Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 451619, 11 pages
doi:10.1155/2010/451619
Research Article
Existence and Nonexistence of Global
Solutions of the Quasilinear Parabolic Equations
with Inhomogeneous Terms
Yasumaro Kobayashi
Faculty of Urban Liberal Arts, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan
Correspondence should be addressed to Yasumaro Kobayashi, [email protected]
Received 20 April 2010; Accepted 14 October 2010
Academic Editor: Abdelkader Boucherif
Copyright q 2010 Yasumaro Kobayashi. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We consider the quasilinear parabolic equation with inhomogeneous term u
t
Δu
m
x
σ
u
p
fx,
ux, 0u
0
x,where0 ≤ fx, u
0
x ∈ CR
N
u
0
x
x ∈ R
N
,
1.1
where 0 ≤ fx, u
0
x ∈ CR
N
, m>0, p>max{1,m},andσ>−2, x :|x|
2
1
1/2
.
For the solution ux, t of 1.1,letT
∗
> 0 be the maximal existence time, that is,
T
∗
: sup
T>0; sup
t∈
r
−2/p−1
|x|<r
dx
x
σ/p−1
≥ c
1
u
0
dx,
lim inf
|x|→∞
x
σ2
u
0
x
p−1
≤ c
in 1.1.Bandleetal.6 study the case m 1, σ 0, and Zeng 8 and Zhang 9 study the
case σ 0. In this paper, we investigate the critical exponents of 1.1 in the case fx
/
≡ 0. Our
results are as follows.
Theorem 1.1. Suppose that N ≥ 3, σ>−2, m>N − 2/N σ, and p>max{1,m}.Put
p
∗
m,σ
:
m
N σ
N − 2
.
1.4
a If p ≤ p
∗
m,σ
, then every nontrivial solution ux, t of 1.1 blows up in finite time.
b If p>p
∗
m,σ
, u
0
x ≤ C
1
x
−2σ/p−m
u
m
∈ L
2
loc
R
N
;
ii for any bounded domain D ⊂ R
N
and for all ψ ∈ C
2
D × 0,T and vanishing on
∂D × 0,T,
τ
0
D
u∂
t
ψ −∇u
m
∇ψ
x
σ
u
0, T;
L
2
loc
Ω, and satisfy
u
t
− Δu
m
≤ v
t
− Δv
m
,
x, t
∈ Ω
T
,
u ≤ v,
x, t
∈ ∂Ω
T
.
2.2
Then u ≤ v for all x, t ∈ Ω
T
x ∈ R
N
,t>0
,
u
x, 0
0
x ∈ R
N
.
3.1
It is clear that the positive solution of Problem 3.1 is a sub-solution of Problem 1.1. If every
positive solution of Problem 3.1 blows up in finite time, then, by Lemma 2.2, every positive
solution of Problem 1.1 also blows up in finite time. Therefore, we only need to consider
Problem 3.1.
The stationary problem of Problem 3.1 is as follows:
Δu
m
x
σ
u
For R>1andT>1, define Q
R,T
≡ B
2R
× 0, 4T,andletΨr, tϕ
R
rη
T
t be a cut-off
function, where ϕ
R
rϕr/R, η
T
tηt/2T. It is easy to check that
−
C
R
≤
dϕ
R
r
dr
≤ 0,
I
R
Q
R,T
x
σ
u
p
Ψ
s
dxdt,
3.4
where s>1 is a positive number to be determined. Then
I
R
Q
R,T
−u∂
t
Ψ
s
∇u
m
∇Ψ
s
m
η
s
T
∇ϕ
s
R
dx dt −
Q
R,T
fΨ
s
dx dt
B
2R
u
x, ·
ϕ
R
r
s
η
T
s
R
dx dt −
Q
R,T
fΨ
s
dx dt
4T
0
|x|2R
u
m
η
s
T
∂ϕ
s
R
∂ν
dS dt.
3.5
Since
R
N
T
B
R
fdxdt ≥ δT.
3.6
Hence, by the definition of ϕ
R
and η
T
, we have
I
R
≤−
4T
2T
B
2R
uϕ
s
R
dη
s
T
dt
dx dt −
4T
|
2
and
Δϕ
R
r
d
2
ϕ
R
r
dr
2
N − 1
r
dϕ
R
r
dr
,
N − 1
R
·
C
R
s
s − 1
ϕ
s−2
R
C
R
2
≤
C
R
2
ϕ
s−2
R
3.9
in B
2R
\ B
4T
2T
B
2R
uΨ
s−1
dx dt
C
R
2
4T
0
B
2R
\B
R
u
m
Ψ
s−2
dx dt − δT.
3.11
Let s be large enough such that s − 1p ≥ s and s − 2p/m ≥ s,andletA
σ
R be as follows:
A
4T
2T
B
2R
uΨ
s−1
dx dt
≤
4T
2T
B
2R
1
4
p
p
x
σ
u
p
Ψ
s−1p
dx dt CT
−p/p−1
4T
2T
B
2R
x
−σ/p−1
dx dt
≤
1
4
I
R
CT
1−p/p−1
A
σ
R
,
3.13
6 Advances in Difference Equations
where 1/p 1/q 1and
C
x
σ
u
mp
Ψ
s−2p
4
q
q
x
−σq
/p
C
q
R
−2q
dx dt
−mσ/p−m
dx dt
≤
1
4
I
R
CTR
−2p/p−m
R
N−mσ/p−m
,
3.14
where p
p/m,1/p
1/q
1. Thus, 3.11 becomes
I
R
≤
1
2
I
R
T
CT
3.16
For N 2, since σ ≥−2, m>0, and p>max{1,m}, we have
2 −
2p mσ
p − m
−
2 σ
m
p − m
≤ 0.
3.17
For N 1, since σ ≥−2,m>0, and p>max{1,m}, we have
1 −
2p mσ
p − m
−p −
1 σ
m
p − m
<
−
2 σ
s
dx dt ≤ CT. 3.20
Thus
2T
T
B
R
x
σ
u
p
dx dt ≤ CT.
3.21
Advances in Difference Equations 7
By the integral mean value theorem, there exists t
1
∈ T, 2T such that
2T
T
B
R
x
1
p
dx ≤ C. 3.23
Since T is a large positive number and a random selection, and ux, t is monotone increasing
to t, there exists a positive number TR > 1 for any fixed R>R
0
such that, for all t>TR,
B
R
x
σ
u
x, t
p
dx ≤ C.
3.24
By the monotone increasing property of ux, t,
B
R
x
σ
ux, t
p
I
∞
R
exists. Thus, for any small ε>0, there exists a large positive constant
which still is denoted by R
0
, such that, for R>R
0
,
lim
t →∞
B
2R
\B
R
x
σ
u
x, t
p
dx ≡ I
∞
2R
− I
∞
satisfying.
i 0 ≤ ξx ≤ 1inR
N
; ξx ≡ 1inB
1
, ξx ≡ 0inB
c
2
;
ii ∂ξ/∂ν 0on∂B
2
\ B
1
;
iii for any α ∈ 0, 1, there exists a positive constant C
α
such that |Δξ|≤C
α
ξ
α
.
Let R and TR be as defined in 3.26 and 3.27. Multiplying 3.1 by ξ
R
xξx/R
and then integrating by parts in R
N
, we have
d
dt
fξ
R
dx.
3.28
8 Advances in Difference Equations
By the definition of ξ
R
x,H
¨
older’s inequality, and 3.27, we have
B
2R
\B
R
u
m
Δξ
R
dx
σ
u
mp
dx
1/p
B
2R
\B
R
x
−σq
/p
ξ
αq
R
dx
1/q
≤
p−m/p
≤ Cε
m/p
R
N−mσ/p−mp−m/p−2
≤ Cε
m/p
,
3.29
where p
p/m,1/p
1/q
1, since
N −
mσ
p − m
p − m
p
− 2
N − 2
p −
fxdx ≥ δ for R>R
0
, 3.28 becomes
F
R
t
≥ G
R
t
− Cε
m/p
δ.
3.31
Thus, let ε be small enough such that Cε
m/p
≤ δ/2, then F
R
t ≥ G
R
tδ/2.
Let t
0
>TR. By making use of H
¨
ξ
R
dx
1/q
≤ G
R
t
1/p
B
2R
x
−σ/p−1
dx
p−1/p
≤ CG
R
t
1/p
A
σ
ds ≤ CA
σ
R
p−1
t
t
0
F
R
s
ds
≤ CA
σ
R
p−1
F
R
t
− F
R
p
ds F
R
t
0
≥ CA
σ
R
−p1
t
t
0
F
R
s
p
ds.
3.34
Let gt
t
t
0
>t
0
such that gt
1
> 0. Since p>1, by solving the differential inequality 3.35 in t
1
,t,
we have
t
t
1
g
s
g
s
p
ds ≥ CA
σ
R
−pp−1
t
t
t
≥
g
t
1
1−p
− C
p − 1
A
σ
R
−pp−1
t − t
1
−1/p−1
.
3.36
Thus, there exists T
1
m
x
σ
u
p
f
x
0
x ∈ R
N
. 4.1
Let vxC
1
x
−s
, where s 2 σ/p − m and the positive constant C
1
satisfies
C
p−m
1
ms
C
m
1
|
x
|
2
1
−ms/2−1
Δ
|
x
|
2
1
−
ms
ms 2
4
C
m
1
|
ms 2
C
m
1
|
x
|
2
x
−ms−4
ms
N − ms − 2
C
m
1
x
−ms−2
ms
ms 2
C
m
σ
v
p
C
2
x
−ms−4
,
4.4
where C
2
msms 2C
m
1
.Thus,iffx ≤ C
2
x
−ms−4
and u
0
x ≤ vx, then v is
a supersolution of Problem 1.1. It is obvious that 0 is s sub-solution of Problem 1.1.
Therefore, by the iterative process and the comparison theorem, Problem 1.1 admits a global
positive solution.
Acknowledgments
This paper was introduced to the author by Professor Kiyoshi Mochizuki in Chuo University.
The author would like to thank him for his proper guidance. The author would also like
to thank Ryuichi Suzuki for useful discussions and friendly encouragement during the
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Advances in Difference Equations 11
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