On the stability of the saddle solution of Allen–Cahn's equation
Michelle Schatzman · Proceedings of the Royal Society of Edinburgh Section A Mathematics · 1995
Letfbe an odd,C2function on [− 1, 1], which vanishes at ± 1, and such thatf′(O) 0 andu↦f(u)/uis increasing. Dang, Fife and Peletier [5] showed that there is a unique solutionuwith values in [−1, 1] of which has the same sign asxy. The linearised operator arounduisBdefined 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.