Propagation of wave front sets of solutions of the Cauchy problem for hyperbolic equations in Gevrey classes
type: Item
instance of: scholarly article
title: Propagation of wave front sets of solutions of the Cauchy problem for hyperbolic equations in Gevrey classes
zbMATH Open document ID: 0631.35052
author: Yoshinori Morimoto
author: Kazuo Taniguchi
published in: Osaka Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: The main purpose of this work is to study the propagation of the wave front set in Gevrey classes \(WF_{G(\kappa)}u\) of the solutions of the Cauchy problem for hyperbolic equations with characteristics of variable multiplicities. The authors reduce this problem to the analysis of the first order pseudo-differential system \({\mathcal L}u(t)=0\), \(u(0)=f\) when the principal symbol of \({\mathcal L}\) is diagonal. They prove that the wave front \(WF_{G(\kappa)}u(t)\) is included in the set \(\Gamma (t,WF_{G(\kappa)}(f))\). Roughly speaking, the set \(\Gamma(t,V)\) is formed by the conic hull of the set of end points at \(\sigma =t\) of all \(\epsilon\)-admissible trajectories \(\gamma(\sigma)\) issued from \((y,\eta)\in V\). The \(\epsilon\)-admissible trajectories represent an \(\epsilon\)-approximation of the so called broken null-bicharacteristics. The proofs are based on a deep and careful examination of Fourier integral operators in Gevrey classes and on their multiproducts.
Mathematics Subject Classification ID: 35L30
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 4025979
zbMATH Keywords: propagation of the wave front set
zbMATH Keywords: Gevrey classes
zbMATH Keywords: Cauchy problem
zbMATH Keywords: characteristics of variable multiplicities
zbMATH Keywords: first order pseudo- differential system
zbMATH Keywords: principal symbol
zbMATH Keywords: broken null-bicharacteristics
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4025979