Long-time behavior for competition-diffusion systems via viscosity comparison

Lia Bronsard; Seong-A Shim

doi:10.3934/dcds.2005.13.561

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2005, Volume 13, Issue 3: 561-581. Doi: 10.3934/dcds.2005.13.561

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Long-time behavior for competition-diffusion systems via viscosity comparison

Lia Bronsard^1, and
Seong-A Shim^2,

1.
Department of Mathematics & Statistics, McMaster University, Hamilton, ON, L8S 4K1
2.
Department of Mathematics, Sungshin Women's University, Seoul, 136-742

Received: August 2004

Revised: January 2005

Published: April 2005

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Abstract

We study the singular limit of competition-diffusion systems in population dynamics when the initial distribution of the solution is not entirely in a domain of attraction for the system. We prove comparison principles in the viscosity sense for the solution and supersolutions to the system. By using travelling wave solutions and the distance function to interfaces, we construct a viscosity supersolution. Finally we study the dynamics of interfaces and the long time behavior of the solution for the systems with large reaction rates.

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Mathematics Subject Classification: 35K57, 53C44.

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