\sum^{k}_{j=1}\sigma (x)^{\mu_ k-j} (\sum_{| \alpha | =k-j}b_{jk\alpha}(t,x)D_ x^{\alpha})D_ t^{m-k} \] while \(t\in (0,T]\), \(x\in {\mathbb{R}}^ n\) and \(\mu\) and \(\mu_ k\) are non-negative integers. It is proven that under certain conditions involving also the lower order terms the Cauchy problem \[ Lu(t,x)=f(t,x)\quad in\quad (0,T]\times {\mathbb{R}}^ n;\quad D^ i_ t u(t,x)=u^ i(x)\quad on\quad {\mathbb{R}}^ n,\quad t=0;\quad i=0,...,m-1 \] is well-posed in some sense.
Mathematics Subject Classification ID: 35L30
Mathematics Subject Classification ID: 35R25
Mathematics Subject Classification ID: 35B30
zbMATH DE Number: 3935609
zbMATH Keywords: Cauchy problem
zbMATH Keywords: weakly hyperbolic operator
zbMATH Keywords: well-posed
reviewed by: Mithat Idemen
MaRDI profile type: Publication
full work available at URL: pjaa.61.66
OpenAlex ID: W2081637169
cites work: Q3943335
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cites work: Q3712838
DOI: 10.3792/PJAA.61.66
description: scientific article; zbMATH DE number 3935609