Multi-dimensional traveling fronts in bistable reaction-diffusion equations

This paper studies traveling front solutions of convex polyhedral shapes in bistable reaction-diffusion equations including the Allen-Cahn equations or the Nagumo equations. By taking the limits of such solutions as the lateral faces go to infinity, we construct a three-dimensional traveling front solution for any given $g\in C^{\infty}(S^{1})$ with $\min_{0\leq \theta\leq 2\pi}g(\theta)=0$.