Time discretization of nonlinear Cauchy problems applying to mixed hyperbolic-parabolic equations
type: Item
instance of: scholarly article
title: Time discretization of nonlinear Cauchy problems applying to mixed hyperbolic-parabolic equations
zbMATH Open document ID: 0859.35077
DOI: 10.1155/S0161171296000683
author: Pierluigi Colli
author: Angelo Favini
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1997-04-09T00:00:00Z
full work available at URL: https://eudml.org/doc/47623
review text: The authors consider the Cauchy evolution problem \[ Lu_{tt}+Bu_t+Au\ni f\quad\text{in }(0,T),\quad u(0)=u_0,\quad u_t(0)=v_0. \] Here \(L:H\to H\) and \(A:V\to V'\) are two linear, bounded selfadjoint operators, where \(V\), \(H\) are Hilbert spaces such that \(V\subset H\) is continuously imbedded and dense. \(B\) is a maximal monotone operator from \(V\) to \(V'\). \(L\) may be degenerate but the sum \(L+B\) is assumed to be coercive in \(H\). The condition on the map \(\alpha I+A\) is to be strongly monotone from \(V\) to \(V'\) for all \(\alpha>0\), where \(I\) denotes the identity in \(H\). The case where \(L=I\) has previously been studied e.g. by \textit{J.-L. Lions} and \textit{W. A. Strauss} [Bull. Soc. Math. Fr. 93, 43-96 (1965; Zbl 0132.10501)]. The authors prove the existence and uniqueness of a variational solution of the given problem. The existence proof is obtained by discretizing the problems with respect to time with the backward Euler method. The Euler approximations are shown to converge with order \(O(\tau^{1/2})\), where \(\tau\) is the time increment.
Mathematics Subject Classification ID: 35L80
Mathematics Subject Classification ID: 35R70
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35A35
zbMATH DE Number: 894080
zbMATH Keywords: maximal monotone operator
zbMATH Keywords: backward Euler method
reviewed by: Rolf Dieter Grigorieff
MaRDI profile type: Publication
OpenAlex ID: W2158637629
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description: scientific article; zbMATH DE number 894080