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.
1
University of British Columbia, Vancouver, Canada
Alexei F. Cheviakov; Michael J. Ward. A two-dimensional metastable flame-front and a degenerate spike-layer problem. Interfaces and free boundaries, Tome 9 (2007) no. 4, pp. 513-547. doi: 10.4171/ifb/175
@article{10_4171_ifb_175,
author = {Alexei F. Cheviakov and Michael J. Ward},
title = {A two-dimensional metastable flame-front and a degenerate spike-layer problem},
journal = {Interfaces and free boundaries},
pages = {513--547},
year = {2007},
volume = {9},
number = {4},
doi = {10.4171/ifb/175},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/175/}
}
TY - JOUR
AU - Alexei F. Cheviakov
AU - Michael J. Ward
TI - A two-dimensional metastable flame-front and a degenerate spike-layer problem
JO - Interfaces and free boundaries
PY - 2007
SP - 513
EP - 547
VL - 9
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.4171/ifb/175/
DO - 10.4171/ifb/175
ID - 10_4171_ifb_175
ER -
%0 Journal Article
%A Alexei F. Cheviakov
%A Michael J. Ward
%T A two-dimensional metastable flame-front and a degenerate spike-layer problem
%J Interfaces and free boundaries
%D 2007
%P 513-547
%V 9
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4171/ifb/175/
%R 10.4171/ifb/175
%F 10_4171_ifb_175
