Microlocal regularity for global solutions of partial differential equations with polynomial coefficients
type: Item
instance of: scholarly article
title: Microlocal regularity for global solutions of partial differential equations with polynomial coefficients
zbMATH Open document ID: 0649.35014
DOI: 10.1007/BF00140753
author: Otto Liess
published in: Annals of Global Analysis and Geometry
publication date: 1987-01-01T00:00:00Z
review text: The author studies regularity questions for solutions \(u\in E^ s({\mathbb{R}}^ n\times I)\) of differential equations \(p(x,t,D_ x,D_ t)u=0\), where I is an open interval in \({\mathbb{R}}\) with \(0\in I\), \(s>1\) is a rational number, \(E^ s({\mathbb{R}}^ n\times I)\) is the space of functions \(f\in C^{\infty}({\mathbb{R}}^ n\times I)\) for which growth type restrictions \(| D_ x^{\alpha}D^ j_ tf(x,t)| \leq c_{\alpha j} \exp (A| x|^ s)\) hold, and \(p(x,t,D_ x,D_ t)\) is of the form \[ p(x,t,D_ x,D_ t)=\sum_{| \alpha | /s+| \beta | /\sigma +j\leq m}a_{\alpha \beta j}(t)x^{\alpha}D_ x^{\beta}D^ j_ t \] with \(s^{- 1}+\sigma^{-1}=1\), \(a_{00m}(t)\equiv 1\) and \(a_{\alpha \beta j}\) real-analytic in I. The statement of the main result involves an analytic wave front set adapted to the study of problems in which growth restritions at infinity appear. A discussion of this wave front set already has been given in an earlier paper of the author [Astérisque 89/90, 163-203 (1981; Zbl 0498.35027)]. Also the main theorem of the present paper has been announced there without proof.
Mathematics Subject Classification ID: 35G05
Mathematics Subject Classification ID: 35A27
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35G10
Mathematics Subject Classification ID: 35A05
zbMATH DE Number: 4059920
zbMATH Keywords: uniqueness
zbMATH Keywords: characteristic Cauchy problems
zbMATH Keywords: polynomial coefficients
zbMATH Keywords: regularity
zbMATH Keywords: analytic wave front set
zbMATH Keywords: growth restritions at infinity
Wikidata QID: Q115395194
reviewed by: Wolfgang Watzlawek
MaRDI profile type: Publication
cites work: Pseudo-differential operators and Gevrey classes
cites work: Uniqueness theorems and wave front sets for solutions of linear differential equations with analytic coefficients
cites work: On the characteristic Cauchy problem for partial differential equations
cites work: Q3963420
cites work: Intersection properties of weak analytically uniform classes of functions
cites work: Necessary and sufficient conditions for propagation of singularities for systems of linear partial differential operators with constant coefficients
cites work: Q5610544
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description: scientific article; zbMATH DE number 4059920