On explicit abstract neutral differential equations with state-dependent delay
type: Item
instance of: scholarly article
title: On explicit abstract neutral differential equations with state-dependent delay
author: Eduardo Hernández
author: Michelle Pierri
published in: Applied Mathematics and Optimization
publication date: 2024-07-01T00:00:00Z
review text: Initial value problems for partial differential equations with dependence on the past time derivative are considered; they take the abstract form \[ \begin{aligned} \dot u (t) & = Au(t) + F[t, \dot u(\sigma(t, u(t)))], \; t \in [0, a], \\\Nu_0 & = \varphi \in C^0([-p, 0]; X), \end{aligned}\] with a Banach space \(X\), and the operator \(A \) is the generator of an analytic semigroup. The consistently bumpy language (singular/plural, past/present etc.), especially in the introduction, did apparently not bother any of the referees whose work is acknowledged.\N\NThree types of solutions are distinguished:\N\begin{itemize}\N\item[1)] Mild solutions (which solve an associated integral equation),\N\N\item[2)] `strict solutions' (which are, roughly speaking, \(C^1 \) w.r. to time on \([0, b]\) for some \( b >0\) (but not necessarily on the past interval \([-p, 0]\), and\N\N\item[3)] \(C^{1 + \alpha}\) solutions (which satisfy \(u(\cdot) \in C^{1+\alpha}(I, X)\) for some interval \(I\) around zero).\N\end{itemize}\NAn important technique is the use of \(L^q\)-Lipschitz functions, roughly meaning that a time-dependent Lipschitz constant (w.r. to the time argument) is of class \(L^q\) w.r. to time. Existence and uniqueness of mild solutions is proved first, using a contraction argument. This is shown under the alternative assumptions \(T(\cdot) \varphi(0) \in C^0( [0, a], X) \) or \(T(\cdot) \varphi(0) \in C^0( [0, a], X_{\gamma}) \), where \(T(\cdot) \) is the semigroup generated by \(A\) and \(X_{\gamma}\) denotes fractional power spaces.\N\NAdditional restrictions on the nonlinearity \(F\) and the delay functional \( \sigma\), together with the condition \(\varphi(0) \in D(A)\), allow to conclude that the mild solutions are also strict. Here the proof is based on earlier work of Hernandez, Fernandes and Wu.\N\NUnder lower smoothness assumptions, but with compactness conditions on the semigroup, a Peano type existence theorem is provided.\N\NFinally, \(C^{1 + \alpha}\) solu…
reviewed by: Bernhard Lani-Wayda
Mathematics Subject Classification ID: 34K43
Mathematics Subject Classification ID: 34K30
Mathematics Subject Classification ID: 34K40
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 7873827
zbMATH Keywords: explicit neutral differential equation
zbMATH Keywords: state-dependent delay
zbMATH Keywords: strict solution
zbMATH Keywords: local and global existence
zbMATH Keywords: uniqueness
zbMATH Keywords: maximal solution
MaRDI profile type: Publication
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DOI: 10.1007/S00245-024-10146-1
description: scientific article; zbMATH DE number 7873827