Saddle solutions of the bistable diffusion equation.

*(English)*Zbl 0764.35048Summary: Stationary solutions of the bistable Cahn-Allen diffusion equation in the plane are constructed, which are positive in quadrants 1 and 3 and negative in the other two quadrants. They are unique and obey certain monotonicity and limiting properties. Solutions of the dynamic problem near this stationary solution generally split the cross-shaped nullset into two disconnected parts which remain bounded away from each other.

##### MSC:

35K55 | Nonlinear parabolic equations |

35B20 | Perturbations in context of PDEs |

35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |

##### Keywords:

existence; uniqueness; stationary solutions; bistable Cahn-Allen diffusion equation; monotonicity; limiting properties; dynamic problem
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\textit{H. Dang} et al., Z. Angew. Math. Phys. 43, No. 6, 984--998 (1992; Zbl 0764.35048)

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