In this paper the existence of global solutions of the parabolic crossdiffusion systems with cooperative reactions is obtained under certain conditions. The uniform boundedness of
W
_{1,2}
norms of the local maximal solution is obtained by using interpolation inequalities and comparison results on differential inequalities.
1. INTRODUCTION
This article deals with the following quasilinear parabolic system in population dynamics which is called cooperative crossdiffusion system.
where
α
_{12}
,
α
_{21}
,
d, a_{i}, b_{i}, c_{i}
are positive constants for
i
= 1, 2. The initial functions
u
_{0}
,
v
_{0}
are not constantly zero. In the system (1.1)
u
and
v
are nonnegative functions which represent the population densities of two species in a cooperative relationship.
d
_{1}
and
d
_{2}
are the
diffusion
rates of the two species, respectively.
a
_{1}
and
a
_{2}
denote the intrinsic growth rates,
b
_{1}
and
c
_{2}
account for intraspecific cooperative pressures,
b
_{2}
and
c
_{1}
are the coefficients for interspecific competitions.
α
_{11}
and
α
_{22}
are usually referred as
selfdiffusion
, and
α
_{12}
,
α
_{21}
are
crossdiffusion
pressures. By adopting the coefficients
α_{ij}
(
i
,
j
= 1, 2) the system (1.1) takes into account the pressures created by mutually interacting species. For more details on the backgrounds of this model, the readers are refered to Okubo and Levin
[7]
.
Pao
[8]
in 2005, and Delgado et al.
[4]
in 2008 have obtained some results on the existence of global solutions of the elliptic crossdiffusion systems with cooperative reactions. In this paper the existence of global solutions of the parabolic crossdiffusion systems with cooperative reactions is obtained under certain conditions. To state results on the system (1.1) we use the following notation throughout this paper.
Notations.
Let Ω be a region in
. The norm in
L_{p}
(Ω) is denoted by ·
_{Lp(Ω)}
, 1 ≤
p
≤ ∞, where 
f

_{Lp(Ω)}
= (
∫
_{Ω}

f
(
x
)
^{p}
dx
)
^{1/p}
, if 1 ≤
p
≤ ∞, and 
f

_{L∞(Ω)}
= sup {
f
(
x
) :
x
∈ Ω}. The usual Sobolev spaces of real valued functions in Ω with exponent
k
≥ 0 are denoted by
, 1 ≤
p
≤ ∞. And
represents the norm in the Sobolev space
.
we shall use the simplified notation ·
_{k,p}
for
and ·
_{p}
for ·
_{Lp(Ω)}
.
The local existence of solutions to (1.1) was established by Amann
[1]
,
[2]
,
[3]
. According to his results the system (1.1) has a unique nonnegative solution
u
(·,
t
),
v
(·,
t
) in
, where
T
∈ (0,∞] is the maximal existence time for the solution
u, v
. The following result is also due to Amann
[2]
.
Theorem 1.1.
Let
u
_{0}
and
v
_{0}
be in
.
The system
(1.1)
possesses a unique nonnegative maximal smooth solution
for
0 ≤ t <
T
, where
p
>
n and
0 <
T
≤ ∞.
If the solution satisfies the estimates
then T
= +∞.
If, in addition, u
_{0}
and
v
_{0}
are in
then
,
and
Here we state the main results of this paper. Throughout this this paper we assume the condition
which means the interspecific competition pressures are greater than the intraspecific cooperative pressures.
Theorem 1.2.
Suppose that the initial functions
u
_{0}
,
v
_{0}
are in
.
Also assume the condition
(1.2).
Let
(
u
(
x,t
),
v
(
x,t
))
be the maximal solution to the system
(1.1)
as in Theorem
1.1
Then there exist positive constant

M0=M0(u01, v01,a1,a2,b1,b2,c1,c2)
such that

sup{u(·,t)1, v(·,t)1:t∈ [0,T)} ≤M0
For the boundedness results of
L
_{2}
and
W
_{1,2}
norms of the maximal solution to the system (1.1) we assume the following condition in Theorem 1.3, Theorem 1.4
Theorem 1.3.
Suppose that the initial functions
u
_{0}
,
v
_{0}
are in
.
Also assume the condition
(1.2)
and
(1.3).
Let
(
u
(
x
,
t
),
v
(
x
,
t
))
be the maximal solution to the system
(1.1)
as in Theorem
1.1.
Then there exist a positive constant
M
_{1}
=
M
_{1}
(
u
_{0}

_{1}
, 
v
_{0}

_{1}
,
d_{i}
,
a_{i}
,
b_{i}
,
c_{i}
,
i
= 1, 2)
such that

sup{u(·,t)2, v(·,t)2:t∈ [0,T)} ≤M1.
Theorem 1.4.
Suppose that the initial functions
u
_{0}
,
v
_{0}
are in
.
Also assume the condition
(1.2)
and
(1.3).
Let
(
u
(
x
,
t
),
v
(
x
,
t
))
be the maximal solution to the system
(1.1)
as in Theorem
1.1.
Then there exist a positive constant
M
_{2}
=
M
_{2}
(
u
_{0}

_{1}
, 
v
_{0}

_{1}
,
d_{i}
,
α_{ij}
,
a_{i}
,
b_{i}
,
c_{i}
,
i
= 1, 2)
such that

sup{u(·,t)1,2, v(·,t)1,2:t∈ [0,T)} ≤M2.
From the results of Theorems 1.2, 1.3 and 1.4 and the Sobolev embedding inequality we have positive constants
M′ = M′
(
d_{i}
,
α_{ij}
,
a_{i}
,
b_{i}
,
c_{i}
,
i
= 1,2)
M = M
(
d_{i}
,
α_{ij}
,
a_{i}
,
b_{i}
,
c_{i}
,
i
= 1,2) such that for the maximal solution (
u, v
) of (1.1) with the conditions (1.2), (1.3)
We also conclude that
T
= + ∞ from Theorem 1.1.
This paper is organized as follows. Section 2 provides preliminaries on differential equations and a few consequences of GagliardoNirenberg interpolation inequality which are necessary for the proofs of Theorems 1.2, 1.3, and 1.4. And Sections 3, 4, and 5 present the proofs of Theorems 1.2, 1.3, and 1.4, respectively.
2. PRELIMINARIES
This section introduce the GagliardoNirenberg interpolation inequality and its consequences. Also some preliminary results on the bounds and comparisons of differential equations and inequalities are provided.
Theorem 2.1
(GagliardoNirenberg interpolation inequality).
Let
be a bounded domain with
∂
Ω
in C^{m}. For every function u in W^{m,r}
(Ω), 1 ≤
q
,
r
≤ ∞
the derivative D^{j}u
, 0 ≤
j
<
m
,
satisfies the inequality
where
for all a in the interval
,
provided one of the following three conditions
:

(i)r≤q,

(ii),or

(iii)andis not a nonnegative integer.
(
The positive constant C depends only on n, m, j, q, r, a.
)
Proof.
We refer the reader to A. Friedman
[5]
or L. Nirenberg
[6]
for the proof of this wellknown calculus inequality.
Corollary 2.1.
There exist positive constants C,
,
and
such that for every function u in
Proof. n
= 1,
m
= 1,
r
= 2,
q
= 1 satisfy the condition (ii) in Theorem 2.1. Letting
j
= 0 in this case the necessary condition on
p, a
for inequality (2.1) becomes
From equation (2.5) if
p
= 4, then
if
then
then if
p
= 2, then
a
= 13. Therefore we have inequalities (2.2), (2.3), (2.4).
Corollary 2.2.
For every function u in
Proof. m
= 2,
r
= 2,
q
= 1 satisfy the condition (ii) in Theorem 2.1.
Theorem 2.2
(Young’s Inequality).
If a and b are nonnegative real numbers and p and q are positive real numbers such that
then
The equality hold if and only if a^{p} = b^{q}
.
Theorem 2.3
(Hölder’s Inequality).
If
are Lebesgue measurable and p, q
∈ [1,∞]
are real numbers such that
then
Lemma 2.1 below presents a few basic inequalities that will be used for the computations in this paper.
Lemma 2.1.
Let x
≥ 0,
y
≥ 0.
Then
Proof
. Inequalities (2.7), (2.8), (2.9) are simply proved.
To show inequalities (2.10), (2.11), let
g
(
x
) = 2
^{k−1}
(
x^{k}
+
y^{k}
) − (
x
+
y
)
^{k}
. Then

g′(x) =k2k−1xk−1−k(x+y)k−1.
Hence the function
g
(
x
) has the critical value 0 at
x = y
which is the minimum value if
k
≥ 1, and the maximum value if 0 <
k
≤ 1. Thus we obtain inequalities (2.10) and (2.11).
Theorem 2.4
(Picard’s local existence and uniqueness theorem).
If f
(
x, t
)
is a continuous realvalued function that satisfies the Lipschitz condition

f(x, t) −f(y, t) ≤Lx−y
in some open rectangle R
= {(
x, t
) 
a
<
x
<
b, c
<
t
<
d
}
that contains the point
(
x
_{0}
,
t
_{0}
),
then the initial value problem
has a unique solution in some closed interval I
= [
t
_{0}
−
ε
,
t
_{0}
+
ε
]
where ε
> 0.
Theorem 2.5.
Let f
(
x
)
be a realvalued differentiable functions defined on an open interval
(
a
,
b
).
Then for every initial point
x
_{0}
in
(
a
,
b
)
a solution of the initial value problem
is either constant or strictly monotone.
Proof
. The conclusion follow from the fact that
f
(
x
(
t
)) never changes sign for the solution
x
(
t
) of the given initial value problem. To see why this is so, suppose that
x
(
t
) is not a constant solution, and
f
(
x
(
t
)) changes sign. Then it would have to be
f
(
x
(
t
_{1}
)) = 0 at some
t
_{1}
> 0 and
f
(
x
(
t
)) ≠ 0 for
t
in the left of
t
_{1}
or right of
t
_{1}
. But it contradict the fact that from Theorem 2.4 the constant solution
y
(
t
) ≡
x
(
t
_{1}
) is a unique solution in some closed interval [
t
_{1}
−
ε
,
t
_{1}
+
ε
], where
ε
> 0.
Corollary 2.3.
Let
c
_{1}
> 0,
p
> 1,
and
c
_{2}
,
c
_{3}
be any real numbers. Then there exists a positive constant M = M
(
x
_{0}
,
p
,
c
_{1}
,
c
_{2}
,
c
_{3}
)
such that the solution of the initial value problem

x′= −c1xp+c2x+c3,x(0) =x0≥ 0
satisfies that
Proof.
The function
f
(
x
) = −
c
_{1}
x^{p}
+
c
_{2}
x
+
c
_{3}
is differentiable functions on
and falls in either of the two cases:

case(a)f(x) ≤ 0 for allx≥ 0

case(b) there exist a positive constantm = m(p,c1,c2,c3) such thatf(m) = 0,f(x) > 0 forxin some interval on the left ofm, andf(x) < 0 for allx>M.
In case (a)
x′
(0) =
f
(
x
_{0}
) ≤ 0, and thus by Theorem 2.5
x
′(
t
) ≤ 0 for all
t
≥ 0. Hence
x
(
t
) ≤
x
_{0}
for all
t
≥ 0. In case (b) if 0 <
x
_{0}
<
m
then the solution
x
(
t
) cannot cross the constant solution
y
(
t
) ≤
m
by Theorem 2.5, and thus
x
(
t
) ≤
m
for all
t
≥ 0. If
x
_{0}
≥
m
then
x′
(0) =
f
(
x
_{0}
) ≤ 0, and thus by Theorem 2.5
x
′(
t
) ≤ 0 for all
t
≥ 0. Hence
x
(
t
) ≤
x
_{0}
for all
t
≥ 0. Therefore in any case there exists a positive constant
M = M
(
x
_{0}
,
p
,
c
_{1}
,
c
_{2}
,
c
_{3}
) such that
x
(
t
) ≤
M
for all
t
≥ 0.
Lemma 2.2
(Gronwall’s inequality and the Comparison Principle for differential equations).
Let a
<
b
≤ ∞,
and ξ
(
t
)
and β
(
t
)
be realvalued continuous functions defined on the interval
[
a, b
].
If ξ
(
t
)
is differentiable in
(
a, b
)
and satisfies the differential inequality

ξ′(t) ≤β(t)ξ(t),t∈ (a,b),
then
ξ
(
t
)
is bounded by the solution of the corresponding differential equation
y
′(
t
) =
β
(
t
)
y
(
t
),
y
(
a
) =
ξ
(
a
),
that is,
for all t
∈ [
a, b
].
And it follows that if in addition
ξ
(
a
) ≤ 0,
then
ξ
(
t
) ≤ 0
for all t
∈ [
a, b
].
Proof.
We refer the reader to
[2]
.
Lemma 2.3.
Let
c
_{1}
> 0,
p
> 1,
and
c
_{2}
,
c
_{3}
be any real numbers. Suppose that two differentiable functions
ϕ
(
t
)
and
x
(
t
)
satisfy

ϕ′ ≤ −c1ϕp+c2ϕ+c3,ϕ(0) =ϕ0

x′ = −c1xp+c2x+c3,x(0) =ϕ0.
Then
And especially, if
ϕ
_{0}
≥ 0
then there exists a positive constant M = M
(
ϕ
_{0}
,
p
,
c
_{1}
,
c
_{2}
,
c
_{3}
)
such that
Proof
. Let
ξ
=
ϕ − x
. Then
ξ
′ =
ϕ
′ −
x
′
≤ −c_{1}(ϕ^{p} − x^{p}) + c_{2}(ϕ − x)
= ξ(−c_{1}η + c_{2}),
where
Here notice that
η
(
t
) is a continuous function using the mean value theorem and the continuities of
ϕ
(
t
) and
x
(
t
). Now, since
ξ
(0) = 0 we conclude that
ξ
(
t
) =
ϕ
(
t
) −
x
(
t
) ≤ 0 for all
t
≥ 0 from Lemma 2.2. And if
ϕ
_{0}
≥ 0, from Corollary 2.3 there exists a positive constant
M = M
(
ϕ
_{0}
,
p
,
c
_{1}
,
c
_{2}
,
c
_{3}
) such that
x
(
t
) ≤
M
for all
t
≥ 0. Thus
ϕ
(
t
) ≤
M
for all
t
≥ 0.
3.L1BOUND OF SOLUTIONS TO (1.1)
Proof of
Theorem 1.2. By taking integration over the interval [0, 1] for the first and second equations in (1.1) we have that
Let
The condition
b
_{1}
c
_{2}
>
b
_{2}
c
_{1}
implies
δ
> 0. It also holds that
Thus it is shown that
from the facts
Using (3.1) we have
and thus
where
From Hölder’s inequality
it follows that
and thus
where
Hence by the Gronwall’s type inequailty in Lemma 2.3 there exists positive constant
M
_{0}
=
M
_{0}
(
u
_{0}

_{1}
, 
v
_{0}

_{1}
,
a
_{1}
,
a
_{2}
,
b
_{1}
,
b
_{2}
,
c
_{1}
,
c
_{2}
) satisfying
for all
t
≥ 0.
4.L2BOUND OF SOLUTIONS TO (1.1)
Proof of
Theorem 1.3. Multiplying the first and second equations in (1.1) by
u
=
u
(
x, t
) and
v
=
v
(
x
,
t
), respectively, and taking integrations over [0, 1] we have that
Using Neumann boundary conditions
and similarly
Using condition (1.3) that
and
, we have
Thus it follows from (4.1) that
By Young’s inequality
holds for any
ε
> 0. And by applying Lemma 2.1 to inequality (2.2) and using the uniform
L
_{1}
boundedness of
u
and
v
from Step 1, we have
where
C
is a positive constant depending only on
a
_{i}
,
b
_{i}
,
c
_{i}
(
i, j
= 1, 2). Thus (4.2) becomes
where
and the constants
and
are depending on
d_{i}
,
a_{i}
,
b_{i}
,
c_{i}
(
i, j
= 1, 2). And by applying Lemma 2.1 to inequality (2.4) and using the uniform
L
_{1}
boundedness of
u
and
v
from Step 1, we have
where
is a positive constant depending only on
a_{i}, b_{i}, c_{i}
(
i, j
= 1, 2). And thus
where
C′
is a positive constant depending only on
a_{i}, b_{i}, c_{i}
(
i, j
= 1, 2). Thus we have
by Lemma 2.1, where
C
_{0}
,
C
_{1}
,
C
_{2}
are positive constants
d_{i}, a_{i}, b_{i}, c_{i}
(
i, j
= 1, 2). Hence by the Gronwall’s type inequailty in Lemma 2.3 we obtain the following
L
_{2}
bound of
u
and
v
such that
where
M
_{1}
is a positive constant depending on 
u
_{0}

_{2}
, 
v
_{0}

_{2}
,
d_{i}, a_{i}, b_{i}, c_{i}
(
i, j
= 1, 2).
5.W1,2BOUND OF SOLUTIONS TO (1.1)
Proof of
Theorem 1.4. To obtain uniform bounds of 
u_{x}

_{2}
and 
v_{x}

_{2}
for the solution of (1.1) let us denote that

P=d1u+α11u2+α12uv,Q=d2v+α21uv+α22v2.
We would show the uniform boundedness of 
P_{x}

_{2}
and 
P_{x}

_{2}
and then obtain the uniform bounds of 
u_{x}

_{2}
and 
v_{x}

_{2}
from it. Here we note from Theorem 1.1 that
for 0 ≤
t
<
T
, and
from the Neumann boundary conditions on
u, v
. Now, multiplying the first equation in (1.1) by
P_{t}
and the second equation by
Q_{t}
, we have
and thus
where
C
_{1}
is a positive constant depending on
d_{i}, α_{ij}, a_{i}, b_{i}, c_{i}
(
i, j
= 1, 2). Here we notice from the condition (1.3) that there exists a positive constant
λ
=
λ
(
α_{i,j}
,
i, j
= 1, 2) satisfying
since
for all
u
≥ 0,
v
≥ 0, if
λ
=
λ
(
α_{ij}
,
i
,
j
= 1, 2) > 0 is small enough.
The terms
in (5.1) ae estimated in terms of
P
and
Q
from inequality (2.7) in lemma 2.1.
Now we observe using Young’s inequality that
hold for any
ε
> 0. Similar estimates are applied to the terms
,
,
,
and so on. Using these inequalities and inequalities (2.8), (2.9) in lemma 2.1 we obtain that
where
C
_{2}
,
C
_{3}
,
C
_{4}
are positive constant depending on
d_{i}, α_{ij}, a_{i}, b_{i}, c_{i}
(
i, j
= 1, 2). Thus we have
for any
ε
> 0. Here we choose a small
ε
> 0 so that
and thus
where
C
_{5}
is a positive constants depending on
d_{i}, α_{ij}, a_{i}, b_{i}, c_{i}
(
i, j
= 1, 2). Now we observe that

P=d1u+α11u2+α12uv≥α11u2,Q=d2v+α21uv+α22v2≥α22v2,
and thus
where
C
_{6}
is a positive constant depending only on
d_{i}, α_{ij} , a_{i}, b_{i}, c_{i}
(
i, j
= 1, 2). Applying the inequalities (2.6) and (2.3) to the function
P
=
d
_{1}
u
+
α
_{11}
u
^{2}
+
α
_{12}
uv
we have
Here using the uniform boundedness of the
L
_{1}
norm of
P
, we have
where
C
_{7}
,
C
_{8}
,
C
_{9}
,
C
_{10}
are positive constants depending on
d_{i}, α_{ij}, a_{i}, b_{i}, c_{i}
(
i, j
= 1, 2). Similar estimates are obtained also for
Q
. Hence we have
where
C
_{11}
,
C
_{12}
,
C
_{13}
are positive constants depending on
d_{i}, α_{ij}, a_{i}, b_{i}, c_{i}
(
i, j
= 1, 2). Hence by the Gronwall’s type inequailty in Lemma 2.3 we obtain the following
W
_{1,2}
bound of
P
and
Q
such that
where
is a positive constant depending on 
u
_{0}

_{2}
, 
v
_{0}

_{2}
,
d_{i}, α_{ij}, a_{i}, b_{i}, c_{i}
(
i, j
= 1, 2).
In order to obtain estimates for
u_{x}
and
v_{x}
, we notice that
where
Here we note that 
A
, the determinant of
A
, is bounded below by the positive constant
d
_{1}
d
_{2}
, and 
A
 is of class
O
(
u
^{2}
+
v
^{2}
) as
u
→ ∞ and
v
→ ∞, we have the inequality

ux + vx ≤C14(Px + Qx) for everyx∈ [0, 1] × [0,∞)
for some constant
C
_{14}
depending only on
d_{i}, α_{ij}
, (
i, j
= 1, 2). Therefore we obtain the following
W
_{1,2}
bound of
u
and
v
such that
where
M
_{2}
is a positive constant depending on 
u
_{0}

_{2}
, 
v
_{0}

_{2}
,
d_{i}, α_{ij}, a_{i}, b_{i}, c_{i}
(
i, j
= 1, 2).
Amann H.
1989
Dynamic theory of quasilinear parabolic equations, III. Global Existence
Math Z.
202
219 
250
DOI : 10.1007/BF01215256
Amann H.
1990
Dynamic theory of quasilinear parabolic equations, II. Reactiondiffusion systems
Differential and Integral Equations
3
(1)
13 
75
Amann H.
1993
Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems
Function spaces, differential operators and nonlinear analysis (1992), TeubnerTexte Math.
Friedrichroda
133
9 
126
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