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

Abstract

Using the method of sub-super-solution, we construct a solution of (−Δ)su−cuz−f(u)=0 on R3 of pyramidal shape. Here (−Δ)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.