Asymptotic behavior of spreading fronts in the anisotropic Allen–Cahn equation on ${\mathbb{R}}^n$

Matano, Hiroshi; Mori, Yoichiro; Nara, Mitsunori

Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 3, pp. 585-626.

Nous considérons le problème de Cauchy pour l'équation d'Allen–Cahn (de moyenne non nulle) anisotropique dans ${\mathbb{R}}^n$ avec $n\ge 2$, et étudions le comportement en temps grand des solutions propageantes. Nous montrons, sous des hypothèses assez faibles sur la donnée initiale, que la solution développe un véritable front de propagation dont la position peut être approchée d'assez près, en temps grand, par une forme de Wulff en expansion. Un tel comportement peut être attendu formellement, et il existe aussi dans la littérature certaines études rigoureuses sur des problèmes analogues. Le principal objectif de cet article est d'établir des résultats d'approximation plus fins que ce qui était connu auparavant. Plus précisément, la distance de Hausdorff entre un ensemble de niveau de la solution et la forme de Wulff en expansion reste bornée uniformément en temps grand. De plus, chaque ensemble de niveau devient en temps fini une hypersurface régulière, quelque soit l'irrégularité de sa configuration initiale, et le mouvement de cette hypersurface est régi (approximativement) par le flot de courbure moyenne anisotropique $V_{\gamma }={\kappa }_{\gamma }+c$, avec une marge d'erreur petite. Nous prouvons aussi la rigidité asymptotique du profil de la solution, c'est-à-dire qu'il converge, à proximité du front et quand le temps tend vers l'infini, vers le profil de l'onde progressive. Une extension au cas anisotropique d'un théorème de type Liouville de Berestycki et Hamel (2007), portant sur les solutions entières de l'équation d'Allen–Cahn, joue un rôle clé dans la preuve de ce dernier résultat, ainsi que de la régularité des ensembles de niveau.

We consider the Cauchy problem for the anisotropic (unbalanced) Allen–Cahn equation on ${\mathbb{R}}^n$ with $n\ge 2$ 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 $V_{\gamma }={\kappa }_{\gamma }+c$ 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.

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