A Two-Dimensional Metastable Flame-Front and a Degenerate Spike-Layer Problem

A formal asymptotic analysis is used to analyze the metastable behavior associated with a nonlocal PDE model describing the upward propagation of a flame-front interface in a vertical channel with a two-dimensional convex cross-section. In a certain asymptotic limit, the flame-front interface assumes a roughly paraboloidal shape whereby the tip of the paraboloid drifts asymptotically exponentially slowly towards the closest point on the wall of the channel. Asymptotic estimates for the exponentially small eigenvalues responsible for this metastable behavior are derived together with an explicit ODE for the slow motion of the tip of the paraboloid. The subsequent slow motion of the tip along the channel wall is also characterized explicitly. The analysis is based on a nonlinear transformation that has the effect of transforming the paraboloidal interface to a spike-layer solution of a specific singularly perturbed quasilinear parabolic problem with a non-differentiable quasilinear term.