Speaker:   Prof. Danielle Hilhorst (Université de Paris-Sud, Orsay)

Title:   The singular limit of the Allen-Cahn equation, once more revisited.

Abstract

Joint work with Matthieu Alfaro and Hiroshi Matano. It is well-known
that some classes of nonlinear diffusion equations give rise to sharp
internal layers (or interfaces) when the diffusion coefficient is
small enough or the reaction coefficient is very large; the motion of
such interfaces is often driven by their curvature. We consider here
the Allen-Cahn equation with a small parameter  eps  and an additional
smooth nonlinearity under Neumann boundary conditions. Our results are
the following:  

  (1) generation of interface: we show, under some mild conditions on
  the initial data, that solutions develop an internal layer near the
  zeros of the initial data within a very short time; furthermore, the
  width of the internal layer is of order  eps ; this estimate is
  optimal and has not been known in higher space dimensions even for
  the homogeneous case; 
  (2) propagation of interface: once the layer is formed, it is
  expected to propagate roughly by the same motion law as the limiting
  interface equation; we show that this is indeed the case and that
  the Hausdorff distance between the limiting interface and the real
  internal layer remains of order  eps  as  t  ranges in a finite
  interval. 
  (3) Finally we extend our results to the case of the FitzHugh-Nagumo
  system.