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

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 $c \geq H^*:=\sup_{0<u \leq \theta} \big(-\frac{1}{u} \int_0^u h(\sigma)\,d\sigma \big)$, 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 $c \geq -\min_{u\in[0,1]} h(u)$. In this note, we close this gap by covering the entire range $c\ge H^*$.