A geometric approach to front propagation problems in bounded domains with Neumann-type boundary conditions
type: Item
instance of: scholarly article
title: A geometric approach to front propagation problems in bounded domains with Neumann-type boundary conditions
zbMATH Open document ID: 1046.35059
DOI: 10.4171/IFB/79
author: Guy Barles
author: Francesca Da Lio
published in: Interfaces and Free Boundaries
publication date: 2004-03-08T00:00:00Z
review text: Summary: We are interested in the asymptotic behavior of the solutions of scaled reaction-diffusion equations in bounded domains, associated with Neumann type boundary conditions, and more precisely in cases when such behavior is described in terms of moving interfaces. A typical example is the case of the Allen-Cahn equation associated with an oblique derivative boundary condition, where the generation of a front moving by mean curvature with an angle boundary condition is shown. In order to establish such results rigourously, we modify and adapt the ``geometrical approach'' introduced by P. E. Souganidis and the first author for solving problems in \(\mathbb{R}^N\): we provide a new definition of weak solution for the global-in-time motion of fronts with curvature-dependent velocities and with angle boundary conditions, which turns out to be equivalent to the level-set approach when there is no fattening phenomenon. We use this definition to obtain the asymptotic behavior of the solutions of a large class of reaction-diffusion equations, including the case of quasilinear ones and \((x, t)\)-dependent reaction terms, but also with any, possibly nonlinear, Neumann boundary conditions.
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35Q72
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 74N20
Mathematics Subject Classification ID: 35K60
zbMATH DE Number: 2052701
zbMATH Keywords: front propagation
zbMATH Keywords: reaction-diffusion equations
zbMATH Keywords: geometrical approach
zbMATH Keywords: level-set approach
zbMATH Keywords: Neumann boundary condition
zbMATH Keywords: angle boundary condition
zbMATH Keywords: viscosity solutions
zbMATH Keywords: flame propagation
zbMATH Keywords: phase transitions
zbMATH Keywords: evolution of populations
zbMATH Keywords: Allen-Cahn equation
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/ifb/79
OpenAlex ID: W1986635690
description: scientific article; zbMATH DE number 2052701