Travelling fronts in cylinders
type: Item
instance of: scholarly article
title: Travelling fronts in cylinders
zbMATH Open document ID: 0799.35073
DOI: 10.1016/S0294-1449(16)30229-3
author: Henri Berestycki
author: Louis Nirenberg
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 1993-06-29T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78290
review text: The authors are concerned with travelling wave solutions in an infinite cylinder \(\Sigma:= \mathbb{R}\times \omega\) with \(\omega\subseteq \mathbb{R}^{n- 1}\) a bounded domain. They consider equations of the form \(\Delta u- \beta(y,c) \partial_{x_ 1} u+ f(u)=0\) in \(\Sigma\) \((x=(x_ 1,y))\) under homogeneous Neumann boundary conditions on \(\partial\Sigma\) and asymptotic conditions \(u(-\infty, \cdot)=0\) and \(u(+\infty, \cdot)=1\). As far as \(\beta\) is concerned, they assume: \(\beta\) continuous on \(\omega\times \mathbb{R}\) and strictly increasing in its second argument, \(\beta(y,c)\to \pm\infty\) as \(c\to \pm\infty\) uniformly for \(y\in\omega\). One may think of \(f\) as being in \(C^ 2([ 0,1])\) with \(f(0)= 0= f(1)\) and \(f'(1)< 0\). Three cases are considered: (A) \(f>0\) on \((0,1)\); (B) \(\exists\theta>0: f|_{[0,\theta]}\equiv 0\) and \(f|_{(\theta,1)} > 0\); (C) \(\exists\theta>0: f|_{(0,\theta)} < 0\) and \(f|_{(\theta,1)} > 0\). In case (A) they show that there exists a \(c^*\in\mathbb{R}\) such that the above problem is solvable, iff \(c\geq c^*\). If \(f'(0)>0\), then the solution is unique modulo translations. For case (B) they obtain a solution \((c,u)\), whereas \(\omega\) convex has to be additionally required in case (C) for that purpose. There are many more significant results in this comprehensive investigation, which extends various classical results from combustion theory as well as the celebrated paper of Kolmogorov, Petrovsky and Piskounov to higher dimensions.
reviewed by: Georg Hetzer
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35K99
Mathematics Subject Classification ID: 80A25
zbMATH DE Number: 220980
zbMATH Keywords: travelling wave solutions
zbMATH Keywords: infinite cylinder
MaRDI profile type: Publication
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OpenAlex ID: W90288657
description: scientific article; zbMATH DE number 220980