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

Danielle Hilhorst; Masato Iida; Masayasu Mimura; Hirokazu Ninomiya

doi:10.3934/dcds.2008.21.233

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2008, Volume 21, Issue 1: 233-244. Doi: 10.3934/dcds.2008.21.233

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

1.
Laboratoire de Mathématiques, CNRS et Université de Paris-Sud XI, F-91405 Orsay Cedex
2.
Department of Mathematics, Faculty of Humanities and Social Sciences, Iwate University, Morioka, 020-8550
3.
Department of Mathematics, School of Science and Technology, Meiji University, 1-1-1 Higashimita, Tamaku, Kawasaki 214-8571
4.
Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu 520-2194

Received: January 2007

Revised: October 2007

Published: January 2008

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Abstract

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.

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Mathematics Subject Classification: Primary: 35K40, 35K57, 35R35; Secondary: 35B40.

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