On the convergence of solutions of multivalued parabolic equations and applications
type: Item
instance of: scholarly article
title: On the convergence of solutions of multivalued parabolic equations and applications
zbMATH Open document ID: 1030.35106
DOI: 10.1016/S0362-546X(03)00093-2
author: Nikolaos S. Papageorgiou
author: Zdzisław Denkowski
author: Stanisław Migórski
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2003-07-30T00:00:00Z
review text: The authors are investigating sequence of parabolic inclusions of the form \[ \partial u_k/\partial_t- \text{div }a_k(x, Du_k))\ni f(t,x)\tag{1} \] in \((0,T)\times \Omega\), where \(\Omega\subset \mathbb{R}^N\) is a bounded domain, \(0< T< \infty\), while the multivalued nonlinear maps \(a_k: \Omega\times \mathbb{R}^N\to 2^{\mathbb{R}^N}\) are measurable in \(x\), with \(a_k(x,\xi)\) maximal monotone in \(\xi\). It is also assumed that certain growth and coercivity conditions are satisfied. Under appropriate hypotheses on data, the authors establish two basic results. The first is an existence result for (1), with an initial condition on \(\Omega\) and with zero condition on the lateral boundary of \((0,T)\times\Omega\). The second result regards the convergence of solutions of (1), as \(k\to\infty\), to the solution of a limit equation, of the same form as (1). An application to an optimal control problem is also discussed.
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35R70
Mathematics Subject Classification ID: 35A05
zbMATH DE Number: 1957311
zbMATH Keywords: \(G\)-convergence
zbMATH Keywords: maximal monotone operator
zbMATH Keywords: coercive operator
zbMATH Keywords: parabolic inclusions
zbMATH Keywords: minimax problem
reviewed by: Constantin Corduneanu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1957311