Duhamel principle and uniformly well-posed Cauchy problems
type: Item
instance of: scholarly article
title: Duhamel principle and uniformly well-posed Cauchy problems
zbMATH Open document ID: 0809.34073
author: Keiichiro Kitagawa
published in: Journal of Mathematics of Kyoto University
publication date: 1994-05-10T00:00:00Z
review text: Mimiking the variation of constants approach to inhomogeneous initial value problems for ordinary differential equations Duhamel's principle yields a solution of a general inhomogeneous evolution equation of the form \(\partial_ t u(t)- A(t)u(t)= F(t)\), \(u(0+)= u_ 0\), (\(F\), \(u_ 0\) given) in terms of the homogeneous solution of a pure initial value problem \((F\equiv 0)\). The author analyzes the underlying relation between the homogeneous \((\text{PCH})_ s\) and the inhomogeneous evolution problem \((\text{PC})_ s\) posed in a Fréchet space \(X\). Here the parameter \(s\in [0,T_ 0[\), \(T_ 0> 0\), denotes the initial time. Based on the concept of uniform well-posedness versus well-posedness, the author demonstrates (under suitable assumptions on \(A(\cdot)\) in terms of the family of semi-norms defining the topology of \(X\)) that, roughly speaking, uniform well-posedness of the problems \((\text{PCH})_ s\) for \(s\in [0,T_ 0[\), is equivalent to well-posedness of the problems \((\text{PC})_ s\) for \(s\in[ 0,T_ 0[\), \(T_ 0> 0\). The result applies to Cauchy problems with time-dependent coefficients.
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 47F05
zbMATH DE Number: 567066
zbMATH Keywords: inhomogeneous evolution equation
zbMATH Keywords: initial value problem
zbMATH Keywords: Fréchet space
zbMATH Keywords: uniform well-posedness
zbMATH Keywords: Cauchy problems
reviewed by: Rainer Picard
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250519181
OpenAlex ID: W1540413764
DOI: 10.1215/KJM/1250519181
description: scientific article; zbMATH DE number 567066