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January  2008, 21(1): 233-244. doi: 10.3934/dcds.2008.21.233

Relative compactness in $L^p$ of solutions of some 2m components competition-diffusion systems

Danielle Hilhorst 1, , Masato Iida 2, , Masayasu Mimura 3, and  Hirokazu Ninomiya 4,

1. 

Laboratoire de Mathématiques, CNRS et Université de Paris-Sud XI, F-91405 Orsay Cedex, France
2. 

Department of Mathematics, Faculty of Humanities and Social Sciences, Iwate University, Morioka, 020-8550, Japan
3. 

Department of Mathematics, School of Science and Technology, Meiji University, 1-1-1 Higashimita, Tamaku, Kawasaki 214-8571, Japan
4. 

Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu 520-2194

Received  January 2007 Revised  October 2007 Published  February 2008

We consider a class of $2m$ components competition-diffusion systems which involve $m$ parabolic equations as well as $m$ ordinary differential equation, and prove the strong convergence in $L^p$ of a subsequence of each component as the reaction coefficient tends to infinity. In the special case of $4$ components the solution of this system converges to that of a Stefan problem.
Keywords: reaction-diffusion system, singular limit, relative compactness in $L^p$, Frechet-Kolmogoroff theorem..
Mathematics Subject Classification: Primary: 35K40, 35K57, 35R35; Secondary: 35B4.
Citation: Danielle Hilhorst, Masato Iida, Masayasu Mimura, Hirokazu Ninomiya. Relative compactness in $L^p$ of solutions of some 2m components competition-diffusion systems. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 233-244. doi: 10.3934/dcds.2008.21.233
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