An abstract doubly nonlinear equation with a measure as initial value
type: Item
instance of: scholarly article
title: An abstract doubly nonlinear equation with a measure as initial value
zbMATH Open document ID: 1160.34055
author: Veli-Matti Hokkanen
author: Sergiu Aizicovici
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-05-04T00:00:00Z
review text: The authors are dealing with nonlinear systems of differential equations of the form \[ \begin{aligned} v'(t)+w(t)+{\mathcal C}(t)u(t)=f(t)\quad & \text{for a.a. }t\in]0,T[,\\ v(t)\in{\mathcal A}(t)u(t),\;w(t)\in{\mathcal B}(t)u(t) \quad &\text{for a.a. }t\in]0,T [,\\ v(0)=v_0,\end{aligned} \] where \(A(t)\) and \(B(t)\) are maximal monotone operators in a real Hilbert space \(V\), such that \(A(t)+B(t)\) is \(V\)-coercive. \(A(t)\) represents the subdifferential of a continuous convex function on a real Banach space \(W\), such that there exists an injection \(i:V\to W\), with \(iV\) dense in \(W\). Under technical assumptions concerning the operators involved in the system and the initial datum, two existence results are obtained and another basic result deals with regularity properties of the solution, showing function spaces to which it belongs. To carry out the proofs, the authors are establishing some auxiliary results (10 Lemmas) that require a delicate and lengthy analysis. An example is treated in detail, at the end of the paper.
reviewed by: Constantin Corduneanu
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 47N20
Mathematics Subject Classification ID: 34G25
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35K90
zbMATH DE Number: 5149565
zbMATH Keywords: nonlinear evolution equation
zbMATH Keywords: maximal monotone operator
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.08.074
OpenAlex ID: W2061096952
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DOI: 10.1016/J.JMAA.2006.08.074
description: scientific article; zbMATH DE number 5149565