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

### Claude-Michel Brauner

School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China, Institut de Mathématiques de Bordeaux, Université de Bordeaux, 33405 Talence Cedex, France### Luca Lorenzi

Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Plesso di Matematica, Università degli Studi di Parma, Parco Area delle Scienze 53/A, I-43124 Parma, Italy### Mingmin Zhang

School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China

## Abstract

In this paper we analyze the stability of the traveling wave solution for an ignition-temperature first-order reaction model of diffusional-thermal combustion in the case of high Lewis numbers ($Le>1$). In contrast to conventional Arrhenius kinetics where the reaction zone is infinitely thin, the reaction zone for stepwise temperature kinetics is of order unity. The system of two parabolic PDEs is characterized by a free interface at which ignition temperature $Θ_{i}$ 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 $m=Θ_{i}/(1−Θ_{i})$ and a perturbation parameter $ε=1/Le$. The main result is the existence of a critical value $m_{c}(ε)$ close to $m_{c}=6$ at which Hopf bifurcation holds for *ε* small enough. Proofs combine spectral analysis and non-standard application of Hurwitz's Theorem with asymptotics as $ε→0$.

## Cite this article

Claude-Michel Brauner, Luca Lorenzi, Mingmin Zhang, Stability analysis and Hopf bifurcation at high Lewis number in a combustion model with free interface. Ann. Inst. H. Poincaré Anal. Non Linéaire 37 (2020), no. 3, pp. 581–604

DOI 10.1016/J.ANIHPC.2020.01.002