  In this paper we consider ignition-temperature, first-order reaction model of
thermo-diffusive combustion that describes dynamics of thick flames arising in
a theory of combustion of hydrogen-oxygen and ethylene-oxygen mixtures. These
flames often assume the shape of propagating curved interfaces that correspond
to level sets of constant temperature. We derive a fully nonlinear equation
that governs dynamics of these level sets under a single assumption of small
curvature. We study this equation for various asymptotic parameter regimes and
discuss the ranges of validity of the corresponding simplified models. Our
theoretical findings are supported by numerical simulations.
