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Internal layers in high-dimensional domains

Published online by Cambridge University Press:  14 November 2011

Kunimochi Sakamoto
Kunimochi Sakamoto
Affiliation:
Department of Mathematics, Hiroshima University, 3-1 Kagamiyama-1, Higashi-Hiroshima 739, Japan
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Abstract

For a system of semilinear elliptic partial differential equations with a small parameter, denned on a bounded multi-dimensional smooth domain, we show the existence of solutions with internal layers. The high-dimensionality of the domain gives rise to quite interesting an outlook in the analysis, dramatically different from that in one-dimensional settings. Our analysis indicates, in a certain situation, an occurrence of an infinite series of bifurcation phenomena accumulating as the small parameter goes to zero. We also present a related free boundary problem with a possible approach to its resolution.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 2 , 1998 , pp. 359 - 401
Copyright
Copyright © Royal Society of Edinburgh 1998

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References

1Chen, X-F.. Generation and propagation of interfaces in reaction-diffusion systems. Trans. Amer. Math. Soc. 334 (1992), 877–913.Google Scholar
2Chen, X-Y.. Dynamics of interfaces in reaction-diffusion systems. Hiroshima Math. J. 21 (1991), 47–83.CrossRefGoogle Scholar
3Pino, M. del. Radially symmetric internal layers in a semilinear elliptic system. Trans. Amer. Math. Soc. 347(1995), 4807–37.Google Scholar
4Fife, P. C.. Semilinear elliptic boundary value problems with small parameters. Arch. Rational Mech. Anal. 52 (1973), 205–32.Google Scholar
5Fife, P. C.. Boundary and interior transition layer phenomena for a pair of second order differential equations. J. Math. Anal. Appl. 54 (1976), 497–521.CrossRefGoogle Scholar
6Fife, P. C. and Greenlee, W. M.. Interior transition layers for elliptic boundary value problems with a small parameter. Russian Math. Surveys 29–4 (1974), 103–31.CrossRefGoogle Scholar
7Ikeda, H.. On the asymptotic solutions for a weakly coupled elliptic boundary value problem with a small parameter. Hiroshima Math. J. 16 (1986), 227–50.Google Scholar
8Ito, M.. A remark on singular perturbation. Hiroshima Math. J. 14 (1984), 619–29.Google Scholar
9Mimura, M., Tabata, M. and Hosono, Y.. Multiple solutions of two-point boundary value problems of Neumann type with a small parameter. SIAM J. Math. Anal. 11 (1980), 613–31.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 Suzuki, H.. Nonexistence of stable Turing patterns with smooth limiting interfacial configurations in higher dimensional spaces (Hokkaido University Mathematics Department Preprint Series 352, 1996, 1–21).Google Scholar
12Sakamoto, K.. Construction and stability analysis of transition layer solutions in reaction-diffusion systems. Tohoku Math. J. 42 (1990), 17–44.Google Scholar
13Suzuki, H.. Asymptotic characterisation of stationary interfacial patterns for reaction diffusion systems. Hokkaido Math. J. (to appear).Google Scholar
14Taira, K.. Analytic Semigroups and Semilinear Initial Boundary Value Problems, London Mathematical Society Lecture Note Series 223 (Cambridge: Cambridge University Press, 1995).CrossRefGoogle Scholar
15Taniguchi, M. and Nishiura, Y.. Instability of planar interfaces in reaction diffusion systems. SIAM. J. Math. Anal. 25 (1994), 99–134.Google Scholar