Linear parabolic evolution equations in \(L^ p\)-spaces
type: Item
instance of: scholarly article
title: Linear parabolic evolution equations in \(L^ p\)-spaces
zbMATH Open document ID: 0568.35047
DOI: 10.1007/BF01762539
author: Gabriella Di Blasio
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1984-01-01T00:00:00Z
review text: The author studies regularity theory for the parabolic initial-boundary value problem \(u_ t(t,x)=A(x;D)u(t,x)+f(t,x),\) \(0<t\leq T\), \(x\in \Omega\), \(B_ i(x;D) u(t,x)=0\), \(1\leq i\leq m\), \(0<t\leq T\), \(x\in \partial \Omega\), \(u(0,x)=u_ 0(x)\). The inhomogeneous term f and the initial data are presumed to lie in \(L^ p\). The problem is studied abstractly as the Cauchy problem \(u'(t)=Au(t)+f(t),\) \(0<t\leq T\), \(u(0)=u_ 0\), where A is the infinitesimal generator of an analytic semigroup in a general Banach space E and \(f\in L^ p(0,T;E)\). A direct approach based upon the theory of interpolation spaces between D(A) and E is used to describe how the regularity of solutions depends on f and \(u_ 0\).
Mathematics Subject Classification ID: 35G10
Mathematics Subject Classification ID: 35K25
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 3906980
zbMATH Keywords: regularity
zbMATH Keywords: Cauchy problem
zbMATH Keywords: infinitesimal generator
zbMATH Keywords: analytic semigroup
zbMATH Keywords: Banach space
zbMATH Keywords: interpolation spaces
reviewed by: Glenn Webb
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3906980