Prevalent behavior of strongly order preserving semiflows

Germán A. Enciso, Morris W. Hirsch, Hal Smith

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

Classical results in the theory of monotone semiflows give sufficient conditions for the generic solution to converge toward an equilibrium or toward the set of equilibria (quasiconvergence). In this paper, we provide new formulations of these results in terms of the measure-theoretic notion of prevalence, developed in Christensen (Israel J. Math., 13, 255-260, 1972) and Hunt et al. (Bull. Am. Math. Soc., 27, 217-238, 1992). For monotone reaction-diffusion systems with Neumann boundary conditions on convex domains, we show the prevalence of the set of continuous initial conditions corresponding to solutions that converge to a spatially homogeneous equilibrium. We also extend a previous generic convergence result to allow its use on Sobolev spaces. Careful attention is given to the measurability of the various sets involved.

Original languageEnglish (US)
Pages (from-to)115-132
Number of pages18
JournalJournal of Dynamics and Differential Equations
Volume20
Issue number1
DOIs
StatePublished - Mar 2008

Fingerprint

Semiflow
Monotone Systems
Converge
Measurability
Convex Domain
Neumann Boundary Conditions
Reaction-diffusion System
Convergence Results
Sobolev Spaces
Monotone
Initial conditions
Formulation
Sufficient Conditions

Keywords

  • Measurability
  • Prevalence
  • Quasi-convergence
  • Reaction-diffusion
  • Strong monotonicity

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

Prevalent behavior of strongly order preserving semiflows. / Enciso, Germán A.; Hirsch, Morris W.; Smith, Hal.

In: Journal of Dynamics and Differential Equations, Vol. 20, No. 1, 03.2008, p. 115-132.

Research output: Contribution to journalArticle

Enciso, Germán A. ; Hirsch, Morris W. ; Smith, Hal. / Prevalent behavior of strongly order preserving semiflows. In: Journal of Dynamics and Differential Equations. 2008 ; Vol. 20, No. 1. pp. 115-132.
@article{08867e318b7f4f3b97a933f6d22f7142,
title = "Prevalent behavior of strongly order preserving semiflows",
abstract = "Classical results in the theory of monotone semiflows give sufficient conditions for the generic solution to converge toward an equilibrium or toward the set of equilibria (quasiconvergence). In this paper, we provide new formulations of these results in terms of the measure-theoretic notion of prevalence, developed in Christensen (Israel J. Math., 13, 255-260, 1972) and Hunt et al. (Bull. Am. Math. Soc., 27, 217-238, 1992). For monotone reaction-diffusion systems with Neumann boundary conditions on convex domains, we show the prevalence of the set of continuous initial conditions corresponding to solutions that converge to a spatially homogeneous equilibrium. We also extend a previous generic convergence result to allow its use on Sobolev spaces. Careful attention is given to the measurability of the various sets involved.",
keywords = "Measurability, Prevalence, Quasi-convergence, Reaction-diffusion, Strong monotonicity",
author = "Enciso, {Germ{\'a}n A.} and Hirsch, {Morris W.} and Hal Smith",
year = "2008",
month = "3",
doi = "10.1007/s10884-007-9084-z",
language = "English (US)",
volume = "20",
pages = "115--132",
journal = "Journal of Dynamics and Differential Equations",
issn = "1040-7294",
publisher = "Springer New York",
number = "1",

}

TY - JOUR

T1 - Prevalent behavior of strongly order preserving semiflows

AU - Enciso, Germán A.

AU - Hirsch, Morris W.

AU - Smith, Hal

PY - 2008/3

Y1 - 2008/3

N2 - Classical results in the theory of monotone semiflows give sufficient conditions for the generic solution to converge toward an equilibrium or toward the set of equilibria (quasiconvergence). In this paper, we provide new formulations of these results in terms of the measure-theoretic notion of prevalence, developed in Christensen (Israel J. Math., 13, 255-260, 1972) and Hunt et al. (Bull. Am. Math. Soc., 27, 217-238, 1992). For monotone reaction-diffusion systems with Neumann boundary conditions on convex domains, we show the prevalence of the set of continuous initial conditions corresponding to solutions that converge to a spatially homogeneous equilibrium. We also extend a previous generic convergence result to allow its use on Sobolev spaces. Careful attention is given to the measurability of the various sets involved.

AB - Classical results in the theory of monotone semiflows give sufficient conditions for the generic solution to converge toward an equilibrium or toward the set of equilibria (quasiconvergence). In this paper, we provide new formulations of these results in terms of the measure-theoretic notion of prevalence, developed in Christensen (Israel J. Math., 13, 255-260, 1972) and Hunt et al. (Bull. Am. Math. Soc., 27, 217-238, 1992). For monotone reaction-diffusion systems with Neumann boundary conditions on convex domains, we show the prevalence of the set of continuous initial conditions corresponding to solutions that converge to a spatially homogeneous equilibrium. We also extend a previous generic convergence result to allow its use on Sobolev spaces. Careful attention is given to the measurability of the various sets involved.

KW - Measurability

KW - Prevalence

KW - Quasi-convergence

KW - Reaction-diffusion

KW - Strong monotonicity

UR - http://www.scopus.com/inward/record.url?scp=43349091042&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=43349091042&partnerID=8YFLogxK

U2 - 10.1007/s10884-007-9084-z

DO - 10.1007/s10884-007-9084-z

M3 - Article

AN - SCOPUS:43349091042

VL - 20

SP - 115

EP - 132

JO - Journal of Dynamics and Differential Equations

JF - Journal of Dynamics and Differential Equations

SN - 1040-7294

IS - 1

ER -