Asymptotically stable equilibria for monotone semiflows

M. W. Hirsch; Hal L. Smith

doi:10.3934/dcds.2006.14.385

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2006, Volume 14, Issue 3: 385-398. Doi: 10.3934/dcds.2006.14.385

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Asymptotically stable equilibria for monotone semiflows

M. W. Hirsch^1, and
Hal L. Smith^2,

1.
Department of Mathematics, University of California, Berkeley, CA 94720-3840
2.
Department of Mathematics and Statistics, Arizona State University, Tempe, AZ, 85287

Received: December 2004

Revised: June 2005

Published: September 2006

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Abstract

Conditions for the existence of a stable equilibrium and for the existence of an asymptotically stable equilibrium for a strongly order preserving semiflow are presented. Analyticity of the semiflow and the compactness of certain subsets of the set of equilibria are required for the latter and yield finiteness of the equilibrium set. Our results are applied to semilinear parabolic partial differential equations and to the classical Kolmogorov competition system with diffusion.

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Mathematics Subject Classification: 37L15, 35K57.

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