báo cáo hóa học: " Non-Newtonian polytropic filtration systems with nonlinear boundary conditions" pot

RESEARC H Open Access
Non-Newtonian polytropic filtration systems with
nonlinear boundary conditions
Wanjuan Du
*
and Zhongping Li
* Correspondence:

College of Mathematic and
Information, China West Normal
University, Nanchong 637002, PR
China
Abstract
This article deals with the global existence and the blow-up of non-Newtonian
polytropic filtration systems with nonlinear boundary conditions. Necessary and
sufficient conditions on the global existence of all positive (weak) solutions are
obtained by constructing various upper and lower solutions.
Mathematics Subject Classification (2000)
35K50, 35K55, 35K65
Keywords: Polytropic filtration systems, Nonlinear boundary conditions, Global exis-
tence, Blow-up
Introduction
In this article, we study the global existen ce and the blow-up of non-Newtonian poly-
tropic filtration systems with nonlinear boundary conditions
(u
k
i
i
)
t
= 

m
i
u
i
(i =1, , n), x ∈ , t > 0,
∇
m
i
u
i
· ν =
n

j=1
u
m
ij
j
(i =1, , n), x ∈ ∂, t > 0
,
u
i
(
x,0
)
= u
i0
(
x
)

> 0
(
i =1, , n
)
, x ∈
¯
,
(1:1)
where

m
i
u
i
=div(|∇u
i
|
m
i
−1
∇u
i
)=
N

j
=1
(|∇u
i
|

m
i
−1
u
ix
j
)
x
j
, ∇
m
i
u
i
=(|∇u
i
|
m
i
−1
u
ix
1
, , |∇u
i
|
m
i
−1
u

ix
N
)
,
Ω ⊂ ℝ
N
is a bounded domain with smooth boundary ∂Ω, ν is the outward normal
vector on the boundary ∂Ω, and the constants k
i
, m
i
>0,m
ij
≥ 0, i, j = 1, , n; u
i0
(x)(i
= 1, , n) are positive C
1
functions, satisfying the compatibility conditions.
The particular feature of the equations in (1.1) is their power- and gradient-depen-
dent diffusibility. Such equations arise in some physical models, such as population
dynamics, chemical reactions, heat transfer, and so on. In particular, equations in (1.1)
maybeusedtodescribethenonstationaryflowsinaporousmediumoffluidswitha
power dependenc e of the tangential stress on the velocity of displacement under poly-
tropic conditions. In this case, the equations in (1.1) are called the non-Newtonian
polytropic filtration equations which have been intensively studied (see [1-4] and the
references therein). For the Neuman problem (1.1), the local existence of solutions in
time have been established; see the monograph [4].
Du and Li Boundary Value Problems 2011, 2011:2
/>© 2011 Du and Li; licensee Springer. This is an Open Access article distributed under the terms o f the Creative Commons Attr ibution

License (http://cr eativeco mmons.org/licenses/b y/2.0), which permits unrestricted use, distribution, and reproduction in any medium,
provided the original work is pro perly cited.
We note that most previous works deal with special cases of (1.1) (see [5-13]). For
example, Sun and Wang [7] studied system (1.1) with n = 1 (the single-equation case)
and showed that all positive (weak) solutions of (1.1) exist globally if and only if m
11
≤
k
1
when k
1
≤ m
1
; and exist g lobally if and only if
m
11
≤
m
1
(k
1
+1
)
m
1
+1
when k
1
>m
1

.In[13],
Wang studied the case n = 2 of (1.1) in one dimension. Recently, Li et al. [5] extended
the results of [13] into more general N-dimensional domain.
On the other hand, for systems involving more than two equations when m
i
=1(i =
1, , n), the special case k
i
=1(i = 1,. , n) (heat equations) is concerned by Wang a nd
Wang [9], and the case k
i
≤ 1(i = 1, , n) (porous medium equations) is discussed in
[12]. In both studies, they obtained the necessary and sufficient conditions to the glo-
bal existence of solutions. The fast-slow diffusi on equations (there exists i(i = 1, , n)
such that k
i
> 1) is studied by Qi et al. [6], and they obtained the necessary and suffi-
cient blow up conditions for the special case Ω = B
R
(0) (the ball centered at the origin
in ℝ
N
with radius R). However, for the general domain Ω,theyonlygavesomesuffi-
cient conditions to the global existence and the blow-up of solutions.
The aim of this article is to study the long-time behavior of solutions to systems (1.1)
and provide a simple criterion of the classification of global existence and nonexistence
of solutions for general powers k
i
m
i

, indices m
ij
, and number n.
Define
b
i
= min{k
i
,
m
i
(k
i
+1)
m
i
+1
}, b
ij
= b
i
δ
ij
, i, j = 1, , n
,
B =(b
i
j
)
n×n

, M =(m
i
j
)
n×n
, A = B − M.
Our main result is
Theorem. All positive solutions of (1.1) exist globally if and only if all of the principal
minor determinants of A are non-negative.
Remark. The conclusion of Theorem covers the results of [5-13]. Moreover, this
article provides the necessary and sufficient conditions to the global existence and the
blow-up of s olutions in the general domain Ω. Therefore, this article improves the
results of [6].
The rest of this article is organized as follows. Some preliminaries will b e given in
next section. The above theorem will be proved in Section 3.
Preliminaries
As is we ll known that degenerate and singular equations need not possess classical
solutions, we give a precise definition of a weak solution to (1.1).
Definition. Let T >0and Q
T
= Ω ×(0,T ]. Avectorfunction(u
1
(x, t), , u
n
(x, t)) is
called a weak upper (or lower) solution to (1.1) in Q
T
if
(i).
u

i
(
x, t
)(
i =1, , n
)
∈ L
∞
(
0, T; W
1,∞
(

))
∩ W
1,2
(
0, T; L
2
(

))
∩ C
(
Q
T
)
;
(ii).(u
1

(x, 0), , u
n
(x, 0)) ≥ (≤)(u
10
(x), , u
n0
(x));
(iii). for any positive functions ψ
i
(i = 1, , n) Î L
1
(0, T; W
1,2
(Ω)) ∩ L
2
(Q
T
), we have


Q
T
[(u
k
i
i
)
t
ψ
i

+ ∇
m
i
u
i
·∇ψ
i
] dxdt ≥ (≤)

T
0

∂
n

j
=1
u
m
ij
j
ψ
i
dsdt (i =1, , n)
.
In particular,(u
1
(x, t), , u
n
(x, t)) is called a weak solution of (1.1) if it is both a weak

upper and a lower solution. For every T < ∞, if (u
1
(x, t), , u
n
(x, t)) is a solution of (1.1)
in Q
T
, then we say that (u
1
(x, t), , u
n
(x, t)) is global.
Du and Li Boundary Value Problems 2011, 2011:2
/>Page 2 of 11
Lemma 2.1 (Comparison Principle.) Assume that u
i0
(i = 1, , n) are positive
C
1
(
¯

)
function s and (u
1
, , u
n
) is any weak solution of (1.1). Also assum e that (u
1
, , u

n
)
≥ (δ, , δ)>0and
(
¯
u
1
, ,
¯
u
n
)
aretheloweranduppersolutionsof(1.1)inQ
T
,respec-
tively, with nonlinear boundary flux
(λ
−

n
j=1
u
−
m
1j
j
, , λ
−

n

j=1
u
−
m
nj
j
)
and
(
¯
λ

n
j=1
¯
u
m
1j
j
, ,
¯
λ

n
j=1
¯
u
m
nj
j

)
, where
0 <λ
−
< 1 <
¯
λ
. Then we have
(
¯
u
1
, ,
¯
u
n
) ≥ (u
1
, , u
n
) ≥ (u
−
1
, , u
−
n
)
in Q
T
.

When n = 2, the proof of Lemma 2.1 is given in [5]. When n >2,theproofis
similar.
For convenience, we denote
0 <λ
−
< 1 <
¯
λ
, which are fixed constants, and let
δ = min
1≤i≤n
{min
¯

u
i0
(x)} >
0
.
In the following, we describe three lemmas, which can be obtained directly from
Lemmas 2.7-2.9 in [6].
Lemma 2.2 Suppose all the principal minor determinants of A are non-neg ative. If A
is irreducible, then for any positive constant c, there exists a =(a
1
, , a
n
)
T
such that A
a ≥ 0 and a

i
>c (i = 1, , n).
Lemma 2.3 Suppose that all the lower-order principal minor determinants of A are
non-negative and A is irreducible. For any positive constant C, there exist large positive
constants L
i
(i = 1, , n) such that
n

j
=1
L
a
ij
j
≥ C (i =1, , n)
.
Lemma 2.4 Suppose that all the lower-order principal minor determinants of A are
non-negative and |A|<0.Then, A is irreducible and, for any positive constant C, there
exists a =(a
1
, , a
n
)
T
, with a
i
>0(i = 1, , n) such that
min{k
i

,
m
i
(k
i
+1)
m
i
+1
}α
i
−
n

j
=1
m
ij
α
j
< −C (i =1, , n)
.
Proof of Theorem
First, we note that if A is reducible, then the full system (1.1) can be reduced to several
sub-systems, independent of each other. Therefore, in the following, we assume that A
is irreducible. In addition, we suppose that k
1
- m
1
≤ k

2
- m
2
≤ ···k
n
- m
n
.
Let
ϕ
m
i
(x)(i =1, , n
)
be the first eigenfunction of
−
m
i
ϕ
m
i
= λϕ
m
i
m
i
(x)in, ϕ
m
i
(x)=0 on ∂


(3:1)
with the first eigenvalue
λ
m
i
, normalized by
||ϕ
m
i
(x)||
∞
=
1
,then
λ
m
i
>
0
,
ϕ
m
i
(x) > 0
in Ω and
ϕ
m
i
(x) ∈ W

1,m
i
+1
0
∩ C
1
(
)
and
∂ϕ
m
i
(x)
∂ν
<
0
on ∂Ω (see [14-16]).
Thus, there exist some positive constants
A
m
i
,
B
m
i
,
C
m
i
, and

D
m
i
such that
A
m
i
≤−
∂ϕ
m
i
(x)
∂ν
≤ B
m
i
, |∇ϕ
m
i
(x)|≥C
m
i
, x ∈ ∂; |∇ϕ
m
i
(x)|≤D
m
i
, x ∈
¯


.
(3:2)
Du and Li Boundary Value Problems 2011, 2011:2
/>Page 3 of 11
We also have


∇ϕ
m
i
(x)


≥ E
m
i
provided
x ∈{x ∈  : dist(x, ∂) ≤ ε
m
i
}
with
E
m
i
=
C
m
i

2
and some positive constant
ε
m
i
. For the fixed
ε
m
i
, there exists a positive constant
F
m
i
such that
ϕ
m
i
(x) ≥ F
m
i
if
x ∈{x ∈  : dist(x, ∂) >ε
m
i
}
.
Proof of the sufficiency. We divide this proof into three different cases.
Case 1. (k
i
<m

i
(i = 1, , n)). Let
¯
u
i
(x, t)=P
i
e
α
i
t
log

(1 − ϕ
m
i
(x))e
(k
i
−m
i
)α
i
t
m
i
+ Q
i

(i =1, , n)

,
(3:3)
where Q
i
satisfies
Q
i
log Q
i
≥
2(m
i
−k
i
)
m
i
(i =1, , n
)
, and constants P
i
, a
i
(i = 1, , n)
remain to be determined. Since
Q
i
log Q
i
≥

2
(
m
i
−
k
i
)
m
i
, by performing direct calculations,
we have
(
¯
u
k
i
i
)
t
≥ k
i
α
i
P
k
i
i
e
k

i
α
i
t

log((1 − ϕ
m
i
(x))e
(k
i
−m
i
)α
i
t
m
i
+ Q
i
)

k
i
+ k
i
α
i
P
k

i
i
e
k
i
α
i
t

log((1 − ϕ
m
i
(x))e
(k
i
−m
i
)α
i
t
m
i
+ Q
i
)

k
i
−1
×

k
i
−m
i
m
i
(1 − ϕ
m
i
(x))e
(k
i
−m
i
)α
i
t
m
i
(1 − ϕ
m
i
(x))e
(k
i
−m
i
)α
i
t

m
i
+ Q
i
≥
k
i
α
i
2
P
k
i
i
e
k
i
α
i
t

log((1 − ϕ
m
i
(x))e
(k
i
−m
i
)α

i
t
m
i
+ Q
i
)

k
i
≥
k
i
α
i
2
P
k
i
i
e
k
i
α
i
t
(log Q
i
)
k

i
,

m
i
¯
u
i
=
N

j=1
⎛
⎜
⎝
P
m
i
i
e
k
i
α
i
t
(−|∇ϕ
m
i
(x)|
m

i
−1
(ϕ
m
i
)
x
j
)
((1 − ϕ
m
i
(x))e
(k
i
−m
i
)α
i
t
m
i
+ Q
i
)
m
i
⎞
⎟
⎠

x
j
≤
λ
m
i
P
m
i
i
e
k
i
α
i
t
Q
m
i
i
in Ω × ℝ
+
. By setting
c
m
i
= C
m
i
if m

i
≥ 1,
c
m
i
= D
m
i
if m
i
< 1, we have one the bound-
ary that
∇
m
i
u
i
· ν ≥
P
m
i
i
c
m
i
−
1
m
i
A

m
i
(1 + Q
i
)
m
i
e
k
i
α
i
t
(i =1, , n),
n

j
=1
¯
u
m
ij
j
≤
n

j
=1
(P
j

log(1 + Q
j
))
m
ij
e

n
j=1
m
ij
α
j
t
(i =1, , n)
.
we have
∇
m
i
¯
u
i
· ν ≥
¯
λ
n

j
=1

¯
u
m
ij
j
(i =1, , n
)
if
P
m
i
i
c
m
i
−1
m
i
A
m
i
(1+Q
i
)
m
i
≥
¯
λ
n


j
=1
(P
j
log(1 + Q
j
))
m
ij
(i =1, , n
)
(3:4)
Du and Li Boundary Value Problems 2011, 2011:2
/>Page 4 of 11
and
k
i
α
i
≥
n

j
=1
m
ij
α
j (i =1, , n)
.

(3:5)
Note that k
i
<m
i
(i = 1, , n). From Lemmas 2.2 and 2.3, we know that inequalities
(3.4) and (3.5) hold for suitable choices of P
i
, a
i
(i = 1, , n). Moreover, if we choose
P
i
, a
i
to be large enough such that
P
i
log Q
i
≥||u
i0
||
∞
, α
i
≥
2λ
m
i

P
m
i
−k
i
i
k
i
Q
m
i
i
(log Q
i
)
k
i
,
then
u
i
(
x,0
)
≥ u
i
0
,
(
¯

u
k
i
i
)
t
≥ 
m
i
¯
u
i
(i =1, , n
)
. Therefore, we have proved that
(
¯
u
1
, ,
¯
u
n
)
is a global upper solution of the system (1.1). The global existence of solu-
tions to the problem (1.1) follows from the comparison principle.
Case 2. (k
i
≥ m
i

(i = 1, , n)). Let
¯
u
i
(x, t)=e
α
i
t
⎛
⎜
⎜
⎝
M +
¯
λ
1
m
i
e
−L
i
ϕ
m
i
e
(k
i
−m
i
)α

i
t
m
i
+1
(2M)
n

j=1
m
ij
m
i
L
−1
i
A
−
1
m
i
i
⎞
⎟
⎟
⎠
(i =1, , n)
,
(3:6)
where

A
i
= A
m
i
C
m
i
−
1
m
i
if m
i
≥ 1,
A
i
= A
m
i
D
m
i
−
1
m
i
if m
i
<1,

ϕ
m
i
,
A
m
i
,
B
m
i
,
C
m
i
are defined
in (3.1) and (3.2), a
i
(i = 1, , n) are positive constants that remain to be determined,
and
M =max
1≤i≤n
{1, ||u
i0
||
∞
}, L
i
=
¯

λ
1
m
i
2

n
j=1
m
ij
m
i
M

n
j=1
m
ij
−m
i
m
i
A
−
1
m
i
i
max


1,
2(k
i
−m
i
)
m
i
+1

.
Since -ye
-y
≥ -e
-1
for any y >0,weknowthat
−L
i
ϕ
m
i
e
(k
i
−m
i
)α
i
t
m

i
+1
e
−L
i
ϕ
m
i
e
(k
i
−m
i
)
α
i
t
m
i
+1
≥−e
−
1
.Thus,for(x, t) Î Ω × ℝ
+
,asimplecomputa-
tion shows that
(
¯
u

k
i
i
)
t
= k
i
α
i
e
k
i
α
i
t
⎛
⎝
M +
¯
λ
1
m
i
e
−L
i
ϕ
m
i
e

(k
i
−m
i
)α
i
t
m
i
+1
(2M)

n
j=1
m
ij
m
i
L
−1
i
A
−
1
m
i
i
⎞
⎠
k

i
+ k
i
e
k
i
α
i
t
⎛
⎝
M +
¯
λ
1
m
i
e
−L
i
ϕ
m
i
e
(k
i
−m
i
)α
i

t
m
i
+1
(2M)

n
j=1
m
ij
m
i
L
−1
i
A
−
1
m
i
i
⎞
⎠
k
i
−1
×
¯
λ
1

m
i
(2M)

n
j=1
m
ij
m
i
L
−1
i
A
−
1
m
i
i
(k
i
− m
i
)α
i
m
i
+1
(−L
i

ϕ
m
i
)e
(k
i
−m
i
)α
i
t
m
i
+1
e
−L
i
ϕ
m
i
e
(k
i
−m
i
)α
i
t
m
i

+1
≥
1
2
k
i
α
i
e
k
i
α
i
t
.
In addition, we have

m
i
¯
u
i
≤
¯
λλ
m
i
(2M)

n

j=1
m
ij
A
−1
i
ϕ
m
i
m
i
e
m
i
α
i
t
e
m
i
(k
i
−m
i
)α
i
t
m
i
+1

e
−L
i
m
i
ϕ
m
i
e
(k
i
−m
i
)α
i
t
m
i
+1
+
¯
λL
i
m
i
(2M)

n
j=1
m

ij
A
−1
i
e
k
i
α
i
t
e
−L
i
m
i
ϕ
m
i
e
(k
i
−m
i
)α
i
t
m
i
+1



∇ϕ
m
i


m
i
+1
≤
¯
λ(λ
m
i
+ L
i
m
i
D
m
i
+1
m
i
)(2M)

n
j=1
m
ij

A
−1
i
e
k
i
α
i
t
.
Du and Li Boundary Value Problems 2011, 2011:2
/>Page 5 of 11
Noting
ϕ
m
i
=0(i =1,2, , n
)
on ∂Ω, we have on the boundary that
∇
m
i
¯
u
i
· ν ≥
¯
λ(2M)
n


j=1
m
ij
e
m
i
(k
i
−m
i
)α
i
t
m
i
+1
,
n

j
=1
¯
u
m
ij
j
≤ (2M)
n

j=1

m
ij
e
n

j=1
m
ij
α
j
t
.
Then, we have
∇
m
i
¯
u
i
· ν ≥
¯
λ
n

j
=1
¯
u
m
ij

j
(i =1, , n
)
if
m
i
(k
i
−1)α
i
m
i
+1
≥
n

j
=1
m
ij
α
j (i =1, , n)
.
(3:7)
From Lemma 2.2, we know that inequalities (3.7) hold for suitable choices of a
i
(i =
1, , n). Moreover, if we choose ∞
i
to be large enough such that

α
i
≥ 2
¯
λ(λ
m
i
+ L
i
m
i
D
m
i
+1
m
i
)(2M)

n
j=1
m
ij
(k
i
A
i
)
−1
,

then
(
¯
u
k
i
i
)
t
≥ 
m
i
¯
u
i
(i =1, , n
)
. Therefore, we have shown that
(
¯
u
1
, ,
¯
u
n
)
is an
upper solution of (1.1) and exists globally. Therefore,
(

u
1
, , u
n
)
≤
(
¯
u
1
, ,
¯
u
n
)
,and
hence the solution (u
1
, , u
n
) of (1.1) exists globally.
Case 3. (k
i
<m
i
(i = 1, , s); k
i
≥ m
i
(i = s + 1, , n)). Let

¯
u
i
(
x, t
)(
i =1, , s
)
be as in
(3.3) and
¯
u
i
(x, t)=e
α
i
t
⎛
⎝
M
i
+
¯
λ
1
m
i
e
−L
i

ϕ
m
i
e
(k
i
−m
i
)α
i
t
m
i
+1
(2M
i
)
k
i
+1
m
i
+1
L
−1
i
A
−
1
m

i
i
⎞
⎠
(i = s +1, , n)
,
where
ϕ
m
i
, and A
i
are as in case 2. By Lemma 2.3, we choose P
i
≥ (log Q
i
)
-1
||u
i0
||
∞
(i
= 1, , s) and M
i
≥ max{1, ||u
i0
||
∞
}(i = s + 1, , n) such that

P
m
i
i
c
m
i
−1
m
i
A
m
i
(1+Q
i
)
m
i
≥
s

j=1
(P
j
log(1 + Q
j
))
m
ij
n


j=s+1
(2M
j
)
m
ij
(i =1, , s),
¯
λ(2M
i
)
m
i
(k
i
+1)
m
i
+1
≥
s

j
=1
(P
j
log(1 + Q
j
))

m
ij
n

j
=s+1
(2M
j
)
m
ij
(i = s +1, , n)
.
(3:8)
Set
L
i
=
¯
λ
1
m
i
2
k
i
+1
m
i
+1

M
k
i
−m
i
m
i
+1
i
A
−
1
m
i
i
max

1,
2(k
i
−m
i
)
m
i
+1

(i = s +1, , n)
.
Du and Li Boundary Value Problems 2011, 2011:2

/>Page 6 of 11
By similar arguments, in cases 1 and 2, we have on the boundary that
∇
m
i
¯
u
i
· ν ≥
P
m
i
i
c
m
i
−
1
m
i
A
m
i
(1+Q
i
)
m
i
e
k

i
α
i
t
(i =1, , s),
∇
m
i
¯
u
i
· ν ≥
¯
λ(2M
i
)
m
i
(k
i
+1)
m
i
+1
e
m
i
(k
i
−1)α

i
t
m
i
+1
(i = s +1, , n),
n

j
=1
¯
u
m
ij
j
≤
s

j
=1
(P
j
log(1 + Q
j
))
m
ij
n

j

=s+1
(2M
j
)
m
ij
e

n
j=1
m
ij
α
j
t
(i =1, , n)
.
Therefore employing (3.8), we see that
∇
m
i
¯
u
i
· ν ≥
¯
λ
n

j

=1
¯
u
m
ij
j
(i =1, , n
)
if we knew
k
i
α
i
≥

n
j=1
m
ij
α
j
(i =1, , s),
m
i
(k
i
−1)α
i
m
i

+1
≥

n
j=1
m
ij
α
j
(i = s +1,, , n)
.
(3:9)
We deduce from Lemma 2.2 that (3.9) holds for suitab le choices of a
i
(i = 1, , n).
Moreover, we can choose a
i
large enough to assure that
α
i
≥
2λ
m
i
P
m
i
−
k
i

i
k
i
Q
m
i
i
(log Q
i
)
k
i
,(i =1, , s),
α
i
≥ 2
¯
λ(λ
m
i
+ L
i
m
i
D
m
i
+1
m
i

)(2M
i
)
m
i
(k
i
+1)
m
i
+1
(k
i
A
i
)
−1
(i = s +1, , n)
,
Then, as in the calculations of cases 1 and 2, we have
(
¯
u
k
i
i
)
t
≥ 
m

i
¯
u
i
(i =1, , n
)
.
We prove that
(
¯
u
1
, ,
¯
u
n
)
is an upper solution of (1.1), so (u
1
, , u
n
) exists globally.
Proof of the necessity.
Without loss of generality, we first assume that all the lower-order principal minor
determinants of A are non-negative, and |A| < 0, for, if not, there exists some lth-
order (1 ≤ l <n) principal minor determinant detA
l × l
of A =(a
ij
)

n×n
which is negative.
Without loss of generality, we may consider that
A
l×l
=
⎛
⎜
⎜
⎝
a
11
a
1l
a
12
a
2l

a
l1
a
ll
⎞
⎟
⎟
⎠
and all of the sth-order (1 ≤ s ≤ l - 1) principal minor determinants detA
s × s
of A

l × l
are non-negative. Then, we consider the following problem:
(w
k
i
i
)
t
= 
m
i
w
i
(i =1, , l),
x ∈ , t > 0,
∇
m
i
w
i
· ν = δ

n
j=l+1
m
ij
n

j=1
w

m
ij
j
(i =1, , l),
x ∈ ∂, t > 0
,
w
i
(
x,0
)
= u
i0
(
x
)(
i =1, , l
)
, x ∈
¯
.
(3:10)
Note that
δ = min
1≤i≤n
{min
¯

u
i0

(x)} >
0
. If we can prove that the solution (w
1
, , w
l
)
of (3.10) blows up in finite time, then (w
1
, w
l
, δ, , δ) is a lower solution of (1.1) that
blows up in finite time. Therefore, the solution of (1.1) blows up in finite time.
We will complete the proof of the necessity of our theorem in three different cases.
Du and Li Boundary Value Problems 2011, 2011:2
/>Page 7 of 11
Case 1. (k
i
<m
i
(i = 1, , n)). Let
u
−
i
= Y
ρ
i
i
and Y
i

= ah
1+
1
m
i
(x)+(b − ct)
−γ
i
(i =1, , n),
(3:11)
where
h(x)=

N
i
=1
x
i
+ Nd +
1
,
d =max
{|
x
||
x ∈
¯

}
,

ρ
i
=
m
i
+
1
γ
i
m
i
−k
i
,
γ
i
=
(m
i
−k
i
)α
i
−1
m
i
,thea
i
are as given in Lemma 2.4 and satisfy
α

i
>
1
m
i
−k
i
,
b =max
1≤i≤n
{1, (
1
2
δ
1
ρ
i
)
−
1
γ
i
}, a = min
1≤i≤n

b
−γ
i
(2Nd +1)
−

1+m
i
m
i
,
⎛
⎝
λ
−
−1
[
(1 + m
i
)ρ
i
N
1
2
2
ρ
i
−1
m
i
]
m
i
(2Nd +1)
⎞
⎠

−
1
m
i
b
−

n
j=1
m
ij
α
j
m
i
⎫
⎪
⎪
⎬
⎪
⎪
⎭
,
c = min
1≤i≤n
{
a
m
i
ρ

m
i
−1
i
(1 +
1
m
i
)
m
i
N
m
i
+1
2
k
i
γ
i
}.
(3:12)
By direct computation for
(x, t) ∈  × (0,
b
c
)
, we have
(u
−

k
i
i
)
t
= ck
i
ρ
i
γ
i
Y
k
i
ρ
i
−1
i
(b − ct)
−(γ
i
+1)
, ∇u
−
i
= aρ
i
(1 +
1
m

i
)Y
ρ
i
−1
i
h
1
m
i
(x)(1, ,1),

m
i
u
−
i
=

N
j=1

(aρ
i
(1 +
1
m
i
))
m

i
N
m
i
−1
2
Y
m
i
(ρ
i
−1)
i
h(x)

x
j
=(aρ
i
(1 +
1
m
i
))
m
i
N
m
i
+1

2
Y
m
i
(ρ
i
−1)
i
+m
i
(ρ
i
− 1)ρ
m
i
i
(a(1 +
1
m
i
))
m
i
+1
N
m
i
+1
2
h

1+
1
m
i
(x)Y
m
i
(ρ
i
−1)−1
i
≥ (aρ
i
(1 +
1
m
i
))
m
i
N
m
i
+1
2
Y
k
i
ρ
i

−1
i
Y
m
i
(ρ
i
−1)−k
i
ρ
i
+1
i
≥ (u
−
k
i
i
)
t
(i =1, , n).
For
(x, t) ∈ ∂ × (0,
b
c
)
, we have
∇
m
i

u
−
i
· ν ≤ (aρ
i
(1 +
1
m
i
))
m
i
N
m
i
2
(2Nd +1)2
m
i
(ρ
i
−1)
(b − ct)
−m
i
(ρ
i
−1)γ
i
=(aρ

i
(1 +
1
m
i
))
m
i
N
m
i
2
(2Nd +1)2
m
i
(ρ
i
−1)
(b − ct)
−(k
i
α
i
+1)
(i =1, , n)
,
n

j
=1

u
−
m
ij
j
=
n

j
=1
Y
m
ij
ρ
j
j
≥
n

j
=1
(b − ct)
−

n
j=1
m
ij
α
j

(i =1, , n).
Thus, by (3.12) and Lemma 2.4, we have
∇
m
i
u
−
i
· ν ≤ λ
−
n

j
=1
u
−
m
ij
j
(i =1, , n)
.
We confirm that (u
1
, , u
n
) is a lower solution of (1.1), which blows up in finite time.
We know by the comparison principle that the solution (u
1
, , u
n

) blows up in finite
time.
Case 2. (k
i
≥ m
i
(i = 1, , n)). Let
d
m
i
= C
m
i
if m
i
<1,
d
m
i
= D
m
i
if m
i
≥ 1. for k
i
≥ m
i
(i
= 1, , n), set

Du and Li Boundary Value Problems 2011, 2011:2
/>Page 8 of 11
u
−
i
=
1
(b−ct)
α
i
e
−aϕ
m
i
(x)
(b−ct)
β
i
,
(3:13)
where a
i
(i = 1, , n) are to determined later and
β
i
=
(k
i
−m
i

)α
i
+1
m
i
+1
,
b =max
1≤i≤n
{1, δ
−
1
α
i
}
,
(3:14)
a = min
1≤i≤n

1, λ
−
1
m
i
(B
m
i
d
m

i
−1
m
i
)
−
1
m
i
b
−

n
j=1
m
ij
α
j
m
i

,
(3:15)
c = min
1≤i≤n

m
i
a
m

i
+1
E
m
i
+1
m
i
k
i
α
i
,
λ
m
i
(k
i
− m
i
)a
m
i
+1
F
m
i
+1
m
i

k
i
α
i

.
(3:16)
By a direct computation, for x Î Ω,0<t <c/b, we obtain that
(u
−
k
i
i
)
t
= k
i
α
i
ce
−ak
i
ϕ
m
i
(x)
(b−ct)
β
i
(b − ct)

−(k
i
α
i
+1)
−
e
−ak
i
ϕ
m
i
(x)
(b−ct)
β
i
ak
i
β
i
cϕ
m
i
(x)
(b−ct)
k
i
α
i
(b−ct)

β
i
+1
≤ k
i
α
i
ce
−ak
i
ϕ
m
i
(x)
(b−ct)
β
i
(b − ct)
−(k
i
α
i
+1)
,

m
i
u
−
i

=
λ
m
i
a
m
i
ϕ
m
i
m
i
e
−am
i
ϕ
m
i
(x)
(b−ct)
β
i
(b−ct)
m
i
(α
i
+β
i
)

+
m
i
a
m
i
+1
e
−am
i
ϕ
m
i
(x)
(b−ct)
β
i
|∇ϕ
m
i
|
m
i
+1
(b−ct)
m
i
(α
i
+β

i
)+β
i
.
(3:17)
If
x ∈{x ∈  : dist(x, ∂) >ε
m
i
}
, we have
ϕ
m
i
≥ F
m
i
, and thus

m
i
u
−
i
≥
λ
m
i
a
m

i
F
m
i
m
i
e
−a
i
m
i
ϕ
m
i
(x)
(b−ct)
β
i
(
b − ct
)
m
i
(α
i
+β
i
)
.
(3:18)

On the other hand, since -ye
-y
≥ -e
-1
for any y > 0, we have
(u
−
k
i
i
)
t
≤ k
i
α
i
ce
−ak
i
ϕ
m
i
(x)
(b−ct)
β
i
(b − ct)
−(k
i
α

i
+1)
≤
k
i
α
i
ce
−am
i
ϕ
m
i
(x)
(b−ct)
β
i
a(k
i
−m
i
)F
m
i
e(b−ct)
m
i
(α
i
+β

i
)
.
(3:19)
We have by (3.16), (3.18), and (3.19) that
(u
−
k
i
i
)
t
≤ 
m
i
u
−
i
(i =1, , n
)
.
If
x ∈{x ∈  : dist(x, ∂) ≤ ε
m
i
}
, then
|
∇ϕ
m

i
|
≥ E
m
i
, and then

m
i
u
−
i
≥
m
i
a
m
i
+1
E
m
i
+1
m
i
e
−ak
i
ϕ
m

i
(x)
(b−ct)
β
i
(
b − ct
)
m
i
(α
i
+β
i
)+β
i
=
m
i
a
m
i
+1
E
m
i
+1
m
i
e

−ak
i
ϕ
m
i
(x)
(b−ct)
β
i
(b − ct)
k
i
α
i
+1
.
(3:20)
It follows from (3.16), (3.17), and (3.20) that
(u
−
k
i
i
)
t
≤ 
m
i
u
−

i
(i =1, , n
)
.
We have on the boundary that
∇
m
i
u
−
i
· ν =
a
m
i
|∇ϕ
m
i
|
m
i
−1
e
−am
i
ϕ
m
i
(x)
(b−ct)

β
i
(−
∂ϕ
m
i
∂ν
)
(b − ct)
m
i
(α
i
+β
i
)
≤
a
m
i
B
m
i
d
m
i
−1
m
i
(b − ct)

m
i
(α
i
+β
i
)
(i =1, , n)
,
n

j
=1
u
−
m
ij
j
=
1
(b − ct)

n
j=1
m
ij
α
j
(i =1,2, , n).
(3:21)

Du and Li Boundary Value Problems 2011, 2011:2
/>Page 9 of 11
Moreover, by (3.14) and Lemma 2.4, we have that
m
i
(α
i
+ β
i
) ≤

n
j
=1
m
ij
α
j
(i =1, , n)
.
(3:22)
(3.15), (3.21), and (3.22) imply that
∇
m
i
u
−
i
· ν ≤ λ
−


n
j=1
u
−
m
ij
j
(i =1, , n
)
. Therefore,
(
u
1
, , u
1
) is a lower solution of (1.1).
For k
i
= m
i
(i = 1, , n), let
u
−
i
=
1
(b−ct)
α
i

e
−aϕ
m
i
(
x
)
(b−ct)
1
m
i
.
(3:23)
For k
i
= m
i
(i = 1, , s)andk
i
>m
i
(i = s + 1, , n), let
¯
u
i
(
x, t
)
as in (3.13) and (3.23).
Using similar arguments as above, we can prove that (

u
1
, , u
n
) is a lower solution of
(1.1). Therefore, (
u
1
, , u
n
) ≤ (u
1
, , u
n
). Consequently, (u
1
, , u
n
) blows up in finite
time.
Case 3. (k
i
<m
i
(i = 1, , s); k
i
≥ m
i
(i = s + 1, , n)). Let
¯

u
i
(
x, t
)(
i =1, , s
)
be as in
(3.11) and
u
−
i
=
1
(
b − ct
)
α
i
e
−aϕ
m
i
(x)
(b−ct)
β
i
(i = s +1, , n)
,
where a

i
’s are to determined later and
β
i
=
(k
i
−m
i
)α
i
+1
m
i
+1
(i = s +1, , n), b = max{1, max
1is
{(
1
2
δ
1
ρ
i
)
−
1
γ
i
}, max

s+1in
{δ
−
1
α
i
}}
,
a = min

min
s+1in
{λ
−
1
m
i
(B
m
i
d
m
i
−1
m
i
)
−
1
m

i
b
−

n
j=1
m
ij
α
j
m
i
}, min
1is
{b
−γ
i
(2Nd +1)
−
1+m
i
m
i
,

λ
−
−1
[
(1+m

i
)ρ
i
N
1
2
2
ρ
i
−1
m
i
]
m
i
(2Nd +1)

−
1
m
i
b
−
−

n
j=1
m
ij
α

j
m
i
}
⎫
⎪
⎬
⎪
⎭
,
c =
min
⎧
⎨
⎩
min
1is
{
a
m
i
ρ
m
i
−1
i
(1+
1
m
i

)
m
i
N
m
i
+1
2
k
i
γ
i
} ,
min
s+1in

m
i
a
m
i
+1
E
m
i
+1
m
i
k
i

α
i
,
λ
m
i
(k
i
−m
i
)a
m
i
+1
F
m
i
+1
m
i
k
i
α
i

.
Based on arguments in cases 1 and 2, we have
(u
−
k

i
i
)
t
≤ 
m
i
u
−
i
(i =1, , n
)
for
(x, t) ∈  × (0,
b
c
)
. Furthermore, for
(x, t) ∈ ∂ × (0,
b
c
)
, we have
∇
m
i
u
−
i
· ν ≤ (aρ

i
(1 +
1
m
i
))
m
i
N
m
i
2
(2Nd +1)2
m
i
(ρ
i
−1)
(b − ct)
−(k
i
α
i
+1)
(i =1, , s)
,
∇
m
i
u

−
i
· ν ≤ a
m
i
B
m
i
d
m
i
−1
m
i
(b − ct)
−m
i
(α
i
+β
i
)
(i = s +1, , n),
n

j
=1
u
−
m

ij
j
≥ (b − ct)
−

n
j=1
m
ij
α
j
(i =1, , n).
Thus,
∇
m
i
u
−
i
· ν ≤ λ
−
n

j
=1
u
−
m
ij
j

(i =1, , n
)
Du and Li Boundary Value Problems 2011, 2011:2
/>Page 10 of 11
holds if
k
i
α
i
+1≤
n

j=1
m
ij
α
j
(i =1, , s),
m
i
(α
i
+ β
i
) ≤
n

j
=1
m

ij
α
j
(i = s +1,, , n)
.
(3:24)
From Lemma 2.4, we know that inequalities (3.24) hold for suitable choices of a
i
(i =
1, , n). We show that (
u
1
, , u
n
) is a lower solution of (1.1). Since (u
1
, , u
n
)blowsup
in finite time, it follows that the solution of (1.1) blows up in finite time.
Acknowledgements
This study was partially supported by the Projects Supported by Scientific Research Fund of SiChuan Provincial
Education Department(09ZC011), and partially supported by the Natural Science Foundation Project of China West
Normal University (07B046).
Authors’ contributions
DW carried out all studies in the paper. LZ participated in the design of the study in the paper.
Competing interests
The authors declare that they have no competing interests.
Received: 9 November 2010 Accepted: 21 June 2011 Published: 21 June 2011
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Cite this article as: Du and Li: Non-Newtonian polytropic filtration systems with nonlinear boundary conditions.
Boundary Value Problems 2011 2011:2.
Du and Li Boundary Value Problems 2011, 2011:2
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