Description:
<jats:p>We consider a parabolic double obstacle problem which is a version of the Allen-Cahn equation<jats:italic>u<jats:sub>t</jats:sub></jats:italic>= Δ<jats:italic>u</jats:italic>—<jats:italic>ϵ</jats:italic><jats:sup>-2</jats:sup><jats:italic>ψ</jats:italic>'(<jats:italic>u</jats:italic>) in<jats:italic>Ω</jats:italic>x (0, ∞), where<jats:italic>Ω</jats:italic>is a bounded domain,<jats:italic>ϵ</jats:italic>is a small constant, and<jats:italic>ψ</jats:italic>is a double well potential; here we take<jats:italic>ψ</jats:italic>such that<jats:italic>ψ</jats:italic>(<jats:italic>u</jats:italic>) = (1 —<jats:italic>u</jats:italic><jats:sup>2</jats:sup>) when |<jats:italic>u</jats:italic>| ≤ 1 and<jats:italic>ψ</jats:italic>(<jats:italic>u</jats:italic>) = ∞ when |<jats:italic>u</jats:italic>| > 1. We study the asymptotic behaviour, as<jats:italic>ϵ</jats:italic>→ 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<jats:italic>ϵ</jats:italic><jats:sup>2</jats:sup>|ln<jats:italic>ϵ</jats:italic>|), the solution takes value 1 in a region<jats:italic>Ω</jats:italic><jats:sup>+</jats:sup><jats:italic><jats:sub>t</jats:sub></jats:italic>and value — 1 in<jats:italic>Ω</jats:italic><jats:sup>-</jats:sup><jats:italic><jats:sub>t</jats:sub></jats:italic>, where the region<jats:italic>Ω</jats:italic>(<jats:italic>Ω</jats:italic><jats:sup>+</jats:sup><jats:italic><jats:sub>t</jats:sub></jats:italic>U<jats:italic>Ω</jats:italic><jats:sup>-</jats:sup><jats:italic><jats:sub>t</jats:sub></jats:italic>) is a thin strip and is contained in either a<jats:italic>O</jats:italic>(<jats:italic>ϵ</jats:italic>|ln<jats:italic>ϵ</jats:italic>|) or<jats:italic>O</jats:italic>(<jats:italic>ϵ</jats:italic>) neighbourhood of a hypersurface<jats:italic>Γ<jats:sub>t</jats:sub></jats:italic>which moves with normal velocity equal to its mean curvature. We also study the asymptotic behaviour, as<jats:italic>t</jats:italic>→ ∞, of the solution in the one-dimensional case. In particular, we prove that the<jats:italic>ω</jats:italic>-limit set consists of a singleton.</jats:p>