Variational characterization of the speed of reaction diffusion fronts for gradient dependent diffusion

We study the asymptotic speed of traveling fronts of the scalar reaction diffusion for positive reaction terms and with a diffusion coefficient depending nonlinearly on the concentration and on its gradient. We restrict our study to diffusion coefficients of the form \(D(u,u_x) = m u^{m-1} u_x^{m(p-...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2017-08
Main Authors: Benguria, R D, Depassier, M C
Format: Article
Language:English
Subjects:
Asymptotic properties
Concentration gradient
Diffusion coefficient
Diffusion rate
Lower bounds
Upper bounds
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We study the asymptotic speed of traveling fronts of the scalar reaction diffusion for positive reaction terms and with a diffusion coefficient depending nonlinearly on the concentration and on its gradient. We restrict our study to diffusion coefficients of the form \(D(u,u_x) = m u^{m-1} u_x^{m(p-2)}\) for which existence and convergence to traveling fronts has been established. We formulate a variational principle for the asymptotic speed of the fronts. Upper and lower bounds for the speed valid for any \(m\ge0, p\ge 1\) are constructed. When \(m=1, p=2\) the problem reduces to the constant diffusion problem and the bounds correspond to the classic Zeldovich Frank-Kamenetskii lower bound and the Aronson-Weinberger upper bound respectively. In the special case \(m(p-1) = 1\) a local lower bound can be constructed which coincides with the aforementioned upper bound. The speed in this case is completely determined in agreement with recent results.
ISSN:2331-8422
DOI:10.48550/arxiv.1706.08197