A permutation characterization of Sturm global attractors of Hamiltonian type
type: Item
instance of: scholarly article
title: A permutation characterization of Sturm global attractors of Hamiltonian type
zbMATH Open document ID: 1233.35042
author: B. Fiedler
author: C. Rocha
author: Matthias Wolfrum
published in: Journal of Differential Equations
publication date: 2011-12-06T00:00:00Z
review text: This paper provides a complete characterization of the global attractors of the equation \(u_t=u_{xx}+f(u)\) defined on an interval with Neumann boundary conditions, under a growth condition on \(f\) ensuring that it defines a dissipative dynamical system, and the assumption, which holds generically, that the equilibria are hyperbolic. In a previous work by the first two authors [Trans. Am. Math. Soc. 352, No. 1, 257--284 (2000; Zbl 0932.37065)], more general equations with \(f\) depending also on \(x\) and \(u_x\) were considered, and it was proved that the global attractor is determined by the Sturm permutation corresponding to \(f\), a permutation in \(S_n\), where \(n\) is the number of equilibria, which is defined solely in terms of the equilibria, that is, solutions of the ODE \(u_{xx}+f(x,u,u_x)=0\) satisfying the boundary conditions. This paper determines which permutations are Sturm permutations corresponding to nonlinearities \(f\) which depend only on \(u\). A purely combinatorial characterization of these permutations is proved.
reviewed by: Guy Katriel
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 37L30
Mathematics Subject Classification ID: 37C70
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 35K58
zbMATH DE Number: 5986525
zbMATH Keywords: Sturm permutation
zbMATH Keywords: one space dimension
zbMATH Keywords: combinatorial characterization
MaRDI profile type: Publication
full work available at URL: j.jde.2011.08.013
OpenAlex ID: W2082578542
Wikidata QID: Q60143385
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DOI: 10.1016/J.JDE.2011.08.013
description: scientific article; zbMATH DE number 5986525