H. Berestycki et al., NONLINEAR DYNAMICS AND METASTABILITY IN A BURGERS TYPE EQUATION (FOR UPWARD PROPAGATING FLAMES), Comptes rendus de l'Academie des sciences. Serie 1, Mathematique, 321(2), 1995, pp. 185-190
This Note concerns a nonlinear parabolic equation describing flame fro
nt evolution in upward propagating flames subject to gravity in vertic
al channels. The results we present here show that the parabolic front
s - which are observed in experiments and in computations - may actual
ly be merely quasi-equilibrium transient states. After an exponentiall
y long time, these parabolic fronts indeed eventually collapse into a
stable configuration in which the inclined front as it were spreads al
ong the wall of the channel. The model, proposed in [10], reduces to t
he following evolution problem: [GRAPHICS] In this Note we give a comp
lete description of the stationary solutions, analyze their stability
properties and discuss the dynamics of this problem for small epsilon.