The Cauchy problem for time-dependent abstract parabolic equations of higher order
type: Item
instance of: scholarly article
title: The Cauchy problem for time-dependent abstract parabolic equations of higher order
zbMATH Open document ID: 0645.34052
DOI: 10.1016/0022-247X(87)90104-1
author: Enrico Obrecht
published in: Journal of Mathematical Analysis and Applications
publication date: 1987-01-01T00:00:00Z
review text: Consider the differential equation in a Banach space (E) \(u^{(n)}+\sum^{n-1}_{k=0}A_ k(t)u^{(k)}=f(t),\) where the operators \(A_ k(t)\) are closed, linear and have dense and constant domains. By assuming parabolicity of equation (E), which means that the operator pencil \(\mu^ n+\sum^{n-1}_{k=0}\mu^ kA_ k(t)\) has an everywhere defined bounded inverse for every \(\mu\) in a sector of the complex plane of half-amplitude greater than \(\pi\) /2 and that inverse decreases in a suitable way at infinite in this sector, and Hölder continuity of the coefficients \(A_ k(t)\) and of the r.h.s. f(t), it is proved that the Cauchy problem for (E) has a unique solution, provided the initial data satisfy conditions of Brézis-Fraenkel type [\textit{H. Brézis}, \textit{L. E. Fraenkel}, J. Funct. Anal. 29, 328-335 (1978; Zbl 0388.46026)]. The results are applicable to Petrovskij parabolic partial differential equations.
Mathematics Subject Classification ID: 34G10
zbMATH DE Number: 4052074
zbMATH Keywords: Banach space
zbMATH Keywords: Cauchy problem
zbMATH Keywords: Petrovskij parabolic partial differential equations
MaRDI profile type: Publication
cites work: A functional with prescribed initial derivatives in different Banach spaces
cites work: Problèmes de Cauchy abstraits et applications à quelques problèmes mixtes
cites work: Some Friedrichs type inequalities in the full Euclidean space
cites work: The underdetermined Cauchy problem in Banach spaces
cites work: Well-Posedness of Higher Order Abstract Cauchy Problems
cites work: Sul problema di Cauchy per le equazioni paraboliche astratte di ordine n
cites work: Evolution operators for higher order abstract parabolic equations
cites work: Existence and Uniqueness of Solutions of Second Order Nonlinear Differential Equations
cites work: Q4764922
cites work: Q3278564
description: scientific article; zbMATH DE number 4052074