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December  2012, 7(4): 781-803. doi: 10.3934/nhm.2012.7.781

Singular limit of an activator-inhibitor type model

1. 

CMI, Université de Provence, 39 rue Frédéric Joliot-Curie 13453 Marseille cedex 13

Received  January 2012 Revised  November 2012 Published  December 2012

We consider a reaction-diffusion system of activator-inhibitor type arising in the theory of phase transition. It appears in biological contexts such as pattern formation in population genetics. The purpose of this work is to prove the convergence of the solution of this system to the solution of a free boundary Problem involving a motion by mean curvature.
Citation: Marie Henry. Singular limit of an activator-inhibitor type model. Networks and Heterogeneous Media, 2012, 7 (4) : 781-803. doi: 10.3934/nhm.2012.7.781
References:
[1]

M. Alfaro, D. Hilhorst and H. Matano, The singular limit of the Allen-Cahn Equation and the Fitzhugh-Nagumo system, J. Differential Equations, 245 (2008), 505-565. doi: 10.1016/j.jde.2008.01.014.

[2]

A. Bonami, D. Hilhorst and E. Logak, Modified Motion by mean curvature: Local existence and uniqueness and qualitative properties, Differential and Integral Equation, 3 (2000), 1371-1392.

[3]

A. Bonami, D. Hilhorst, E. Logak and M. Mimura, Singular limit of a chemotaxis growth model, Advances in Differential Equations, 6 (2001), 1173-1218.

[4]

X. Chen, Generation and propagation of interfaces in reaction-diffusion systems, Transactions of the American Mathematical society, 32 (1992), 877-913. doi: 10.2307/2154487.

[5]

P. C. Fife and L. Hsiao, The generation and propagation of internal layers, Nonlinear Analysis TMA, 12 (1988), 19-41. doi: 10.1016/0362-546X(88)90010-7.

[6]

M. Henry, D. Hilhorst and R. Schätzle, Convergence to a viscosity solution for an advection-reaction-diffusion equation arising from a chemotaxis-growth model, Hiroshima Math. Journal, 29 (1999), 591-630.

[7]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," American Mathematical Society, 1968.

[8]

E. Logak, Singular limit of reaction-diffusion systems and modified motion by mean curvature, Roy. Soc. Edinburgh. Sect. A, 132 (2002), 951-973. doi: 10.1017/S0308210500001955.

[9]

Y. Nishiura and M. Mimura, Layer oscillations in reaction-diffusion systems, SIAM J. Appl. Math., 49 (1989), 481-514. doi: 10.1137/0149029.

[10]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Springer-Verlag, 1984. doi: 10.1007/978-1-4612-5282-5.

[11]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations," Springer-Verlag, 1994.

show all references

References:
[1]

M. Alfaro, D. Hilhorst and H. Matano, The singular limit of the Allen-Cahn Equation and the Fitzhugh-Nagumo system, J. Differential Equations, 245 (2008), 505-565. doi: 10.1016/j.jde.2008.01.014.

[2]

A. Bonami, D. Hilhorst and E. Logak, Modified Motion by mean curvature: Local existence and uniqueness and qualitative properties, Differential and Integral Equation, 3 (2000), 1371-1392.

[3]

A. Bonami, D. Hilhorst, E. Logak and M. Mimura, Singular limit of a chemotaxis growth model, Advances in Differential Equations, 6 (2001), 1173-1218.

[4]

X. Chen, Generation and propagation of interfaces in reaction-diffusion systems, Transactions of the American Mathematical society, 32 (1992), 877-913. doi: 10.2307/2154487.

[5]

P. C. Fife and L. Hsiao, The generation and propagation of internal layers, Nonlinear Analysis TMA, 12 (1988), 19-41. doi: 10.1016/0362-546X(88)90010-7.

[6]

M. Henry, D. Hilhorst and R. Schätzle, Convergence to a viscosity solution for an advection-reaction-diffusion equation arising from a chemotaxis-growth model, Hiroshima Math. Journal, 29 (1999), 591-630.

[7]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," American Mathematical Society, 1968.

[8]

E. Logak, Singular limit of reaction-diffusion systems and modified motion by mean curvature, Roy. Soc. Edinburgh. Sect. A, 132 (2002), 951-973. doi: 10.1017/S0308210500001955.

[9]

Y. Nishiura and M. Mimura, Layer oscillations in reaction-diffusion systems, SIAM J. Appl. Math., 49 (1989), 481-514. doi: 10.1137/0149029.

[10]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Springer-Verlag, 1984. doi: 10.1007/978-1-4612-5282-5.

[11]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations," Springer-Verlag, 1994.

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