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Development of interfaces in ℝN

Published online by Cambridge University Press:  14 November 2011

P. de Mottoni  and
M. Schatzman
P. de Mottoni
Affiliation:
Dipartimento di Matematica, Universitá di Roma – Tor Vergata, I-00173 ROMA –, Italy
M. Schatzman
Affiliation:
Laboratoire d'Analyse Numérique, Université Lyon 1, 43 bd du 11 novembre 1918, 69622 Villeurbanne Cedex, France
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Abstract

Consider the reaction-diffusion equation in ℝN × ℝ+: ut − h2 Δu + Φ(u) = 0, where Φ is the derivative of a bistable even potential, and h is a small parameter. If the initial data have a smooth noncritical zero set, we prove that an interface appears in time O(log (h−1)), and that the solution stays close to it for at least time O(1/√h).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 116 , Issue 3-4 , 1990 , pp. 207 - 220
Copyright
Copyright © Royal Society of Edinburgh 1990

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References

1Bronsard, L.. Reaction-diffusion equations and motion by mean curvature. (Ph.D. Thesis, N.Y.U., 1988).Google Scholar
2Carr, J. and Pego, R. L.. Metastable patterns in solutions of ut = ϵ2uxx − f(u). Comm. Pure Appl. Math. 42 (1989), 523–576.CrossRefGoogle Scholar
3Danilov, V. G. and Subochev, P. Yu.. Exact one and two phase wave-like solutions of semilinear parabolic equations (preprint, Steklov Institute, Moscow, 1988).Google Scholar
4Fife, P. C.. Nonlinear diffusive waves (CBMS Conference at Little Cottonwood Canyon, Utah 1987, CMBS Conference Series, 1989).Google Scholar
5Fife, P. C. and Ling, Xiao. The generation and propagation of internal layers. Nonlinear Anal. 12 (1988), 19–41.CrossRefGoogle Scholar
6Fife, P. C. and McLeod, J. B., The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch. Rational Mech. Anal. 65 (1977), 335–361.CrossRefGoogle Scholar
7Fusco, G. and Hale, J. K.. Slow-Motion Manifolds, Dormant Instability, and Singular Perturbations. J. Dynam. Differ. Equat., 1 (1989), 75–94.CrossRefGoogle Scholar
8Fusco, G.. A Geometric Approach to the Dynamics of ut = ε2uxx + f(u) for small ε. (preprint, June, 1989).Google Scholar
9I, Ya.. Kanel'. On the stabilization of solutions of the Cauchy problem for the equations arising in the theory of combustion. Math. Sb. 59 (1965), 398–413.Google Scholar
10de Mottoni, P. and Schatzman, M.. Geometrical Evolution of developed interfaces (preprint, Equipe d'Analyse Numérique Lyon-Saint-Etienne 1989).Google Scholar
11Rubinstein, J., Sternberg, P. and Keller, J. B.. Fast reaction, slow diffusion and curve shortening. S.I.A.M. J. Appl. Math. 49 (1989), 116–133.CrossRefGoogle Scholar