# American Institute of Mathematical Sciences

February  2012, 5(1): 147-158. doi: 10.3934/dcdss.2012.5.147

## A relation between cross-diffusion and reaction-diffusion

 1 Graduate School of Science and Engineering for Research, University of Toyama, 3190 Gofuku, Toyama 930-8555, Japan

Received  March 2009 Revised  January 2010 Published  February 2011

Reaction-diffusion system approximations to a cross-diffusion system are investigated. Iida and Ninomiya~[Recent Advances on Elliptic and Parabolic Issues, 145--164 (2006)] proposed a semilinear reaction-diffusion system with a small parameter and showed that the limit equation takes the form of a weakly coupled cross-diffusion system provided that solutions of both the reaction-diffusion and the cross-diffusion systems are sufficiently smooth. In this paper, the results are extended to a more general cross-diffusion problem involving strongly coupled systems. It is shown that a solution of the problem can be approximated by that of a semilinear reaction-diffusion system without any assumptions on the solutions. This indicates that the mechanism of cross-diffusion might be captured by reaction-diffusion interaction.
Citation: Hideki Murakawa. A relation between cross-diffusion and reaction-diffusion. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 147-158. doi: 10.3934/dcdss.2012.5.147
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