A new class of doubly nonlinear evolution equations
type: Item
instance of: scholarly article
title: A new class of doubly nonlinear evolution equations
zbMATH Open document ID: 1073.35142
author: Masayasu Aso
author: Nobuyuki Kenmochi
author: Takeshi Fukao
published in: Taiwanese Journal of Mathematics
publication date: 2004-09-28T00:00:00Z
review text: The authors consider an evolution equation of the form: \[ u'(t)+K(t,\theta(t))+G(t,u(t))=f(t) \text{ in }V^{*} \text{ for a.e. }t\in[0,T] u(0)=u_0, \] where for each \(t\in[0,T],\) \(K(t,.)\) is a weakly continuous operator from a reflexive Banach space \(V\) into its dual space \(V^{*},\) \(G(t,.)\) is a weakly continuous operator from a Hilbert space \(H\) into \(V^{*}\), where \(V\) is densely and compactly imbedded in \(H\), \(f\) is a given source function and \(u_0\) is an initial data. It is assumed that \(\theta(t)=\partial\psi^{t}(u(t))\) in \(V\) for a.e. \(t\in[0,T]\), where \(\{\psi^{t}\}\) is a family of proper lower semicontinuous and convex functions on the space \(V^{*}\) and \(\partial\psi^{t}\) is the subdifferential of \(\psi^{t}\) from \(V^{*}\) into \(V\). By using a time discretization scheme, the authors give an existence result for the above problem which complements previous results on other types of doubly nonlinear evolution equations by other authors such as \textit{P. Colli} and \textit{A. Visintin} [Commun. Partial Differ. Equations 15, No. 5, 737--756 (1990; Zbl 0707.34053)], \textit{N. Kenmochi} and \textit{I. Pawlow} [Nonlinear Anal. 10, 1181--1202 (1986; Zbl 0635.35043)] and \textit{E. Maitre} and \textit{P. Witomski} [Nonlinear Anal. 50, No. 2(A), 223--250 (2002; Zbl 1001.35091)].
Mathematics Subject Classification ID: 35K90
Mathematics Subject Classification ID: 47J35
Mathematics Subject Classification ID: 35K65
zbMATH DE Number: 2104156
zbMATH Keywords: doubly nonlinear evolution equations
zbMATH Keywords: pseudo-monotone operator
zbMATH Keywords: transmission problems
zbMATH Keywords: degenerate parabolic equations
zbMATH Keywords: Stefan problem
reviewed by: Behzad Djafari Rouhani
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.11650/twjm/1500558460
OpenAlex ID: W4206407742
DOI: 10.11650/TWJM/1500558460
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description: scientific article; zbMATH DE number 2104156