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

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 ($\mathrm{Le}>1$). 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 $\epsilon =1/\mathrm{Le}$. The main result is the existence of a critical value $m^c(\epsilon )$ close to $m^c=6$ at which Hopf bifurcation holds for $\epsilon$ small enough. Proofs combine spectral analysis and non-standard application of Hurwitz Theorem with asymptotics as $\epsilon \to 0$.