Ancient shrinking spherical interfaces in the Allen–Cahn flow

Konstantinos T. Gkikas; Manuel del Pino

doi:10.1016/j.anihpc.2017.03.005

JournalsaihpcVol. 35, No. 1pp. 187–215

Ancient shrinking spherical interfaces in the Allen–Cahn flow

Konstantinos T. Gkikas
Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile
Manuel del Pino
Departamento de Ingeniería Matemática, and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile

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Abstract

We consider the parabolic Allen–Cahn equation in $R^{n}$ , $n \geq 2$ ,

u_{t} = Δ u + (1 - u^{2}) u in R^{n} \times (- \infty, 0] .

We construct an ancient radially symmetric solution $u (x, t)$ with any given number $k$ of transition layers between $- 1$ and $+ 1$ . At main order they consist of $k$ time-traveling copies of $w$ with spherical interfaces distant $O (lo g ∣ t ∣)$ one to each other as $t \to - \infty$ . These interfaces are resemble at main order copies of the shrinking sphere ancient solution to mean the flow by mean curvature of surfaces: $∣ x ∣ = - 2 (n - 1) t$ . More precisely, if $w (s)$ denotes the heteroclinic 1-dimensional solution of $w^{″} + (1 - w^{2}) w = 0$ $w (\pm \infty) = \pm 1$ given by $w (s) = tanh (\frac{s}{2})$ we have

u (x, t) \approx j = 1 \sum k (- 1)^{j - 1} w (∣ x ∣ - ρ_{j} (t)) - \frac{1}{2} (1 + (- 1)^{k}) as t \to - \infty

where

ρ_{j} (t) = - 2 (n - 1) t + \frac{1}{2} (j - \frac{k + 1}{2}) lo g (\frac{∣ t ∣}{lo g ∣ t ∣}) + O (1), j = 1, \dots, k .

Cite this article

Konstantinos T. Gkikas, Manuel del Pino, Ancient shrinking spherical interfaces in the Allen–Cahn flow. Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 1, pp. 187–215

DOI 10.1016/J.ANIHPC.2017.03.005