# Stability analysis and Hopf bifurcation at high Lewis number in a combustion model with free interface

## TL;DR

This paper investigates the stability of traveling wave solutions in a thermo-diffusive combustion model with high Lewis numbers, identifying conditions for Hopf bifurcation through spectral analysis and asymptotic methods.

## Contribution

It introduces a bifurcation analysis for combustion models at high Lewis numbers, establishing the existence of Hopf bifurcation near a critical parameter value.

## Key findings

- Hopf bifurcation occurs at a critical parameter near m^c=6 for small epsilon.
- Spectral analysis and Hurwitz Theorem are used to prove bifurcation.
- Critical value m^c(epsilon) approaches 6 as epsilon approaches zero.

## Abstract

In this paper we analyze the stability of the traveling wave solution for an ignition-temperature, first-order reaction model of thermo-diffusive combustion, in the case of high Lewis numbers (${\rm Le} >1$). The system of two parabolic PDEs is characterized by a free interface at which ignition temperature $\Theta_i$ is reached. We turn the model to a fully nonlinear problem in a fixed domain. When the Lewis number is large, we define a bifurcation parameter $m=\Theta_i/(1-\Theta_i)$ and a perturbation parameter $\varepsilon= 1/{\rm Le}$. The main result is the existence of a critical value $m^c(\varepsilon)$ close to $m^c=6$ at which Hopf bifurcation holds for $\varepsilon$ small enough. Proofs combine spectral analysis and non-standard application of Hurwitz Theorem with asymptotics as $\varepsilon\to 0$.