We consider the Cauchy problem for the anisotropic (unbalanced) Allen–Cahn equation on R n with n≥2 and study the large time behavior of the solutions with spreading fronts. We show, under very mild assumptions on the initial data, that the solution develops a well-formed front whose position is closely approximated by the expanding Wulff shape for all large times. Such behavior can naturally be expected on a formal level and there are also some rigorous studies in the literature on related problems, but we will establish approximation results that are more refined than what has been known before. More precisely, the Hausdorff distance between the level set of the solution and the expanding Wulff shape remains uniformly bounded for all large times. Furthermore, each level set becomes a smooth hypersurface in finite time no matter how irregular the initial configuration may be, and the motion of this hypersurface is approximately subject to the anisotropic mean curvature flow V γ =κ γ +c with a small error margin. We also prove the eventual rigidity of the solution profile at the front, meaning that it converges locally to the traveling wave profile everywhere near the front as time goes to infinity. In proving this last result as well as the smoothness of the level surfaces, an anisotropic extension of the Liouville type theorem of Berestycki and Hamel (2007) for entire solutions of the Allen–Cahn equation plays a key role.
|Original language||English (US)|
|Number of pages||42|
|Journal||Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire|
|State||Published - May 1 2019|
Bibliographical noteFunding Information:
This research was supported in part by JSPS KAKENHI 16H02151 (to H.M.), NSF DMS-1620316, DMS-1516978 (to Y.M.), and JSPS KAKENHI 16K05220 (to M.N.).
- Anisotropic Allen–Cahn equation
- Anisotropic mean curvature flow
- Spreading front
- Wulff shape