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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]
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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

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