Interface dynamics of the porous medium equation with a bistable reaction term

Matthieu Alfaro (1) , Danielle Hilhorst (2)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay

We consider a degenerate partial differential equation arising in population dynamics, namely the porous medium equation with a bistable reaction term. We study its asymptotic behavior as a small parameter, related to the thickness of a diffuse interface, tends to zero. We prove the rapid formation of transition layers which then propagate. We prove the convergence to a sharp interface limit whose normal velocity, at each point, is that of the underlying degenerate travelling wave.

Domaines: Equations aux dérivées partielles [math.AP]

Matthieu Alfaro, Danielle Hilhorst. Interface dynamics of the porous medium equation with a bistable reaction term. Asymptotic Analysis, 2012, 76 (1), pp.35-48. ⟨10.3233/ASY-2011-1067⟩. ⟨hal-00609087⟩