A metastable flame-front interface and Carrier's problem

Abstract

Metastable dynamics for a nonlocal PDE modeling the upwards propagation of a flame-front interface in a vertical channel is analyzed in the one-dimensional case where the channel cross-section is taken to be the slab - 1 < x < 1 . In a certain asymptotic limit, the interface assumes a roughly concave parabolic shape, and the tip of the parabola drifts asymptotically exponentially slowly towards the boundary of the domain. In contrast to previous analyses that studied this behavior by transforming the governing nonlocal PDE to a convection–diffusion equation, a novel nonlinear transformation is introduced that transforms the problem to a singularly perturbed quasilinear PDE. The steady-state problem for this transformed PDE, for which the parabolic interface shape maps onto a one-spike solution, is closely related to a class of two-point boundary value problems with seemingly spurious solutions studied initially by G. Carrier in 1968. Rigorous and formal asymptotic results for a one-spike solution to this transformed PDE are obtained together with a formal metastability analysis of certain time-dependent solutions.