On a sharp Levi condition in Gevrey classes for some infinitely degenerate hyperbolic equations and its necessity
type: Item
instance of: scholarly article
title: On a sharp Levi condition in Gevrey classes for some infinitely degenerate hyperbolic equations and its necessity
zbMATH Open document ID: 1028.35089
author: Haruhisa Ishida
author: Karen Yagdjian
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2004-01-26T00:00:00Z
review text: The authors consider the Cauchy problem in Gevrey class for a second-order linear weakly hyperbolic operator \(L= \partial^2_t- \sum a_{jk}(t) \partial_j\partial_k- \sum a_j(t)\partial_j\) with time dependent coefficients in \(C^1\) class. Let \(\lambda(t)\in C^1([0,T])\) be a weight function satisfying \(\lambda(0)= \lambda'(0)= 0\), \(\lambda'(t)> 0\), \(t> 0\) and for \(t\in [0,T]\) \[ c_0{\lambda(t)\over \Lambda(t)}\leq {\lambda'(t)\over \lambda(t)}\leq c_1{\lambda'(t)\over \Lambda(t)}, \] where \(\Lambda(t)= \int^t_0 \lambda(\tau) d\tau\). They prove that, if the coefficients of \(L\) satisfy for \(t\in [0,T]\) \[ d_0\lambda(t)^2 |\xi|^2\leq \sum a_{jk}(t) \xi_j\xi_k\leq d_1\lambda(t)^2 |\xi|^2, \] \[ |\partial_t \sum a_{jk}(t) \xi_j\xi_k|\leq d_3\lambda(t)^3 \Lambda(t)^{-s/(s- 1)}|\xi|^2, \] \[ \max_{j=1,2,\dots,n} |\partial^k_t a_j(t)|\leq d_1\lambda(t)^{-(k+1)s/(s+ 1)}\quad (k= 0,1), \] then the Cauchy problem for \(L\) is well posed in Gevrey class of index \(s\).
Mathematics Subject Classification ID: 35L15
Mathematics Subject Classification ID: 35L80
zbMATH DE Number: 1833276
zbMATH Keywords: well-posedness
zbMATH Keywords: Cauchy problem
zbMATH Keywords: Gevrey class
zbMATH Keywords: second-order linear weakly hyperbolic operator
reviewed by: Kunihiko Kajitani
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1145476338
OpenAlex ID: W2017415355
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DOI: 10.2977/PRIMS/1145476338
description: scientific article; zbMATH DE number 1833276