An extended Cauchy-Kovalevskaya problem and its solution in associated spaces
type: Item
instance of: scholarly article
title: An extended Cauchy-Kovalevskaya problem and its solution in associated spaces
zbMATH Open document ID: 1041.35002
DOI: 10.4171/ZAA/1126
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2002-01-01T00:00:00Z
review text: The authors are interested in solving operator equations of the form \[ u(t,x)= F_1(t,x,u)+ \int^t_0 F_2(\tau, x,u,\nabla_xu)\,d\tau. \] Both operators are supposed to be associated to an (elliptic) operator \(G\), i.e., \(Gu= 0\) implies \(G(F_ju)= 0\) for each fixed \(t\). The ellipticity of \(G\) with time-independent coefficients is reasonable because the authors assume interior estimates for the solutions of \(Gu= 0\). Solutions of the above operator equation possess the property of conical evolution. This allows to define a suitable Banach space. Some Lipschitz properties of \(F_1\) and \(F_2\) guarantee the existence of a unique solution in this Banach space. The solutions exist locally in \(t\).
Mathematics Subject Classification ID: 35A10
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 35A07
zbMATH DE Number: 1925968
zbMATH Keywords: abstract operator equation
zbMATH Keywords: interior estimates
zbMATH Keywords: conical evolution of solutions
zbMATH Keywords: Cauchy problem
author: Kiyoshi Asano
author: Wolfgang Tutschke
reviewed by: Michael Reissig
MaRDI profile type: Publication
cites work: A note on the abstract Cauchy-Kowalewski theorem
cites work: Q3984200
full work available at URL: https://doi.org/10.4171/zaa/1126
OpenAlex ID: W2021011876
description: scientific article; zbMATH DE number 1925968