Some entire solutions of the Allen-Cahn equation
type: Item
instance of: scholarly article
title: Some entire solutions of the Allen-Cahn equation
zbMATH Open document ID: 1052.35095
author: Yukitaka Fukao
author: Yoshihisa Morita
author: Hirokazu Ninomiya
published in: Taiwanese Journal of Mathematics
publication date: 2004-09-28T00:00:00Z
review text: The paper deals with entire solutions of a bistable reaction-diffusion equation with Nagumo type nonlinearity, the so called Allen-Cahn equation. The entire solutions are the solutions defined for all \((x,t)\in (\mathbb{R}\times \mathbb{R}).\) The authors show the existence of an entire solution which behaves as two traveling front solutions coming from both sides of the \(x\)-axis and annihilating in a finite time, using the explicit expression of the traveling front and the comparison theorem. They also show the existence of an entire solution emanating from the unstable standing pulse solution and converging to the pair of diverging traveling fronts as the time tends to infinity. Then in terms of the comparison principle a rather general result is proved on the existence of an unstable set of an unstable equilibrium to apply to the present case.
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 2104150
zbMATH Keywords: entire solution
zbMATH Keywords: traveling front
zbMATH Keywords: reaction diffusion equation
zbMATH Keywords: annihilation
zbMATH Keywords: diverging fronts
zbMATH Keywords: Nagumo type nonlinearity
reviewed by: Lubomira Softova
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.11650/twjm/1500558454
OpenAlex ID: W1509215099
DOI: 10.11650/TWJM/1500558454
description: scientific article; zbMATH DE number 2104150