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July  2007, 8(1): 39-44. doi: 10.3934/dcdsb.2007.8.39

Fast reaction limit and long time behavior for a competition-diffusion system with Dirichlet boundary conditions

E. C.M. Crooks 1, , E. N. Dancer 2, and  Danielle Hilhorst 3,

1. 

Mathematical Institute, 24-29 St. Giles', Oxford OX1 3LB, United Kingdom
2. 

School of Mathematics and Statistics, University of Sydney, N.S.W. 2006, Australia
3. 

CNRS and Laboratoire de Mathématiques, Université Paris-Sud 11, Bat. 425, F-91405 Orsay, France

Received  November 2005 Revised  February 2006 Published  April 2007

We consider a two-component competition-diffusion system with equal diffusion coefficients and inhomogeneous Dirichlet boundary conditions. Provided certain conditions on a limit problem hold and provided that the competition rate is sufficiently large, all non-negative solutions of the system converge to a stationary solution of the system as $ t \rightarrow \infty$.
Keywords: boundary-value problem, long-time behaviour, spatial segregation., singular limit, Competition-diffusion system.
Mathematics Subject Classification: 35K50, 35B40, 35K57, 92D2.
Citation: E. C.M. Crooks, E. N. Dancer, Danielle Hilhorst. Fast reaction limit and long time behavior for a competition-diffusion system with Dirichlet boundary conditions. Discrete and Continuous Dynamical Systems - B, 2007, 8 (1) : 39-44. doi: 10.3934/dcdsb.2007.8.39
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