Asymptotics for a parabolic double obstacle problem

Abstract

We consider a parabolic double obstacle problem which is a version of the Allen-Cahn equationut= Δu—ϵ-2ψ'(u) inΩx (0, ∞), whereΩis a bounded domain,ϵis a small constant, andψis a double well potential; here we takeψsuch thatψ(u) = (1 —u2) when |u| ≤ 1 andψ(u) = ∞ when |u| > 1. We study the asymptotic behaviour, asϵ→ 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 orderϵ2|lnϵ|), the solution takes value 1 in a regionΩ+tand value — 1 inΩ-t, where the regionΩ(Ω+tUΩ-t) is a thin strip and is contained in either aO(ϵ|lnϵ|) orO(ϵ) neighbourhood of a hypersurfaceΓtwhich moves with normal velocity equal to its mean curvature. We also study the asymptotic behaviour, ast→ ∞, of the solution in the one-dimensional case. In particular, we prove that theω-limit set consists of a singleton.