Some properties of solutions of the Cauchy problem for evolution equations
type: Item
instance of: scholarly article
title: Some properties of solutions of the Cauchy problem for evolution equations
zbMATH Open document ID: 1252.35173
DOI: 10.1134/S0012266112030093
author: S. G. Pyatkov
author: M. V. Uvarova
published in: Differential Equations
publication date: 2012-08-13T00:00:00Z
review text: The authors study solvability of the Cauchy problem \[ \begin{aligned} u_t-Lu &=f,\qquad t\in [0,T],\quad T<\infty,\\ u|_{t=0} &=u_0,\end{aligned} \] where \(L : E\to E\) is the generator of an analytic semigroup in a Banach space \(E\). In contrast to many other papers on the subject, the authors assume that the Cauchy data \(u_0\) and the right-hand side \(f\) as \(t\to 0\) are singular in some sense: \(u_0\) belongs to some negative space (space of distributions) and \(f\) belongs to the space \(L_p\) with weight \(t^{\alpha}\). Sharp weighted estimates for the solutions and existence theorems in the case of arbitrary weight function and singular initial data are obtained. The results are used to study initial boundary value problems for parabolic equations with operator-valued coefficients.
reviewed by: Yakov Yakubov
Mathematics Subject Classification ID: 35K90
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 47D06
zbMATH DE Number: 6065264
zbMATH Keywords: analytic semigroup
zbMATH Keywords: Sobolev-Besov spaces
zbMATH Keywords: UMD spaces
zbMATH Keywords: \(R\)-bounded set
zbMATH Keywords: parabolic equations with operator-valued coefficients
MaRDI profile type: Publication
cites work: Équations différentielles abstraites
cites work: Navier­Stokes Equations with Nonhomogeneous Dirichlet Data
cites work: Semilinear parabolic equations involving measures and low regularity data
cites work: Maximal regularity for evolution equations in weighted \(L_p\)-spaces
cites work: Q3872868
cites work: Commutativite de deux foncteurs d'interpolation et applications
cites work: \(\mathcal R\)-boundedness, pseudodifferential operators, and maximal regularity for some classes of partial differential operators
cites work: ℛ-boundedness, Fourier multipliers and problems of elliptic and parabolic type
cites work: Optimal \(L^{p}\)- \(L^{q}\)-estimates for parabolic boundary value problems with inhomogeneous data
cites work: Sommes d'opérateurs linéaires et équations différentielles opérationnelles
cites work: Q5538553
cites work: Maximal regular boundary value problems in Banach-valued weighted space
cites work: Maximal regular boundary value problems in Banach-valued function spaces and applications
cites work: Q4493625
full work available at URL: s0012266112030093
OpenAlex ID: W1970174556
description: scientific article; zbMATH DE number 6065264