Generation and propagation of interfaces in reaction-diffusion systems
Authors: Chen Xinfu 
Published: June 2012

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:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row u Subscript t Superscript epsilon Baseline minus epsilon normal upper Delta u Superscript epsilon Baseline equals StartFraction 1 Over epsilon EndFraction f left-parenthesis u Superscript epsilon Baseline comma upsilon Superscript epsilon Baseline right-parenthesis identical-to StartFraction 1 Over epsilon EndFraction left-bracket u Superscript epsilon Baseline left-parenthesis 1 minus u Superscript epsilon Baseline 2 Baseline right-parenthesis minus upsilon Superscript epsilon Baseline right-bracket comma 2nd Row upsilon Subscript t Superscript epsilon Baseline minus normal upper Delta upsilon Superscript epsilon Baseline equals u Superscript epsilon Baseline minus gamma upsilon Superscript epsilon EndLayout">

{

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

.