Traveling Front Solutions of Dimension n Generate Entire Solutions of Dimension \((n-1)\) in Reaction–Diffusion Equations as the Speeds Go to Infinity

Abstract

Multidimensional traveling front solutions and entire solutions of reaction–diffusion equations have been studied intensively. To study the relationship between multidimensional traveling front solutions and entire solutions, we study the reaction–diffusion equation with a bistable nonlinear term. It is well known that there exist multidimensional traveling front solutions with every speed that is greater than the speed of a one-dimensional traveling front solution connecting two stable equilibria. In this paper, we show that the limit of the n-dimensional multidimensional traveling front solutions as the speeds go to infinity generates an entire solution of the same reaction–diffusion equation in the \((n-1)\)-dimensional space.