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

Abstract

Letfbe an odd,C2function on [− 1, 1], which vanishes at ± 1, and such thatf′(O) < 0,f′ (±1) > 0 andu↦f(u)/uis increasing. Dang, Fife and Peletier [5] showed that there is a unique solutionuwith values in [−1, 1] ofwhich has the same sign asxy. The linearised operator arounduisBdefined byIt 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.