On the stability of the saddle solution of Allen-Cahn’s equation. (English) Zbl 0852.35020

It is known (by a result of H. Dang, P. C. Fife and L. A. Peletier) that if \(f\) is odd and of class \(C^2\) in \([0, 1]\), \(f(\pm 1)= 0\), \(f'(0)< 0\), \(f'(\pm 1)> 0\) and \(f(u)/u\) is an increasing function then there exists a unique solution \(u\) with values in \([- 1, 1]\) of the equation \(- \Delta u+ f(u)= 0\), which has the same sign as \(xy\). The author considers here the linearized operator around \(u\), i.e., \[ Bv= -\Delta v+ f'(u) v,\quad D(B)= H^2(\mathbb{R}). \] It is proved that the spectrum of \(B\) contains at least one negative eigenvalue. Moreover, it is shown that the eigenfunction corresponding to any strictly negative eigenvalue of \(B\) has the symmetries of the square. In the case of Allen-Cahn’s nonlinearity \((f(u)=2u^3-2u)\) the author proves that there is exactly one negative eigenvalue of \(B\).
Reviewer: E.Minchev (Sofia)


35B35 Stability in context of PDEs
35J60 Nonlinear elliptic equations
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