{'fraction_non_alphanumeric': 0.06825232678386763, 'fraction_numerical': 0.010341261633919338, 'mean_word_length': 3.84, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Asymptotics for a parabolic double obstacle problem

We consider a parabolic double obstacle problem which is a version of the Allen-Cahn equation ut = Δu — ϵ-2ψ'(u) in Ω x (0, ∞), where Ω is a bounded domain, ϵ is a small constant, and ψ is a double well potential; here we take ψ such that ψ(u) = (1 — u2) when |u| ≤ 1 and ψ(u) = ∞ when |u| > 1. We study the asymptotic behaviour, as ϵ → 0, of the solution of the double obstacle problem. Under some natural restrictions on the initial data, we show that after a short time (of order ϵ2|ln ϵ|), the solution takes value 1 in a region Ω+t and value — 1 in Ω-t, where the region Ω(Ω+t U Ω-t) is a thin strip and is contained in either a O(ϵ|ln ϵ|) or O(ϵ) neighbourhood of a hypersurface Γt which moves with normal velocity equal to its mean curvature. We also study the asymptotic behaviour, as t → ∞, of the solution in the one-dimensional case. In particular, we prove that the ω-limit set consists of a singleton.