Traveling waves for combustion reaction-diffusion-convection equations: the full range of wave speeds

Abstract

We consider traveling wave solutions to a reaction--diffusion--convection equation with a combustion-type reaction term. While a necessary condition for the existence of traveling waves is $c\ge H^{\ast }:={\sup }_{0<u\le \theta }(-\frac{1}{u}{\int }_0^{u}h(\sigma )d\sigma )$, where $c$ denotes the wave speed, $\theta \in (0,1)$ the ignition threshold, and $h$ the convective term, the available results in \cite{DZ25,MM03} establish existence and nonexistence only under the restriction $c\ge -{\min }_{u\in [0,1]}h(u)$. In this note, we close this gap by covering the entire range $c\ge H^{\ast }$.