A relation between cross-diffusion and reaction-diffusion

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

A relation between cross-diffusion and reaction-diffusion

Hideki Murakawa ^1,

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

Abstract
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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.

Keywords: Reaction-diffusion system, Cross-diffusion system, Approximation..

Mathematics Subject Classification: 35K55, 35K5.

Citation: Hideki Murakawa. A relation between cross-diffusion and reaction-diffusion. Discrete and 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), 301-322. doi: 10.1137/S0036141003427798.
[2]	L. Chen and A. Jüngel, Analysis of a parabolic cross-diffusion population model without self-diffusion, J. Differential Equations, 224 (2006), 39-59. doi: 10.1016/j.jde.2005.08.002.
[3]	M. E. Gurtin, Some mathematical models for population dynamics that lead to segregation, Quart. Appl. Math, 32 (1974), 1-9.
[4]	M. Iida, M. Mimura and H. Ninomiya, Diffusion, cross-diffusion and competitive interaction, J. Math. Biol., 53 (2006), 617-641. doi: 10.1007/s00285-006-0013-2.
[5]	M. Iida and H. Ninomiya, A reaction-diffusion approximation to a cross-diffusion system, in "Recent Advances on Elliptic and Parabolic Issues" (eds. M. Chipot and H. Ninomiya), World Scientific, (2006), 145-164. doi: 10.1142/9789812774170_0007.
[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), 1387-1401. doi: 10.1016/j.jmaa.2005.11.065.
[7]	E. H. Kerner, Further considerations on the statistical mechanics of biological associations, Bull. Math. Biophys., 21 (1959), 217-255. doi: 10.1007/BF02476361.
[8]	H. Murakawa, Reaction-diffusion system approximation to degenerate parabolic systems, Nonlinearity, 20 (2007), 2319-2332. doi: 10.1088/0951-7715/20/10/003.
[9]	H. Murakawa, A solution of nonlinear diffusion problems by semilinear reaction-diffusion systems, Kybernetika, 45 (2009), 580-590.
[10]	H. Murakawa, Discrete-time approximation to nonlinear degenerate parabolic problems using a semilinear reaction-diffusion system,, preprint., ().
[11]	A. Okubo and S. A. Levin, "Diffusion and Ecological Problems: Modern Perspectives. Second Edition," Interdisciplinary Applied Mathematics, 14, Springer-Verlag, New York, 2001.
[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), 245-273. doi: 10.1016/j.jde.2004.01.004.
[13]	N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theor. Biol., 79 (1979), 83-99. doi: 10.1016/0022-5193(79)90258-3.
[14]	R. Temam, "Navier-Stokes Equation Theory and Numerical Analysis," AMS Chelsea Publishing, Providence, RI., 2001.

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), 301-322. doi: 10.1137/S0036141003427798.
[2]	L. Chen and A. Jüngel, Analysis of a parabolic cross-diffusion population model without self-diffusion, J. Differential Equations, 224 (2006), 39-59. doi: 10.1016/j.jde.2005.08.002.
[3]	M. E. Gurtin, Some mathematical models for population dynamics that lead to segregation, Quart. Appl. Math, 32 (1974), 1-9.
[4]	M. Iida, M. Mimura and H. Ninomiya, Diffusion, cross-diffusion and competitive interaction, J. Math. Biol., 53 (2006), 617-641. doi: 10.1007/s00285-006-0013-2.
[5]	M. Iida and H. Ninomiya, A reaction-diffusion approximation to a cross-diffusion system, in "Recent Advances on Elliptic and Parabolic Issues" (eds. M. Chipot and H. Ninomiya), World Scientific, (2006), 145-164. doi: 10.1142/9789812774170_0007.
[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), 1387-1401. doi: 10.1016/j.jmaa.2005.11.065.
[7]	E. H. Kerner, Further considerations on the statistical mechanics of biological associations, Bull. Math. Biophys., 21 (1959), 217-255. doi: 10.1007/BF02476361.
[8]	H. Murakawa, Reaction-diffusion system approximation to degenerate parabolic systems, Nonlinearity, 20 (2007), 2319-2332. doi: 10.1088/0951-7715/20/10/003.
[9]	H. Murakawa, A solution of nonlinear diffusion problems by semilinear reaction-diffusion systems, Kybernetika, 45 (2009), 580-590.
[10]	H. Murakawa, Discrete-time approximation to nonlinear degenerate parabolic problems using a semilinear reaction-diffusion system,, preprint., ().
[11]	A. Okubo and S. A. Levin, "Diffusion and Ecological Problems: Modern Perspectives. Second Edition," Interdisciplinary Applied Mathematics, 14, Springer-Verlag, New York, 2001.
[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), 245-273. doi: 10.1016/j.jde.2004.01.004.
[13]	N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theor. Biol., 79 (1979), 83-99. doi: 10.1016/0022-5193(79)90258-3.
[14]	R. Temam, "Navier-Stokes Equation Theory and Numerical Analysis," AMS Chelsea Publishing, Providence, RI., 2001.

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