Regularity of mild solutions for fractional abstract Cauchy problem with order \({\alpha \in (1, 2)}\)
type: Item
instance of: scholarly article
title: Regularity of mild solutions for fractional abstract Cauchy problem with order \({\alpha \in (1, 2)}\)
zbMATH Open document ID: 1348.34020
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 2016-01-05T00:00:00Z
review text: The author considers the fractional linear differential equation \[ D_t^{\alpha}u(t)+Au(t)=f(t) \] with \(t\in (0,T]\), \(u(0)\), \(u'(0)\) given and where the solution takes values in a Banach space \(X\). Here, \(D_t^{\alpha}\) denotes the fractional \(t\)-derivative of order \(\alpha\) and \(\alpha \in (1,2)\). Thus, the equation lies between a pure diffusion equation and a pure wave equation. \(A\) is linear and admits an analytic resolvent operator. The function \(f\) satisfies at least \(f\in L^1((0,T);X)\). Mild solutions are given by the resolvent operating on the initial values and \(f\). The paper gives conditions under which the mild solution satisfies additional regularity, at least on an interval \([\epsilon, T]\). The conditions involve the \(\alpha\) and \(p\) where \(f\in L^p ((0,T];X)\) and in some cases conditions on the given initial values.
Mathematics Subject Classification ID: 34A08
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 33E12
zbMATH DE Number: 6526494
zbMATH Keywords: fractional Cauchy problem
zbMATH Keywords: analytic resolvents
zbMATH Keywords: mild solutions
zbMATH Keywords: strong solutions
author: Ya-Ning Li
reviewed by: Stig-Olof Londen
MaRDI profile type: Publication
full work available at URL: s00033-015-0577-z
OpenAlex ID: W1211242801
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DOI: 10.1007/S00033-015-0577-Z
description: scientific article; zbMATH DE number 6526494