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Asymptotics for a Parabolic Double Obstacle Problem

Xinfu Chen and Charles M. Elliott
Proceedings: Mathematical and Physical Sciences
Vol. 444, No. 1922 (Mar. 8, 1994), pp. 429-445
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/52532
Page Count: 17
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Asymptotics for a Parabolic Double Obstacle Problem
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Abstract

We consider a parabolic double obstacle problem which is a version of the Allen-Cahn equation ut = Δ u - ε -2ψ (u) in Ω × (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 ε 2|ln ε|), the solution takes value 1 in a region Ω t+ and value -1 in Ω t-, where the region $\Omega \backslash $t+∪ Ω t-) is a thin strip and is contained in either a O(ε|ln ε|) or O(ε ) neighbourhood of a hypersurface Γ t 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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