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, yasumaro@hkg.odn.ne.jp
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 ut Δum xσ up fx,
ux, 0 u0 x, where 0 ≤ fx, u0 x ∈ CRN , m > 0, p > max{1, m}, and σ > −2,
x : |x|2 1
1/2
. In this paper, we investigate the critical exponents of this equation.
1. Introduction
We consider the quasi-linear parabolic equation with inhomogeneous term
ut Δum xσ up fx
ux, 0 u0 x
x ∈ RN , t > 0 ,
1.1
x ∈ RN ,
where 0 ≤ fx, u0 x ∈ CRN , m > 0, p > max{1, m}, and σ > −2, x : |x|2 11/2 .
For the solution ux, t of 1.1, let T ∗ > 0 be the maximal existence time, that is,
∗
T : sup T > 0; sup u·, t∞ < ∞ .
t∈0,T 1.2
If T ∗ ∞, we say that ux, t is a global solution; if T ∗ < ∞, we say that ux, t blows up in
finite time.
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For quasi-linear parabolic equations, the authors of 1–5 and so on. study the
homogeneous equations i.e., fx ≡ 0 in 1.1. Baras and Kersner 1 proved that 1.1
with m 1 and fx ≡ 0 has a global solution, two constants c1 and c2 depending on N and
p exist such that
lim inf r
−2/p−1
r →∞
dx
|x|<r
lim infx
|x| → ∞
≥ c1
xσ/p−1
σ2
p−1
u0 x
u0 dx,
1.3
≤ c2 .
Mochizuki and Mukai 2 and Qi 4 study the case m > 0, σ 0, Pinsky 3 studies the case
m 1, σ > −2, and Suzuki 5 studies the case m ≥ 1, −∞ < σ < ∞. The following two results
are proved by them:
∗
, then every nontrivial solution ux, t of 1.1 blows up in finite time;
1 if p ≤ pm,σ
∗
2 if p > pm,σ
, then 1.1 has a global solution for some initial value u0 x,
∗
∗
where pm,σ
m 2 σ/N for N ≥ 2, σ > −2 and for N 1, σ > −1, pm,σ
m 1 for N 1,
∗
σ ≤ −1. This pm,σ is called the critical exponent.
On the other hand, 6–9 and so on. study the inhomogeneous equations i.e., fx /
≡0
in 1.1. Bandle et al. 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
∗
:
pm,σ
mN σ
.
N−2
1.4
∗
a If p ≤ pm,σ
, then every nontrivial solution ux, t of 1.1 blows up in finite time.
∗
, u0 x ≤ C1 x−2σ/p−m , and fx ≤ C2 x−m2σ/p−m−4 , then 1.1 has
b If p > pm,σ
a global solution for some constants C1 and C2 .
Theorem 1.2. Suppose that N 1, 2, σ ≥ −2, m > 0, and p > max{1, m}. Then every nontrivial
solution ux, t of 1.1 blows up in finite time.
Remark 1.3. Theorems 1.1 and 1.2 are the extension of the results of 8. If we put σ 0 in
these theorems, the same results as Theorem 1 in 8 are obtained.
We will prove Theorem 1.1a and b in Sections 3 and 4, respectively. The proof of
Theorem 1.2 is included in the proof of Theorem 1.1a.
In the following, R and T are two given positive real numbers greater than 1. C is a
positive constant independent of R and T , and its value may change from line to line.
2. Preliminaries
In this section, we first give the definition of a solution for Problem 1.1 and then cite the
comparison theorem and a known result.
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Definition 2.1. A continuous function u ux, t is called a solution of Problem 1.1 in QT ≡
RN × 0, T if the following holds:
i ∇x um ∈ L2loc RN ;
ii for any bounded domain D ⊂ RN and for all ψ ∈ C2 D × 0, T and vanishing on
∂D × 0, T ,
τ 0
σ p
u∂t ψ − ∇u ∇ψ x u ψ fψ dx dt m
D
D
τ
ux, ·ψx, ·0 dx,
2.1
for all τ ∈ 0, T .
Lemma 2.2 the comparison theorem. Let u, v ∈ C0, T ; L2loc Ω, ∇um , ∇vm ∈ L2 0, T ;
L2loc Ω, and satisfy
ut − Δum ≤ vt − Δvm ,
u ≤ v,
x, t ∈ ΩT ,
x, t ∈ ∂ΩT .
2.2
Then u ≤ v for all x, t ∈ ΩT , where Ω is a bounded domain in RN with smooth boundary ∂Ω or
Ω RN and ΩT Ω × 0, T .
Lemma 2.3 the monotonicity property. Let ux be a nonnegative sub-solution to the stationary
problems of Problem 1.1. Then the positive solution ux, t with initial data ux is monotone
increasing to t.
3. Proof of Theorem 1.1(a)
We first consider the following problem:
ut Δum xσ up fx
ux, 0 0
x ∈ RN , t > 0 ,
x ∈ RN .
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:
Δum xσ up fx 0
x ∈ RN .
3.2
It is obvious that 0 is a sub-solution of Problem 3.2 and does not satisfy Problem 3.2. Thus,
by making use of Lemmas 2.2 and 2.3, the positive solution of Problem 3.1 is monotone
increasing to t.
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We argue by contradiction. Assume that Problem 3.1 has a global positive solution
∗
.
for p ≤ pm,σ
Let ϕr and ηt be two functions in C∞ 0, ∞, and satisfy
i 0 ≤ ϕr ≤ 1 in 0, ∞; ϕr ≡ 1 in 0, 1, ϕr ≡ 0 in 2, ∞; −C ≤ ϕ r ≤ 0,
|ϕr| ≤ C;
ii 0 ≤ ηt ≤ 1 in 0, ∞; ηt ≡ 1 in 0, 1, ηt ≡ 0 in 2, ∞; −C ≤ η t ≤ 0.
For R > 1 and T > 1, define QR,T ≡ B2R × 0, 4T , and let Ψr, t ϕR rηT t be a cut-off
function, where ϕR r ϕr/R, ηT t ηt/2T . It is easy to check that
d2 ϕ r C
R
,
≤
dr 2 R2
C dϕR r
− ≤
≤ 0,
R
dr
−
dηT t
C
≤
≤ 0.
2T
dt
3.3
Let
xσ up Ψs dxdt,
IR 3.4
QR,T
where s > 1 is a positive number to be determined. Then
IR −u∂t Ψ ∇u ∇Ψ − fΨ dx dt s
m
s
s
QR,T
B2R
−
QR,T
B2R
−
QR,T
RN
uϕsR
dηTs
dx dt −
dt
4T 0
Since
dηTs
dx dt dt
QR,T
∇um ηTs ∇ϕsR dx dt
−
fΨs dx dt
QR,T
4T
ux, ·ϕR rs ηT ·s 0 dx
uϕsR
4T
ux, ·Ψr, ·s 0 dx
|x|2R
um ηTs
3.5
QR,T
um ηTs ΔϕsR dx dt
fΨ dx dt s
4T
0
QR,T
fΨs dx dt
QR,T
∂ϕsR
dS dt.
∂ν
fxdx > 0, there exist δ > 0 and R0 > 1 such that
−
ηTs
B2R
fϕsR dx dt
≥
BR
fxdx ≥ δ for R > R0 :
2T T
fdx dt ≥ δT.
3.6
BR
Hence, by the definition of ϕR and ηT , we have
IR ≤ −
4T 2T
B2R
uϕsR
dηTs
dx dt −
dt
4T 0
B2R \BR
um ηTs ΔϕsR dx dt − δT.
3.7
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s−2
2
Since ΔϕsR sϕs−1
R ΔϕR ss − 1ϕR |∇ϕR | and
∇ϕR 2 d2 ϕR r N − 1 dϕR r
,
ΔϕR r r
dr
dr 2
dϕR r
dr
2
,
3.8
2
C
C
≤ 2 ϕs−2
R
R R
3.9
we obtain from 3.3 that
s
Δϕ ≤ sϕs−1
R
R
C N−1 C
·
R
R
R2
ss − 1ϕs−2
R
in B2R \ BR and
dηTs
dηT
C
C
sηTs−1
≥ −sηTs−1
≥ − ηTs−1
dt
dt
2T
T
3.10
in 2T, 4T . Thus, 3.7 becomes
C
IR ≤
T
4T uΨ
2T
s−1
B2R
C
dx dt 2
R
4T 0
B2R \BR
um Ψs−2 dx dt − δT.
3.11
Let s be large enough such that s − 1p ≥ s and s − 2p/m ≥ s, and let Aσ R be as follows:
Aσ R ⎧
⎨RN−σ/p−1
⎩logR 1
σ <N p−1 ,
σ ≥N p−1 .
3.12
Then, by making use of Young’s inequality, we have
C
T
4T 2T
≤
uΨs−1 dx dt
B2R
4T 2T
≤
≤
1
4
B2R
4T 0
4q
1
σ p s−1p
−σq/p q −q
u
Ψ
C
T
dx dt
x
x
4p p
q
xσ up Ψs dx dt CT −p/p−1
B2R
1
IR CT 1−p/p−1 Aσ R,
4
4T 2T
x−σ/p−1 dx dt
B2R
3.13
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where 1/p 1/q 1 and
C
R2
4T B2R \BR
0
≤
≤
4T B2R \BR
0
≤
um Ψs−2 dx dt
1
4
4T 0
1
4q
σ mp s−2p
Ψ
x−σq /p Cq R−2q
x u
p
q
4 p
xσ up Ψs dx dt CR−2p/p−m
4T 0
B2R
B2R \BR
dx dt
3.14
x−mσ/p−m dx dt
1
IR CT R−2p/p−m RN−mσ/p−m ,
4
where p p/m, 1/p 1/q 1. Thus, 3.11 becomes
IR ≤
1
IR T CT −p/p−1 Aσ R CRN−2pmσ/p−m − δ .
2
3.15
For N ≥ 3, since σ > −2, 1/p 1/q 1, and max{1, m} < p ≤ mN σ/N − 2, we have
N−
2p mσ N − 2p − N σm
≤ 0.
p−m
p−m
3.16
For N 2, since σ ≥ −2, m > 0, and p > max{1, m}, we have
2−
2p mσ −2 σm
≤ 0.
p−m
p−m
3.17
For N 1, since σ ≥ −2, m > 0, and p > max{1, m}, we have
1−
2p mσ −p − 1 σm −2 σm
<
≤ 0.
p−m
p−m
p−m
3.18
Let T ≥ Aσ Rp−1/p such that T −p/p−1 Aσ R ≤ 1, then
IR ≤ CT,
3.19
that is,
4T 0
xσ up Ψs dx dt ≤ CT.
3.20
xσ up dx dt ≤ CT.
3.21
B2R
Thus
2T T
BR
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By the integral mean value theorem, there exists t1 ∈ T, 2T such that
2T T
xσ up dx dt T
BR
3.22
xσ ux, t1 p dx ≤ CT,
BR
that is,
xσ ux, t1 p dx ≤ C.
3.23
BR
Since T is a large positive number and a random selection, and ux, t is monotone increasing
to t, there exists a positive number T R > 1 for any fixed R > R0 such that, for all t > T R,
xσ ux, tp dx ≤ C.
3.24
BR
By the monotone increasing property of ux, t, BR xσ ux, tp dx also is increasing to t. This,
combined with 3.24, yields that the limit IR∞ exists such that
IR∞
xσ ux, tp dx ≤ C.
≡ lim
t→∞
3.25
BR
Since ux, t is nonnegative, IR∞ is monotone increasing to R. This, combined with 3.25,
yields that limR → ∞ IR∞ exists. Thus, for any small ε > 0, there exists a large positive constant
which still is denoted by R0 , such that, for R > R0 ,
lim
t→∞
B2R \BR
∞
− IR∞ < ε.
xσ ux, tp dx ≡ I2R
3.26
Hence, by similar argument as that in 3.24, there exists a large positive number T R > 1
such that
xσ ux, tp dx < ε, ∀t > T R.
3.27
B2R \BR
On the other hand, we argue as in 6, 10. Let ξx ∈ C2 RN be a positive function
satisfying.
i 0 ≤ ξx ≤ 1 in RN ; ξx ≡ 1 in B1 , ξx ≡ 0 in B2c ;
ii ∂ξ/∂ν 0 on ∂B2 \ B1 ;
iii for any α ∈ 0, 1, there exists a positive constant Cα such that |Δξ| ≤ Cα ξα .
Let R and T R be as defined in 3.26 and 3.27. Multiplying 3.1 by ξR x ξx/R
and then integrating by parts in RN , we have
d
dt
RN
uξR dx B2R \BR
um ΔξR dx RN
xσ up ξR dx RN
fξR dx.
3.28
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By the definition of ξR x, Hölder’s inequality, and 3.27, we have
C α
m
u ΔξR dx ≤ 2
um ξRα dx
B2R \BR
R B2R \BR
Cα
≤ 2
R
Cα
≤ 2
R
σ mp
B2R \BR
x u
1/p dx
B2R \BR
αq
x−σq /p ξR dx
m/p x u dx
3.29
p−m/p
σ p
B2R \BR
1/q
B2R \BR
x
−mσ/p−m
dx
≤ Cεm/p RN−mσ/p−mp−m/p−2 ≤ Cεm/p ,
where p p/m, 1/p 1/q 1, since
N−
p−m
N − 2p − N σm
mσ
−2
≤ 0.
p−m
p
p
3.30
Let FR t RN uξR dx and GR t RN xσ up ξR dx. Then, by making use of 3.29 and
fxdx ≥ δ for R > R0 , 3.28 becomes
RN
FR t ≥ GR t − Cεm/p δ.
3.31
Thus, let ε be small enough such that Cεm/p ≤ δ/2, then FR t ≥ GR t δ/2.
Let t0 > T R. By making use of Hölder’s inequality, we obtain that
FR t ≤
xσ up ξR dx
RN
1/p RN
x−σq/p ξR dx
1/q
p−1/p
−σ/p−1
1/p
≤ GR t
x
3.32
dx
B2R
≤ CGR t1/p Aσ Rp−1/p ,
where 1/p 1/q 1. Thus, we obtain that
t
t0
FR sp ds ≤ CAσ Rp−1
t
GR sds ≤ CAσ Rp−1
t0
≤ CAσ Rp−1 FR t − FR t0 .
t
t0
FR sds
3.33
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Since FR t ≥ 0 for all t ≥ 0, we have
FR t ≥ CAσ R
−p1
t
FR sp ds FR t0 t0
≥ CAσ R−p1
t
3.34
FR sp ds.
t0
Let gt t
t0
FR sp ds, then
gt FR tp ≥ CAσ R−pp−1 gtp .
3.35
Let t1 > t0 such that gt1 > 0. Since p > 1, by solving the differential inequality 3.35 in t1 , t,
we have
t
t1
gs
ds ≥ CAσ R−pp−1
gsp
t
ds,
t1
gt1−p ≤ gt1 1−p − C p − 1 Aσ R−pp−1 t − t1 ,
3.36
−1/p−1
gt ≥ gt1 1−p − C p − 1 Aσ R−pp−1 t − t1 .
Thus, there exists T1 with t1 < T1 ≤ t1 Cp − 1−1 Aσ Rpp−1 gt1 1−p , such that limt↑T1 gt ∞, which implies that gt and then u blow up in finite time. It contradicts our assumption.
Therefore, every positive solution of Problem 3.1 blows up in finite time. Hence, every
positive solution of Problem 1.1 blows up in finite time.
4. Proof of Theorem 1.1(b)
In this section, we prove that for p > mN σ/N − 2, there exist some fx and u0 x, such
that Problem 1.1 admits a global positive solution.
We first consider the stationary problem of Problem 1.1 as follows:
Δum xσ up fx 0
x ∈ RN .
4.1
Let vx C1 x−s , where s 2 σ/p − m and the positive constant C1 satisfies
p−m
C1
m2 σ N − 2p − N σm
msN − ms − 2 > 0.
2
p−m
4.2
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Then, we have
−Δvm −ms/2−1 ms m 2
C1 |x| 1
Δ |x|2 1
2
−
−ms/2−2 2
msms 2 m 2
C1 |x| 1
∇ |x|2 1 4
4.3
NmsC1m x−ms−2 − msms 2C1m |x|2 x−ms−4
msN − ms − 2C1m x−ms−2 msms 2C1m x−ms−4 .
p−m
Since C1
msN − ms − 2 and −ms − 2 σ − ps, we have
−Δvm C1 xσ−ps C2 x−ms−4 xσ vp C2 x−ms−4 ,
p
4.4
where C2 msms 2C1m . Thus, if fx ≤ C2 x−ms−4 and u0 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
preparation of this paper.
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