  We prove the existence of a family of travelling wave solutions in a variant
of the $\textit{Zeldovich-Frank-Kamenetskii (ZFK) equation}$, a
reaction-diffusion equation which models the propagation of planar laminar
premixed flames in combustion theory. Our results are valid in an asymptotic
regime which corresponds to a reaction with high activation energy, and provide
a rigorous and geometrically informative counterpart to formal asymptotic
results that have been obtained for similar problems using $\textit{high
activation energy asymptotics}$. We also go beyond the existing results by (i)
proving smoothness of the minimum wave speed function $\overline c(\epsilon)$,
where $0< \epsilon \ll 1$ is the small parameter, and (ii) providing an
asymptotic series for a flat slow manifold which plays a role in the
construction of travelling wave solutions for non-minimal wave speeds $c >
\overline c(\epsilon)$. The analysis is complicated by the presence of an
exponential nonlinearity which leads to two different scaling regimes as
$\epsilon \to 0$, which we refer to herein as the
$\textit{convective-diffusive}$ and $\textit{diffusive-reactive}$ zones. The
main idea of the proof is to use the geometric blow-up method to identify and
characterise a $(c,\epsilon)$-family of heteroclinic orbits which traverse both
of these regimes, and correspond to travelling waves in the original ZFK
equation. More generally, our analysis contributes to a growing number of
studies which demonstrate the utility of geometric blow-up approaches to the
study dynamical systems with singular exponential nonlinearities.
