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April  2005, 13(3): 561-581. doi: 10.3934/dcds.2005.13.561

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, Canada
2. 

Department of Mathematics, Sungshin Women's University, Seoul, 136-742, South Korea

Received  August 2004 Revised  January 2005 Published  May 2005

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.
Keywords: convergence, viscosity super-solutions, dynamics of interfaces., comparison principles, distance function, Competition-diffusion system, travelling wave solutions.
Mathematics Subject Classification: 35K57, 53C4.
Citation: Lia Bronsard, Seong-A Shim. Long-time behavior for competition-diffusion systems via viscosity comparison. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 561-581. doi: 10.3934/dcds.2005.13.561
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