# Global Solutions in the Species Competitive Chemotaxis System with Inequal Diffusion Rates

Authors: Huaihuo Cao
Journal: Discrete Dynamics in Nature and Society
 (2016)

---

## Abstract

This paper is devoted to studying the two-species competitive chemotaxis system with signal-dependent chemotactic sensitivities and inequal diffusion ratesu
1
t
=
Δ
u
1
-
∇
·
u
1
χ
1
v
∇
v
+
μ
1
u
1
1
-
u
1
-
a
1
u
2, x
∈
Ω
,
t
>
0, u
2
t
=
Δ
u
2
-
∇
·
u
2
χ
2
v
∇
v
+
μ
2
u
2
1
-
a
2
u
1
-
u
2, x
∈
Ω
,
t
>
0, v
t
=
τ
Δ
v
-
γ
v
+
u
1
+
u
2, x
∈
Ω
,
t
>
0, under homogeneous Neumann boundary conditions in a bounded and regular domain Ω
⊂
R
n
(
n
≥
1
). If the nonnegative initial date (
u
10
,
u
20
,
v
0
)
∈
(
C
1
(
Ω
¯
)
)
3 and v
0
∈
(
v
_
,
v
¯
) where the constants v
¯
>
v
_
≥
0, the system possesses a unique global solution that is uniformly bounded under some suitable assumptions on the chemotaxis sensitivity functions χ
1
(
v
), χ
2
(
v
) and linear chemical production function -
γ
v
+
u
1
+
u
2.


---

## Body

## 1. Introduction

In this paper, we consider the following two-species competitive chemotaxis system with signal-dependent chemotactic sensitivities and inequal diffusion rates:(1)
u
1
t
=
Δ
u
1
-
∇
·
u
1
χ
1
v
∇
v
+
μ
1
u
1
1
-
u
1
-
a
1
u
2
,
x
∈
Ω
,
t
>
0
,
u
2
t
=
Δ
u
2
-
∇
·
u
2
χ
2
v
∇
v
+
μ
2
u
2
1
-
a
2
u
1
-
u
2
,
x
∈
Ω
,
t
>
0
,
v
t
=
τ
Δ
v
-
γ
v
+
u
1
+
u
2
,
x
∈
Ω
,
t
>
0
,
∂
u
1
∂
ν
=
∂
u
2
∂
ν
=
∂
v
∂
ν
=
0
,
x
∈
∂
Ω
,
t
>
0
,
u
1
x
,
0
=
u
10
x
,
u
2
x
,
0
=
u
20
x
,
v
x
,
0
=
v
0
x
,
x
∈
Ω
,where Ω is a bounded and regular domain in R
n
(
n
≥
1
) and ν is the outward unit normal vector of the boundary ∂
Ω. u
1
(
x
,
t
) and u
2
(
x
,
t
) represent the populations densities, and both populations reproduce themselves and mutually compete with the other, according to the classical Lotka-Volterra kinetics [1], and the diffusion rates of the populations are 1. v
(
x
,
t
) denotes the concentration of the chemoattractant, and the diffusion rate of the chemical substance is strictly less than 1 (i.e., 0
<
τ
<
1). μ
1, μ
2, a
1, a
2, and γ are positive parameters, where μ
1, μ
2 are the growth coefficients and a
1, a
2 are the competitive degradation rates of population, respectively. The chemotactic sensitivity function χ
i
(
v
)
∈
W
loc
⁡
1
,
∞
(
R
+
)
∩
C
1
(
R
+
) for i
=
1,2, which is assumed to be positive. From a biological point of view, when χ
i
(
v
)
>
0, populations exhibit a tendency to move towards higher signal concentrations (chemoattraction), while conversely the choice χ
i
(
v
)
<
0 leads to a model for chemorepulsion, where populations prefer to move away from the chemical in question [2]. Denote h
(
u
1
,
u
2
,
v
)
=
-
γ
v
+
u
1
+
u
2 representing the balance between the production of the chemical substance by the populations themselves and its natural degradation (see [3] for details).The classical chemotaxis model was first introduced by Keller and Segel using a mathematical model of two parabolic equations to describe the aggregation ofDictyostelium discoideum as well as a soil-living amoeba, in the early 1970s [4]. After the pioneering works of Keller and Segel, a large amount of chemotaxis models has been used to model the phenomena for population dynamics or gravitational collapse, among others. Winkler [5] studied the chemotaxis system(2)
u
t
=
Δ
u
-
∇
·
u
χ
v
∇
v
,
x
∈
Ω
,
t
>
0
,
v
t
=
Δ
v
-
v
+
u
,
x
∈
Ω
,
t
>
0
,under homogeneous Neumann boundary conditions in a smooth bounded domain Ω
⊂
R
n. It was proved that the chemotactic collapse was absented for any nonnegative initial date u
(
·
,
0
)
∈
C
0
(
Ω
¯
) and v
(
·
,
0
)
∈
W
1
,
r
(
Ω
) with some r
>
n; the corresponding initial-boundary value problem possessed a unique global uniformly bounded solution. Tello and Winkler [6] considered the parabolic-parabolic-elliptic system(3)
u
t
=
Δ
u
-
χ
1
∇
·
u
∇
w
+
μ
1
u
1
-
u
-
a
1
v
,
x
∈
Ω
,
t
>
0
,
v
t
=
Δ
v
-
χ
2
∇
·
v
∇
w
+
μ
2
v
1
-
a
2
u
-
v
,
x
∈
Ω
,
t
>
0
,
-
Δ
w
=
-
λ
w
+
u
+
v
,
x
∈
Ω
,
t
>
0under homogeneous Neumann boundary conditions in a bounded domain Ω
⊂
R
n
(
n
≥
1
) with smooth boundary. Given the suitable positive parameters a
1, a
2 and χ
1, χ
2, they showed that all solutions stabilized towards a uniquely determined spatially homogeneous positive steady state within a certain nonempty range of the logistic growth coefficients μ
1 and μ
2. Negreanu and Tello [7] studied a two-species chemotaxis system with nondiffusive chemoattractant:(4)
u
t
=
Δ
u
-
∇
·
u
χ
1
w
∇
w
+
μ
1
u
1
-
u
,
x
∈
Ω
,
t
>
0
,
v
t
=
Δ
v
-
∇
·
v
χ
2
w
∇
w
+
μ
2
v
1
-
v
,
x
∈
Ω
,
t
>
0
,
w
t
=
h
u
,
v
,
w
,
x
∈
Ω
,
t
>
0
,under suitable boundary and initial conditions in an n-dimensional open and bounded domain Ω for n
≥
1. They considered the case of positive chemosensitivities and chemical production function h increasing as the concentration of the species u, v increasing. The paper proved the global existence and uniform boundedness of solutions, and the asymptotic stability of the spatially homogeneous steady state was a consequence of the growth of h
,
χ
i and the size of μ
i for i
=
1,2.Reviewing the recent studies, Zhang and Li [8] considered the following fully parabolic system:(5)
u
t
=
Δ
u
-
∇
·
u
χ
1
w
∇
w
+
μ
1
u
1
-
u
-
a
1
v
,
x
∈
Ω
,
t
>
0
,
v
t
=
Δ
v
-
∇
·
v
χ
2
w
∇
w
+
μ
2
v
1
-
a
2
u
-
v
,
x
∈
Ω
,
t
>
0
,
w
t
=
Δ
w
-
w
+
u
+
v
,
x
∈
Ω
,
t
>
0
,under homogeneous Neumann boundary conditions in a smooth bounded domain Ω
⊂
R
n
(
n
≥
1
). By extending the method in [5] (see also [9, 10]), the first step is to estimate some associated weighted functions which depend on signal density, and the second step is to obtain L
∞-bounds of solutions from L
p-bounds using the variation-of-constants representation and a series of standard semigroup arguments (see [2, 5, 11, 12]). They proved that, if the nonnegative initial date (
u
(
·
,
0
)
,
v
(
·
,
0
)
)
∈
(
C
0
(
Ω
¯
)
)
2 and w
(
·
,
0
)
∈
W
1
,
r
(
Ω
) for some r
>
n, the system possesses a unique global solution that is uniformly bounded under some appropriate conditions on the coefficients μ
1, μ
2 and the chemotaxis sensitivity functions χ
1
(
w
), χ
2
(
w
).Inspired by the foregoing research, the main purpose of this paper is to consider the existence of global solution for the two-species competitive chemotaxis model (1) with inequal diffusion rates. This paper is organized as follows: In Section 2, we formulate the main results of this paper by means of the theorem and establish some preliminaries which are important for our proofs. In Section 3, we firstly consider the local existence of solutions and then proceed with the extensibility criterion. Finally, under some appropriate conditions, we prove that the solutions are uniformly bounded in time using an iterative method.


## 2. Preliminaries and Main Results

For convenience, we denote thatg
1
(
u
1
,
u
2
)
=
1
-
u
1
-
a
1
u
2, g
2
(
u
1
,
u
2
)
=
1
-
a
2
u
1
-
u
2, and they are quasi-monotonic decrease in R
+
2. Denote h
(
u
1
,
u
2
,
v
)
=
-
γ
v
+
u
1
+
u
2.In order to establish the global existence and uniform boundedness of solutions to (1), we need to make some restrictive conditions throughout this paper in h
(
u
1
,
u
2
,
v
) and χ
i
(
v
)
(
i
=
1,2
):(i) Let the initial datev
0 satisfy 0
≤
v
_
<
v
0
<
v
¯, where v
_ and v
¯ are some positive constants.(ii) There exist positive constantsκ
1 and κ
2 such that (6)
γ
v
χ
i
v
≤
κ
i
for
v
_
≤
v
≤
v
¯
,
i
=
1,2
.(iii) There exist positive constantsκ
01 and κ
02 such that (7)
0
<
κ
0
i
≤
χ
i
v
e
x
p
∫
v
_
v
χ
i
s
d
s
for
v
_
≤
v
,
i
=
1,2
.(iv) Assume that(8)
h
u
1
,
u
2
,
v
_
≥
0
,
h
u
1
,
u
2
,
v
¯
<
0
,
f
o
r
0
≤
u
1
≤
u
¯
1
,
0
≤
u
2
≤
u
¯
2
,where(9)
u
¯
1
≔
f
1
v
¯
max
⁡
κ
1
1
-
τ
+
μ
1
κ
01
+
μ
1
-
1
,
u
10
L
∞
Ω
,
u
¯
2
≔
f
2
v
¯
max
⁡
κ
2
1
-
τ
+
μ
2
κ
02
+
μ
2
-
1
,
u
20
L
∞
Ω
,for f
i
(
v
¯
) defined by(10)
f
i
v
¯
=
e
x
p
1
1
-
τ
∫
v
_
v
¯
χ
i
s
d
s
,
i
=
1,2
.(v) Finally, for some technical reasons, we also assume that(11)
χ
i
2
v
+
1
-
τ
χ
i
′
v
≤
0
,
i
=
1,2
.We illustrate the validity to above assumptions with the following generalized example [7].Example 1.
We take the chemosensitivity functionsχ
i
(
v
)
=
α
i
/
(
β
i
+
v
)
(
i
=
1,2
) for positive constants α
i
,
β
i fulfilling(12)
α
i
≤
1
-
τ
,
4
γ
<
β
i
≤
8
γ
,
i
=
1,2
.Clearly, (11) holds. Take a lower bound v
_
=
0 and upper bound v
¯ is to be defined later. Moreover, the initial dates u
10 and u
20 are satisfied as(13)
u
10
L
∞
Ω
≤
κ
1
1
-
τ
+
μ
1
κ
01
+
μ
1
-
1
,
u
20
L
∞
Ω
≤
κ
2
1
-
τ
+
μ
2
κ
02
+
μ
2
-
1
.Then, consider the following.(1)
Condition (6) is equivalent to γ
α
i
(
v
/
(
β
i
+
v
)
)
≤
κ
i, so choosing κ
i
=
γ
α
i
(
v
¯
/
(
β
i
+
v
¯
)
)
(
i
=
1,2
)
.
(2)
Taking positive constantsκ
0
i
=
α
i
β
i
-
α
i
/
(
β
i
+
v
¯
)
1
-
α
i
(
i
=
1,2
), then condition (7) holds since(14)
κ
0
i
≤
α
i
β
i
+
v
e
α
i
∫
0
v
1
/
β
i
+
s
d
s
=
α
i
β
i
+
v
1
-
α
i
1
β
i
α
i
fori
=
1,2
.
(3)
We notice thath
(
0,0
,
0
)
=
0 as well as(15)
h
u
¯
1
,
u
¯
2
,
v
¯
=
u
¯
1
+
u
¯
2
-
γ
v
¯
=
e
α
1
/
1
-
τ
∫
0
v
¯
1
/
β
1
+
s
d
s
κ
1
/
1
-
τ
+
μ
1
κ
01
+
μ
1
+
e
α
2
/
1
-
τ
∫
0
v
¯
1
/
β
2
+
s
d
s
κ
2
/
1
-
τ
+
μ
2
κ
02
+
μ
2
-
γ
v
¯
.
A sufficient condition for the second inequality in (8) holding is(16)
e
α
1
/
1
-
τ
∫
0
v
¯
1
/
β
1
+
s
d
s
κ
1
/
1
-
τ
+
μ
1
κ
01
+
μ
1
<
γ
2
v
¯
,
e
α
2
/
1
-
τ
∫
0
v
¯
1
/
β
2
+
s
d
s
κ
2
/
1
-
τ
+
μ
2
κ
02
+
μ
2
<
γ
2
v
¯
,
and, for simplicity, let us takeμ
i
=
κ
i
/
(
1
-
τ
)
(
i
=
1,2
) and then derive(17)
v
¯
>
max
⁡
β
1
γ
/
4
β
1
-
1
,
β
2
γ
/
4
β
2
-
1
.Up to now, the above all restrictive conditions are verified, which implies that conditions (6)–(11) are sufficient to ensure the global existence of solutions.Remark 2.
In literature [7], the chemical production function h
(
u
1
,
u
2
,
v
)
=
-
v
+
u
1
+
u
2, and the chemotactic sensitivity χ
i
(
v
)
=
γ
i
/
(
1
+
γ
i
v
)
(
i
=
1,2
).In this paper, the purpose is to study the global existence and uniform boundedness of solutions to (1) applying an iterative method. The main results are stated by the following theorem.Theorem 3.
Under assumptions (6)–(11), for any initial date (
u
10
,
u
20
,
v
0
)
∈
(
C
1
(
Ω
¯
)
)
3 satisfying the homogeneous Neumann boundary conditions and u
10
≥
0, u
20
≥
0, there exists a unique solution to (1):(18)
u
1
,
u
2
,
v
∈
L
p
0
,
∞
;
W
2
,
p
Ω
∩
W
1
,
p
0
,
∞
;
L
p
Ω
3for any p
>
n. Moreover the solution is uniformly bounded; that is,(19)
u
1
L
∞
Ω
+
u
2
L
∞
Ω
+
v
L
∞
Ω
≤
C
,where C
(<∞) is positive constant.The proof of Theorem3 is split into several steps. Step one: we start to consider the local existence of solutions and then proceed with the extensibility criterion. Step two: under some appropriate conditions, we prove that the solutions are uniformly bounded in time.


## 3. Global Existence of Solutions

We will first be devoted to dealing with local-in-time existence and uniqueness of a nonnegative solution for (1). The corresponding conclusions are written by Lemma 4.


### 3.1. Local Existence of Solutions

Lemma 4.
Given positive initial date(
u
10
,
u
20
,
v
0
)
∈
(
C
1
(
Ω
¯
)
)
3 and the same assumptions as in Theorem 3, there exists a small enough time T
>
0 and a unique triple of nonnegative functions (
u
1
,
u
2
,
v
)
∈
(
L
p
(
[
0
,
T
]
;
W
2
,
p
(
Ω
)
)
∩
W
1
,
p
(
[
0
,
T
]
;
L
p
(
Ω
)
)
)
3
(
for
p
>
n
) such that (
u
1
,
u
2
,
v
) is a solution of (1) in Ω
×
(
0
,
T
). Moreover, we have v
≥
v
_.Proof.
Introduce the change of variablesu
~
1 and u
~
2 given by(20)
u
1
=
f
1
v
u
~
1
,
u
2
=
f
2
v
u
~
2
,where the function f
i
(
v
) is defined by(21)
f
i
v
=
e
x
p
1
1
-
τ
∫
v
_
v
χ
i
s
d
s
,
i
=
1,2
.A direct calculation yields(22)
u
1
t
=
f
1
′
v
τ
Δ
v
+
h
u
1
,
u
2
,
v
u
~
1
+
f
1
v
u
~
1
t
,
Δ
u
1
=
f
1
′
′
v
u
~
1
∇
v
2
+
2
f
1
′
v
∇
u
~
1
·
∇
v
+
f
1
′
v
u
~
1
Δ
v
+
f
1
v
Δ
u
~
1
,
∇
·
u
1
χ
1
v
∇
v
=
f
1
′
v
u
~
1
χ
1
v
∇
v
2
+
f
1
v
χ
1
v
∇
u
~
1
·
∇
v
+
f
1
v
u
~
1
χ
1
′
v
∇
v
2
+
f
1
v
u
~
1
χ
1
v
Δ
v
;similarly,(23)
u
2
t
=
f
2
′
v
τ
Δ
v
+
h
u
1
,
u
2
,
v
u
~
2
+
f
2
v
u
~
2
t
,
Δ
u
2
=
f
2
′
′
v
u
~
2
∇
v
2
+
2
f
2
′
v
∇
u
~
2
·
∇
v
+
f
2
′
v
u
~
2
Δ
v
+
f
2
v
Δ
u
~
2
,
∇
·
u
2
χ
2
v
∇
v
=
f
2
′
v
u
~
2
χ
2
v
∇
v
2
+
f
2
v
χ
2
v
∇
u
~
2
·
∇
v
+
f
2
v
u
~
2
χ
2
′
v
∇
v
2
+
f
2
v
u
~
2
χ
2
v
Δ
v
,where(24)
f
i
′
v
=
χ
i
v
1
-
τ
f
i
v
,
f
i
′
′
v
=
χ
i
2
v
1
-
τ
2
+
χ
i
′
v
1
-
τ
f
i
v
,
i
=
1,2
.Substitute (20), (22), and (23) into (1) and denote that(25)
L
i
v
u
~
i
=
u
~
i
t
-
Δ
u
~
i
-
1
+
τ
1
-
τ
χ
i
v
∇
v
·
∇
u
~
i
,
G
i
u
~
1
,
u
~
2
,
v
=
-
x
i
v
1
-
τ
h
f
1
v
u
~
1
,
f
2
v
u
~
2
,
v
+
μ
i
g
i
f
1
v
u
~
1
,
f
2
v
u
~
2
,for i
=
1,2
. Then system (1) becomes(26)
L
1
v
u
~
1
=
u
~
1
G
1
u
~
1
,
u
~
2
,
v
+
u
~
1
τ
1
-
τ
2
χ
1
2
v
+
1
-
τ
χ
1
′
v
∇
v
2
,
x
∈
Ω
,
t
>
0
,
L
2
v
u
~
2
=
u
~
2
G
2
u
~
1
,
u
~
2
,
v
+
u
~
2
τ
1
-
τ
2
χ
2
2
v
+
1
-
τ
χ
2
′
v
∇
v
2
,
x
∈
Ω
,
t
>
0
,
v
t
=
τ
Δ
v
+
h
f
1
v
u
~
1
,
f
2
v
u
~
2
,
v
,
x
∈
Ω
,
t
>
0
,with boundary conditions ∂
u
~
1
/
∂
ν
=
∂
u
~
2
/
∂
ν
=
∂
v
/
∂
ν
=
0, x
∈
∂
Ω, t
>
0, and initial conditions u
~
1
(
x
,
0
)
≔
u
~
10
(
x
)
=
u
10
(
x
)
/
f
1
(
v
0
(
x
)
), u
~
2
(
x
,
0
)
≔
u
~
20
(
x
)
=
u
20
(
x
)
/
f
2
(
v
0
(
x
)
)
,
v
(
x
,
0
)
=
v
0
(
x
)
,
x
∈
Ω.
For sufficiently smallT
>
0, in the space [
L
p
(
[
0
,
T
]
;
W
2
,
p
(
Ω
)
)
∩
W
1
,
p
(
[
0
,
T
]
;
L
p
(
Ω
)
)
]
2
(
for
p
>
n
), we implement a fixed point argument. Taking v
∈
C
(
[
0
,
T
]
;
C
1
(
Ω
¯
)
) satisfying v
∈
(
v
_
,
v
¯
) and |
∇
v
|
<
C, we define u
~
1
,
u
~
2 in (
0
,
T
) as the unique solution to the question(27)
L
1
v
u
~
1
=
u
~
1
G
1
u
~
1
,
u
~
2
,
v
+
u
~
1
τ
1
-
τ
2
χ
1
2
v
+
1
-
τ
χ
1
′
v
∇
v
2
i
n
Ω
×
0
,
T
,
L
2
v
u
~
2
=
u
~
2
G
2
u
~
1
,
u
~
2
,
v
+
u
~
2
τ
1
-
τ
2
χ
2
2
v
+
1
-
τ
χ
2
′
v
∇
v
2
i
n
Ω
×
0
,
T
;for the details, we refer the reader to [13, Remark48.3]. Notice that the right-hand side terms of (27) are the multiplicative for u
~
1 and u
~
2, applying the maximum principle to verify that both u
~
1 and u
~
2 are nonnegative.
Since condition (11), we easily see that the regular functions u
~
1 and u
~
2 satisfy(28)
L
1
v
u
~
1
≤
u
~
1
G
1
u
~
1
,
u
~
2
,
v
,
L
2
v
u
~
2
≤
u
~
2
G
2
u
~
1
,
u
~
2
,
v
,and the right-hand side terms of (27) are quasi-decreased; looking upon the lower-solution u
_
1
=
u
_
2
=
0, one may construct upper-solutions u
¯
1 and u
¯
2 such that(29)
0
≤
u
1
≤
u
¯
1
,
0
≤
u
2
≤
u
¯
2
,
t
<
T
.It follows that we apply Theorem 2.1 of [7] to get a unique nonnegative solution.
We solve the parabolic equation(30)
v
t
=
τ
Δ
v
+
h
f
1
v
u
~
1
,
f
2
v
u
~
2
,
v
.Thanks to the essential estimations of parabolic equations and the embedding theorems, we apply the Schauder fixed point theorem to obtain the local existence of solution v
(
x
,
t
). The smoothness of h
(
f
1
(
v
)
u
~
1
,
f
2
(
v
)
u
~
2
,
v
) ensures the uniqueness of solution v
(
x
,
t
).
To provev
≥
v
_, in view of assumption (8) and the monotone of h
(
u
1
,
u
2
,
v
), we have 0
≤
h
(
0,0
,
v
_
)
≤
h
(
u
1
,
u
2
,
v
_
). It follows that v
_ is a lower-solution for the equation(31)
v
t
-
τ
Δ
v
=
h
u
1
,
u
2
,
v
i
n
Ω
×
0
,
T
,
∂
v
∂
ν
=
0
o
n
∂
Ω
×
0
,
T
,
v
x
,
0
=
v
0
x
i
n
Ω
.The proof of Lemma 4 is completed.About the above solution(
u
1
,
u
2
,
v
), we have the following extensibility criterion:The solution is extended to the interval
(
0
,
T
max
⁡
),where
T
max
⁡
has the following property:
(32)
If
T
max
⁡
<
∞
,
t
h
e
n
lim
sup
t
→
T
max
⁡
u
1
·
,
t
L
∞
Ω
+
u
2
·
,
t
L
∞
Ω
+
v
·
,
t
L
∞
Ω
=
∞
.


### 3.2. Uniform Boundedness of Solutions

To obtain some a priori estimates, we need some technical lemmas. The followingL
1-estimate of solution is first given.Lemma 5.
For allt
∈
(
0
,
T
m
a
x
), the solutions to (1) satisfy the following estimates:(33)
u
1
L
1
Ω
≤
max
⁡
Ω
,
u
10
L
1
Ω
≤
M
,
u
2
L
1
Ω
≤
max
⁡
Ω
,
u
20
L
1
Ω
≤
M
,
v
L
1
Ω
≤
max
⁡
v
0
L
1
Ω
,
2
M
γ
-
1
,where M
=
max
⁡
{
|
Ω
|
,
u
10
L
1
Ω
,
u
20
L
1
Ω
}.Proof.
Integrating the first equation of (1) over Ω, we have (34)
d
d
t
∫
Ω
u
1
d
x
=
μ
1
∫
Ω
u
1
-
u
1
2
-
a
1
u
1
u
2
d
x
≤
μ
1
∫
Ω
u
1
-
u
1
2
d
x
.Using Cauchy inequality yields(35)
d
d
t
∫
Ω
u
1
d
x
≤
μ
1
∫
Ω
1
-
u
1
d
x
≤
μ
1
Ω
-
μ
1
∫
Ω
u
1
d
x
.By the Gronwall Lemma, we derive(36)
u
1
x
,
t
L
1
Ω
≤
max
⁡
Ω
,
u
10
x
L
1
Ω
≤
M
;a similar estimate holds for u
2, which leads to (37)
u
2
x
,
t
L
1
Ω
≤
max
⁡
Ω
,
u
20
x
L
1
Ω
≤
M
.
Integrating the third equation of (1) over Ω, we have(38)
d
d
t
∫
Ω
v
d
x
=
∫
Ω
-
γ
v
+
u
1
+
u
2
d
x
=
-
γ
∫
Ω
v
d
x
+
∫
Ω
u
1
+
u
2
d
x
≤
-
γ
∫
Ω
v
d
x
+
2
M
;it implies(39)
v
L
1
Ω
≤
max
⁡
v
0
L
1
Ω
,
2
M
γ
-
1
.This proves the lemma.Lemma 6.
Lettingp
>
1 and under assumption (11), then the following estimates hold:(40)
d
d
t
∫
Ω
u
1
p
f
1
p
1
-
p
d
x
≤
1
-
p
1
-
τ
p
-
1
/
p
∫
Ω
u
1
p
χ
1
v
f
1
p
1
-
p
h
u
1
,
u
2
,
v
d
x
+
p
μ
1
∫
Ω
u
1
p
f
1
p
1
-
p
g
1
u
1
,
u
2
d
x
,
(41)
d
d
t
∫
Ω
u
2
p
f
2
p
1
-
p
d
x
≤
1
-
p
1
-
τ
p
-
1
/
p
∫
Ω
u
2
p
χ
2
v
f
2
p
1
-
p
h
u
1
,
u
2
,
v
d
x
+
p
μ
2
∫
Ω
u
2
p
f
2
p
1
-
p
g
2
u
1
,
u
2
d
x
,where f
i
p
(
v
) is defined by(42)
f
i
p
v
=
exp
⁡
1
1
-
τ
p
-
1
/
p
∫
v
_
v
χ
i
s
d
s
,
i
=
1,2
.Proof.
It is easy to check(43)
f
1
p
′
v
=
1
1
-
τ
p
-
1
/
p
χ
1
v
f
1
p
v
,
f
2
p
′
v
=
1
1
-
τ
p
-
1
/
p
χ
2
v
f
2
p
v
.Then, for any p
>
1, we have(44)
d
d
t
∫
Ω
u
1
p
f
1
p
1
-
p
d
x
=
p
∫
Ω
u
1
p
-
1
u
1
t
f
1
p
1
-
p
d
x
+
1
-
p
∫
Ω
u
1
p
f
1
p
-
p
f
1
p
′
v
t
d
x
=
p
∫
Ω
u
1
p
-
1
f
1
p
1
-
p
Δ
u
1
-
∇
·
u
1
χ
1
v
∇
v
+
τ
1
-
p
p
-
τ
p
-
1
u
1
χ
1
v
Δ
v
d
x
+
1
-
p
1
-
τ
p
-
1
/
p
∫
Ω
u
1
p
f
1
p
1
-
p
χ
1
v
h
u
1
,
u
2
,
v
d
x
+
p
μ
1
∫
Ω
u
1
p
f
1
p
1
-
p
g
1
u
1
,
u
2
d
x
≔
I
1
+
I
2
+
I
3
.Clearly, if I
1
≤
0, then (40) is a consequence of (44). We in the following verify that the result holds.
In fact(45)
I
1
=
p
∫
Ω
u
1
p
-
1
f
1
p
1
-
p
Δ
u
1
-
∇
·
u
1
χ
1
v
∇
v
+
τ
1
-
p
p
-
τ
p
-
1
∇
·
u
1
χ
1
v
∇
v
-
∇
u
1
χ
1
v
·
∇
v
d
x
=
p
∫
Ω
u
1
p
-
1
f
1
p
1
-
p
∇
·
f
1
p
∇
u
1
f
1
p
-
1
-
τ
1
-
p
p
-
τ
p
-
1
χ
1
v
∇
u
1
·
∇
v
+
u
1
χ
1
′
v
∇
v
2
d
x
=
p
1
-
p
∫
Ω
u
1
p
-
2
f
1
p
2
-
p
f
1
p
∇
u
1
f
1
p
-
1
2
-
τ
u
1
χ
1
v
∇
u
1
·
∇
v
+
u
1
χ
1
′
v
∇
v
2
f
1
p
p
-
τ
p
-
1
d
x
=
p
1
-
p
∫
Ω
u
1
p
-
2
f
1
p
3
-
p
∇
u
1
f
1
p
-
1
2
d
x
+
τ
p
p
-
1
p
-
τ
p
-
1
∫
Ω
u
1
p
-
1
f
1
p
1
-
p
χ
1
v
∇
u
1
·
∇
v
d
x
+
τ
p
p
-
1
p
-
τ
p
-
1
∫
Ω
u
1
p
f
1
p
1
-
p
χ
1
′
v
∇
v
2
d
x
≔
J
1
+
J
2
+
J
3
.
Since∇
u
1
=
∇
(
f
1
p
(
u
1
f
1
p
-
1
)
), multiplying both sides by χ
1
(
v
)
∇
v, we get(46)
χ
1
v
∇
u
1
·
∇
v
=
f
1
p
χ
1
v
∇
u
1
f
1
p
-
1
·
∇
v
+
1
1
-
τ
p
-
1
/
p
u
1
χ
1
2
v
∇
v
2
,and by ε-Yöung’s inequality(47)
χ
1
v
∇
u
1
f
1
p
-
1
·
∇
v
≤
p
-
τ
p
-
1
τ
f
1
p
u
1
∇
u
1
f
1
p
-
1
2
+
τ
p
-
τ
p
-
1
u
1
f
1
p
χ
1
2
v
∇
v
2
.Combining (46) and (47) and substituting it into J
2, we derive(48)
J
2
≤
p
p
-
1
∫
Ω
u
1
p
-
2
f
1
p
3
-
p
∇
u
1
f
1
p
-
1
2
d
x
+
τ
p
p
-
1
τ
+
p
p
-
τ
p
-
1
2
∫
Ω
u
1
p
f
1
p
1
-
p
χ
1
2
v
∇
v
2
d
x
.Inserting (48) into (45) yields(49)
I
1
≤
τ
p
p
-
1
p
-
τ
p
-
1
∫
Ω
τ
+
p
p
-
τ
p
-
1
χ
1
2
v
+
χ
1
′
v
u
1
p
f
1
p
1
-
p
∇
v
2
d
x
=
τ
p
p
-
1
p
-
τ
p
-
1
∫
Ω
1
1
-
τ
1
-
τ
2
τ
+
1
-
τ
p
χ
1
2
v
+
χ
1
′
v
u
1
p
f
1
p
1
-
p
∇
v
2
d
x
≤
τ
p
p
-
1
p
-
τ
p
-
1
∫
Ω
1
1
-
τ
χ
1
2
v
+
χ
1
′
v
u
1
p
f
1
p
1
-
p
∇
v
2
d
x
;we deduce from assumption (11) that I
1
≤
0 for any p
>
1, and the proof ends. In the same way, we achieve (41). This completes the proof of Lemma 6.In view ofv
∈
C
(
(
0
,
T
max
⁡
)
;
C
1
(
Ω
)
) and initial date v
_
<
v
0
<
v
¯, we consider T
∗
>
0 defined by(50)
T
∗
≔
T
m
a
x
,
if
v
≤
v
¯
in
0
,
T
max
⁡
,
T
v
,
if
v
L
∞
Ω
T
max
⁡
>
v
¯
,where T
v
<
T
max
⁡ satisfies that v
L
∞
(
Ω
T
v
)
≤
v
¯ and v
L
∞
(
Ω
T
v
+
δ
)
>
v
¯ for any δ
>
0.Lemma 7.
Lettingp
>
1 and for any t
<
T
∗, then the solutions to (1) satisfy(51)
1
p
-
1
d
d
t
∫
Ω
u
1
p
f
1
p
1
-
p
d
x
≤
-
κ
01
+
μ
1
∫
Ω
u
1
p
+
1
f
1
p
-
p
d
x
+
κ
1
1
-
τ
+
p
p
-
1
μ
1
∫
Ω
u
1
p
f
1
p
1
-
p
d
x
,
(52)
1
p
-
1
d
d
t
∫
Ω
u
2
p
f
2
p
1
-
p
d
x
≤
-
κ
02
+
μ
2
∫
Ω
u
2
p
+
1
f
2
p
-
p
d
x
+
κ
2
1
-
τ
+
p
p
-
1
μ
2
∫
Ω
u
2
p
f
2
p
1
-
p
d
x
.Proof.
In order to prove Lemma7, we consider the terms I
2 and I
3 appearing in the proof of Lemma 6.
By the nonnegativity foru
1
,
u
2 and assumption (6), we obtain(53)
-
h
u
1
,
u
2
,
v
≤
-
u
1
+
κ
1
χ
1
v
.Multiplying (53) by u
1
p
f
1
p
1
-
p
χ
1
(
v
) and integrating over Ω yield(54)
-
∫
Ω
u
1
p
f
1
p
1
-
p
χ
1
v
h
u
1
,
u
2
,
v
d
x
≤
-
∫
Ω
u
1
p
+
1
f
1
p
1
-
p
χ
1
v
d
x
+
κ
1
∫
Ω
u
1
p
f
1
p
1
-
p
d
x
.By the monotonic property of exponential function and assumption (7), we have (55)
0
<
κ
01
≤
χ
1
v
exp
⁡
∫
v
_
v
χ
1
s
d
s
≤
χ
1
v
f
1
p
v
.It follows that (54) becomes(56)
-
∫
Ω
u
1
p
f
1
p
1
-
p
χ
1
v
h
u
1
,
u
2
,
v
d
x
≤
-
κ
01
∫
Ω
u
1
p
+
1
f
1
p
-
p
d
x
+
κ
1
∫
Ω
u
1
p
f
1
p
1
-
p
d
x
.Multiplying (56) by (
p
-
1
)
/
(
1
-
τ
(
(
p
-
1
)
/
p
)
), we get(57)
I
2
≤
p
-
1
1
-
τ
p
-
1
/
p
-
κ
01
∫
Ω
u
1
p
+
1
f
1
p
-
p
d
x
+
κ
1
∫
Ω
u
1
p
f
1
p
1
-
p
d
x
≤
p
-
1
-
κ
01
∫
Ω
u
1
p
+
1
f
1
p
-
p
d
x
+
κ
1
1
-
τ
∫
Ω
u
1
p
f
1
p
1
-
p
d
x
.For the term I
3, we easily achieve(58)
I
3
≤
p
μ
1
∫
Ω
u
1
p
f
1
p
1
-
p
d
x
-
∫
Ω
u
1
p
+
1
f
1
p
-
p
d
x
≤
p
-
1
μ
1
∫
Ω
u
1
p
f
1
p
1
-
p
d
x
-
∫
Ω
u
1
p
+
1
f
1
p
-
p
d
x
+
μ
1
∫
Ω
u
1
p
f
1
p
1
-
p
d
x
.Combining (44), (57), and (58) leads to(59)
d
d
t
∫
Ω
u
1
p
f
1
p
1
-
p
d
x
≤
p
-
1
-
κ
01
∫
Ω
u
1
p
+
1
f
1
p
-
p
d
x
+
κ
1
1
-
τ
∫
Ω
u
1
p
f
1
p
1
-
p
d
x
+
p
-
1
μ
1
∫
Ω
u
1
p
f
1
p
1
-
p
d
x
-
∫
Ω
u
1
p
+
1
f
1
p
-
p
d
x
+
μ
1
∫
Ω
u
1
p
f
1
p
1
-
p
d
x
,which implies that (51) holds. The same arguments are used to achieve (52). The proof ends.To establish the uniform boundedness of solutions, we apply an iterative method based on theL
p-n
o
r
m of solutions inspired by the Moser-Alikakos type iterations (see [10, 13]).Lemma 8.
For anyt
<
T
∗, then the solutions to (1) are bounded by(60)
u
1
L
∞
Ω
≤
f
1
v
¯
max
⁡
κ
1
1
-
τ
+
μ
1
κ
01
+
μ
1
-
1
,
u
10
L
∞
Ω
,
(61)
u
2
L
∞
Ω
≤
f
2
v
¯
max
⁡
κ
2
1
-
τ
+
μ
2
κ
02
+
μ
2
-
1
,
u
20
L
∞
Ω
.Proof.
Denote∫
Ω
‍
u
1
p
f
1
p
1
-
p
d
x by Ψ
(
t
). Taking arbitrary positive constant ρ
>
0, then it holds that(62)
Ψ
t
=
∫
f
1
p
≤
ρ
κ
01
+
μ
1
u
1
u
1
p
f
1
p
1
-
p
d
x
+
∫
f
1
p
>
ρ
κ
01
+
μ
1
u
1
u
1
p
f
1
p
1
-
p
d
x
≤
ρ
κ
01
+
μ
1
∫
f
1
p
≤
ρ
κ
01
+
μ
1
u
1
u
1
p
+
1
f
1
p
-
p
d
x
+
ρ
κ
01
+
μ
1
1
-
p
∫
f
1
p
>
ρ
κ
01
+
μ
1
u
1
u
1
d
x
≤
ρ
κ
01
+
μ
1
∫
Ω
u
1
p
+
1
f
1
p
-
p
d
x
+
ρ
κ
01
+
μ
1
1
-
p
∫
Ω
u
1
d
x
.Hence(63)
-
κ
01
+
μ
1
∫
Ω
u
1
p
+
1
f
1
p
-
p
d
x
≤
-
1
ρ
Ψ
t
+
ρ
-
p
κ
01
+
μ
1
1
-
p
∫
Ω
u
1
d
x
.Substituting (63) into (51), we have(64)
1
p
-
1
d
d
x
Ψ
t
≤
κ
1
1
-
τ
+
p
p
-
1
μ
1
-
1
ρ
Ψ
t
+
ρ
-
p
κ
01
+
μ
1
1
-
p
∫
Ω
u
1
d
x
.Take 1
/
ρ
>
κ
1
/
(
1
-
τ
)
+
(
p
/
(
p
-
1
)
)
μ
1. Noticing Lemma 5 and initial date u
10
∈
C
1
(
Ω
¯
), we apply the Gronwall Lemma to obtain(65)
Ψ
t
≤
max
⁡
ρ
-
p
κ
01
+
μ
1
1
-
p
1
ρ
-
κ
1
1
-
τ
-
p
p
-
1
μ
1
-
1
∫
Ω
u
1
d
x
,
Ψ
0
;taking limits leads to(66)
lim
p
→
∞
⁡
Ψ
1
/
p
t
≤
max
⁡
ρ
-
1
κ
01
+
μ
1
-
1
,
u
10
L
∞
Ω
.Since(67)
Ψ
t
=
∫
Ω
u
1
p
f
1
p
1
-
p
v
d
x
≥
f
1
p
1
-
p
v
¯
∫
Ω
u
1
p
d
x
,combining (66) and (67) and taking limits as ρ
-
1
→
κ
1
/
(
1
-
τ
)
+
μ
1, it is found that (60) holds. The same processes are used to prove (61), which ends the proof of Lemma 8.Remark 9.
In the above arguments, we used well such fact: LetΩ be bounded and regular domain of R
n. If u
L
p
(
Ω
)
≤
C, where C is some positive constant and independent of p, then u
L
p
(
Ω
)
→
u
L
∞
(
Ω
) as p
→
∞.To end the arguments of uniform boundedness of solutions, it follows to prove thatT
∗
=
T
m
a
x and T
m
a
x
=
+
∞ by contradiction.As in Lemmas4–8, we define u
^
1
,
u
^
2 and v
^ for the unique solutions in (
0
,
T
^
m
a
x
) to(68)
u
^
1
t
=
Δ
u
^
1
-
∇
·
u
^
1
χ
1
v
^
∇
v
+
μ
1
u
^
1
g
1
u
^
1
,
u
^
2
,
x
∈
Ω
,
t
>
0
,
u
^
2
t
=
Δ
u
^
2
-
∇
·
u
^
2
χ
2
v
^
∇
v
^
+
μ
2
u
2
g
2
u
^
1
,
u
^
2
,
x
∈
Ω
,
t
>
0
,
v
^
t
=
τ
Δ
v
^
+
h
^
u
^
1
,
u
^
2
,
v
^
,
x
∈
Ω
,
t
>
0
,
∂
u
^
1
∂
ν
=
∂
u
^
2
∂
ν
=
∂
v
^
∂
ν
=
0
,
x
∈
∂
Ω
,
t
>
0
,
u
^
1
x
,
0
=
u
10
x
,
u
^
2
x
,
0
=
u
20
x
,
v
^
x
,
0
=
v
0
x
,
x
∈
Ω
,where T
^
max
⁡ is defined in the same fashion as (32) and the truncation h
^
(
u
1
,
u
2
,
v
) is defined as follows:(69)
h
^
u
1
,
u
2
,
v
≔
h
u
1
,
u
2
,
v
,
if
v
≤
v
¯
,
h
u
1
,
u
2
,
v
¯
,
if
v
>
v
¯
.Then it is easy to see that(70)
u
^
1
L
∞
Ω
≤
f
1
v
¯
max
⁡
κ
1
1
-
τ
+
μ
1
κ
01
+
μ
1
-
1
,
u
10
L
∞
Ω
,
u
^
2
L
∞
Ω
≤
f
2
v
¯
max
⁡
κ
2
1
-
τ
+
μ
2
κ
02
+
μ
2
-
1
,
u
20
L
∞
Ω
,as far as v
^
≤
v
¯.By analogy withT
∗, we define(71)
T
^
∗
≔
T
^
m
a
x
,
if
v
^
≤
v
¯
i
n
0
,
T
^
max
⁡
,
T
^
v
^
,
if
v
^
L
∞
Ω
T
^
max
⁡
>
v
¯
,where T
^
v
^
<
T
^
m
a
x satisfies that v
^
L
∞
(
Ω
T
^
v
^
)
≤
v
¯ and v
^
L
∞
(
Ω
T
^
v
^
+
δ
)
>
v
¯ for any δ
>
0. To prove T
^
∗
=
T
^
m
a
x, we suppose (72)
T
^
∗
<
T
^
max
⁡
;then h
^
(
u
^
1
,
u
^
2
,
v
^
)
≤
h
^
(
u
¯
1
,
u
¯
2
,
v
¯
)
=
h
(
u
¯
1
,
u
¯
2
,
v
¯
)
<
0. It follows to easily check that v
¯ is a strict upper-solution to(73)
v
^
t
-
τ
Δ
v
^
=
h
^
u
^
1
,
u
^
2
,
v
^
i
n
Ω
×
0
,
T
^
∗
,
∂
v
^
∂
ν
=
0
o
n
∂
Ω
×
0
,
T
^
∗
,
v
^
x
,
0
=
v
0
x
i
n
Ω
.This impliesv
^
(
x
,
t
)
<
v
¯
,
∀
x
∈
Ω
,
t
∈
(
0
,
T
^
∗), which contradicts (72) and proves T
^
∗
=
T
^
max
⁡ as well as T
^
max
⁡
=
∞.Thanks tov
^
≤
v
¯, we show that (
u
^
1
,
u
^
2
,
v
^
) is also a solution to (6). So we have(74)
T
∗
=
T
max
⁡
≥
T
^
max
⁡
=
∞
.The proof of Theorem 3 is completed.


## 3.1. Local Existence of Solutions

Lemma 4.
Given positive initial date(
u
10
,
u
20
,
v
0
)
∈
(
C
1
(
Ω
¯
)
)
3 and the same assumptions as in Theorem 3, there exists a small enough time T
>
0 and a unique triple of nonnegative functions (
u
1
,
u
2
,
v
)
∈
(
L
p
(
[
0
,
T
]
;
W
2
,
p
(
Ω
)
)
∩
W
1
,
p
(
[
0
,
T
]
;
L
p
(
Ω
)
)
)
3
(
for
p
>
n
) such that (
u
1
,
u
2
,
v
) is a solution of (1) in Ω
×
(
0
,
T
). Moreover, we have v
≥
v
_.Proof.
Introduce the change of variablesu
~
1 and u
~
2 given by(20)
u
1
=
f
1
v
u
~
1
,
u
2
=
f
2
v
u
~
2
,where the function f
i
(
v
) is defined by(21)
f
i
v
=
e
x
p
1
1
-
τ
∫
v
_
v
χ
i
s
d
s
,
i
=
1,2
.A direct calculation yields(22)
u
1
t
=
f
1
′
v
τ
Δ
v
+
h
u
1
,
u
2
,
v
u
~
1
+
f
1
v
u
~
1
t
,
Δ
u
1
=
f
1
′
′
v
u
~
1
∇
v
2
+
2
f
1
′
v
∇
u
~
1
·
∇
v
+
f
1
′
v
u
~
1
Δ
v
+
f
1
v
Δ
u
~
1
,
∇
·
u
1
χ
1
v
∇
v
=
f
1
′
v
u
~
1
χ
1
v
∇
v
2
+
f
1
v
χ
1
v
∇
u
~
1
·
∇
v
+
f
1
v
u
~
1
χ
1
′
v
∇
v
2
+
f
1
v
u
~
1
χ
1
v
Δ
v
;similarly,(23)
u
2
t
=
f
2
′
v
τ
Δ
v
+
h
u
1
,
u
2
,
v
u
~
2
+
f
2
v
u
~
2
t
,
Δ
u
2
=
f
2
′
′
v
u
~
2
∇
v
2
+
2
f
2
′
v
∇
u
~
2
·
∇
v
+
f
2
′
v
u
~
2
Δ
v
+
f
2
v
Δ
u
~
2
,
∇
·
u
2
χ
2
v
∇
v
=
f
2
′
v
u
~
2
χ
2
v
∇
v
2
+
f
2
v
χ
2
v
∇
u
~
2
·
∇
v
+
f
2
v
u
~
2
χ
2
′
v
∇
v
2
+
f
2
v
u
~
2
χ
2
v
Δ
v
,where(24)
f
i
′
v
=
χ
i
v
1
-
τ
f
i
v
,
f
i
′
′
v
=
χ
i
2
v
1
-
τ
2
+
χ
i
′
v
1
-
τ
f
i
v
,
i
=
1,2
.Substitute (20), (22), and (23) into (1) and denote that(25)
L
i
v
u
~
i
=
u
~
i
t
-
Δ
u
~
i
-
1
+
τ
1
-
τ
χ
i
v
∇
v
·
∇
u
~
i
,
G
i
u
~
1
,
u
~
2
,
v
=
-
x
i
v
1
-
τ
h
f
1
v
u
~
1
,
f
2
v
u
~
2
,
v
+
μ
i
g
i
f
1
v
u
~
1
,
f
2
v
u
~
2
,for i
=
1,2
. Then system (1) becomes(26)
L
1
v
u
~
1
=
u
~
1
G
1
u
~
1
,
u
~
2
,
v
+
u
~
1
τ
1
-
τ
2
χ
1
2
v
+
1
-
τ
χ
1
′
v
∇
v
2
,
x
∈
Ω
,
t
>
0
,
L
2
v
u
~
2
=
u
~
2
G
2
u
~
1
,
u
~
2
,
v
+
u
~
2
τ
1
-
τ
2
χ
2
2
v
+
1
-
τ
χ
2
′
v
∇
v
2
,
x
∈
Ω
,
t
>
0
,
v
t
=
τ
Δ
v
+
h
f
1
v
u
~
1
,
f
2
v
u
~
2
,
v
,
x
∈
Ω
,
t
>
0
,with boundary conditions ∂
u
~
1
/
∂
ν
=
∂
u
~
2
/
∂
ν
=
∂
v
/
∂
ν
=
0, x
∈
∂
Ω, t
>
0, and initial conditions u
~
1
(
x
,
0
)
≔
u
~
10
(
x
)
=
u
10
(
x
)
/
f
1
(
v
0
(
x
)
), u
~
2
(
x
,
0
)
≔
u
~
20
(
x
)
=
u
20
(
x
)
/
f
2
(
v
0
(
x
)
)
,
v
(
x
,
0
)
=
v
0
(
x
)
,
x
∈
Ω.
For sufficiently smallT
>
0, in the space [
L
p
(
[
0
,
T
]
;
W
2
,
p
(
Ω
)
)
∩
W
1
,
p
(
[
0
,
T
]
;
L
p
(
Ω
)
)
]
2
(
for
p
>
n
), we implement a fixed point argument. Taking v
∈
C
(
[
0
,
T
]
;
C
1
(
Ω
¯
)
) satisfying v
∈
(
v
_
,
v
¯
) and |
∇
v
|
<
C, we define u
~
1
,
u
~
2 in (
0
,
T
) as the unique solution to the question(27)
L
1
v
u
~
1
=
u
~
1
G
1
u
~
1
,
u
~
2
,
v
+
u
~
1
τ
1
-
τ
2
χ
1
2
v
+
1
-
τ
χ
1
′
v
∇
v
2
i
n
Ω
×
0
,
T
,
L
2
v
u
~
2
=
u
~
2
G
2
u
~
1
,
u
~
2
,
v
+
u
~
2
τ
1
-
τ
2
χ
2
2
v
+
1
-
τ
χ
2
′
v
∇
v
2
i
n
Ω
×
0
,
T
;for the details, we refer the reader to [13, Remark48.3]. Notice that the right-hand side terms of (27) are the multiplicative for u
~
1 and u
~
2, applying the maximum principle to verify that both u
~
1 and u
~
2 are nonnegative.
Since condition (11), we easily see that the regular functions u
~
1 and u
~
2 satisfy(28)
L
1
v
u
~
1
≤
u
~
1
G
1
u
~
1
,
u
~
2
,
v
,
L
2
v
u
~
2
≤
u
~
2
G
2
u
~
1
,
u
~
2
,
v
,and the right-hand side terms of (27) are quasi-decreased; looking upon the lower-solution u
_
1
=
u
_
2
=
0, one may construct upper-solutions u
¯
1 and u
¯
2 such that(29)
0
≤
u
1
≤
u
¯
1
,
0
≤
u
2
≤
u
¯
2
,
t
<
T
.It follows that we apply Theorem 2.1 of [7] to get a unique nonnegative solution.
We solve the parabolic equation(30)
v
t
=
τ
Δ
v
+
h
f
1
v
u
~
1
,
f
2
v
u
~
2
,
v
.Thanks to the essential estimations of parabolic equations and the embedding theorems, we apply the Schauder fixed point theorem to obtain the local existence of solution v
(
x
,
t
). The smoothness of h
(
f
1
(
v
)
u
~
1
,
f
2
(
v
)
u
~
2
,
v
) ensures the uniqueness of solution v
(
x
,
t
).
To provev
≥
v
_, in view of assumption (8) and the monotone of h
(
u
1
,
u
2
,
v
), we have 0
≤
h
(
0,0
,
v
_
)
≤
h
(
u
1
,
u
2
,
v
_
). It follows that v
_ is a lower-solution for the equation(31)
v
t
-
τ
Δ
v
=
h
u
1
,
u
2
,
v
i
n
Ω
×
0
,
T
,
∂
v
∂
ν
=
0
o
n
∂
Ω
×
0
,
T
,
v
x
,
0
=
v
0
x
i
n
Ω
.The proof of Lemma 4 is completed.About the above solution(
u
1
,
u
2
,
v
), we have the following extensibility criterion:The solution is extended to the interval
(
0
,
T
max
⁡
),where
T
max
⁡
has the following property:
(32)
If
T
max
⁡
<
∞
,
t
h
e
n
lim
sup
t
→
T
max
⁡
u
1
·
,
t
L
∞
Ω
+
u
2
·
,
t
L
∞
Ω
+
v
·
,
t
L
∞
Ω
=
∞
.


## 3.2. Uniform Boundedness of Solutions

To obtain some a priori estimates, we need some technical lemmas. The followingL
1-estimate of solution is first given.Lemma 5.
For allt
∈
(
0
,
T
m
a
x
), the solutions to (1) satisfy the following estimates:(33)
u
1
L
1
Ω
≤
max
⁡
Ω
,
u
10
L
1
Ω
≤
M
,
u
2
L
1
Ω
≤
max
⁡
Ω
,
u
20
L
1
Ω
≤
M
,
v
L
1
Ω
≤
max
⁡
v
0
L
1
Ω
,
2
M
γ
-
1
,where M
=
max
⁡
{
|
Ω
|
,
u
10
L
1
Ω
,
u
20
L
1
Ω
}.Proof.
Integrating the first equation of (1) over Ω, we have (34)
d
d
t
∫
Ω
u
1
d
x
=
μ
1
∫
Ω
u
1
-
u
1
2
-
a
1
u
1
u
2
d
x
≤
μ
1
∫
Ω
u
1
-
u
1
2
d
x
.Using Cauchy inequality yields(35)
d
d
t
∫
Ω
u
1
d
x
≤
μ
1
∫
Ω
1
-
u
1
d
x
≤
μ
1
Ω
-
μ
1
∫
Ω
u
1
d
x
.By the Gronwall Lemma, we derive(36)
u
1
x
,
t
L
1
Ω
≤
max
⁡
Ω
,
u
10
x
L
1
Ω
≤
M
;a similar estimate holds for u
2, which leads to (37)
u
2
x
,
t
L
1
Ω
≤
max
⁡
Ω
,
u
20
x
L
1
Ω
≤
M
.
Integrating the third equation of (1) over Ω, we have(38)
d
d
t
∫
Ω
v
d
x
=
∫
Ω
-
γ
v
+
u
1
+
u
2
d
x
=
-
γ
∫
Ω
v
d
x
+
∫
Ω
u
1
+
u
2
d
x
≤
-
γ
∫
Ω
v
d
x
+
2
M
;it implies(39)
v
L
1
Ω
≤
max
⁡
v
0
L
1
Ω
,
2
M
γ
-
1
.This proves the lemma.Lemma 6.
Lettingp
>
1 and under assumption (11), then the following estimates hold:(40)
d
d
t
∫
Ω
u
1
p
f
1
p
1
-
p
d
x
≤
1
-
p
1
-
τ
p
-
1
/
p
∫
Ω
u
1
p
χ
1
v
f
1
p
1
-
p
h
u
1
,
u
2
,
v
d
x
+
p
μ
1
∫
Ω
u
1
p
f
1
p
1
-
p
g
1
u
1
,
u
2
d
x
,
(41)
d
d
t
∫
Ω
u
2
p
f
2
p
1
-
p
d
x
≤
1
-
p
1
-
τ
p
-
1
/
p
∫
Ω
u
2
p
χ
2
v
f
2
p
1
-
p
h
u
1
,
u
2
,
v
d
x
+
p
μ
2
∫
Ω
u
2
p
f
2
p
1
-
p
g
2
u
1
,
u
2
d
x
,where f
i
p
(
v
) is defined by(42)
f
i
p
v
=
exp
⁡
1
1
-
τ
p
-
1
/
p
∫
v
_
v
χ
i
s
d
s
,
i
=
1,2
.Proof.
It is easy to check(43)
f
1
p
′
v
=
1
1
-
τ
p
-
1
/
p
χ
1
v
f
1
p
v
,
f
2
p
′
v
=
1
1
-
τ
p
-
1
/
p
χ
2
v
f
2
p
v
.Then, for any p
>
1, we have(44)
d
d
t
∫
Ω
u
1
p
f
1
p
1
-
p
d
x
=
p
∫
Ω
u
1
p
-
1
u
1
t
f
1
p
1
-
p
d
x
+
1
-
p
∫
Ω
u
1
p
f
1
p
-
p
f
1
p
′
v
t
d
x
=
p
∫
Ω
u
1
p
-
1
f
1
p
1
-
p
Δ
u
1
-
∇
·
u
1
χ
1
v
∇
v
+
τ
1
-
p
p
-
τ
p
-
1
u
1
χ
1
v
Δ
v
d
x
+
1
-
p
1
-
τ
p
-
1
/
p
∫
Ω
u
1
p
f
1
p
1
-
p
χ
1
v
h
u
1
,
u
2
,
v
d
x
+
p
μ
1
∫
Ω
u
1
p
f
1
p
1
-
p
g
1
u
1
,
u
2
d
x
≔
I
1
+
I
2
+
I
3
.Clearly, if I
1
≤
0, then (40) is a consequence of (44). We in the following verify that the result holds.
In fact(45)
I
1
=
p
∫
Ω
u
1
p
-
1
f
1
p
1
-
p
Δ
u
1
-
∇
·
u
1
χ
1
v
∇
v
+
τ
1
-
p
p
-
τ
p
-
1
∇
·
u
1
χ
1
v
∇
v
-
∇
u
1
χ
1
v
·
∇
v
d
x
=
p
∫
Ω
u
1
p
-
1
f
1
p
1
-
p
∇
·
f
1
p
∇
u
1
f
1
p
-
1
-
τ
1
-
p
p
-
τ
p
-
1
χ
1
v
∇
u
1
·
∇
v
+
u
1
χ
1
′
v
∇
v
2
d
x
=
p
1
-
p
∫
Ω
u
1
p
-
2
f
1
p
2
-
p
f
1
p
∇
u
1
f
1
p
-
1
2
-
τ
u
1
χ
1
v
∇
u
1
·
∇
v
+
u
1
χ
1
′
v
∇
v
2
f
1
p
p
-
τ
p
-
1
d
x
=
p
1
-
p
∫
Ω
u
1
p
-
2
f
1
p
3
-
p
∇
u
1
f
1
p
-
1
2
d
x
+
τ
p
p
-
1
p
-
τ
p
-
1
∫
Ω
u
1
p
-
1
f
1
p
1
-
p
χ
1
v
∇
u
1
·
∇
v
d
x
+
τ
p
p
-
1
p
-
τ
p
-
1
∫
Ω
u
1
p
f
1
p
1
-
p
χ
1
′
v
∇
v
2
d
x
≔
J
1
+
J
2
+
J
3
.
Since∇
u
1
=
∇
(
f
1
p
(
u
1
f
1
p
-
1
)
), multiplying both sides by χ
1
(
v
)
∇
v, we get(46)
χ
1
v
∇
u
1
·
∇
v
=
f
1
p
χ
1
v
∇
u
1
f
1
p
-
1
·
∇
v
+
1
1
-
τ
p
-
1
/
p
u
1
χ
1
2
v
∇
v
2
,and by ε-Yöung’s inequality(47)
χ
1
v
∇
u
1
f
1
p
-
1
·
∇
v
≤
p
-
τ
p
-
1
τ
f
1
p
u
1
∇
u
1
f
1
p
-
1
2
+
τ
p
-
τ
p
-
1
u
1
f
1
p
χ
1
2
v
∇
v
2
.Combining (46) and (47) and substituting it into J
2, we derive(48)
J
2
≤
p
p
-
1
∫
Ω
u
1
p
-
2
f
1
p
3
-
p
∇
u
1
f
1
p
-
1
2
d
x
+
τ
p
p
-
1
τ
+
p
p
-
τ
p
-
1
2
∫
Ω
u
1
p
f
1
p
1
-
p
χ
1
2
v
∇
v
2
d
x
.Inserting (48) into (45) yields(49)
I
1
≤
τ
p
p
-
1
p
-
τ
p
-
1
∫
Ω
τ
+
p
p
-
τ
p
-
1
χ
1
2
v
+
χ
1
′
v
u
1
p
f
1
p
1
-
p
∇
v
2
d
x
=
τ
p
p
-
1
p
-
τ
p
-
1
∫
Ω
1
1
-
τ
1
-
τ
2
τ
+
1
-
τ
p
χ
1
2
v
+
χ
1
′
v
u
1
p
f
1
p
1
-
p
∇
v
2
d
x
≤
τ
p
p
-
1
p
-
τ
p
-
1
∫
Ω
1
1
-
τ
χ
1
2
v
+
χ
1
′
v
u
1
p
f
1
p
1
-
p
∇
v
2
d
x
;we deduce from assumption (11) that I
1
≤
0 for any p
>
1, and the proof ends. In the same way, we achieve (41). This completes the proof of Lemma 6.In view ofv
∈
C
(
(
0
,
T
max
⁡
)
;
C
1
(
Ω
)
) and initial date v
_
<
v
0
<
v
¯, we consider T
∗
>
0 defined by(50)
T
∗
≔
T
m
a
x
,
if
v
≤
v
¯
in
0
,
T
max
⁡
,
T
v
,
if
v
L
∞
Ω
T
max
⁡
>
v
¯
,where T
v
<
T
max
⁡ satisfies that v
L
∞
(
Ω
T
v
)
≤
v
¯ and v
L
∞
(
Ω
T
v
+
δ
)
>
v
¯ for any δ
>
0.Lemma 7.
Lettingp
>
1 and for any t
<
T
∗, then the solutions to (1) satisfy(51)
1
p
-
1
d
d
t
∫
Ω
u
1
p
f
1
p
1
-
p
d
x
≤
-
κ
01
+
μ
1
∫
Ω
u
1
p
+
1
f
1
p
-
p
d
x
+
κ
1
1
-
τ
+
p
p
-
1
μ
1
∫
Ω
u
1
p
f
1
p
1
-
p
d
x
,
(52)
1
p
-
1
d
d
t
∫
Ω
u
2
p
f
2
p
1
-
p
d
x
≤
-
κ
02
+
μ
2
∫
Ω
u
2
p
+
1
f
2
p
-
p
d
x
+
κ
2
1
-
τ
+
p
p
-
1
μ
2
∫
Ω
u
2
p
f
2
p
1
-
p
d
x
.Proof.
In order to prove Lemma7, we consider the terms I
2 and I
3 appearing in the proof of Lemma 6.
By the nonnegativity foru
1
,
u
2 and assumption (6), we obtain(53)
-
h
u
1
,
u
2
,
v
≤
-
u
1
+
κ
1
χ
1
v
.Multiplying (53) by u
1
p
f
1
p
1
-
p
χ
1
(
v
) and integrating over Ω yield(54)
-
∫
Ω
u
1
p
f
1
p
1
-
p
χ
1
v
h
u
1
,
u
2
,
v
d
x
≤
-
∫
Ω
u
1
p
+
1
f
1
p
1
-
p
χ
1
v
d
x
+
κ
1
∫
Ω
u
1
p
f
1
p
1
-
p
d
x
.By the monotonic property of exponential function and assumption (7), we have (55)
0
<
κ
01
≤
χ
1
v
exp
⁡
∫
v
_
v
χ
1
s
d
s
≤
χ
1
v
f
1
p
v
.It follows that (54) becomes(56)
-
∫
Ω
u
1
p
f
1
p
1
-
p
χ
1
v
h
u
1
,
u
2
,
v
d
x
≤
-
κ
01
∫
Ω
u
1
p
+
1
f
1
p
-
p
d
x
+
κ
1
∫
Ω
u
1
p
f
1
p
1
-
p
d
x
.Multiplying (56) by (
p
-
1
)
/
(
1
-
τ
(
(
p
-
1
)
/
p
)
), we get(57)
I
2
≤
p
-
1
1
-
τ
p
-
1
/
p
-
κ
01
∫
Ω
u
1
p
+
1
f
1
p
-
p
d
x
+
κ
1
∫
Ω
u
1
p
f
1
p
1
-
p
d
x
≤
p
-
1
-
κ
01
∫
Ω
u
1
p
+
1
f
1
p
-
p
d
x
+
κ
1
1
-
τ
∫
Ω
u
1
p
f
1
p
1
-
p
d
x
.For the term I
3, we easily achieve(58)
I
3
≤
p
μ
1
∫
Ω
u
1
p
f
1
p
1
-
p
d
x
-
∫
Ω
u
1
p
+
1
f
1
p
-
p
d
x
≤
p
-
1
μ
1
∫
Ω
u
1
p
f
1
p
1
-
p
d
x
-
∫
Ω
u
1
p
+
1
f
1
p
-
p
d
x
+
μ
1
∫
Ω
u
1
p
f
1
p
1
-
p
d
x
.Combining (44), (57), and (58) leads to(59)
d
d
t
∫
Ω
u
1
p
f
1
p
1
-
p
d
x
≤
p
-
1
-
κ
01
∫
Ω
u
1
p
+
1
f
1
p
-
p
d
x
+
κ
1
1
-
τ
∫
Ω
u
1
p
f
1
p
1
-
p
d
x
+
p
-
1
μ
1
∫
Ω
u
1
p
f
1
p
1
-
p
d
x
-
∫
Ω
u
1
p
+
1
f
1
p
-
p
d
x
+
μ
1
∫
Ω
u
1
p
f
1
p
1
-
p
d
x
,which implies that (51) holds. The same arguments are used to achieve (52). The proof ends.To establish the uniform boundedness of solutions, we apply an iterative method based on theL
p-n
o
r
m of solutions inspired by the Moser-Alikakos type iterations (see [10, 13]).Lemma 8.
For anyt
<
T
∗, then the solutions to (1) are bounded by(60)
u
1
L
∞
Ω
≤
f
1
v
¯
max
⁡
κ
1
1
-
τ
+
μ
1
κ
01
+
μ
1
-
1
,
u
10
L
∞
Ω
,
(61)
u
2
L
∞
Ω
≤
f
2
v
¯
max
⁡
κ
2
1
-
τ
+
μ
2
κ
02
+
μ
2
-
1
,
u
20
L
∞
Ω
.Proof.
Denote∫
Ω
‍
u
1
p
f
1
p
1
-
p
d
x by Ψ
(
t
). Taking arbitrary positive constant ρ
>
0, then it holds that(62)
Ψ
t
=
∫
f
1
p
≤
ρ
κ
01
+
μ
1
u
1
u
1
p
f
1
p
1
-
p
d
x
+
∫
f
1
p
>
ρ
κ
01
+
μ
1
u
1
u
1
p
f
1
p
1
-
p
d
x
≤
ρ
κ
01
+
μ
1
∫
f
1
p
≤
ρ
κ
01
+
μ
1
u
1
u
1
p
+
1
f
1
p
-
p
d
x
+
ρ
κ
01
+
μ
1
1
-
p
∫
f
1
p
>
ρ
κ
01
+
μ
1
u
1
u
1
d
x
≤
ρ
κ
01
+
μ
1
∫
Ω
u
1
p
+
1
f
1
p
-
p
d
x
+
ρ
κ
01
+
μ
1
1
-
p
∫
Ω
u
1
d
x
.Hence(63)
-
κ
01
+
μ
1
∫
Ω
u
1
p
+
1
f
1
p
-
p
d
x
≤
-
1
ρ
Ψ
t
+
ρ
-
p
κ
01
+
μ
1
1
-
p
∫
Ω
u
1
d
x
.Substituting (63) into (51), we have(64)
1
p
-
1
d
d
x
Ψ
t
≤
κ
1
1
-
τ
+
p
p
-
1
μ
1
-
1
ρ
Ψ
t
+
ρ
-
p
κ
01
+
μ
1
1
-
p
∫
Ω
u
1
d
x
.Take 1
/
ρ
>
κ
1
/
(
1
-
τ
)
+
(
p
/
(
p
-
1
)
)
μ
1. Noticing Lemma 5 and initial date u
10
∈
C
1
(
Ω
¯
), we apply the Gronwall Lemma to obtain(65)
Ψ
t
≤
max
⁡
ρ
-
p
κ
01
+
μ
1
1
-
p
1
ρ
-
κ
1
1
-
τ
-
p
p
-
1
μ
1
-
1
∫
Ω
u
1
d
x
,
Ψ
0
;taking limits leads to(66)
lim
p
→
∞
⁡
Ψ
1
/
p
t
≤
max
⁡
ρ
-
1
κ
01
+
μ
1
-
1
,
u
10
L
∞
Ω
.Since(67)
Ψ
t
=
∫
Ω
u
1
p
f
1
p
1
-
p
v
d
x
≥
f
1
p
1
-
p
v
¯
∫
Ω
u
1
p
d
x
,combining (66) and (67) and taking limits as ρ
-
1
→
κ
1
/
(
1
-
τ
)
+
μ
1, it is found that (60) holds. The same processes are used to prove (61), which ends the proof of Lemma 8.Remark 9.
In the above arguments, we used well such fact: LetΩ be bounded and regular domain of R
n. If u
L
p
(
Ω
)
≤
C, where C is some positive constant and independent of p, then u
L
p
(
Ω
)
→
u
L
∞
(
Ω
) as p
→
∞.To end the arguments of uniform boundedness of solutions, it follows to prove thatT
∗
=
T
m
a
x and T
m
a
x
=
+
∞ by contradiction.As in Lemmas4–8, we define u
^
1
,
u
^
2 and v
^ for the unique solutions in (
0
,
T
^
m
a
x
) to(68)
u
^
1
t
=
Δ
u
^
1
-
∇
·
u
^
1
χ
1
v
^
∇
v
+
μ
1
u
^
1
g
1
u
^
1
,
u
^
2
,
x
∈
Ω
,
t
>
0
,
u
^
2
t
=
Δ
u
^
2
-
∇
·
u
^
2
χ
2
v
^
∇
v
^
+
μ
2
u
2
g
2
u
^
1
,
u
^
2
,
x
∈
Ω
,
t
>
0
,
v
^
t
=
τ
Δ
v
^
+
h
^
u
^
1
,
u
^
2
,
v
^
,
x
∈
Ω
,
t
>
0
,
∂
u
^
1
∂
ν
=
∂
u
^
2
∂
ν
=
∂
v
^
∂
ν
=
0
,
x
∈
∂
Ω
,
t
>
0
,
u
^
1
x
,
0
=
u
10
x
,
u
^
2
x
,
0
=
u
20
x
,
v
^
x
,
0
=
v
0
x
,
x
∈
Ω
,where T
^
max
⁡ is defined in the same fashion as (32) and the truncation h
^
(
u
1
,
u
2
,
v
) is defined as follows:(69)
h
^
u
1
,
u
2
,
v
≔
h
u
1
,
u
2
,
v
,
if
v
≤
v
¯
,
h
u
1
,
u
2
,
v
¯
,
if
v
>
v
¯
.Then it is easy to see that(70)
u
^
1
L
∞
Ω
≤
f
1
v
¯
max
⁡
κ
1
1
-
τ
+
μ
1
κ
01
+
μ
1
-
1
,
u
10
L
∞
Ω
,
u
^
2
L
∞
Ω
≤
f
2
v
¯
max
⁡
κ
2
1
-
τ
+
μ
2
κ
02
+
μ
2
-
1
,
u
20
L
∞
Ω
,as far as v
^
≤
v
¯.By analogy withT
∗, we define(71)
T
^
∗
≔
T
^
m
a
x
,
if
v
^
≤
v
¯
i
n
0
,
T
^
max
⁡
,
T
^
v
^
,
if
v
^
L
∞
Ω
T
^
max
⁡
>
v
¯
,where T
^
v
^
<
T
^
m
a
x satisfies that v
^
L
∞
(
Ω
T
^
v
^
)
≤
v
¯ and v
^
L
∞
(
Ω
T
^
v
^
+
δ
)
>
v
¯ for any δ
>
0. To prove T
^
∗
=
T
^
m
a
x, we suppose (72)
T
^
∗
<
T
^
max
⁡
;then h
^
(
u
^
1
,
u
^
2
,
v
^
)
≤
h
^
(
u
¯
1
,
u
¯
2
,
v
¯
)
=
h
(
u
¯
1
,
u
¯
2
,
v
¯
)
<
0. It follows to easily check that v
¯ is a strict upper-solution to(73)
v
^
t
-
τ
Δ
v
^
=
h
^
u
^
1
,
u
^
2
,
v
^
i
n
Ω
×
0
,
T
^
∗
,
∂
v
^
∂
ν
=
0
o
n
∂
Ω
×
0
,
T
^
∗
,
v
^
x
,
0
=
v
0
x
i
n
Ω
.This impliesv
^
(
x
,
t
)
<
v
¯
,
∀
x
∈
Ω
,
t
∈
(
0
,
T
^
∗), which contradicts (72) and proves T
^
∗
=
T
^
max
⁡ as well as T
^
max
⁡
=
∞.Thanks tov
^
≤
v
¯, we show that (
u
^
1
,
u
^
2
,
v
^
) is also a solution to (6). So we have(74)
T
∗
=
T
max
⁡
≥
T
^
max
⁡
=
∞
.The proof of Theorem 3 is completed.



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*Source: 5015246-2016-12-08.xml*