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

## TL;DR

This paper develops a variational framework to analyze the asymptotic speed of reaction-diffusion fronts with nonlinear, gradient-dependent diffusion coefficients, providing bounds and exact speeds in special cases.

## Contribution

It introduces a variational principle for front speed in reaction-diffusion equations with gradient-dependent diffusion, extending classical bounds to nonlinear, gradient-dependent cases.

## Key findings

- Established bounds for front speed for all m ≥ 0, p ≥ 1.
- Recovered classical bounds in the case m=1, p=2.
- Determined exact front speed when m(p-1)=1.

## Abstract

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.