Cauchy problem for SG-hyperbolic equations with constant multiplicities
type: Item
instance of: scholarly article
title: Cauchy problem for SG-hyperbolic equations with constant multiplicities
zbMATH Open document ID: 0935.35106
author: Sandro Coriasco
author: Luigi Rodino
published in: Ricerche di Matematica
publication date: 2000-05-14T00:00:00Z
review text: The goal of the authors is to study Cauchy problems for so-called SG-hyperbolic linear operators with constant multiplicities. This means that the principal symbol has \(\mu\) distinct real roots \(\lambda_j=\lambda_j(t;x,\xi)\) of constant multplicity which belong to \(C^\infty(J,SG^{(1,1)}_l)\) and satisfy \[ |\lambda_j (t;x,\xi)-\lambda_k(t;x,\xi)|\geq C\langle \xi\rangle\langle x\rangle. \] By \(SG^{(1,1)}_l\) the authors denote the symbol space \[ \{a= a(x,\xi)\in C^\infty(\mathbb{R}^n\times \mathbb{R}^n) : |\partial^\alpha_\xi \partial^\beta_x a(x,\xi)|\leq C\langle \xi\rangle^{1-|\alpha|}\langle x\rangle^{1-|\beta|}\}. \] They explain the symbolic calculus, Sobolev spaces and the notion of a pseudodifferential operator; the notion of ellipticity; the notion of parametrix; the notion of a Fourier integral operator and the notion of hyperbolicity basing on the SG symbol classes. The SG-hyperbolic operators with constant multiplicities have to satisfy some Levi conditions. Finally a reduction procedure is proposed which transfers the starting Cauchy problem to a Cauchy problem for a system \[ \partial_tU - i K(t)U = F(t),\quad U(0) = U_0, \] with diagonal principal part, this means, that a perfect diagonalization modulo lower order terms exists. This reduction bases on a well-known result of Mizohata. These considerations generate a lot of interesting questions. What about considerations in Gelfand-Shilov classes? What about considerations for the weakly hyperbolic case with characteristic roots of variable multiplicity?
Mathematics Subject Classification ID: 35L80
Mathematics Subject Classification ID: 35A17
Mathematics Subject Classification ID: 35S30
zbMATH DE Number: 1370119
zbMATH Keywords: symbolic calculus
zbMATH Keywords: Levi conditions
zbMATH Keywords: perfect diagonalization modulo lower order terms
reviewed by: Michael Reissig
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1370119