A fully nonlinear equation for the flame front in a quasi-steady combustion model

We revisit the Near Equidiffusional Flames (NEF) model introduced by Matkowsky and Sivashinsky in 1979 and consider a simplified, quasi-steady version of it. This simplification allows, near the planar front, an explicit derivation of the front equation. The latter is a pseudodifferential fully nonlinear parabolic equation of the fourth-order. First, we study the (orbital) stability of the null solution. Second, introducing a parameter $\epsilon$, we rescale both the dependent and independent variables and prove rigourously the convergence to the solution of the Kuramoto-Sivashinsky equation as $\epsilon\to 0$.