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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)

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