### Mathematics in Engineering

2021, Issue 5:1-36. doi: 10.3934/mine.2021037
Research article

# Semilinear integro-differential equations, Ⅱ: one-dimensional and saddle-shaped solutions to the Allen-Cahn equation

• Received: 16 June 2020 Accepted: 03 September 2020 Published: 18 September 2020
• This paper addresses saddle-shaped solutions to the semilinear equation $L_K u = f(u)$ in $\mathbb{R}^{2m}$, where $L_K$ is a linear elliptic integro-differential operator with a radially symmetric kernel $K$, and $f$ is of Allen-Cahn type. Saddle-shaped solutions are doubly radial, odd with respect to the Simons cone $\{(x', x'') \in \mathbb{R}^m \times \mathbb{R}^m \, : \, |x'| = |x''|\}$, and vanish only in this set. We establish the uniqueness and the asymptotic behavior of the saddle-shaped solution. For this, we prove a Liouville type result, the one-dimensional symmetry of positive solutions to semilinear problems in a half-space, and maximum principles in "narrow" sets. The existence of the solution was already proved in part Ⅰ of this work.

Citation: Juan-Carlos Felipe-Navarro, Tomás Sanz-Perela. Semilinear integro-differential equations, Ⅱ: one-dimensional and saddle-shaped solutions to the Allen-Cahn equation[J]. Mathematics in Engineering, 2021, 3(5): 1-36. doi: 10.3934/mine.2021037

### Related Papers:

• This paper addresses saddle-shaped solutions to the semilinear equation $L_K u = f(u)$ in $\mathbb{R}^{2m}$, where $L_K$ is a linear elliptic integro-differential operator with a radially symmetric kernel $K$, and $f$ is of Allen-Cahn type. Saddle-shaped solutions are doubly radial, odd with respect to the Simons cone $\{(x', x'') \in \mathbb{R}^m \times \mathbb{R}^m \, : \, |x'| = |x''|\}$, and vanish only in this set. We establish the uniqueness and the asymptotic behavior of the saddle-shaped solution. For this, we prove a Liouville type result, the one-dimensional symmetry of positive solutions to semilinear problems in a half-space, and maximum principles in "narrow" sets. The existence of the solution was already proved in part Ⅰ of this work.

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