Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space

Abstract

<p style='text-indent:20px;'>The Allen–Cahn–Nagumo equation is a reaction-diffusion equation with a bistable nonlinearity. This equation appears to be simple, however, it includes a rich behavior of solutions. The Allen–Cahn–Nagumo equation features a solution that constantly maintains a certain profile and moves with a constant speed, which is referred to as a traveling wave solution. In this paper, the entire solution of the Allen–Cahn–Nagumo equation is studied in multi-dimensional space. Here an entire solution is meant by the solution defined for all time including negative time, even though it satisfies a parabolic partial differential equation. Especially, this equation admits traveling wave solutions connecting two stable states. It is known that there is an entire solution which behaves as two traveling wave solutions coming from both sides in one dimensional space and annihilating in a finite time and that this one-dimensional entire solution is unique up to the shift. Namely, this entire solution is symmetric with respect to some point. There is a natural question whether entire solutions coming from all directions in the multi-dimensional space are radially symmetric or not. To answer this question, radially asymmetric entire solutions will be constructed by using super-sub solutions.