The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system

Matthieu Alfaro (1) , Danielle Hilhorst (2, 3) , Hiroshi Matano (4)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier
2 Laboratoire d'Analyse Numérique
3 LM-Orsay - Laboratoire de Mathématiques d'Orsay
4 GSMS - Graduate School of Mathematical Sciences

We consider an Allen-Cahn type equation with a bistable nonlinearity associated to a double-well potential whose well-depths can be slightly unbalanced, and where the coefficient of the nonlinear reaction term is very small. Given rather general initial data, we perform a rigorous analysis of both the generation and the motion of interface. More precisely we show that the solution develops a steep transition layer within a small time, and we present an optimal estimate for its width. We then consider a class of reaction-diffusion systems which includes the FitzHugh-Nagumo system as a special case. Given rather general initial data, we show that the first component of the solution vector develops a steep transition layer and that all the results mentioned above remain true for this component.

Domaines: Equations aux dérivées partielles [math.AP]

Matthieu Alfaro, Danielle Hilhorst, Hiroshi Matano. The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system. Journal of Differential Equations, 2008, 245, pp.505-565. ⟨10.1016/j.jde.2008.01.014⟩. ⟨hal-00174999v2⟩