Global existence, uniqueness, and continuous dependence for a semilinear initial value problem
type: Item
instance of: scholarly article
title: Global existence, uniqueness, and continuous dependence for a semilinear initial value problem
zbMATH Open document ID: 1029.34048
DOI: 10.1016/S0022-247X(03)00126-4
author: F. Periago
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-05-27T00:00:00Z
review text: The author considers the semilinear initial value problem \[ u'(t)+ Au(t)= f(t, u(t)),\quad t\geq 0,\quad u(0)+ u_0,\tag{IVP} \] in a Banach space \(X\) in which \(f:[0,\infty)\times X\to X\) is continuous in \(t\) and satisfies a Lipschitz-type condition in \(x\), \(A\) is a closed linear operator whose spectrum lies in a certain sector of the complex plane and whose resolvent satisfies \[ \|(z- A)^{-1}\|\leq M|z|^\gamma\tag{1} \] for all \(z\) outside that sector, where \(\gamma\in (-1,0)\). (Compare condition (1) to the inequality \(\|(z- A)^{-1}\|\leq M|z|^{-1}\), which has been used elsewhere in the literature.) Along with some growth conditions, the author proves global existence and uniqueness of a mild solution to (IVP) and also demonstrates that this solution depends continuously on the initial value. Examples are given which consider the application of these ideas to parabolic partial differential equations, fractional differential equations and the Korteweg-de Vries operator. This article is based on the work of \textit{F. Periago} and \textit{B. Straub} [J. Evol. Equ. 2, No. 1, 41-68 (2002; Zbl 1005.47015)].
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 34A12
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 1917783
zbMATH Keywords: global existence
zbMATH Keywords: uniqueness
zbMATH Keywords: continuous dependence
zbMATH Keywords: semilinear initial value problem
zbMATH Keywords: Korteweg-de Vries operator
reviewed by: Daniel C. Biles
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1917783