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Stability and characteristic wavelength of planar interfaces in the large diffusion limit of the inhibitor

Published online by Cambridge University Press:  14 November 2011

Masaharu Taniguchi  and
Yasumasa Nishiura
Masaharu Taniguchi
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-01, Japan
Yasumasa Nishiura
Affiliation:
Laboratory of Nonlinear Studies and Computation, Research Institute for Electronic Science, Hokkaido University, Sapporo 060, Japan
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Abstract

A characteristic wavelength and its parametric dependency are studied for planar interfaces of activator-inhibitor systems as well as their stability in two-dimensional space. When an unstable planar interface is slightly perturbed in a random way, it develops with a characteristic wavelength, that is, the fastest-growing one. A natural question is to ask under what conditions this characteristic wavelength remains finite and approaches a positive definite value as the width of interface, say ε, tends to zero. In this paper, we show that the fastest-growing wavelength has a positive limit value as ε tends to zero for the system:

This is a fundamental fact for stuyding the domain size of patterns in higher-space dimensions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 1 , 1996 , pp. 117 - 145
Copyright
Copyright © Royal Society of Edinburgh 1996

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References

1Chen, X. Y.. Dynamics of interfaces in reaction diffusion systems. Hiroshima Math. J. 21 (1991), 47–83.CrossRefGoogle Scholar
2Chen, X., Hilhorst, D. and Logak, E.. Asymptotic behavior of solutions of an Allen-Cahn equation with a non-local term (Preprint).Google Scholar
3Fife, P. C.. Boundary and interior transition layer phenomena for pairs of second-order differential equations. J. Math. Anal. Appl. 54 (1976), 497–521.CrossRefGoogle Scholar
4Fife, P. C. and Hsiao, L.. The generation and propagation of internal layers. Nonlinear Anal. 12 (1988), 19–41.CrossRefGoogle Scholar
5Hilhorst, D., Nishiura, Y. and Mimura, M.. A free boundary problem arising in some reacting-diffusing systems. Proc. Roy. Soc. Edinburgh Sect. A 118 (1991), 355–78.CrossRefGoogle Scholar
6Ito, M.. A remark on singular perturbation methods. Hiroshima Math. J. 14 (1985), 619–29.CrossRefGoogle Scholar
7Mimura, M., Tabata, M. and Hosono, Y.. Multiple solutions of two-point boundar) value problems of Neumann type with a small parameter. SIAM J. Math. Anal. 11 (1980), 613–31.CrossRefGoogle Scholar
8Nishiura, Y.. Singular limit approach to stability and bifurcation for bistable reaction diffusion system. Rocky Mountain J. Math. 21 (1991), 727–67.CrossRefGoogle Scholar
9Nishiura, Y.. Coexistence of Infinitely Many Stable Solutions in Reaction Diffusion Systems in the Singular Limit, Dynamics Reported (New Series) 3 (1994), 25–103, eds C. K. R. T. Jones, L. Kirchgraber and H. O. Walther (Berlin: Springer).Google Scholar
10Nishiura, Y. and Fujii, H.. Stability of singularly perturbed solutions to systems of reaction-diffusion equations. SIAM J. Math. Anal. 18 (1987), 1726–70.CrossRefGoogle Scholar
11Nishiura, Y. and Mimura, M.. Layer oscillations in reaction-diffusion systems. SIAM J. Appl. Math. 49(1989), 481–514.CrossRefGoogle Scholar
12Sakamoto, K.. Construction and stability analysis of transition layer solutions in reaction-diffusion systems. Tohoku Math. J. 42 (1990), 17–44.CrossRefGoogle Scholar
13Taniguchi, M. and Nishiura, Y.. Instability of planar interfaces in reaction-diffusion systems. SIAM J. Math. Anal. 25 (1994), 99–134.CrossRefGoogle Scholar