Branching of singularities for degenerate hyperbolic operators
type: Item
instance of: scholarly article
title: Branching of singularities for degenerate hyperbolic operators
zbMATH Open document ID: 0555.35077
author: Kazuo Amano
author: Gen Nakamura
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1984-01-01T00:00:00Z
review text: Let \(P(t,x,D_ t,D_ x)=\sum A_{k,\alpha}(t,x)D^ k_ tD_ x^{\alpha},\) \(k+| \alpha | \leq m\), be an m-th order linear hyperbolic partial differential operator with \(C^{\infty}\) coefficients defined for -T\(\leq t\leq T\) and \(x\in R^ n\). The following conditions are assumed: 1) the principal symbol \(P_ m(t,x,\tau,\xi)=\prod (\tau - t^{\ell}\lambda_ j(t,x,\xi)),\) \(1\leq j\leq m\), of P is smoothly factorizable, with \(\ell\) a given positive integer, 2) for \(j\neq k\), \(| \lambda_ j(t,x,\xi)-\lambda_ k(t,x,\xi)| \geq c| \xi |,\) \(c>0\) constant, \(3)\quad A_{k,\alpha}(t,x)=t^{i\ell +i+k- m}B_{k,\alpha}(t,x)\) for each (k,\(\alpha)\) such that \(| \alpha | =i\) and \(i\ell +i+k-m\geq 0\), with \(B_{k,\alpha}\) smooth. The authors generalize and complete previous results about the branching of singularities for solutions to the Cauchy problem: \(Pu=0\) in \(]s,T]\times R^ n,\) \(D^ k_ tu|_{t=s}=g_ k\), \(0\leq k\leq m- 1\), \(g_ k\) given distributions with compact support in \(R^ n\). By constructing suitable parametrices for the Cauchy problem, they obtain sufficient conditions of branching of singularities.
Mathematics Subject Classification ID: 35L30
Mathematics Subject Classification ID: 35L80
Mathematics Subject Classification ID: 35L67
Mathematics Subject Classification ID: 35A20
zbMATH DE Number: 3884629
zbMATH Keywords: propagation of singularities
zbMATH Keywords: branching of singularities
zbMATH Keywords: Cauchy problem
zbMATH Keywords: parametrices
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195181607
OpenAlex ID: W1971140277
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DOI: 10.2977/PRIMS/1195181607
description: scientific article; zbMATH DE number 3884629