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

Using the method of sub-super-solution, we construct a solution of (-Delta)(s)u - 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 Calm 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 resealed mollified pyramid. In the main estimate we use an expansion of the fractional Laplacian in the Fermi coordinates. (C) 2016 Elsevier Inc. All rights reserved.