On the principle of linearized stability for variational inequalities
type: Item
instance of: scholarly article
title: On the principle of linearized stability for variational inequalities
zbMATH Open document ID: 0642.49006
DOI: 10.1007/BF01446434
author: Pavol Quittner
published in: Mathematische Annalen
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164507
review text: We study the Lyapunov stability of stationary solutions of parabolic variational inequalities of the form \[ (du(t)/dt+F(u(t)),v-u(t))\geq 0,\quad v\in K,\quad u(t)\in K,\quad u(0)=u_ 0, \] where K is a closed convex subset of a real Hilbert space. We derive conditions on \(F(u_ 0)\), \(F'(u_ 0)\) and K, which are sufficient for the stability or the instability of a stationary solution \(u_ 0\). In some special cases, these conditions can be written in terms of eigenvalues of the corresponding variational inequality \(0\neq u\in K_ 0:\) \((F'(u_ 0)u- \lambda u,v-u)\geq 0,v\in K_ 0,\) where \(K_ 0\) is the closure of the set \(\cup _{\alpha >0}\alpha (K-u_ 0)\), and then they are analogous to those occurring in the principle of linearized stability for semilinear parabolic equations. An application with F being a semilinear elliptic differential operator is given.
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35K85
zbMATH DE Number: 4046495
zbMATH Keywords: Lyapunov stability
zbMATH Keywords: parabolic variational inequalities
zbMATH Keywords: principle of linearized stability
MaRDI profile type: Publication
cites work: Propriétés régularisantes de certains semi-groupes non linéaires
cites work: Q3748648
cites work: Bifurcation points and eigenvalues of inequalities of reaction-diffusion type.
OpenAlex ID: W40762328
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description: scientific article; zbMATH DE number 4046495