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
References:
[1]

L. Chen and A. Jüngel, Analysis of a multidimensional parabolic population model with strong cross-diffusion,, SIAM J. Math. Anal., 36 (2006), 301.  doi: 10.1137/S0036141003427798.  Google Scholar

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T. Kadota and K. Kuto, Positive steady states for a prey-predator model with some nonlinear diffusion terms,, J. Math. Anal. Appl., 323 (2006), 1387.  doi: 10.1016/j.jmaa.2005.11.065.  Google Scholar

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E. H. Kerner, Further considerations on the statistical mechanics of biological associations,, Bull. Math. Biophys., 21 (1959), 217.  doi: 10.1007/BF02476361.  Google Scholar

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H. Murakawa, Reaction-diffusion system approximation to degenerate parabolic systems,, Nonlinearity, 20 (2007), 2319.  doi: 10.1088/0951-7715/20/10/003.  Google Scholar

[9]

H. Murakawa, A solution of nonlinear diffusion problems by semilinear reaction-diffusion systems,, Kybernetika, 45 (2009), 580.   Google Scholar

[10]

H. Murakawa, Discrete-time approximation to nonlinear degenerate parabolic problems using a semilinear reaction-diffusion system,, preprint., ().   Google Scholar

[11]

A. Okubo and S. A. Levin, "Diffusion and Ecological Problems: Modern Perspectives. Second Edition,", Interdisciplinary Applied Mathematics, 14 (2001).   Google Scholar

[12]

P. Y. H. Pang and M. X. Wang, Strategy and stationary pattern in a three-species predator-prey model,, J. Differential Equations, 200 (2004), 245.  doi: 10.1016/j.jde.2004.01.004.  Google Scholar

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N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species,, J. Theor. Biol., 79 (1979), 83.  doi: 10.1016/0022-5193(79)90258-3.  Google Scholar

[14]

R. Temam, "Navier-Stokes Equation Theory and Numerical Analysis,", AMS Chelsea Publishing, (2001).   Google Scholar

show all references

References:
[1]

L. Chen and A. Jüngel, Analysis of a multidimensional parabolic population model with strong cross-diffusion,, SIAM J. Math. Anal., 36 (2006), 301.  doi: 10.1137/S0036141003427798.  Google Scholar

[2]

L. Chen and A. Jüngel, Analysis of a parabolic cross-diffusion population model without self-diffusion,, J. Differential Equations, 224 (2006), 39.  doi: 10.1016/j.jde.2005.08.002.  Google Scholar

[3]

M. E. Gurtin, Some mathematical models for population dynamics that lead to segregation,, Quart. Appl. Math, 32 (1974), 1.   Google Scholar

[4]

M. Iida, M. Mimura and H. Ninomiya, Diffusion, cross-diffusion and competitive interaction,, J. Math. Biol., 53 (2006), 617.  doi: 10.1007/s00285-006-0013-2.  Google Scholar

[5]

M. Iida and H. Ninomiya, A reaction-diffusion approximation to a cross-diffusion system,, in, (2006), 145.  doi: 10.1142/9789812774170_0007.  Google Scholar

[6]

T. Kadota and K. Kuto, Positive steady states for a prey-predator model with some nonlinear diffusion terms,, J. Math. Anal. Appl., 323 (2006), 1387.  doi: 10.1016/j.jmaa.2005.11.065.  Google Scholar

[7]

E. H. Kerner, Further considerations on the statistical mechanics of biological associations,, Bull. Math. Biophys., 21 (1959), 217.  doi: 10.1007/BF02476361.  Google Scholar

[8]

H. Murakawa, Reaction-diffusion system approximation to degenerate parabolic systems,, Nonlinearity, 20 (2007), 2319.  doi: 10.1088/0951-7715/20/10/003.  Google Scholar

[9]

H. Murakawa, A solution of nonlinear diffusion problems by semilinear reaction-diffusion systems,, Kybernetika, 45 (2009), 580.   Google Scholar

[10]

H. Murakawa, Discrete-time approximation to nonlinear degenerate parabolic problems using a semilinear reaction-diffusion system,, preprint., ().   Google Scholar

[11]

A. Okubo and S. A. Levin, "Diffusion and Ecological Problems: Modern Perspectives. Second Edition,", Interdisciplinary Applied Mathematics, 14 (2001).   Google Scholar

[12]

P. Y. H. Pang and M. X. Wang, Strategy and stationary pattern in a three-species predator-prey model,, J. Differential Equations, 200 (2004), 245.  doi: 10.1016/j.jde.2004.01.004.  Google Scholar

[13]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species,, J. Theor. Biol., 79 (1979), 83.  doi: 10.1016/0022-5193(79)90258-3.  Google Scholar

[14]

R. Temam, "Navier-Stokes Equation Theory and Numerical Analysis,", AMS Chelsea Publishing, (2001).   Google Scholar

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