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 (−∆)su − cuz − f(u) = 0 on R of pyramidal shape. Here (−∆)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.