The Cauchy problem for hyperbolic operators of strong type
type: Item
instance of: scholarly article
title: The Cauchy problem for hyperbolic operators of strong type
zbMATH Open document ID: 0826.35066
DOI: 10.1215/S0012-7094-94-07511-X
author: Seiichiro Wakabayashi
author: Tatsuo Nishitani
author: Kunihiko Kajitani
published in: Duke Mathematical Journal
publication date: 1995-11-23T00:00:00Z
review text: Let \(P(x,D) = P_m (x,D) + P_{m - 1} (x,D) + \cdots + P_0(x)\) be a differential operator of order \(m\) in the variables \(x = (x_1, \ldots, x_n)\) (here \(x_1\) can be interpreted as a time variable), \(P_j\) being its homogeneous part of order \(j\). The authors are concerned with the \(C^\infty\) well posedness of the Cauchy problem for \(P\) with respect to the hyperplanes \(x_1 = \text{const}\) in the case of multiple characteristics. For sake of simplicity we shall assume that the multiplicity of the characteristic roots of \(P_m\) is \(m\). The celebrated Mizohata and Ivrii-Petkov theorems tell us that, for arbitrary \(m\), if the Cauchy problem for \(P\) is well posed, then \(P_m\) is hyperbolic and, if \(P_m\) vanishes at \(\rho\) of order \(m\), then \(P_j\) vanishes at \(\rho\) of order \([m - 2j]_+\). Moreover it is known that in general these conditions are not enough to ensure the well posedness of the Cauchy problem for \(P\). When they are, we say that \(P\) is strongly hyperbolic. The authors of the present paper prove that a certain class of hyperbolic operators with characteristics of higher multiplicity are strongly hyperbolic. It is too long to give details on the assumptions here, but one can roughly describe the class considered by the authors saying that the difference of any two characteristic roots of the operator is bounded from below by a time function vanishing on the characteristic manifold, or, equivalently, that the propagation cone of (a localization of) \(P\) at a multiple point is transverse to the multiple characteristic variety. The method of proof, which contains also a propagation of singularities result, consists in proving first a microlocal a priori estimate in spaces with a pseudodifferential weight function. It is worth noting that the class considered by the authors falls presumably very short of the class of all strongly hyperbolic operators; a discussion of this issue is given in the introductory section of the paper.
Mathematics Subject Classification ID: 35L30
Mathematics Subject Classification ID: 35L80
zbMATH DE Number: 706692
zbMATH Keywords: \(C^ \infty\) well posedness of the Cauchy problem
zbMATH Keywords: multiple characteristics
reviewed by: Antonio Bove
MaRDI profile type: Publication
cites work: Necessary and sufficient conditions for the well-posedness of the Cauchy problem for a class of hyperbolic operators with triple characteristics
cites work: Q3921589
cites work: Spatially inhomogeneous pseudodifferential operators, I
cites work: Weighted distribution spaces and pseudodifferential operators
cites work: An example of a weakly hyperbolic Cauchy problem not well posed in \(C^\infty\)
cites work: Q3240445
cites work: Q3671563
cites work: Q3671564
cites work: Q3736112
cites work: The Cauchy problem for differential equations with double characteristics
cites work: NECESSARY CONDITIONS FOR THE CAUCHY PROBLEM FOR NON-STRICTLY HYPERBOLIC EQUATIONS TO BE WELL-POSED
cites work: Q4153110
cites work: Q4188185
cites work: The Cauchy problem for effectively hyperbolic equations (general cases)
cites work: Primitive extensions of rank 3 of \(2^n\cdot GL(n,2)\)
cites work: On linear partial differential equations with analytic coefficients unique continuation of data
cites work: The hyperbolic Cauchy problem
cites work: Q4306515
cites work: The Cauchy problem for effectively hyperbolic operators
cites work: Some remarks on the Cauchy problem
cites work: Q3783984
cites work: Local energy integrals for effectively hyperbolic operators. I
cites work: Local energy integrals for effectively hyperbolic operators. II
cites work: Hyperbolic operators with symplectic multiple characteristics
cites work: Propagation of singularities for hyperbolic operators with transverse propagation cone
cites work: A Note on Hyperbolic Polynomials.
cites work: The Cauchy problem for non-effectively hyperbolic differential operators and their product
cites work: Necessary and sufficient conditions for the hyperbolicity of polynomials with hyperbolic principal part
cites work: Singularities of solutions of the caught problem for symmetric hyperbolic systems
cites work: Remarks on hyperbolic polynomials
cites work: Q3730690
description: scientific article; zbMATH DE number 706692