On certain nonlinear differential equations of second order in time
type: Item
instance of: scholarly article
title: On certain nonlinear differential equations of second order in time
zbMATH Open document ID: 0597.35074
author: Kenji Maruo
published in: Osaka Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: The author considers initial-value problems associated with the following equation in a Hilbert space \(H:\) \((d^ 2/dt^ 2)u(t) + \partial\phi(u(t)) + \partial I_ K(u(t)) \ni f(t,u(t))\) where \(\phi\) is a coercive proper convex l.s.c. functional on a dense subspace of H; the sub-differential \(\partial \phi\) is assumed single valued and \(I_ K\) is the indicator function of a closed convex of H. First, he proves the existence of a global solution (in an adequate weak sense) for some specific data. When \(\partial \phi\) is a positive self- adjoint linear operator in H, he obtains an existence theorem for energy conserving solutions. Restricting to a more specific class of energy conserving solutions, he gets existence and uniqueness. These results extend previous ones obtained by M. Schatzman. Two applications are briefly cited.
Mathematics Subject Classification ID: 35L10
Mathematics Subject Classification ID: 35L15
Mathematics Subject Classification ID: 35L20
Mathematics Subject Classification ID: 35L70
zbMATH DE Number: 3962310
zbMATH Keywords: nonlinear differential equations
zbMATH Keywords: initial-value problems
zbMATH Keywords: equation in a Hilbert space H
zbMATH Keywords: existence of a global solution
zbMATH Keywords: energy conserving solutions
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3962310