This paper is concerned with the asymptotic behavior, as
ε
↘
0
\varepsilon \searrow 0
, of the solution
(
u
ε
,
v
ε
)
({u^\varepsilon },{v^\varepsilon })
of the second initial-boundary value problem of the reaction-diffusion system:
\[
{
u
t
ε
−
ε
Δ
u
ε
=
1
ε
f
(
u
ε
,
υ
ε
)
≡
1
ε
[
u
ε
(
1
−
u
ε
2
)
−
υ
ε
]
,
υ
t
ε
−
Δ
υ
ε
=
u
ε
−
γ
υ
ε
\left \{ {\begin {array}{*{20}{c}} {u_t^\varepsilon - \varepsilon \Delta {u^\varepsilon } = \frac {1}{\varepsilon }f({u^\varepsilon },{\upsilon ^\varepsilon }) \equiv \frac {1}{\varepsilon }[{u^\varepsilon }(1 - {u^{\varepsilon 2}}) - {\upsilon ^\varepsilon }],} \hfill \\ {\upsilon _t^\varepsilon - \Delta {\upsilon ^\varepsilon } = {u^\varepsilon } - \gamma {\upsilon ^\varepsilon }} \hfill \\ \end {array} } \right .
\]
where
γ
>
0
\gamma > 0
is a constant. When
v
∈
(
−
2
3
/
9
,
2
3
/
9
)
v \in ( - 2\sqrt 3 /9,2\sqrt 3 /9)
,
f
f
is bistable in the sense that the ordinary differential equation
u
t
=
f
(
u
,
v
)
{u_t} = f(u,v)
has two stable solutions
u
=
h
−
(
v
)
u = {h_ - }(v)
and
u
=
h
+
(
v
)
u = {h_ + }(v)
and one unstable solution
u
=
h
0
(
v
)
u = {h_0}(v)
, where
h
−
(
v
)
,
h
0
(
v
)
{h_ - }(v), {h_0}(v)
, and
h
+
(
v
)
{h_ + }(v)
are the three solutions of the algebraic equation
f
(
u
,
v
)
=
0
f(u,v) = 0
. We show that, when the initial data of
v
v
is in the interval
(
−
2
3
/
9
,
2
3
/
9
)
( - 2\sqrt 3 /9,2\sqrt 3 /9)
, the solution
(
u
ε
,
v
ε
)
({u^\varepsilon },{v^\varepsilon })
of the system tends to a limit
(
u
,
v
)
(u,v)
which is a solution of a free boundary problem, as long as the free boundary problem has a unique classical solution. The function
u
u
is a "phase" function in the sense that it coincides with
h
+
(
v
)
{h_ + }(v)
in one region
Ω
+
{\Omega _ + }
and with
h
−
(
v
)
{h_ - }(v)
in another region
Ω
−
{\Omega _ - }
. The common boundary (free boundary or interface) of the two regions
Ω
−
{\Omega _ - }
and
Ω
+
{\Omega _ + }
moves with a normal velocity equal to
V
(
v
)
\mathcal {V}(v)
, where
V
(
∙
)
\mathcal {V}( \bullet )
is a function that can be calculated. The local (in time) existence of a unique classical solution to the free boundary problem is also established. Further we show that if initially
u
(
∙
,
0
)
−
h
0
(
v
(
∙
,
0
)
)
u( \bullet , 0) - {h_0}(v( \bullet , 0))
takes both positive and negative values, then an interface will develop in a short time
O
(
ε
|
ln
ε
|
)
O(\varepsilon |\ln \varepsilon |)
near the hypersurface where
u
(
x
,
0
)
−
h
0
(
v
(
x
,
0
)
)
=
0
u(x,0) - {h_0}(v(x,0)) = 0
.