An implicit nonlinear time dependent equation has a solution
type: Item
instance of: scholarly article
title: An implicit nonlinear time dependent equation has a solution
zbMATH Open document ID: 0755.49001
DOI: 10.1016/0022-247X(91)90364-6
author: Veli-Matti Hokkanen
published in: Journal of Mathematical Analysis and Applications
publication date: 1992-09-27T00:00:00Z
review text: It will be shown that the differential equation \(v'(t)+w(t)+{\mathcal C}(t)u(t)=f(t)\) for a.e. \(t\in]0,T[\), \(v(t)\in{\mathcal A}(t)u(t)\), \(w(t)\in{\mathcal B}(t)u(t)\), for a.e. \(t\in]0,T[\), and \(v(0)=v_ 0\), has a solution \((u,v,w)\). The operators \({\mathcal A}(t)\) and \({\mathcal B}(t)\) are maximal monotone from a real Hilbert space \(V\) to its dual such that \({\mathcal A}(t)+{\mathcal B}(t)\) are \(V\)-coercive. A linear compact injection \(i\) embeds \(V\) to a real Banach space \(W\) and each \({\mathcal A}(t)\) is the strongly-monotone subdifferential of a continuous convex function \(\phi(t,.)\) on \(W\). The function \(f\) is square integrable. The functions \({\mathcal C}(t): V\to W^*\) are Lipschitzian as \(V^*\)-valued functions. As example an implicit nonlinear parabolic PDE with time dependent coefficients is considered. The result implies the existence of its weak solution.
Mathematics Subject Classification ID: 49J15
Mathematics Subject Classification ID: 47N20
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 49J27
Mathematics Subject Classification ID: 47H05
zbMATH DE Number: 57295
zbMATH Keywords: maximal monotone operators
zbMATH Keywords: real Hilbert space
zbMATH Keywords: strongly-monotone subdifferential
zbMATH Keywords: implicit nonlinear parabolic PDE
zbMATH Keywords: time dependent coefficients
zbMATH Keywords: weak solution
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 57295