Group actions on strongly monotone dynamical systems
type: Item
instance of: scholarly article
title: Group actions on strongly monotone dynamical systems
zbMATH Open document ID: 0642.58031
DOI: 10.1007/BF01457499
author: Janusz Mierczyński
author: Peter Poláčik
published in: Mathematische Annalen
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164491
review text: It is well known that second order parabolic partial differential equations satisfying the strong comparison principle generate strongly monotone flows on appropriate function spaces. For such systems, almost all points whose trajectories are relatively compact are quasiconvergent, that is, their \(\omega\)-limit sets contain only equilibria. The purpose of the paper is to investigate strongly monotone flows on which a compact connected group acts monotonely and equivariantly. The main results state, loosely speaking, that almost all trajectories eventually symmetrize. More precisely, assume for simplicity that all points \(x\in X\) have relatively compact trajectories. Then the following holds: (1) every stable equilibrium is symmetric; (2) if X is a separable Banach space then the set Y of points whose \(\omega\)-limit sets contain only symmetric equilibria is residual in X; (3) if the flow is order- compact then Y is open and dense in X.
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 37C80
zbMATH DE Number: 4046751
zbMATH Keywords: second order parabolic partial differential equations
zbMATH Keywords: strong comparison principle
zbMATH Keywords: strongly monotone flows
zbMATH Keywords: equivariance
zbMATH Keywords: group action
zbMATH Keywords: strongly ordered space
zbMATH Keywords: symmetry
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4046751