We consider a parabolic double obstacle problem which is a version of the Allen-Cahn equation
u
t
= Δ
u
—
ϵ
-2
ψ
'(
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 —
u
2
) 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
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
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.
Asymptotic behaviour and propagation properties of the one-dimensional flow of gas in a porous medium
