Ancient shrinking spherical interfaces in the Allen–Cahn flow

del Pino, Manuel; Gkikas, Konstantinos T.

doi:10.1016/j.anihpc.2017.03.005

del Pino, Manuel ; Gkikas, Konstantinos T.

Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 1, pp. 187-215.

Résumé

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 (\log | 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 | = \sqrt{- 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}{\sqrt{2}})

we have

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

where

ρ_{j} (t) = \sqrt{- 2 (n - 1) t} + \frac{1}{\sqrt{2}} (j - \frac{k + 1}{2}) \log (\frac{| t |}{\log | t |}) + O (1), j = 1, \dots, k .

MR Zbl

DOI : 10.1016/j.anihpc.2017.03.005

Mots clés : Nonlinear parabolic equation, Allen–Cahn equation, Ancient solutions

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     author = {del Pino, Manuel and Gkikas, Konstantinos T.},
     title = {Ancient shrinking spherical interfaces in the {Allen{\textendash}Cahn} flow},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {187--215},
     publisher = {Elsevier},
     volume = {35},
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     year = {2018},
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     zbl = {1421.35195},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2017.03.005/}
}

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del Pino, Manuel; Gkikas, Konstantinos T. Ancient shrinking spherical interfaces in the Allen–Cahn flow. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 1, pp. 187-215. doi : 10.1016/j.anihpc.2017.03.005. http://www.numdam.org/articles/10.1016/j.anihpc.2017.03.005/

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