Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-22T01:26:34.640Z Has data issue: false hasContentIssue false
  • English
  • Français

On the stability of the saddle solution of Allen–Cahn's equation

Published online by Cambridge University Press:  14 November 2011

M. Schatzman
M. Schatzman
Affiliation:
Analyse Numérique, U.R.A. 740 du C.N.R.S., Université Lyon 1-Claude-Bernard, 69622 Villeurbanne CEDEX, France e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Let f be an odd, C2 function on [− 1, 1], which vanishes at ± 1, and such that f′(O) < 0, f′ (±1) > 0 and u ↦ f(u)/u is increasing. Dang, Fife and Peletier [5] showed that there is a unique solution u with values in [−1, 1] of

which has the same sign as xy. The linearised operator around u is B defined by

It is proved here that the spectrum of B contains at least one negative eigenvalue, that all eigenfunctions corresponding to negative eigenvalues have the symmetries of the square, and that for Allen–Cahn's nonlinearity (f(u) = 2u3 − 2u), there is exactly one negative eigenvalue.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 6 , 1995 , pp. 1241 - 1275
Copyright
Copyright © Royal Society of Edinburgh 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Blowey, J. and Elliott, C.. The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy. Part I: Numerical Analysis. European J. Appl. Math. 2 (1991), 233−79.CrossRefGoogle Scholar
2Chen, Y.-G., Giga, Y. and Goto, S.. Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Differential Geom. 33 (1991), 749−86.Google Scholar
3Coddington, E. A. and Levinson, N.. Theory of Ordinary Differential Equations (New York: MacGraw-Hill, 1955).Google Scholar
4Cycon, H. L., Froese, R. G., Kirsch, W. and Simon, B.. Schrodinger Operators with Applications to Quantum Mechanics and Global Geometry (Berlin: Springer, 1986).Google Scholar
5Dang, H., Fife, P. C. and Peletier, L. A.. Saddle solutions of the bistable diffusion equation (Preprint, 1991).Google Scholar
6Evans, L. C., Soner, H. M. and Souganidis, P. E.. Phase transition and generalized motion by mean curvature. Comm. Pure Appl. Math. 45 (1992), 1097−123.Google Scholar
7Evans, L. C. and Spruck, J.. Motion of level sets by mean curvature I. J. Differential Geom. 33 (1991), 635−81.Google Scholar
8Halpern, L. and Schatzman, M.. Artificial boundary conditions for incompressible viscous flow. SIAM J. Math. Anal. 20 (1989), 308−53.Google Scholar
9Henry, D.. Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics 840 (Berlin: Springer, 1981).Google Scholar
10Kato, T.. Perturbation Theory for Linear Operators (Berlin: Springer, 1966).Google Scholar
11Mottoni, P. de and Schatzman, M.. Geometrical evolution of developed interfaces. Trans. Amer. Math. Soc. 347 (1995) 1533−89.CrossRefGoogle Scholar
12Nataf, F.. Approximation paraxiale pour les fluides incompressibles (Doctorat de l'Ecole Polytechnique, February 1989).Google Scholar
13Nikiforov, A. and Ouvarov, V.. Fonctions Speciales de la Physique Mathematique (Moscow: Mir, 1983).Google Scholar
14Reed, M. and Simon, B.. Methods of Modern Mathematical Physics IV: Analysis of Operators (New York: Academic Press, 1978).Google Scholar
15Reed, M. and Simon, B.. Methods of Modern Mathematical Physics I: Functional Analysis, revised edn (New York: Academic Press, 1980).Google Scholar
16Yosida, K.. Functional Analysis, 4th edn (Berlin: Springer, 1974).Google Scholar