# Blow-Up and Global Existence for a Quasilinear Parabolic System

Authors: Chunchen Wu
Journal: Discrete Dynamics in Nature and Society
 (2014)

---

## Abstract

The problem of solutions to a class of quasilinear coupling parabolic system was studied. By constructing weak upper-solutions and weak lower-solutions, we obtain the global existence and blow-up of solutions under appropriate conditions.


---

## Body

## 1. Introduction and Main Result

In this paper, we consider global existence of nonnegative solutions for a class of nonlocal degenerate quasilinear parabolic system as follows:(1)
u
t
=
Δ
u
m
+
u
p
1
∥
v
∥
α
p
2
,
v
t
=
Δ
v
n
+
v
q
1
∥
w
∥
β
q
2
,
w
t
=
Δ
w
h
+
w
r
1
∥
u
∥
γ
r
2
,
(
x
,
t
)
∈
Ω
×
(
0
,
T
)
,
u
(
x
,
t
)
=
v
(
x
,
t
)
=
w
(
x
,
t
)
=
0
,
(
x
,
t
)
∈
∂
Ω
×
(
0
,
T
)
,
u
(
x
,
0
)
=
u
0
(
x
)
,
v
(
x
,
0
)
=
v
0
(
x
)
,
w
(
x
,
0
)
=
w
0
(
x
)
,
x
∈
Ω
,
where Ω
∈
R
N is bounded region; Ω has smooth boundary ∂
Ω; the parameters m
,
n
,
h
>
1,  α
,
β
,
γ
≥
1,  p
1, q
1, and r
1
≥
0,  p
2
,
q
2
,
r
2
>
0; the initial functions u
0, v
0, and w
0 are nonnegative and bounded, and ∥
·
∥
α
α
=
∫
Ω
‍
|
·
|
α
d
x.Quasilinear parabolic system is the model for many problems in the scientific field, for example, gas flow model in some seepage medium, some biological population growth model. In recent years, there are many papers to investigate the nonlinear parabolic equation and many excellent results are obtained (see [1–10] and the references cited therein). In this paper, we expand the equation of [10] into 3 and discuss the global existence and blow-up of the solutions for problem (1), and the main results of this paper are the following.Theorem 1.
If one of the following conditions holds, system (1) has global solutions:(1)
m
>
p
1, n
>
q
1, h
>
r
1, and  p
2
q
2
r
2
<
(
m
-
p
1
)
(
n
-
q
1
)
(
h
-
r
1
);
(2)
m
>
p
1, n
>
q
1, h
>
r
1, and p
2
q
2
r
2
=
(
m
-
p
1
)
(
n
-
q
1
)
(
h
-
r
1
), and the magnitude of the region Ω is sufficiently small;
(3)
Ifm
≤
p
1,  n
≤
q
1, or h
≤
r
1, or if m
>
p
1, n
>
q
1
,
h
>
r
1, and p
2
q
2
r
2
>
(
m
-
p
1
)
(
n
-
q
1
)
(
h
-
r
1
), and the initial data u
0, v
0, and w
0 are sufficiently small.Theorem 2.
If one of the following conditions holds, the solution of system (1) blows up in finite time:(1)
m
>
p
1, n
>
q
1,  h
>
r
1, and  p
2
q
2
r
2
>
(
m
-
p
1
)
(
n
-
q
1
)
(
h
-
r
1
), and the initial data u
0, v
0, and w
0 are sufficiently large;
(2)
m
≤
p
1,   n
≤
q
1, or h
≤
r
1, and the initial data u
0, v
0, and w
0 are sufficiently large.


## 2. Proof of Global Existence

As we know, nonlocal degenerate quasilinear parabolic system may not have classical solutions. Similar to the proof in [10] (see page 388-389), we can obtain that system (1) has nonnegative weak upper-solutions and nonnegative weak lower-solutions, and the following comparison principle holds.Lemma 3 (comparison principle).
Suppose(
u
¯
,
v
¯
,
w
¯
),  (
u
_
,
v
_
,
w
_
) are the nonnegative weak upper-solutions and nonnegative weak lower-solutions of system (1), if
(2)
u
¯
,
v
¯
,
w
¯
≥
δ
>
0
,
or
u
_
,
v
_
,
w
_
≥
δ
>
0
,
one has (
u
_
(
x
,
t
)
,
v
_
(
x
,
t
)
,
w
_
(
x
,
t
)
)
≤
(
u
¯
(
x
,
t
)
,
v
¯
(
x
,
t
)
,
w
¯
(
x
,
t
)
) for every (
x
,
t
)
∈
Ω
×
(
0
,
T
).Therefore, in order to prove Theorem1, we only show that, for all T
>
0, there exists a positive bounded weak upper-solutions. Let φ
(
x
) be the unique positive solution of linear elliptic equation as follows:
(3)
-
Δ
φ
=
1
,
x
∈
Ω
;
φ
(
x
)
=
0
,
x
∈
∂
Ω
.DenoteC
=
max
⁡
x
∈
Ω
φ
(
x
), then 0
≤
φ
(
x
)
≤
C. Define u
¯
(
x
,
t
)
,
v
¯
(
x
,
t
)
,
w
¯
(
x
,
t
) as follows:
(4)
u
¯
=
(
k
(
φ
(
x
)
+
1
)
)
l
1
,
v
¯
=
(
k
(
φ
(
x
)
+
1
)
)
l
2
,
w
¯
=
(
k
(
φ
(
x
)
+
1
)
)
l
3
,
where l
1
,
l
2
,
l
3
<
1 and satisfy m
l
1
,
n
l
2
,
h
l
3
<
1. k is a positive constant to be determined suitably. For all T
>
0, u
¯, v
¯, and w
¯ are bounded, and u
¯
≥
k
l
1
,
v
¯
≥
k
l
2, and w
¯
≥
k
l
3. By deducing, we have
(5)
u
¯
t
-
Δ
u
¯
m
=
-
k
m
l
1
[
m
l
1
(
m
l
1
-
1
)
(
φ
+
1
)
m
l
1
-
2
|
∇
φ
|
2
+
m
l
1
(
φ
+
1
)
m
l
1
-
1
Δ
φ
]
≥
m
l
1
(
C
+
1
)
m
l
1
-
1
k
m
l
1
,
u
¯
p
1
∥
v
¯
∥
α
p
2
=
(
k
(
φ
+
1
)
)
l
1
p
1
∥
(
k
(
φ
+
1
)
)
l
2
∥
α
p
2
≤
k
l
1
p
1
+
l
2
p
2
(
C
+
1
)
l
1
p
1
+
l
2
p
2
|
Ω
|
p
2
/
α
.
Similarly
(6)
v
¯
t
-
Δ
v
¯
n
≥
n
l
2
(
C
+
1
)
n
l
2
-
1
k
n
l
2
,
v
¯
q
1
∥
w
¯
∥
β
q
2
≤
k
l
2
q
1
+
l
3
q
2
(
C
+
1
)
l
2
q
1
+
l
3
q
2
|
Ω
|
q
2
/
β
,
(7)
w
¯
t
-
Δ
w
¯
h
≥
h
l
3
(
C
+
1
)
h
l
3
-
1
k
h
l
3
,
w
¯
r
1
∥
u
¯
∥
γ
r
2
≤
k
l
3
r
1
+
l
1
r
2
(
C
+
1
)
l
3
r
1
+
l
1
r
2
|
Ω
|
r
2
/
γ
.Denote(8)
C
1
=
(
|
Ω
|
p
2
/
α
m
l
1
(
C
+
1
)
l
1
p
1
+
l
2
p
2
-
m
l
1
+
1
)
1
/
(
m
l
1
-
l
1
p
1
-
l
2
p
2
)
;
C
2
=
(
|
Ω
|
q
2
/
β
n
l
2
(
C
+
1
)
l
2
q
1
+
l
3
q
2
-
n
l
2
+
1
)
1
/
(
n
l
2
-
l
2
q
1
-
l
3
q
2
)
;
C
3
=
(
|
Ω
|
r
2
/
γ
h
l
3
(
C
+
1
)
l
3
r
1
+
l
1
r
2
-
h
l
3
+
1
)
1
/
(
h
l
3
-
l
3
r
1
-
l
1
r
2
)
.
(1) If m
>
p
1, n
>
q
1, h
>
r
1, and p
2
q
2
r
2
<
(
m
-
p
1
)
(
n
-
q
1
)
(
h
-
r
1
), there exist positive constants l
1, l
3 such that
(9)
h
-
r
1
r
2
>
l
1
l
3
>
p
2
q
2
(
m
-
p
1
)
(
n
-
q
1
)
and m
l
1
,
n
l
2
,
h
l
3
<
1. And there exists positive constant l
2 such that
(10)
l
1
l
2
>
p
2
m
-
p
1
,
l
2
l
3
>
q
2
n
-
q
1
.
Thus,
(11)
p
2
l
2
<
(
m
-
p
1
)
l
1
,
q
2
l
3
<
(
n
-
q
1
)
l
2
,
r
2
l
1
<
l
3
(
h
-
r
1
)
.
From (11), we can choose k sufficiently large, such that k
>
max
⁡
{
C
1
,
C
2
,
C
3
} and
(12)
(
k
(
φ
(
x
)
+
1
)
)
l
1
≥
u
0
(
x
)
,
(
k
(
φ
(
x
)
+
1
)
)
l
2
≥
v
0
(
x
)
,
(
k
(
φ
(
x
)
+
1
)
)
l
3
≥
w
0
(
x
)
.
By (4)–(12) we obtain that (
u
¯
(
x
,
t
)
,
v
¯
(
x
,
t
)
,
w
¯
(
x
,
t
)
) is the weak upper-solutions of system (1).(2) Ifm
>
p
1, n
>
q
1, h
>
r
1, and p
2
q
2
r
2
=
(
m
-
p
1
)
(
n
-
q
1
)
(
h
-
r
1
), there exist positive constants l
1, l
3 such that
(13)
h
-
r
1
r
2
=
l
1
l
3
=
p
2
q
2
(
m
-
p
1
)
(
n
-
q
1
)
.
And there exists positive constant l
2 such that
(14)
l
1
l
2
=
p
2
m
-
p
1
,
l
2
l
3
=
q
2
n
-
q
1
.
Therefore,
(15)
p
2
l
2
=
(
m
-
p
1
)
l
1
,
q
2
l
3
=
(
n
-
q
1
)
l
2
,
r
2
l
1
=
l
3
(
h
-
r
1
)
.Without loss of generality, we assume that the domain we discuss is contained in a sufficiently large ballB; denote φ
B
(
x
) is the unique positive solution of the following linear elliptic equation:
(16)
-
Δ
φ
=
1
,
x
∈
B
;
φ
(
x
)
=
0
,
x
∈
∂
B
.
Let C
0
=
max
⁡
x
∈
B
φ
B
(
x
); hence C
≤
C
0. Suppose that |
Ω
| is sufficiently small such that
(17)
|
Ω
|
<
min
⁡
{
(
m
l
1
C
0
+
1
)
α
/
p
2
,
(
n
l
2
C
0
+
1
)
β
/
q
2
,
(
h
l
3
C
0
+
1
)
γ
/
r
2
}
.
In addition, we choose k large enough, such that k satisfies (12); then from (5), (6), and (12)–(17) we obtain that (
u
¯
(
x
,
t
)
,
v
¯
(
x
,
t
)
,
w
¯
(
x
,
t
)
) is the weak upper-solutions of system (1).(3) At last, ifm
≤
p
1,  n
≤
q
1, or h
≤
r
1, or if m
>
p
1, n
>
q
1, h
>
r
1, and p
2
q
2
r
2
>
(
m
-
p
1
)
(
n
-
q
1
)
(
h
-
r
1
), there exist positive constants l
1, l
2, and l
3 such that
(18)
p
2
l
2
>
(
m
-
p
1
)
l
1
,
q
2
l
3
>
(
n
-
q
1
)
l
2
,
r
2
l
1
>
l
3
(
h
-
r
1
)
,
m
l
1
,
n
l
2
,
h
l
3
<
1
.
Therefore, from (18), we can choose k small enough,such that k
<
min
⁡
{
C
1
,
C
2
,
C
3
}. Furthermore, if u
0, v
0, and w
0 are sufficiently small to satisfy (12), then by (5), (6), (12), and (18) we obtain that (
u
¯
(
x
,
t
)
,
v
¯
(
x
,
t
)
,
w
¯
(
x
,
t
)
) is the weak upper-solutions of system (1).This completes the proof of Theorem1.


## 3. Proof of Blow-Up

In this section we will prove Theorem2, so we construct blow-up positive weak lower-solutions of system (1). Let λ
1, ϕ
(
x
) be the first eigenvalue and corresponding eigenfunction of the eigenvalue problem as follows:
(19)
-
Δ
ϕ
(
x
)
=
λ
ϕ
(
x
)
,
x
∈
Ω
;
ϕ
(
x
)
=
0
,
x
∈
∂
Ω
.
Then λ
1
>
0. Standardized ϕ
(
x
), such that ϕ
(
x
)
|
Ω
>
0 and max
⁡
x
∈
Ω
ϕ
(
x
)
=
1, then (
∂
ϕ
/
∂
n
)
|
∂
Ω
<
0, where n is the outer normal direction of ∂
Ω, suppose Ω
1
⊂
⊂
Ω is a compact subset of Ω, if any solution (
u
,
v
,
w
) blows up in Ω
1, also (
u
,
v
,
w
) blows up in Ω.Define functions(20)
u
_
(
x
,
t
)
=
(
s
(
t
)
ϕ
(
x
)
)
l
1
,
v
_
(
x
,
t
)
=
(
s
(
t
)
ϕ
(
x
)
)
l
2
,
w
_
(
x
,
t
)
=
(
s
(
t
)
ϕ
(
x
)
)
l
3
,
where l
1, l
2, and l
3 satisfy m
l
1
,
n
l
2
,
h
l
3
>
1, and s
(
t
) is the solution of the initial value problem as follows:
(21)
s
′
(
t
)
=
k
s
d
,
s
(
0
)
=
b
.
Here k
,
b
>
0,   d
>
1 is to be determined suitably. It is clear that s
(
t
)
≥
b and s
(
t
) are unbounded in finite time, and
(22)
u
_
(
x
,
t
)
≥
(
ρ
b
)
l
1
>
0
,
v
_
(
x
,
t
)
≥
(
ρ
b
)
l
2
>
0
,
w
_
(
x
,
t
)
≥
(
ρ
b
)
l
3
>
0
,
x
∈
Ω
1
,
t
>
0
,
where ρ
=
min
⁡
Ω
¯
ϕ
(
x
)
>
0; by computing we obtain that
(3)
Δ
u
_
m
+
u
_
p
1
∥
v
_
∥
α
p
2
=
s
m
l
1
[
m
l
1
(
m
l
1
-
1
)
ϕ
m
l
1
-
2
|
∇
ϕ
|
2
+
m
l
1
ϕ
m
l
1
-
1
Δ
ϕ
]
+
s
l
1
p
1
+
l
2
p
2
ϕ
l
1
p
1
∥
ϕ
l
2
∥
α
p
2
≥
-
λ
1
m
l
1
s
m
l
1
ϕ
m
l
1
+
C
1
s
l
1
p
1
+
l
2
p
2
=
l
1
s
l
1
-
1
ϕ
l
1
(
C
1
s
l
1
p
1
+
l
2
p
2
-
l
1
+
1
ϕ
-
l
1
l
1
-
λ
1
m
s
m
l
1
-
l
1
+
1
ϕ
m
l
1
-
l
1
+
C
1
s
l
1
p
1
+
l
2
p
2
-
l
1
+
1
ϕ
-
l
1
l
1
)
≥
l
1
s
l
1
-
1
ϕ
l
1
C
1
l
1
s
m
l
1
-
l
1
+
1
×
(
-
λ
1
m
l
1
C
1
+
s
l
1
p
1
+
l
2
p
2
-
m
l
1
)
,
u
_
t
=
l
1
s
l
1
-
1
ϕ
l
1
s
′
(
t
)
.
Similarly
(24)
Δ
v
_
n
+
v
_
q
1
∥
w
_
∥
β
q
2
≥
l
2
s
l
2
-
1
ϕ
l
2
C
2
l
2
s
n
l
2
-
l
2
+
1
(
-
λ
1
n
l
2
C
2
+
s
l
2
q
1
+
l
3
q
2
-
n
l
2
)
,
v
_
t
=
l
2
s
l
2
-
1
ϕ
l
2
s
′
(
t
)
,
Δ
w
_
h
+
w
_
r
1
∥
u
_
∥
γ
r
2
≥
l
3
s
l
3
-
1
ϕ
l
3
C
3
l
3
s
h
l
3
-
l
3
+
1
(
-
λ
1
h
l
3
C
3
+
s
l
3
r
1
+
l
1
r
2
-
h
l
3
)
,
w
_
t
=
l
3
s
l
3
-
1
ϕ
l
3
s
′
(
t
)
,
where
(25)
C
1
≡
ϕ
l
1
p
1
∥
ϕ
l
2
∥
α
p
2
,
C
2
≡
ϕ
l
2
q
1
∥
ϕ
l
3
∥
β
q
2
,
C
3
≡
ϕ
l
3
r
1
∥
ϕ
l
1
∥
γ
r
2
.
(1) If m
>
p
1, n
>
q
1, h
>
r
1, and p
2
q
2
r
2
>
(
m
-
p
1
)
(
n
-
q
1
)
(
h
-
r
1
), there exist positive constants l
1, l
2, and l
3 satisfying m
l
1
,
n
l
2
,
h
l
3
>
1 and
(26)
p
2
l
2
>
(
m
-
p
1
)
l
1
,
q
2
l
3
>
(
n
-
q
1
)
l
2
,
r
2
l
1
>
l
3
(
h
-
r
1
)
.
Choose
(27)
k
=
min
⁡
{
C
1
l
1
,
C
2
l
2
,
C
3
l
3
}
,
d
=
min
⁡
{
m
l
1
-
l
1
+
1
,
n
l
2
-
l
2
+
1
,
h
l
3
-
l
3
+
1
}
,
b
=
max
⁡
{
(
1
+
λ
1
m
l
1
C
1
)
1
/
(
l
1
p
1
+
l
2
p
2
-
m
l
1
)
,
(
1
+
λ
1
n
l
2
C
2
)
1
/
(
l
2
q
1
+
l
3
q
2
-
n
l
2
)
,
(
1
+
λ
1
h
l
3
C
3
)
1
/
(
l
3
r
1
+
l
1
r
2
-
h
l
3
)
}
.
From (26)-(27) and m
,
n
,
h
>
1, we have k
>
0,  d
>
1, and b
>
1. Suppose that u
0, v
0, and w
0 are sufficiently large to satisfy
(28)
u
0
(
x
)
≥
(
b
ϕ
)
l
1
,
v
0
(
x
)
≥
(
b
ϕ
)
l
2
,
w
0
(
x
)
≥
(
b
ϕ
)
l
3
.
From (20)–(28),we obtain that (u
_
,
v
_
,
w
_) is positive weak lower-solutions of system (1) in Ω
1.(2) Ifm
≤
p
1, n
≤
q
1, or h
≤
r
1, (26) still holds, the results are obtained by the same methods.Thus completes the proof of Theorem2.



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*Source: 821503-2014-02-13.xml*