Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 109546, 19 pages
doi:10.1155/2012/109546
Research Article
Blow-Up Analysis for a Quasilinear Degenerate
Parabolic Equation with Strongly Nonlinear Source
Pan Zheng, Chunlai Mu, Dengming Liu,
Xianzhong Yao, and Shouming Zhou
School of Mathematics and Statistics, Chongqing University, Chongqing 401331, China
Correspondence should be addressed to Pan Zheng, zhengpan52@sina.com
Received 8 January 2012; Accepted 26 April 2012
Academic Editor: Yong Hong Wu
Copyright q 2012 Pan Zheng et al. 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 investigate the blow-up properties of the positive solution of the Cauchy problem for a
quasilinear degenerate parabolic equation with strongly nonlinear source ut div|∇um |p−2 ∇ul uq , x, t ∈ RN × 0, T , where N ≥ 1, p > 2 , and m, l, q > 1, and give a secondary critical exponent
on the decay asymptotic behavior of an initial value at infinity for the existence and nonexistence
of global solutions of the Cauchy problem. Moreover, under some suitable conditions we prove
single-point blow-up for a large class of radial decreasing solutions.
1. Introduction
In this paper, we consider the following Cauchy problem to a quasilinear degenerate
parabolic equation with strongly nonlinear source
ut div |∇um |p−2 ∇ul uq ,
ux, 0 u0 x,
x, t ∈ RN × 0, T ,
1.1
x∈R ,
N
where N ≥ 1, p > 2, m , l , q > 1, and the initial data u0 x is nonnegative bounded and
continuous.
Equation 1.1 has been suggested as a mathematical model for a variety of physical
problems see 1. For instance, it appears in the non-Newtonian fluids and is a nonlinear
form of heat equation. Moreover, it can also be used to model the nonlinear heat propagation
in a reaction medium see 2.
2
Abstract and Applied Analysis
One of the particular features of problem 1.1 is that the equation is degenerate at
points where u 0 or ∇u 0. Hence, there is no classical solution in general and we introduce
the following definition of weak solution see 3, 4.
Definition 1.1. A nonnegative measurable function ux, t defined in RN × 0, T is called a
weak solution of the Cauchy problem 1.1 if for every bounded open set Ω with smooth
p
boundary ∂Ω, u ∈ Cloc Ω × 0, T , um , ul ∈ Lloc 0, T ; W 1,p Ω, and
Ω
uϕ dx t t0
Ω
m p−2
−uϕt |∇u |
t q
∇u · ∇ϕ dx dt u ϕ dx dt ux, t0 ϕx, t0 dx
l
t0
Ω
Ω
1.2
for all 0 ≤ t0 ≤ t ≤ T and all test functions ϕ ∈ C01 Ω × 0, T . Moreover,
lim
t → t0
Ω
ux, tηxdx Ω
ux, t0 ηxdx
1.3
for any ηx ∈ C01 Ω.
Under some suitable assumptions, the existence, uniqueness and regularity of a weak
solution to the Cauchy problem 1.1 and their variants have been extensively investigated
by many authors see 5–7 and the references therein.
The first goal of this paper is to study the blow-up behavior of solution ux, t of 1.1
when the initial data u0 x has slow decay near x ∞. For instance, in the following case
−a
u0 x ∼
M|x|
with M > 0, a ≥ 0,
1.4
we investigate the existence of global and nonglobal solutions for the Cauchy problem 1.1
in terms of M and a. In recent years, many authors have studied the properties of solutions
to the Cauchy problem 1.1 and their variants see 8–17 and the references therein. In
particular, J.-S. Guo and Y. Y. Guo 18 obtained the secondary critical exponent for the
following porous medium type equation in high dimensions:
ut Δum up ,
x, t ∈ RN × 0, T ,
ux, 0 u0 x,
x ∈ RN ,
1.5
where p > 1, m > 1 or max{0, 1 − 2/N} < m < 1, u0 x is nonnegative bounded and
∗
m2/N, there exists a secondary critical exponent
continuous, and proved that for p > pm
a∗ 2/p − m such that the solution ux, t of 1.5 blows up in finite time for the initial data
u0 x, which behaves like |x|−a at x ∞ if a ∈ 0, a∗ , and there exists a global solution for
the initial data u0 x, which behaves like |x|−a at x ∞ if a ∈ a∗ , N. Here, we say that the
solution blows up in finite time; it means that there exists T ∈ 0, ∞ such that u·, t
L∞ < ∞
for all t ∈ 0, T , but limt → T − sup u., t
L∞ ∞.
Abstract and Applied Analysis
3
Mu et al. 19 studied the secondary critical exponent for the following p-Laplacian
equation with slow decay initial values:
ut div |∇u|p−2 ∇u uq ,
ux, 0 u0 x,
x, t ∈ RN × 0, T ,
x ∈ RN ,
1.6
where p > 2, q > 1, and showed that, for q > qc∗ p − 1 p/N, there exists a secondary
critical exponent a∗c p/q 1 − p such that the solution ux, t of 1.6 blows up in finite
time for the initial data u0 x which behaves like |x|−a at x ∞ if a ∈ 0, a∗c , and there exists
a global solution for the initial data u0 x, which behaves like |x|−a at x ∞ if a ∈ a∗c , N.
Recently, Mu et al. 20 also investigated the secondary critical exponent for the doubly
degenerate parabolic equation with slow decay initial values and obtained similar results.
On the other hand, in this paper, we will also consider single-point blow-up for the
Cauchy problem 1.1. It is interesting to study the set of blow-up points and the behavior of
the solution ux, t at the blow-up point.
In order to investigate single-point blow-up for the Cauchy problem 1.1, we
introduce the concept of the blow-up point.
Definition 1.2. A point x ∈ Ω is called a blow-up point if there exists a sequence xn , tn such
that xn → x, tn → T − and uxn , tn → ∞ as n → ∞, where T is blow-up time.
In recent years, some authors also studied single-point blow-up for the Cauchy
problem to nonlinear parabolic equations see 21, 22 and the references therein by different
methods. In particular, when p 2, l 1 and N 1, the Cauchy problem 1.1 has been
investigated by Weissler in 23, and the author obtained that the solution blows up only at a
single point. Galaktionov and Posashkov 24 studied the single-point blow-up and gave the
upper and lower bound near the blow-up point for the Cauchy problem 1.1 when p > 2 and
m l 1. Recently, when p > 2 and m l, Mu and Zeng 25 extended Galaktionov’s results
to the doubly degenerate parabolic equation. For more works about single-point blow-up, we
refer to 26, 27, where the parabolic systems have been considered.
Motivated by the above works, based on a modification of the energy methods,
comparison principle, and regularization methods used in 15, 19, 21, 24, we investigate the
secondary critical exponent and single-point blow-up for the Cauchy problem 1.1. Before
stating the results of the secondary critical exponent, we start with some notations as follows.
Let Cb RN be the space of all bounded continuous functions in RN . For a ≥ 0, we
define
a
a
N
| φx ≥ 0, lim sup |x| φx < ∞ ,
Φ φx ∈ Cb R
|x| → ∞
1.7
a
N
Φa φx ∈ Cb R
| φx ≥ 0, lim inf |x| φx > 0 .
|x| → ∞
Moreover, we denote
p
qc∗ l m p − 2 ,
N
a∗c Our main results of this paper are stated as follows.
p
.
q−l−m p−2
1.8
4
Abstract and Applied Analysis
Theorem 1.3. For N ≥ 2, p > 2, m > 1, l > 1, and q > qc∗ l mp − 2 p/N, suppose that
u0 x ∈ Φa for some a ∈ 0, a∗c ; then the solution ux, t of the Cauchy problem 1.1 blows up in
finite time.
Theorem 1.4. For N ≥ 2, p > 2, m > 1, l > 1, and q > qc∗ l mp − 2 p/N, suppose that
u0 x λϕx for some λ > 0 and ϕx ∈ Φa for some a ∈ a∗c , N; then there is λ0 λ0 ϕ > 0
such that the solution ux, t of the Cauchy problem 1.1 exists globally for all t > 0, and if λ < λ0 ,
one has
||ux, t||∞ ≤ Ct−aβ ,
∀t > 0,
1.9
where β 1/al mp − 2 − 1 p, C > 0.
Remark 1.5. When p > 2, N ≥ 2 and q > qc∗ , we have qc∗ > 1 and 0 < a∗c < N.
Remark 1.6. It follows from Theorems 1.3 and 1.4 that the number a∗c p/q − l − mp − 2
gives another cut-off between the blow-up case and the global existence case. Therefore, the
number a∗c is a new secondary critical exponent of the Cauchy problem 1.1. Unfortunately,
in the critical case a a∗c , we do not know whether the solution of 1.1 exists globally or
blows up in finite time.
Remark 1.7. When m l 1 or m l > 1, the results of Theorems 1.3 and 1.4 are consistent
with those in 19, 20, respectively.
Remark 1.8. In 28, Afanas’eva and Tedeev also established the Fujita type results for 1.1
with m l. In particular, if u0 x ∼ |x|−a , 0 < a < N, they obtained that if q < mp − 1 p/a,
then every nontrivial solution blows up in finite time, and if q > mp − 1 p/a, then the
solution exists globally for a small initial data u0 x. We note that when m l in 1.1, if
q > mp − 1 p/N and 0 < a < p/q − mp − 1, then 0 < a < N and q < mp − 1 p/a,
while if q > mp − 1 p/N and p/q − mp − 1 < a < N, then q > mp − 1 p/a.
Therefore, the results of Theorems 1.3 and 1.4 coincide with those in 28.
Finally, we also consider single-point blow-up for a large number of radial decreasing
solutions of the Cauchy problem 1.1 and give upper bound of the radial solution ur, t in
a small neighborhood of the point x, t, where x 0, t T . We assume that the initial data
u0 x u0 r satisfies the following condition:
H u0 x u0 r ≥ 0 for r > 0, u0 0 > 0, and u0 r ∈ C1 R1 , u0 0 0, and u0 r ≤ 0
for r > 0, M0 sup u0 r < ∞, K0 sup |u0 r| < ∞.
Theorem 1.9. Let N ≥ 1, p > 2, m > 1, l > 1, and q > l mp − 2, and let condition (H) hold. In
addition, assume that the initial function u0 x u0 r satisfies
q
u0 r · r N o1 as r −→ ∞,
⎧ p−2 ⎫
⎪
⎨ um
⎬
ul0 ⎪
p−2
0 −
∈ 0, inf
.
λ0 q
⎪
r>0, u0 r>0⎪
p − 2 N 1 2
ru0
⎩
⎭
1.10
1.11
Abstract and Applied Analysis
5
Let T be the blow-up time; then one has
ur, t ≤ Cr −p/q−l−mp−2 ,
1.12
r, t ∈ R1 × 0, T ,
where
C
q−l−m p−2
lmp−2 1/p−1
p
−p−1/q−l−mp−2
1/p−1
λ0
1.13
> 0,
that is, there is single-point blow-up at point x 0.
Remark 1.10. By 1.11, the best upper estimate 1.12 obtained by our method has the
following form:
⎡
1/p−1 ⎤−p−1/q−l−mp−2
q−l−m p−2
p−2
⎦
ur, t ≤ ⎣ · r −p/q−l−mp−2
1/p−1 p−2
p
−
2
1
2
N
p lm
1.14
in R1 × 0, T . But, we do not give the lower bound estimate of the radial solution ur, t in a
small neighborhood of the point x, t, where x 0, t T .
Remark 1.11. When m l 1 or m l > 1, the results of Theorem 1.9 are consistent with those
in 24, 25, respectively. For 1 < q < l mp − 2, in 29, the authors obtained the results of
global blow-up to arbitrary compactly supported initial data.
Remark 1.12. From Theorem 1.9, we obtain the same decay exponent as that of Theorem 1.2
in 29 by different methods. Moreover, it is interesting to see that the decay exponent of the
upper estimate of Theorem 1.9 is also the same as the secondary critical exponent of Theorems
1.3 and 1.4.
This paper is organized as follows. In Section 2, by using the energy method, we
will obtain a blow-up condition and prove Theorem 1.3. In Section 3, using the comparison
principle, we can construct a global supersolution to prove Theorem 1.4. Finally, we consider
the single-point blow-up under some suitable conditions and prove Theorem 1.9 in Section 4.
2. Blow-Up Case
By using the energy method, we will obtain a blow-up condition corresponding to 1.1.
Therefore, we need to select a suitable test function as follows:
ψε x Aε e−ε|x| ,
Aε RN
1
e−ε|x| dx
∞
ωN 0
εN
,
e−r r N−1 dr ds
∇ψε x −εψε
x
.
|x|
2.1
6
Abstract and Applied Analysis
Proof of Theorem 1.3. Suppose that ux, t is the solution of the Cauchy problem 1.1 and T is
the blow-up time. Let
1
Et s
RN
where 0 < s < 1/p, p > 2, Then, Et ∈ C0, T E t u
s−1
ψε ut dx RN
u
s−1
RN
− ls − 1
u
sl−3
t ∈ 0, T ,
us x, tψε xdx,
C1 0, T and
m p−2
ψε div |∇u |
uqs−1 ψε dx
RN
ψε |∇u |
us−1 ψε |∇um |p−2 ∇ul
2
|∇u| dx ε
RN
∇u dx l
m p−2
2.2
RN
x
dx
|x|
2.3
uqs−1 ψε dx
RN
≥ lm
p−2
ulmp−2s−p−1 ψε |∇u|p dx
1 − s
RN
− lm
p−2
ε
u
lmp−2s−p
ψε |∇u|
p−1
dx uqs−1 ψε dx.
RN
RN
Using Young’s inequality, we have
ε
u
lmp−2s−p
RN
ψε |∇u|
p−1
p−1
dx ≤
p
εp
p
RN
ulmp−2s−p−1 ψε |∇u|p dx
2.4
u
lmp−2s−1
ψε dx.
RN
Since 0 < s < 1/p, it follows from 2.3 and 2.4 that
E t ≥
uqs−1 ψε dx −
RN
lmp−2 εp
p
By q > qc∗ l mp − 2 p/N > l mp − 2 > 1,
we obtain
RN
RN
ψε xdx 1, and Hölder’s inequality,
ulmp−2s−1 ψε
≤
uqs−1 ψε dx
RN
2.5
ulmp−2s−1 ψε dx.
RN
lmp−2s−1/qs−1
ulmp−2s−1 ψε dx RN
q−lmp−2/qs−1
ψε
dx
2.6
lmp−2s−1/qs−1
.
Abstract and Applied Analysis
7
Therefore, by 2.5 and 2.6, we have
dE
≥
dt
u
qs−1
lmp−2s−1/qs−1
ψε dx
RN
×
u
qs−1
q−l−mp−2/qs−1
ψε dx
RN
2.7
lmp−2 εp
.
−
p
Applying Jensen’s inequality, we obtain
uqs−1 ψε dx
q−l−mp−2/qs−1
≥
RN
us ψε dx
q−l−mp−2/s
.
2.8
RN
Thus, it follows from 2.7 and 2.8 that
dE 1
≥
dt
2
qs−1/s
s
u ψε dx
RN
1 qs−1/s qs−1/s
s
E
t
2
2.9
as long as
1
Et ≥
s
2lmp−2 εp
p
s/q−l−mp−2
∀t ∈ 0, T .
2.10
Hence, if E0 satisfies
1
E0 ≥
s
2lmp−2 εp
p
s/q−l−mp−2
C0 ,
2.11
q − 1 q−1/s
s
> 0.
2
2.12
then Et increases and remains below C0 for all t ∈ 0, T .
And by 2.9 we have
−s/q−1
,
Et ≥ Eq−1/s 0 − C1 t
where C1 Therefore, from 2.11 and 2.12, we obtain that ux, t blows up in finite time T 1/C1 Eq−1/s 0: and get an estimate on the blow-up time T of the solution ux, t as follows:
2
T≤
q−1
2lmp−2 εp
p
1−q/q−l−mp−2
.
2.13
8
Abstract and Applied Analysis
Finally, it remains to verify the blow-up condition 2.11. Since u0 x ∈ Φa for some a ∈
0, a∗c , there exist two positive constants M and R0 > 1 such that u0 x ≥ M|x|−a for all
|x| ≥ R0 , and we have
1
E0 s
RN
us0 xψε xdx
M s Aε
>
s
|x|≥R0
Ms Aε −Nas
ε
s
|x|−as e−ε|x| dx
|y|≥εR0
2.14
as −|y|
y e dy.
By the definition of Aε , 0 < a < a∗c , we can choose 0 < ε ≤ 1/R0 so small such that 2.11
holds. The proof of Theorem 1.3 is complete.
3. Global Existence
In this section, we shall prove Theorem 1.4 by constructing a global supersolution. To do this,
we introduce the radially symmetric self-similar solution UM,a x, t to the following Cauchy
problem:
ut div |∇um |p−2 ∇ul ,
x, t ∈ RN × 0, ∞,
ux, 0 u0 x M|x|−a ,
3.1
3.2
x ∈ RN .
It is well known that the existence and uniqueness of the solution of 3.1 have been well
established see 7. By symmetric properties of 3.1, the solution UM,a x, t is given by the
following form
UM,a x, t t−aβ fM r,
with r 1
|x|
,
, β β
a lm p−2 −1 p
t
3.3
where the positive function fM is the solution of the problem
N − 1 p−2 m m p−2 l l
fM
fM
r βrfM
r aβfM r 0,
fM fM r
fM r ≥ 0,
r ≥ 0,
fM
0
0,
r > 0,
3.4
lim r fM r M.
a
r → ∞
We shall prove the existence of solution fM r to 3.4 by the following ordinary differential
equation, and furthermore we obtain the nonincreasing property of the solution fM r.
Abstract and Applied Analysis
9
Firstly, given a fixed η > 0, we consider the following Cauchy problem:
N − 1 p−2 m m p−2 l g
g l r βrg r aβgr 0,
g g r
g0 η,
r > 0,
3.5
g 0 0.
According to the standard of the Cauchy problem for ODE and the methods used in 7, 30,
we can obtain that the solution gr of the Cauchy problem 3.5 is positive, and gr → 0 as
r → ∞; moreover,
lim r a gr M
3.6
r → ∞
for some M Mη > 0.
Secondly, we shall prove that there exists a one-to-one correspondence between M ∈
0, ∞ and η ∈ 0, ∞. Indeed, this can be seen from the following relation:
1−l−m p−2
σ 3.7
,
gη r ηg1 η r , σ p
where g1 r is the solution of 3.5 for η 1. Then,
M η η1−aσ M1,
with M1 lim r a g1 r.
r → ∞
3.8
Therefore, we can deduce that, for each M > 0, there exists a positive, bounded, and global
solution fM r satisfying 3.4.
Finally, we shall prove that the solution gr is non-increasing, that is, fM r is also
non-increasing. To do this, we need the following lemmas.
Lemma 3.1. Let gr be the solution of 3.5; then
lim
r →0
p−2 g l r
N g m r
r
− aβg0.
3.9
Proof. Integrating the 3.5 over 0, ε with ε > 0, we have
ε
ε
ε
N − 1 m p−2 l m p−2 l g
g dr βrg dr aβg dr 0.
ε g g r
0
0
0
3.10
Dividing by ε and taking ε → 0 in 3.10, we obtain
⎡ p−2 ⎤
m g ε
g l ε N − 1 p−2 ⎢
⎥
m g l ε⎦ − lim aβgε,
lim ⎣
g ε
ε→0
ε→0
ε
ε
which implies that 3.9 holds. The proof of Lemma 3.1 is complete.
3.11
10
Abstract and Applied Analysis
Lemma 3.2. If there exists r0 ∈ 0, ∞ such that gr0 0, then gr 0 for all r ≥ r0 .
Proof. We shall prove by contradiction. Assuming that Lemma 3.2 does not hold, it is easy to
see that there exists ε > 0 such that
g r > 0 in r0 , r0 ε.
gr > 0,
3.12
Multiplying 3.5 by r N−1 and integrating over r0 , r with r ∈ r0 , r0 ε, we obtain
r
r
p−2 g l βr N gr Nβr N−1 grdr −
aβr N−1 grdr.
r N−1 g m r0
3.13
r0
It follows from 3.12 and 3.13 that
βr gr ≤
N
r
Nβr
N−1
grdr −
r0
r
aβr N−1 grdr ≤
r0
N − aβ
gr r N − r0N ,
N
3.14
equivalently,
N−a
1≤
N
1−
r0
r
N .
3.15
Letting r → r0 in 3.15, we obtain the inequality 1 ≤ 0, which is a contradiction. The proof
of Lemma 3.2 is complete.
Lemma 3.3. The solution gr of 3.5 is monotone nonincreasing in 0, ∞.
Proof. Our method is based on the contradiction argument. Suppose that, for some r0 > 0,
g r0 > 0, by Lemma 3.1, there exists r1 ∈ 0, r0 such that
g r1 0 ,
m p−2 l g
r1 ≥ 0.
g 3.16
By Lemma 3.2, we have gr1 > 0. Using the similar argument in Lemma 3.1, we obtain
lim
r → r1
p−2 N g m r
g l r
r − r1
−aβgr1 < 0,
3.17
which is a contradiction with 3.14. The proof of Lemma 3.3 is complete.
Next, we apply the monotone properties of fM r to obtain the condition on the global
existence of the solution to 1.1.
Proof of Theorem 1.4. We prove Theorem 1.4 by the following steps.
Abstract and Applied Analysis
11
Step 1. Since ϕx ∈ Φa , there exists a constant K > 0 such that
ϕx ≤ K1 |x|−a
3.18
∀x ∈ RN .
Taking M > K and the self-similar solution UM,a x, t of 3.1 defined as 3.3, since
limr → ∞ r a fM r M > K, there exists a positive constant R0 such that
r a fM r > K
3.19
for r ≥ R0 .
Setting γ fMR0 min{fM r | r ∈ 0, R0 } > 0, it is easy to verify that ϕx ≤ UM,a x, t
−aβ
for all x ∈ RN , where t0 ∈ 0, 1 and t0 γ > ϕ
∞ .
Let λ > 0; then wx, t λUM,a x, λlmp−2−1 t t0 is the solution of the following
problem
wt div |∇wm |p−2 ∇wl ,
x, t ∈ RN × 0, ∞,
wx, 0 λUM,a x, t0 ≥ λϕx,
3.20
x ∈ RN .
Taking η fM 0 and noting that fM r is non-increasing, we have
−aβ
.
wx, t
∞ ηλ λlmp−2−1 t t0
3.21
Step 2. Set vx, t Atwx, Bt, where At and Bt are solutions of the following
problem:
!
"−aq−1/almp−2−1p
Aq t,
A t ηq−1 λq−1 λlmp−2−1 Bt t0
B t Almp−2−1 t
A0 1,
t ∈ 0, ∞,
3.22
t ∈ 0, ∞,
B0 0.
By a direct calculation, we obtain that vx, t satisfies
vt ≥ div |∇vm |p−2 ∇vl vq ,
x, t ∈ RN × 0, ∞,
vx, 0 wx, 0 λUM,a x, t0 ≥ λϕx,
3.23
x∈R .
N
Step 3. We shall prove that there exists a positive constant λ0 λ0 ϕ such that the
problem 3.22 has a global solution At, Bt with At bounded in 0, T if λ ∈ 0, λ0 .
According to the standard theory of ODE, the local existence and uniqueness of solution
At, Bt of 3.22 hold. By 3.22, we have A t > 0, At > 1 for t > 0; furthermore, the
solution is continuous as long as the solution exists and At is finite.
12
Abstract and Applied Analysis
From 3.22, when At exists in 0, t, then Bt is uniquely defined by
Bt t
3.24
Almp−2−1 sds.
0
Since p > 2 and At is increasing, we obtain
Bs s
Almp−2−1 τdτ ≥ Almp−2−1 0s s
3.25
∀s ∈ 0, t.
0
By 3.22, 3.25, and a > a∗c p/q − l − mp − 2, it follows that
1−A
1−q
t q − 1 ηq−1 λq−1
t !
λlmp−2−1 Bs t0
"−aq−1/almp−2−1p
ds
0
≤ q − 1 ηq−1 λq−1
t λlmp−2−1 s t0
−aq−1/almp−2−1p
ds
3.26
0
q − 1 ηq−1 λq−l−mp−2
1−aq−1/almp−2−1p
≤ .
t
a q−1 / a lm p−2 −1 p −1 0
Let λ0 λ0 ϕ be a positive constant defined by
# $
q − 1 a l m p − 2 − 1 p q−1 q−l−mp−2 1−aq−1/almp−2−1p 1
η λ0
t0
.
2
a q−l−m p−2 −p
3.27
Then from 3.26, q > qc∗ > l mp − 2 > 1 and a > a∗c p/q − l − mp − 2, we have
1 ≤ At ≤ 21/q−1 for any λ ∈ 0, λ0 , as long as At exists globally.
On the other hand, by 3.22 and 3.25, we have
t ≤ Bt ≤ 2p−2/q−1 t
3.28
∀t ≥ 0.
Therefore, Bt is also global.
Step 4. For any λ ∈ 0, λ0 , where λ0 λ0 ϕ is defined as 3.27, the solution ux, t
of 1.1 with initial value u0 x λϕx exists globally, and ux, t ≤ vx, t in RN × 0, ∞.
Moreover, there exists a positive constant C such that
−aβ
≤ Ct−aβ
u·, t
∞ ≤ v., t
∞ ≤ 21/q−1 ηλ λlmp−2−1 Bt t0
The proof of Theorem 1.4 is complete.
∀t > 0.
3.29
Abstract and Applied Analysis
13
4. Single Point Blow-Up
In this section, under some suitable assumptions, we shall prove that the blow-up set consists
of the single point x 0. Moreover, we also give the upper estimate of the solution ux, t in
a small neighborhood of the point x, t, where x 0, t T .
First, we suppose that the solution is radially symmetric, that is, depending only on
r |x| at a given time t > 0. Therefore, we study the following problem:
!
"
uq ,
ut r −N−1 r N−1 |um r |p−2 ul
r r
ur, 0 u0 r,
r N−1 |um r |p−2 ul 0,
r
r, t ∈ 0, ∞ × 0, ∞,
4.1
r ∈ 0, ∞,
r, t ∈ {0} × 0, ∞.
Proof of Theorem 1.9. It is based on the method in 21. The main idea is to apply the maximum
principle to the auxiliary function ωi r, t, which is defined in 4.7, and to show that ωi is
small enough in 0, i × 0, T . Then by integrating the obtained inequality and taking limit as
i → ∞, one can get upper bound of the solution ur, t. Therefore, we divide the proof into
the following steps.
Step 1. Since problem 1.1 has no classical solution, we will construct the weak
solution by means of regularization of the degenerate equation.
Now define a strictly monotone sequence {εi }, εi > 0 for all i 1, 2, 3, . . ., such that
εi −→ 0,
as i −→ ∞.
4.2
Then, the weak solution ur, t is the limit function of the solution of the following regularized
problem see 31:
p−2/2 q
2
2
l
u
ε
ε
ui ,
um
ui t r −N−1 r N−1
i
i
i
i r
r
r N−1
um
i
2
r
ui r, 0 u0 r,
r ∈ 0, i,
ui i, t u0 i,
t ∈ 0, T ,
εi2
p−2/2 uli εi 0,
r
r, t ∈ 0, i × 0, T ,
r
4.3
r, t ∈ {0} × 0, T .
By the standard methods used in 1, 32, the uniform estimates for the passage to the
limit which do not depend on εi are established. Therefore, for any fixed εi > 0, we may
assume that, for all sufficiently large i, the function ui r, t satisfies the following conditions:
|ui | ≤ M1 ,
|ui r | ≤ M2
in Qi,T 0, i × 0, T ,
4.4
where M1 , M2 do not depend on i, and
ui r, tr |r0 0 ∀t ∈ 0, T .
4.5
14
Abstract and Applied Analysis
Moreover, by using condition H and the maximum principle in 33, we have
ui r, tr ≤ 0 in Qi,T .
4.6
Step 2. Set
p−2/2 q
2
2
uli λ0 r N ui ,
ε
wi r, t r N−1 um
i
i r
4.7
r
where λ0 is given in 1.11.
By a direct calculation, we find that wi r, t satisfies the following parabolic equation:
wi t ai wi rr bi wi r ci wi di ei ,
4.8
where
ai lul−1
i
um
i
2
r
εi2
p−2/2
p−4/2 2
uli
m p − 2 um−1
εi2
um
um
i r
i r,
i
r
p−2/2 l1 − Nul−1
m i
l−2
ll − 1ui ui r p − 2 uli
bi ui r
r
r
p−4/2 m1 − Num−1
2
i
m
2
m−2
mm − 1ui ui r ,
·
ui r εi
r
2
um
i r
4.9
εi2
q−1 #
ci qui
q−1
di Aqr N−1 1 − λ0 N 1ui ,
$
p − 1 − λ0 p − 2 N 1 2 ,
"p−4/2
! m 2
with A ≡ − p − 2 εi2 uli
,
ui r εi2
r
4.10
4.11
4.12
p−4/2 2
ei p − 2 r N−1
uli
εi2
um
um
i r
i r
r
"
! qm−3
× − m q − 2 mqrλ0 ui
ui r 2 m−11−λ0 Nq1−λ0 N 1 muqm−2 ui r
r N−1
um
i
2
r
εi2
p−2/2 ! ql−3
− l q − 2 lqrλ0 ui
ui r 2
"
l − 11−λ0 Nq1−λ0 N 1 luql−2 ui r
# $
2q−1
λ0 q λ0 p − 2 N 1 2 − p − 2 r N ui .
4.13
Since N ≥ 1, p > 2, m > 1, l > 1, and q > l mp − 2, it follows from 1.11 and 4.6 that
ei ≤ 0.
Now we consider the coefficients ci and di in Qi,T . By 4.6 and p > 2, we have
"p−4/2
! m 2
A − p − 2 εi2 uli
≥ 0.
ui r εi2
r
4.14
Abstract and Applied Analysis
15
It follows from 4.4 that |A| Oεiν as i → ∞, where ν min{2, p − 1}, and we obtain the
following estimate:
sup ci ≤ M3 ,
sup di ≤ M4 iN−1 εiν
in Qi,T ,
4.15
where M3 , M4 are positive constants, which are independent of i.
Therefore, we have the parabolic differential inequality
wi t ≤ ai wi rr bi wi r ci wi di ,
in Qi,T ,
4.16
where ci , di satisfy 4.15.
Next, we consider the function wi r, t on the parabolic boundary of Qi,T . At first, it is
q
easy to see that wi 0, t 0 for all t ∈ 0, T . By 1.10, we have wi i, t ≤ λ0 iN u0 i o1, as
i → ∞ for all t ∈ 0, T . Finally, it follows from 1.11 that
wi r, 0 r N−1
um
0
2
r
εi2
p−2/2 q
ul0 λ0 r N u0
r
! "
p−2 ul λ0 ruq ≤ 0
≤ r N−1 um
0
0 r
0
r
4.17
∀r ∈ 0, i.
Hence, for all sufficiently large i, there exists γi sup wi > 0 on the parabolic boundary
of Qi,T and γi o1 as i → ∞.
In order to estimate wi r, t in Qi,T , we study the following ODE:
dwi
M3 wi M4 iN−1 εiν ,
dt
t > 0,
4.18
wi 0 γi ,
which has the solution
wi t M4 iN−1 εiν
M4 iN−1 ν
γi eM3 t −
εi ,
M3
M3
t > 0.
4.19
Taking the sequence εi such that
iN−1 εiν −→ 0
as i −→ ∞,
4.20
it is obvious that
wi t ≤ wi T αi
∀t ∈ 0, T ,
4.21
where
αi −→ 0
as i −→ ∞.
4.22
16
Abstract and Applied Analysis
Setting
zr, t wi r, t − wi t,
4.23
we have zr, t ≤ 0 on the parabolic boundary, and zr, t satisfies the following parabolic
inequality:
"
!
zt ≤ ai zrr bi zr ci z − M3 − ci wi di − M4 iN−1 εiν ,
in Qi,T .
4.24
It follows from 4.15 that
zt ≤ ai zrr bi zr ci z,
4.25
in Qi,T .
By the maximum principle Chapter II, 33, we obtain that zr, t ≤ 0 in Qi,T , that is,
wi r, t ≤ wi t ≤ wi T αi
4.26
in Qi,T .
Step 3. For large i and ui r, tr ∈ −M2 , 0, we have the following estimate
um
i
2
r
εi2
p−2/2 p−2 l uli ≥ um
ui − δ i ,
i r
r
4.27
r
where
0 < δi O εiν −→ 0 as i −→ ∞.
4.28
By 4.7 and 4.26, we have
p−2 m uli
ui r q
ui
r
≤
r 1−N αi δi
q
ui
4.29
− λ0 r,
namely,
l−1m−1p−2−q
lmp−2 ui
ui r |ui r |p−2 ≤
r 1−N αi δi
q
ui
− λ0 r.
4.30
Letting Fb b|b|−p−2/p−1 , we deduce that
lm
p−2
1/p−1
lmp−2−q−p1/p−1
ui
ui r
≤F
r 1−N αi δi
q
ui
− λ0 r .
4.31
Abstract and Applied Analysis
17
For arbitrary t0 ∈ 0, T , r0 ∈ supp ur, t0 ≡ {r > 0 | ur, t0 > 0}, we denote μ0 ux0 , t0 > 0.
By 4.6 and the uniform convergence ui r, t0 → ur, t0 as i → ∞, r ∈ 0, r0 , we have
ui r, t0 ≥ μ0 /2, r ∈ 0, r0 for all sufficiently large i ≥ i0 . Therefore, from 4.31 we obtainly
for t t0 , i > i0 ,
lm
p−2
1/p−1
lmp−2−q−p1/p−1
ui
ui r
≤F
2q r 1−N αi δi
q
μ0
− λ0 r .
4.32
Integrating the above inequality over interval r1 , r0 , where r1 > 0, we obtain
−
r0 q 1−N
αi δi
p − 1 −q−l−mp−2/p−1 r0
2 r
|r1 ≤
F
− λ0 r dr.
ui
q
q−l−m p−2
μ0
r1
lmp−2
1/p−1 4.33
Letting i → ∞, by 4.22, 4.28, and the uniform convergence ui r, t0 → ur, t0 as i → ∞,
r ∈ 0, r0 , we have the following estimate:
"
p − 1 ! −q−l−mp−2/p−1
u
−
r0 , t0 − u−q−l−mp−2/p−1 r1 , t0 q−l−m p−2
r0 q 1−N
αi δi
2 r
≤ lim
F
− λ0 r dr
q
i→∞ r
μ0
1
lmp−2
1/p−1 1/p−1 p
− λ0
4.34
− 1 p/p−1
p/p−1
r0
.
− r1
p
Setting r1 → 0, from 4.34, we obtain the upper estimate
−p−1/q−l−mp−2
q − l − m p − 2 1/p−1
−p/q−l−mp−2
r0
,
ur0 , t0 ≤
p−2 1/p−1 λ0
p lm
4.35
where r0 , t0 ∈ R1 × 0, T . The proof of Theorem 1.9 is complete.
Acknowledgments
The authors would like to thank the anonymous referees for their valuable comments on this
work. The second author is supported in part by NSF of China 11071266 and in part by the
Natural Science Foundation Project of CQ CSTC2010BB9218. This work is supported by the
Fundamental Research Funds for the Central University, Project No CDJXS 11 10 00 40.
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