Polyhedral entire solutions in reaction-diffusion equations

Abstract

This paper studies polyhedral entire solutions to a bistable reaction-diffusion equation in $R^n$. We consider a pyramidal traveling front solution to the same equation in $R^{n+1}$. As the speed goes to infinity, its projection converges to an n-dimensional polyhedral entire solution. Conversely, as the time goes to −∞, an n-dimensional polyhedral entire solution gives n-dimensional pyramidal traveling front solutions. The result in this paper suggests a correlation between traveling front solutions and entire solutions in general reaction-diffusion equations or systems.