Traveling wave solutions for bistable fractional Allen-Cahn equations with a pyramidal front

TL;DR

This paper constructs pyramidal-shaped traveling wave solutions for a fractional Allen-Cahn equation with bistable nonlinearity using sub-super-solution methods, expanding the understanding of solutions with complex geometries in fractional PDEs.

Contribution

It introduces a novel method to construct pyramidal traveling wave solutions for fractional Allen-Cahn equations using sub-super-solution techniques and fractional Laplacian expansions.

Findings

Existence of pyramidal traveling wave solutions is established.

Method extends to fractional Laplacian of order 1/2<s<1.

Provides new insights into complex solution geometries in fractional PDEs.

Abstract

Using the method of sub-super-solution, we construct a solution of $(-\Delta)^su-cu_z-f(u)=0$ on $\R^3$ of pyramidal shape. Here $(-\Delta)^s$ is the fractional Laplacian of sub-critical order $1/2<s<1$ and $f$ is a bistable nonlinearity. Hence, the existence of a traveling wave solution for the parabolic fractional Allen-Cahn equation with pyramidal front is asserted. The maximum of planar traveling wave solutions in various directions gives a sub-solution. A super-solution is roughly defined as the one-dimensional profile composed with the signed distance to a rescaled mollified pyramid. In the main estimate we use an expansion of the fractional Laplacian in the Fermi coordinates.