A uniqueness result for a semilinear reaction-diffusion system
Authors: Escobedo M. , Herrero M. A. 
Published: June 2010

Let

(
u
(
t
,
x
)
,
v
(
t
,
x
)
)

(u(t,x),v (t,x))

and

(

u
¯

(
t
,
x
)
,

v
¯

(
t
,
x
)
)

(\bar u(t,x),\bar v (t,x))

be two nonnegative classical solutions of (S)
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row 1st Column u Subscript t Baseline equals normal upper Delta u plus v Superscript p Baseline comma 2nd Column a m p semicolon p greater-than 0 2nd Row 1st Column v Subscript t Baseline equals normal upper Delta v plus u Superscript q Baseline comma 2nd Column a m p semicolon q greater-than 0 EndLayout">

{

u
t

=
Δ
u
+

v
p

,

a
m
p
;

p
>
0

v
t

=
Δ
v
+

u
q

,

a
m
p
;

q
>
0

\left \{ {\begin {array}{*{20}{c}} {{u_t} = \Delta u + {v ^p},} & {p > 0} \\ {{v _t} = \Delta v + {u^q},} & {q > 0} \\ \end {array} } \right .

\]
in some strip

S
T

=
(
0
,
T
)
×

R

N

{S_T} = (0,T) \times {\mathbb {R}^N}

, where

0
>
T
≤
∞

0 > T \leq \infty

, and suppose that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u left-parenthesis 0 comma x right-parenthesis equals ModifyingAbove u With bar left-parenthesis 0 comma x right-parenthesis comma v left-parenthesis 0 comma x right-parenthesis equals ModifyingAbove v With bar left-parenthesis 0 comma x right-parenthesis comma">

u
(
0
,
x
)
=

u
¯

(
0
,
x
)
,

v
(
0
,
x
)
=

v
¯

(
0
,
x
)
,

u(0,x) = \bar u(0,x),\quad v (0,x) = \bar v (0,x),

\]
where

u
(
0
,
x
)

u(0,x)

and

v
(
0
,
x
)

v (0,x)

are continuous, nonnegative, and bounded real functions, one of which is not identically zero. Then one has
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u left-parenthesis t comma x right-parenthesis equals ModifyingAbove u With bar left-parenthesis t comma x right-parenthesis comma v left-parenthesis t comma x right-parenthesis equals ModifyingAbove v With bar left-parenthesis t comma x right-parenthesis in upper S Subscript upper T Baseline period">

u
(
t
,
x
)
=

u
¯

(
t
,
x
)
,

v
(
t
,
x
)
=

v
¯

(
t
,
x
)

in

S
T

.

u(t,x) = \bar u(t,x),\quad v (t,x) = \bar v (t,x)\quad {\text {in}}\;{S_T}.

\]
If

p
q
≥
1

pq \geq 1

, the result is also true if

u
(
0
,
x
)
=
v
(
0
,
x
)
=
0

u(0,x) = v (0,x) = 0

. On the other hand, when

0
>
p
q
>
1

0 > pq > 1

, the set of solutions of (S) with zero initial values is given by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u left-parenthesis t semicolon s right-parenthesis equals c 1 left-parenthesis t minus s right-parenthesis Subscript plus Superscript left-parenthesis p plus 1 right-parenthesis slash left-parenthesis 1 minus p q right-parenthesis Baseline comma v left-parenthesis t semicolon s right-parenthesis equals c 2 left-parenthesis t minus s right-parenthesis Subscript plus Superscript left-parenthesis q plus 1 right-parenthesis slash left-parenthesis 1 minus p q right-parenthesis Baseline comma">

u
(
t
;
s
)
=

c
1

(
t
−
s

)
+

(
p
+
1
)

/

(
1
−
p
q
)

,

v
(
t
;
s
)
=

c
2

(
t
−
s

)
+

(
q
+
1
)

/

(
1
−
p
q
)

,

u(t;s) = {c_1}(t - s)_ + ^{(p + 1)/(1 - pq)},\quad v (t;s) = {c_2}(t - s)_ + ^{(q + 1)/(1 - pq)},

\]
where

0
≤
s
≤
t
,

c
1

0 \leq s \leq t,{c_1}

and

c
2

{c_2}

are two positive constants depending only on

p
p

and

q
q

, and

(
ξ

)
+

=
max
{
ξ
,
0
}

(\xi )_+ = \max \{ \xi , 0 \}

.