Asymptotics for a parabolic double obstacle problem

Abstract
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.

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