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

Hirokazu Ninomiya

doi:10.3934/dcds.2020364

Article Contents

2021, Volume 41, Issue 1: 395-412. Doi: 10.3934/dcds.2020364

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

Hirokazu Ninomiya

School of Interdisciplinary Mathematical Sciences, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo 164-8525, Japan

Received: December 31, 2019

Revised: August 31, 2020

Early access: October 2020

Published: January 2021

The author was partially supported by JSPS KAKENHI Grant Numbers JP16KT0022 and JP20H01816

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Abstract

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.

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Mathematics Subject Classification: Primary:35K57, 35C07;Secondary:35B40, 35B06.

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Figure 1. The case when $ p_+(t) $ is large. The dashing curve in the gray region indicates the set of $ H( \boldsymbol{x})=p_+(t) $. The gray region indicates (iii). The region surrounded by the solid curve is $ K_0 $. One can observe that the curvature is getting small as $ p_\pm $ goes to infinity

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