Propagation of solutions of a nonlinear functional differential equation
type: Item
instance of: scholarly article
title: Propagation of solutions of a nonlinear functional differential equation
zbMATH Open document ID: 0726.34053
author: Janet Dyson
author: R. Villella-Bressan
published in: Differential and Integral Equations
publication date: 1991-01-01T00:00:00Z
review text: The authors extend their previous results in Rend. Semin. Mat. Univ. Padova 80, 55-63 (1988; Zbl 0671.34066), on the propagation of solutions of seminlinear delay equations to the nonlinear functional differential equation \(\dot x(t)=f(x(t))+g(x_ t),x(0)=h\in X\), \(x_ 0\in L^ p=L^ p(-r,0;X)\), where X is a Hilbert space, \(-f+\alpha I: D(f)\subseteq X\to X\) is m-accretive for some \(\alpha >0\) and g: \(L^ p\to X\) satisfies a Lipschitz condition. Let \(T_ f(t)\) and \(T_ A(t)\) denote the semigroups generated by f and by \(A\{\phi,h\}=\{\phi ',f(h)+g(\phi)\},\) \(D(A)=\{\{\phi,h\},\quad \phi \in W^{1,p}(- r,0;X),\quad \phi (0)=h\in D(f)\}.\) The authors prove that if \(\{X_ t\}_{t\geq 0}\) is a family of closed subspaces of X and \(T_ f(t)\) propagates \(\{X_ t\}\), then \(T_ A(t)\) propagates \(\{Y_ t\}\), where \(Y_ t\subseteq L^ p\times X\) is constructed from \(X_ t\).
Mathematics Subject Classification ID: 34K99
Mathematics Subject Classification ID: 34K30
zbMATH DE Number: 4198518
zbMATH Keywords: seminlinear delay equations
zbMATH Keywords: nonlinear functional differential equation
zbMATH Keywords: Hilbert space
zbMATH Keywords: semigroups
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4198518