Construction of the evolution operator of parabolic type
type: Item
instance of: scholarly article
title: Construction of the evolution operator of parabolic type
zbMATH Open document ID: 1020.47027
author: Tosinobu Muramatu
author: Takio Tojima
published in: Osaka Journal of Mathematics
publication date: 2001-01-01T00:00:00Z
reviewed by: Jürgen Voigt
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 35K90
Mathematics Subject Classification ID: 34G10
zbMATH DE Number: 1625450
zbMATH Keywords: evolution equation
zbMATH Keywords: parabolic
zbMATH Keywords: evolution family
zbMATH Keywords: propagator
zbMATH Keywords: fundamental solution
MaRDI profile type: Publication
review text: The authors show the existence of an evolution family for the linear differential equation NEWLINE\[NEWLINE{du(t)\over dt}+A(t)u(t) =f(t)\quad (a<t<b)NEWLINE\]NEWLINE of parabolic type in a Banach space \(X\). The operators \(A(t)\) have \((t\)-independent) domain \(Y=D(A(t))\), a Banach space. Besides more traditional assumptions, the new hypothesis is that for NEWLINE\[NEWLINE\omega(h): =\sup\biggl\{\bigl \|A(t+h)- A(t)\bigr\|_{Y\to X}; \;a\leq t\leq b-h\biggr\},NEWLINE\]NEWLINE the function \(\omega (h)/h\) is integrable at 0. This generalises previous results where \(t\mapsto A(t)\in {\mathcal L}(Y,X)\) is Hölder continuous, or where \(\omega(h) |\log h|/h\) is integrable at 0.
description: scientific article; zbMATH DE number 1625450