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July  2006, 14(3): 385-398. doi: 10.3934/dcds.2006.14.385

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, United States
2. 

Department of Mathematics and Statistics, Arizona State University, Tempe, AZ, 85287, United States

Received  December 2004 Revised  June 2005 Published  December 2005

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.
Keywords: analytic semiflow, asymptotically stable equilibria, Order preserving semiflow, Kolmogorov competition system..
Mathematics Subject Classification: 37L15, 35K5.
Citation: M. W. Hirsch, Hal L. Smith. Asymptotically stable equilibria for monotone semiflows. Discrete & Continuous Dynamical Systems, 2006, 14 (3) : 385-398. doi: 10.3934/dcds.2006.14.385
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