Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 898546, 6 pages
http://dx.doi.org/10.1155/2013/898546
Research Article
Asymptotic Behavior of Solutions to Fast Diffusive
Non-Newtonian Filtration Equations Coupled by Nonlinear
Boundary Sources
Wang Zejia,1 Wang Shunti,2 and Zhang Chengbin2
1
2
College of Mathematics and Information Science, Jiangxi Normal University, Nanchang 330022, China
Institute of Mathematics, Jilin University, Changchun 130012, China
Correspondence should be addressed to Wang Zejia; wangzj1979@gmail.com
Received 6 February 2013; Accepted 27 March 2013
Academic Editor: Sining Zheng
Copyright © 2013 Wang Zejia 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.
This paper is concerning the asymptotic behavior of solutions to the fast diffusive non-Newtonian filtration equations coupled
by the nonlinear boundary sources. We are interested in the critical global existence curve and the critical Fujita curve, which
are used to describe the large-time behavior of solutions. It is shown that the above two critical curves are both the same for the
multidimensional problem we considered.
1. Introduction
In this paper, we study the non-Newtonian filtration equations coupled by the nonlinear boundary sources
𝜕𝑢
= div (|∇𝑢|𝑝−2 ∇𝑢) ,
𝜕𝑡
𝜕V
= div (|∇V|𝑞−2 ∇V) ,
𝜕𝑡
(1)
(𝑥, 𝑡) ∈ (R𝑁 \ 𝐵1 (0)) × (0, 𝑇) ,
|∇𝑢|𝑝−2 ∇𝑢 ⋅ ]⃗ = V𝛼 (𝑥, 𝑡) ,
|∇V|𝑞−2 ∇V ⋅ ]⃗ = 𝑢𝛽 (𝑥, 𝑡) ,
(𝑥, 𝑡) ∈ 𝜕𝐵1 (0) × (0, 𝑇) ,
𝑢 (𝑥, 0) = 𝑢0 (𝑥) ,
V (𝑥, 0) = V0 (𝑥) ,
(2)
𝑥 ∈ R𝑁 \ 𝐵1 (0) ,
(3)
where 1 < 𝑝, 𝑞 < 2, 𝛼, 𝛽 ≥ 0, 𝑁 ≥ 2, 𝐵1 (0) is the unit ball
in R𝑁 with boundary 𝜕𝐵1 (0), ]⃗ is the inward normal vector
on 𝜕𝐵1 (0), and 𝑢0 (𝑥) and V0 (𝑥) are nonnegative, suitably
smooth, and bounded functions satisfying the appropriate
compatibility conditions.
The system (1)–(3) can be used to describe the models in
population dynamics, chemical reactions, heat propagation,
and so on. It is well known that the classical solutions do
not exist because the equations in (1) are degenerate in
{(𝑥, 𝑡); ∇𝑢(𝑥, 𝑡) = 0}, while the local existence and the
comparison principle of the weak solutions can be obtained;
see [1, 2]. In this paper, we investigate the asymptotic behavior
of solutions to the system (1)–(3), including blowup in a finite
time and global existence in time.
Since the beginning work on critical exponent done by
Fujita in [3], there are a lot of Fujita type results established
for various equations; see the survey papers [4, 5] and the
references therein and also the papers [6–9]. We recall some
results on the fast diffusion case. In [10], the authors obtained
the critical exponents for the single one-dimensional fast
diffusive equation on (0, +∞) × (0, +∞), that is,
𝜕 󵄨󵄨󵄨󵄨 𝜕𝑢 󵄨󵄨󵄨󵄨𝑝−2 𝜕𝑢
𝜕𝑢
=
(󵄨 󵄨
),
𝜕𝑡 𝜕𝑥 󵄨󵄨󵄨 𝜕𝑥 󵄨󵄨󵄨 𝜕𝑥
𝑥 > 0, 𝑡 > 0,
󵄨󵄨 𝜕𝑢 󵄨󵄨𝑝−2 𝜕𝑢
󵄨󵄨 󵄨󵄨
(0, 𝑡) = 𝑢𝛼 (0, 𝑡) ,
󵄨󵄨 󵄨󵄨
󵄨󵄨 𝜕𝑥 󵄨󵄨 𝜕𝑥
𝑢 (𝑥, 0) = 𝑢0 (𝑥) ,
𝑡 > 0,
(4)
𝑥 > 0.
They showed that the global critical exponent is 𝛼0 = 2(𝑝 −
1)/𝑝 and the critical Fujita exponent is 𝛼𝑐 = 2(𝑝 − 1).
2
Abstract and Applied Analysis
After that, the corresponding results on the coupled 𝑝-laplace
equations can be found in the paper [11], in which the authors
consider multiple equations coupled by boundary sources.
For the system,
𝜕 󵄨󵄨󵄨󵄨 𝜕𝑢 󵄨󵄨󵄨󵄨𝑝−2 𝜕𝑢
𝜕𝑢
=
(󵄨 󵄨
),
𝜕𝑡 𝜕𝑥 󵄨󵄨󵄨 𝜕𝑥 󵄨󵄨󵄨 𝜕𝑥
𝜕V
𝜕 󵄨󵄨󵄨󵄨 𝜕V 󵄨󵄨󵄨󵄨𝑞−2 𝜕V
=
(󵄨 󵄨
),
𝜕𝑡 𝜕𝑥 󵄨󵄨󵄨 𝜕𝑥 󵄨󵄨󵄨 𝜕𝑥
𝑢 (𝑥, 0) = 𝑢0 (𝑥) ,
(5)
𝑝−2
𝑡 > 0,
V (𝑥, 0) = V0 (𝑥) ,
|∇𝑢|
∇𝑢 ⋅ ]⃗ = 𝑢 (𝑥, 𝑡) ,
𝑢 (𝑥, 0) = 𝑢0 (𝑥) ,
𝑢 (𝑥, 0) = 𝑢0 (𝑥) ,
V (𝑥, 0) = V0 (𝑥) ,
(8)
𝑥 ∈ R𝑁 \ 𝐵1 (0) ,
(9)
2. Main Results and Their Proofs
Theorem 1. The critical global existence curve and the critical
Fujita curve for the system (1)–(3) with 𝑁 ≥ 2 are the same
one, that is, the curve given by
𝑥 > 0,
(𝑥, 𝑡) ∈ (R𝑁 \ 𝐵1 (0)) × (0, 𝑇) ,
𝛼
(𝑥, 𝑡) ∈ 𝜕𝐵1 (0) × (0, 𝑇) ,
In this section, we first state our main results and then prove
them. Our main results are as follows.
the results in [11] are that the critical global existence curve is
𝛼𝛽 = 4(𝑝−1)(𝑞−1)/(𝑝𝑞), and when 𝛼𝛽 > 4(𝑝−1)(𝑞−1)/(𝑝𝑞),
the critical Fujita curve is min{𝑙1 − 𝑘1 , 𝑙2 − 𝑘2 } = 0, where
𝑘1 , 𝑘2 , 𝑙1 , and 𝑙2 are constants depending on 𝑝, 𝑞, 𝛼, and 𝛽.
It can be seen by some computations that the critical Fujita
curve is on the strictly right of the critical global existence
curve.
As for the multidimensional problem, the single equation
case of (1)–(3) was discussed in [12]; that is,
𝜕𝑢
= div (|∇𝑢|𝑝−2 ∇𝑢) ,
𝜕𝑡
|∇V|𝑞−2 ∇V ⋅ ]⃗ = 𝑢𝛽 (𝑥, 𝑡) ,
with 𝜆 1 > 𝑝 − 𝑁, 𝜆 2 > 𝑞 − 𝑁, 𝑁 ≥ 1.
We will state our main results and prove them in the next
section.
𝑥 > 0, 𝑡 > 0,
󵄨󵄨 𝜕𝑢 󵄨󵄨𝑝−2 𝜕𝑢
󵄨󵄨 󵄨󵄨
(0, 𝑡) = V𝛼 (0, 𝑡) ,
󵄨󵄨 󵄨󵄨
󵄨󵄨 𝜕𝑥 󵄨󵄨 𝜕𝑥
󵄨󵄨 𝜕V 󵄨󵄨𝑞−2 𝜕𝑢
󵄨󵄨 󵄨󵄨
(0, 𝑡) = 𝑢𝛽 (0, 𝑡) ,
󵄨󵄨 󵄨󵄨
󵄨󵄨 𝜕𝑥 󵄨󵄨
𝜕𝑥
|∇𝑢|𝑝−2 ∇𝑢 ⋅ ]⃗ = V𝛼 (𝑥, 𝑡) ,
𝑥 ∈ R𝑁 \ 𝐵1 (0)
(6)
with 1 < 𝑝 < 2, 𝛼 ≥ 0, 𝑁 ≥ 2. It was shown that the critical
global exponent and the critical Fujita exponent both are 𝛼 =
𝑝 − 1.
Motivated by the papers mentioned above, the aim of this
paper is to study the asymptotic behavior of solutions to the
system (1)–(3). We show that the phenomenon that the two
critical exponents for multi-dimensional equation coincide
also occurs in the coupled equations.
Furthermore, by virtue of the radial symmetry of the
exterior domain of the unit ball, we note that the above result
can be extended to the following more general problems:
𝜕
(|𝑥|𝜆 1 𝑢) = div (|𝑥|𝜆 1 |∇𝑢|𝑝−2 ∇𝑢) ,
𝜕𝑡
(𝑥, 𝑡) ∈ (R𝑁 \ 𝐵1 (0)) × (0, 𝑇) ,
(7)
4 (𝑝 − 1) (𝑞 − 1)
.
𝑝𝑞
(10)
Namely, if 𝛼𝛽 ≤ 4(𝑝 − 1)(𝑞 − 1)/(𝑝𝑞), then all nonnegative
solutions of the system (1)–(3) exist globally in time, while if
𝛼𝛽 > 4(𝑝 − 1)(𝑞 − 1)/(𝑝𝑞), then there exist both blow-up
solutions for large initial data and global existent solution for
small initial data.
Theorem 2. Assume that 𝜆 1 > 𝑝 − 𝑁, 𝜆 2 > 𝑞 − 𝑁, and 𝑁 ≥
1. Then both the critical global existence curve and the critical
Fujita curve for the system (7)–(9) are the curve given by
𝛼𝛽 =
(𝑥, 𝑡) ∈ 𝜕𝐵1 (0) × (0, 𝑇) ,
𝜕
(|𝑥|𝜆 2 V) = div (|𝑥|𝜆 2 |∇V|𝑞−2 ∇V) ,
𝜕𝑡
𝛼𝛽 =
4 (𝑝 − 1) (𝑞 − 1)
.
𝑝𝑞
(11)
Since the proofs of Theorems 1 and 2 are similar, we
shall give the proof of Theorem 2 only. We note that the
equations in (1) and (7) are degenerate at the points where
∇𝑢(𝑥, 𝑡) = 0, and classical solutions may not exist generally.
It is therefore necessary to consider weak solutions in the
distribution sense.
To prove Theorem 2, we first establish some preliminary
results. Let 𝑟 = |𝑥| and consider the problem:
̃ 󵄨󵄨󵄨 𝜕𝑢 󵄨󵄨󵄨𝑝−2 𝜕𝑢
𝜆
𝜕 󵄨󵄨󵄨󵄨 𝜕𝑢 󵄨󵄨󵄨󵄨𝑝−2 𝜕𝑢
𝜕𝑢
=
(󵄨󵄨 󵄨󵄨
) + 1 󵄨󵄨󵄨 󵄨󵄨󵄨
,
𝜕𝑡 𝜕𝑟 󵄨󵄨 𝜕𝑟 󵄨󵄨 𝜕𝑟
𝑟 󵄨󵄨 𝜕𝑟 󵄨󵄨 𝜕𝑟
̃ 󵄨󵄨󵄨 𝜕V 󵄨󵄨󵄨𝑞−2 𝜕V
𝜆
𝜕V
𝜕 󵄨󵄨󵄨󵄨 𝜕V 󵄨󵄨󵄨󵄨𝑞−2 𝜕V
=
(󵄨󵄨 󵄨󵄨
) + 2 󵄨󵄨󵄨 󵄨󵄨󵄨
,
𝜕𝑡 𝜕𝑟 󵄨󵄨 𝜕𝑟 󵄨󵄨 𝜕𝑟
𝑟 󵄨󵄨 𝜕𝑟 󵄨󵄨 𝜕𝑟
󵄨󵄨 𝜕𝑢 󵄨󵄨𝑝−2 𝜕𝑢
󵄨󵄨 󵄨󵄨
(1, 𝑡) = V𝛼 (1, 𝑡) ,
󵄨󵄨 󵄨󵄨
󵄨󵄨 𝜕𝑟 󵄨󵄨 𝜕𝑟
󵄨󵄨󵄨 𝜕V 󵄨󵄨󵄨𝑞−2 𝜕V
𝛽
󵄨󵄨 󵄨󵄨
󵄨󵄨 𝜕𝑟 󵄨󵄨 𝜕𝑟 (1, 𝑡) = 𝑢 (1, 𝑡) ,
󵄨 󵄨
𝑢 (𝑟, 0) = 𝑢0 (𝑟) ,
𝑟 > 1,
𝑡>0
(12)
(13)
𝑡 > 0,
V (𝑟, 0) = V0 (𝑟) ,
𝑟 > 1,
(14)
Abstract and Applied Analysis
3
̃ = 𝜆 + 𝑁 − 1 > 𝑝 − 1, 𝜆
̃ >
where 1 < 𝑝, 𝑞 < 2, 𝜆
1
1
2
𝑞 − 1, 𝛼, 𝛽 ≥ 0, 𝑢0 , V0 are nonnegative, suitably smooth, and
bounded functions. It is not difficult to check that the solution
(𝑢, V)(𝑟, 𝑡) of the system (12)–(14) is also the solution of the
system (7)–(9) if 𝑢0 (𝑥) and V0 (𝑥) are radially symmetrical. We
now study the asymptotic behavior of solutions to the system
(12)–(14).
On the other hand,
󵄨󵄨 𝜕𝑢 󵄨󵄨𝑝−2 𝜕𝑢
𝑝−1
󵄨 󵄨
−󵄨󵄨󵄨 󵄨󵄨󵄨
(1, 𝑡) = 𝑀1 e(𝑝−1)(𝐿 1 +𝐽1 )𝑡
󵄨󵄨 𝜕𝑟 󵄨󵄨 𝜕𝑟
= (𝐾 + 1)𝛼 e2(𝑝−1)𝐿 1 𝑡/𝑝 ,
V𝛼 (1, 𝑡) = e𝛼𝐿 2 𝑡 (𝐾 + 1)𝛼
Proposition 3. If 𝛼𝛽 ≤ 4(𝑝−1)(𝑞−1)/𝑝𝑞, then all nonnegative
solutions of the system (12)–(14) exist globally in time.
= (𝐾 + 1)𝛼 e2(𝑝−1)𝐿 1 𝑡/𝑝
Proof of Proposition 3. We prove this proposition by constructing a kind of global upper solutions. Let
󵄨󵄨 𝜕𝑢 󵄨󵄨𝑝−2 𝜕𝑢
󵄨 󵄨
= −󵄨󵄨󵄨 󵄨󵄨󵄨
(1, 𝑡) .
󵄨󵄨 𝜕𝑥 󵄨󵄨 𝜕𝑥
𝑢 (𝑟, 𝑡) = e𝐿 1 𝑡 (𝐾 + e−𝑀1 𝜂1 ) ,
𝜂1 = (𝑟 − 1) e𝐽1 𝑡 ,
𝑟 > 1, 𝑡 > 0,
V (𝑟, 𝑡) = e𝐿 2 𝑡 (𝐾 + e−𝑀2 𝜂2 ) ,
𝜂2 = (𝑟 − 1) e𝐽2 𝑡 ,
𝑟 > 1, 𝑡 > 0,
(15)
where 𝑀1 = (𝐾 + 1)𝛼/(𝑝−1) , 𝑀2 = (𝐾 + 1)𝛽/(𝑞−1) , 𝐽1 = 𝐿 1 (2 −
𝑝)/𝑝, 𝐽2 = 𝐿 2 (2 − 𝑞)/𝑞, 𝐿 2 = 2(𝑝 − 1)𝐿 1 /(𝑝𝛼), and 𝐾 and 𝐿 1
are large constants satisfying that
󵄩 󵄩 󵄩 󵄩 2−𝑝 2−𝑞
𝐾 > max {1, 󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩∞ , 󵄩󵄩󵄩V0 󵄩󵄩󵄩∞ ,
,
},
𝑝e
𝑞e
𝑝
𝑞
𝑝 (𝑝 − 1) 𝑒𝑀1
𝛼𝑝𝑞 (𝑞 − 1) 𝑒𝑀2
𝐿 1 > max {
}.
,
𝑝 − 2 + 𝑝𝑒𝐾 2 (𝑝 − 1) (𝑞 − 2 + 𝑞𝑒𝐾)
(16)
Obviously, we have 𝑢(𝑟, 0) ≥ 𝑢0 (𝑟) and V(𝑟, 0) ≥ V0 (𝑟) for
𝑟 > 1, and a direct computation yields that
󵄨󵄨 𝜕𝑢 󵄨󵄨𝑝−2 𝜕𝑢
𝐽1 𝑡
𝑝−1
󵄨󵄨 󵄨󵄨
= −𝑀1 e(𝑝−1)(𝐿 1 +𝐽1 )𝑡 e−(𝑝−1)(𝑟−1)𝑀1 e < 0,
󵄨󵄨 󵄨󵄨
󵄨󵄨 𝜕𝑟 󵄨󵄨 𝜕𝑟
𝐽1
𝜕 󵄨󵄨󵄨󵄨 𝜕𝑢 󵄨󵄨󵄨󵄨𝑝−2 𝜕𝑢
𝑝
(󵄨󵄨 󵄨󵄨
) = (𝑝 − 1) 𝑀1 e((𝑝−1)𝐿 1 +𝑝𝐽1 )𝑡 e−(𝑝−1)(𝑟−1)𝑀1 e 𝑡
𝜕𝑟 󵄨󵄨 𝜕𝑟 󵄨󵄨 𝜕𝑟
𝑝
≤ (𝑝 − 1) 𝑀1 e((𝑝−1)𝐿 1 +𝑝𝐽1 )𝑡 .
−𝑦
Noticing that −𝑦e
−1
≥ −e
(17)
for 𝑦 > 0, we have
+ 𝐽1 e
𝐿1𝑡
𝐽1 𝑡 −𝑀1 (𝑟−1)e𝐽1 𝑡
(−𝑀1 (𝑟 − 1) e e
−𝑀1 (𝑟−1)e𝐽1 𝑡
≥ 𝐿 1e
(𝐾 + e
≥ (𝐿 1 −
𝐽1
) 𝐾e𝐿 1 𝑡 .
𝐾e
𝐿 1 𝑡 −1
) − 𝐽1 e
Similarly, we have
̃ 󵄨󵄨󵄨 𝜕V 󵄨󵄨󵄨𝑞−2 𝜕V
𝜆
𝜕V
𝜕 󵄨󵄨󵄨󵄨 𝜕V 󵄨󵄨󵄨󵄨𝑞−2 𝜕V
≥
(󵄨󵄨 󵄨󵄨
) + 2 󵄨󵄨󵄨 󵄨󵄨󵄨
,
𝜕𝑡 𝜕𝑟 󵄨󵄨 𝜕𝑟 󵄨󵄨 𝜕𝑟
𝑟 󵄨󵄨 𝜕𝑟 󵄨󵄨 𝜕𝑟
󵄨󵄨 𝜕V 󵄨󵄨𝑞−2 𝜕V
𝑞−1
󵄨 󵄨
−󵄨󵄨󵄨 󵄨󵄨󵄨
(1, 𝑡) = 𝑀2 e(𝑞−1)(𝐿 2 +𝐽2 )𝑡
󵄨󵄨 𝜕𝑟 󵄨󵄨 𝜕𝑟
(21)
= (𝐾 + 1)𝛽 e4(𝑝−1)(𝑞−1)𝐿 1 𝑡/(𝑝𝑞𝛼) .
Since that 𝛼𝛽 ≤ 4(𝑝 − 1)(𝑞 − 1)/(𝑝𝑞), then
𝑢𝛽 (1, 𝑡) = e𝛽𝐿 1 𝑡 (𝐾 + 1)𝛽 ≤ (𝐾 + 1)𝛽 e4(𝑝−1)(𝑞−1)𝐿 1 𝑡/(𝑝𝑞𝛼) .
(22)
This indicates that
󵄨󵄨 𝜕V 󵄨󵄨𝑞−2 𝜕V
󵄨 󵄨
−󵄨󵄨󵄨 󵄨󵄨󵄨
(1, 𝑡) ≥ 𝑢𝛽 (1, 𝑡) .
󵄨󵄨 𝜕𝑟 󵄨󵄨 𝜕𝑟
(23)
Noticing the global existence in time of (𝑢, V), we get that
the solution of the problem (12)–(14) exists globally by the
comparison principle. The proof is complete.
Proposition 4. If 𝛼𝛽 > 4(𝑝 − 1)(𝑞 − 1)/(𝑝𝑞), then the
nonnegative nontrivial solutions of the system (12)–(14) blow
up in finite time for large initial data.
Proof of Proposition 4. The proposition is proved by constructing a kind of lower blow-up solutions. Set
𝐽1 𝑡
𝜕𝑢
= 𝐿 1 e𝐿 1 𝑡 (𝐾 + e−𝑀1 (𝑟−1)e )
𝜕𝑡
𝐿1𝑡
(20)
𝑢 (𝑟, 𝑡) = (𝑇 − 𝑡)−𝑘1 𝑓1 (𝜉) ,
)
𝜉 = (𝑟 − 1) (𝑇 − 𝑡)−𝑙1 ,
(18)
e
̃ > 0,
Due to the choice of 𝐿 1 and (𝑝 − 1)𝐿 1 + 𝑝𝐽1 = 𝐿 1 , 𝜆
1
then we get
𝜕𝑢
𝜕 󵄨󵄨󵄨󵄨 𝜕𝑢 󵄨󵄨󵄨󵄨𝑝−2 𝜕𝑢
≥
(󵄨 󵄨
)
𝜕𝑡 𝜕𝑟 󵄨󵄨󵄨 𝜕𝑟 󵄨󵄨󵄨 𝜕𝑟
(19)
̃ 󵄨󵄨󵄨 𝜕𝑢 󵄨󵄨󵄨𝑝−2 𝜕𝑢
𝜆
𝜕 󵄨󵄨󵄨󵄨 𝜕𝑢 󵄨󵄨󵄨󵄨𝑝−2 𝜕𝑢
1󵄨
󵄨
(󵄨 󵄨
) + 󵄨󵄨 󵄨󵄨
.
≥
𝜕𝑟 󵄨󵄨󵄨 𝜕𝑟 󵄨󵄨󵄨 𝜕𝑟
𝑟 󵄨󵄨 𝜕𝑟 󵄨󵄨 𝜕𝑟
𝑟 > 1, 𝑡 > 0,
V (𝑟, 𝑡) = (𝑇 − 𝑡)−𝑘2 𝑓2 (𝜂) ,
𝜂 = (𝑟 − 1) (𝑇 − 𝑡)−𝑙2 ,
(24)
𝑟 > 1, 𝑡 > 0,
where 𝑇 > 0, and
𝑘1 =
𝑘2 =
(𝑞 − 1) (2 (𝑝 − 1) + 𝛼𝑝)
,
𝛼𝛽𝑝𝑞 − 4 (𝑝 − 1) (𝑞 − 1)
(𝑝 − 1) (2 (𝑞 − 1) + 𝛽𝑞)
,
𝛼𝛽𝑝𝑞 − 4 (𝑝 − 1) (𝑞 − 1)
𝑙1 =
𝑙2 =
1 + (2 − 𝑝) 𝑘1
,
𝑝
1 + (2 − 𝑞) 𝑘2
.
𝑞
(25)
4
Abstract and Applied Analysis
Due to 1 < 𝑝, 𝑞 < 2, 𝛼𝛽 > 4(𝑝 − 1)(𝑞 − 1)/(𝑝𝑞), we see that
𝑘1 , 𝑙1 , 𝑘2 , 𝑙2 > 0 and
𝑘1 + 1 = (𝑝 − 1) 𝑘1 + 𝑝𝑙1 ,
(𝑝 − 1) (𝑘1 + 𝑙1 ) = 𝑘2 𝛼,
𝑘2 + 1 = (𝑞 − 1) 𝑘2 + 𝑞𝑙2 ,
(𝑞 − 1) (𝑘2 + 𝑙2 ) = 𝑘1 𝛽.
(26)
We claim that (𝑢, V) is a lower solution to the problem
(12)–(14) with 𝑢0 (𝑟) ≥ 𝑢(𝑟, 0), V0 (𝑟) ≥ V(𝑟, 0) if the following
inequalities hold:
̃ (𝑇 − 𝑡)𝑙1 󵄨 󸀠 󵄨𝑝−2 󸀠
󸀠
𝜆
󵄨𝑝−2
󵄨
󵄨󵄨𝑓 (𝜉)󵄨󵄨 𝑓 (𝜉)
(󵄨󵄨󵄨󵄨𝑓1󸀠 (𝜉)󵄨󵄨󵄨󵄨 𝑓1󸀠 (𝜉)) + 1
󵄨󵄨 1 󵄨󵄨
1
𝑟
−
𝑙1 𝜉𝑓1󸀠
that is,
(29)
(30)
Note that
(31)
for 𝑟 > 1.
𝑝/(𝑝−2)
− (2𝐴 1 )
).
(41)
󸀠
󸀠󸀠
𝜉, 𝜂 ≥ 0,
(32)
then (27)– (30) hold provided that
󸀠
󵄨𝑝−2
󵄨
̃ 𝑇𝑙 󵄨󵄨󵄨󵄨𝑓󸀠 (𝜉)󵄨󵄨󵄨󵄨𝑝−2 𝑓󸀠 (𝜉) − 𝑘 𝑓 (𝜉) ≥ 0,
(󵄨󵄨󵄨󵄨𝑓1󸀠 (𝜉)󵄨󵄨󵄨󵄨 𝑓1󸀠 (𝜉)) + 𝜆
1 1󵄨 1
1 1
1
󵄨
(33)
󵄨𝑝−2
󵄨
−󵄨󵄨󵄨󵄨𝑓1󸀠 (0)󵄨󵄨󵄨󵄨 𝑓1󸀠 (0) ≤ 𝑓2𝛼 (0) ,
Then, the inequalities in (40) are valid provided that, for 0 <
𝜉 < 𝐴 1 /𝐵1 ,
(𝑝 − 1) 𝐵1
̃ 𝑇𝑙1 (𝐴 + 𝐵 𝜉) ,
̃ 𝐴 𝑇𝑙1 ≥ 𝜆
≥ 2𝜆
1 1
1
1
1
2−𝑝
(42)
𝑝
(𝑝 − 1) 𝑝𝑝−1 𝐵1
(2 − 𝑝)
(34)
󸀠
󵄨
󵄨𝑞−2
(󵄨󵄨󵄨󵄨𝑓2󸀠 (𝜂)󵄨󵄨󵄨󵄨 𝑓2󸀠 (𝜂))
𝑝
≥ 𝑘1 .
1/𝑝
̃ 𝑇𝑙2 󵄨󵄨󵄨󵄨𝑓󸀠 (𝜂)󵄨󵄨󵄨󵄨𝑞−2 𝑓󸀠 (𝜂) − 𝑘 𝑓 (𝜂) ≥ 0,
+𝜆
2
2 2
2
󵄨 2
󵄨
󵄨𝑞−2
󵄨
𝛽
−󵄨󵄨󵄨󵄨𝑓2󸀠 (0)󵄨󵄨󵄨󵄨 𝑓2󸀠 (0) ≤ 𝑓1 (0) .
(35)
𝑝/(𝑝−2)
− (2𝐴 1 )
− (2𝐴 2 )
𝑞/(𝑞−2)
So, we choose 𝐵1 = 𝑘1 (2 − 𝑝)/[(𝑝 − 1)𝑝𝑝−1 ]1/𝑝 , and 𝐴 1 is
small enough that
𝐴1 ≤
(36)
Take
𝑞/(𝑞−2)
𝑝
(𝑝 − 1) 𝑝𝑝−1 𝐵1
𝑝 − 1 󵄨󵄨 󸀠 󵄨󵄨𝑝−2 󸀠󸀠
𝑝/(𝑝−2)
󵄨󵄨𝑓1 (𝜉)󵄨󵄨 𝑓1 (𝜉) =
(𝐴 1 + 𝐵1 𝜉)
,
𝑝
󵄨
2 󵄨
(2 − 𝑝)
𝑝/(𝑝−2)
𝑓1 (𝜉)󸀠 , 𝑓2 (𝜂) ≤ 0,
𝑓1 (𝜉)󸀠󸀠 , 𝑓2 (𝜂) ≥ 0,
𝑓2 (𝜂) = ((𝐴 2 + 𝐵2 𝜂)
𝑝𝐵1
2/(𝑝−2)
,
(𝐴 + 𝐵1 𝜉)
𝑝−2 1
𝑘1 𝑓1 (𝜉) = 𝑘1 ((𝐴 1 + 𝐵1 𝜉)
So if we assume that 𝑓1 , 𝑓2 satisfy
𝑓1 (𝜉) = ((𝐴 1 + 𝐵1 𝜉)
𝑝 − 1 󸀠󸀠
𝑝 − 1 2𝑝𝐵12
(4−𝑝)/(𝑝−2)
(𝐴 + 𝐵1 𝜉)
,
𝑓1 (𝜉) =
2
2 (𝑝 − 2)2 1
̃ 𝑇𝑙1 𝑓󸀠 (𝜉) = −𝜆
̃ 𝑇𝑙1
−𝜆
1
1
1
̃ (𝑇 − 𝑡)𝑙1
𝜆
1
̃ 𝑇𝑙1 ,
̃ (𝑇 − 𝑡)𝑙1 ≤ 𝜆
≤𝜆
1
1
𝑟
𝑝/(𝑝−2)
(40)
In the first place, we compute each term in (40) as follows:
󵄨𝑞−2
󵄨
𝛽
−󵄨󵄨󵄨󵄨𝑓2󸀠 (0)󵄨󵄨󵄨󵄨 𝑓2󸀠 (0) ≤ 𝑓1 (0) .
𝑓1 (𝜉) , 𝑓2 (𝜂) ≥ 0,
𝑝 − 1 󸀠󸀠
̃ 𝑇𝑙1 𝑓󸀠 (𝜉) ,
𝑓 (𝜉) ≥ − 𝜆
1
1
2 1
𝑝 − 1 󵄨󵄨 󸀠 󵄨󵄨𝑝−2 󸀠󸀠
󵄨󵄨𝑓 (𝜉)󵄨󵄨 𝑓1 (𝜉) ≥ 𝑘1 𝑓1 (𝜉) .
2 󵄨 1 󵄨
(𝜂) − 𝑘2 𝑓2 (𝜂) ≥ 0,
̃ (𝑇 − 𝑡)𝑙1
𝜆
2
̃ 𝑇𝑙2 ,
≤𝜆
2
𝑟
(39)
󸀠
1 󵄨󵄨 󸀠 󵄨󵄨𝑝−2 󸀠
(󵄨󵄨󵄨𝑓1 (𝜉)󵄨󵄨󵄨 𝑓1 (𝜉)) ≥ 𝑘1 𝑓1 (𝜉) ,
2
(28)
̃ (𝑇 − 𝑡)𝑙2 󵄨 󸀠 󵄨𝑞−2 󸀠
󸀠
𝜆
󵄨
󵄨𝑞−2
󵄨󵄨𝑓 (𝜂)󵄨󵄨 𝑓 (𝜂)
(󵄨󵄨󵄨󵄨𝑓2󸀠 (𝜂)󵄨󵄨󵄨󵄨 𝑓2󸀠 (𝜂)) + 2
󵄨󵄨 2
󵄨󵄨
2
𝑟
−
󸀠
1 󵄨󵄨 󸀠 󵄨󵄨𝑝−2 󸀠
̃ 𝑇𝑙1 󵄨󵄨󵄨󵄨𝑓󸀠 (𝜉)󵄨󵄨󵄨󵄨𝑝−2 𝑓󸀠 (𝜉) ,
(󵄨󵄨󵄨𝑓1 (𝜉)󵄨󵄨󵄨 𝑓1 (𝜉)) ≥ − 𝜆
1
1
󵄨 1 󵄨
2
(𝜉) − 𝑘1 𝑓1 (𝜉) ≥ 0,
󵄨𝑝−2
󵄨
−󵄨󵄨󵄨󵄨𝑓1󸀠 (0)󵄨󵄨󵄨󵄨 𝑓1󸀠 (0) ≤ 𝑓2𝛼 (0) ,
𝑙2 𝜂𝑓2󸀠
(27)
where 0 < 𝐴 1 , 𝐴 2 < 1, 𝐵1 , 𝐵2 > 0 are the constants to be
determined. It is clear that the above 𝑓1 , 𝑓2 satisfy (32). Now,
we verify that 𝑓1 , 𝑓2 satisfy (33)–(36) in the distribution sense.
We claim that (33) is valid for 0 < 𝜉 < 𝐴 1 /𝐵1 . In fact, we only
need to verify that
𝐵1 (𝑝 − 1)
.
̃𝜆 𝑇𝑙1 (2 − 𝑝)
1
(43)
Similarly, choose 𝐵2 , 𝐴 2 , satisfying that
) ,
+
) ,
+
𝜉 ≥ 0,
(37)
𝜂 ≥ 0,
(38)
𝑞
𝐵2 = (𝑘2
1/𝑞
(2 − 𝑞)
)
(𝑞 − 1) 𝑞𝑞−1
to get the validity of (35).
,
𝐴2 ≤
𝐵2 (𝑞 − 1)
̃ 𝑇𝑙2 (2 − 𝑞)
𝜆
2
(44)
Abstract and Applied Analysis
5
Next, we verify that 𝑓1 and 𝑓2 given by (37) and (38) also
satisfy the boundary conditions (34) and (36):
𝑝−1
𝑝𝐵1 2/(𝑝−2)
󵄨𝑝−2
󵄨
)
𝐴
−󵄨󵄨󵄨󵄨𝑓1󸀠 (0)󵄨󵄨󵄨󵄨 𝑓1󸀠 (0) = (
2−𝑝 1
=(
𝑓2𝛼
(0) =
𝑝𝐵1 𝑝−1 2(𝑝−1)/(𝑝−2)
,
) 𝐴1
2−𝑝
𝑞/𝑞−2
(𝐴 2
(45)
𝛼
)
𝑞𝛼/(𝑞−2)
= (1 − 2𝑞/𝑞−2 ) 𝐴 2
󸀠
̃ 󵄨 󵄨𝑞−2
(𝑟𝜆 2 󵄨󵄨󵄨󵄨̃V󸀠 󵄨󵄨󵄨󵄨 ̃V󸀠 ) = 0,
𝑟 > 1,
󵄨𝑝−2
󵄨
−󵄨󵄨󵄨󵄨̃V󸀠 (1)󵄨󵄨󵄨󵄨 ̃V󸀠 (1) = 𝑢̃𝛽 (1) ,
(53)
which implies that
̃ 1 /(𝑝−1)
𝛼/(𝑝−1) −𝜆
𝑢̃󸀠 = −𝐴 2
𝑟
̃ 2 /(𝑞−1)
𝛽/(𝑞−1) −𝜆
̃V󸀠 = −𝐴 1
,
𝑟
,
.
𝑟>1
(54)
with 𝐴 1 = 𝑢̃(1), 𝐴 2 = ̃V(1). Integrating the above equalities
yields
From (34), we need
(
󸀠
̃ 󵄨 󵄨𝑝−2
(𝑟𝜆 1 󵄨󵄨󵄨󵄨𝑢̃󸀠 󵄨󵄨󵄨󵄨 𝑢̃󸀠 ) = 0,
󵄨𝑝−2
󵄨
−󵄨󵄨󵄨󵄨𝑢̃󸀠 (1)󵄨󵄨󵄨󵄨 𝑢̃󸀠 (1) = ̃V𝛼 (1) ,
𝑞/𝑞−2 𝛼
− (2𝐴 2 )
A direct calculation shows that 𝑢̃(𝑟), ̃V(𝑟) should satisfy
𝛼 𝑞𝛼/(𝑞−2)
𝑝𝐵1 𝑝−1 2(𝑝−1)/(𝑝−2)
) 𝐴1
≤ (1 − 2𝑞/(𝑞−2) ) 𝐴 2
.
2−𝑝
𝛼/(𝑝−1)
(46)
𝐴𝜎1 , where 𝜎 is to be determined. Then rewriting the
Set 𝐴 2 =
last inequality, we get
2(𝑝−1)/(𝑝−2) −𝑞𝛼𝜎/(𝑞−2)
𝐴1
𝛼
≤ (1 − 2𝑞/(𝑞−2) ) (
𝐴2
𝑢̃ (𝑟) = (𝐴 1 −
̃ / (𝑝 − 1) − 1
𝜆
1
𝛼/(𝑝−1)
+
2 − 𝑝 𝑝−1
) , (47)
𝑝𝐵1
𝐴2
̃
̃ / (𝑝 − 1) − 1
𝜆
1
𝑟1−𝜆 1 /(𝑝−1) ,
(55)
𝛽/(𝑞−1)
𝐴1
̃V (𝑟) = (𝐴 2 −
which can be obtained by choosing 𝐴 1 > 0 small enough, if
̃ / (𝑞 − 1) − 1
𝜆
2
𝛽/(𝑞−1)
2 (𝑝 − 1) 𝑞𝛼𝜎
−
> 0.
𝑝−2
𝑞−2
)
+
(48)
𝐴1
̃ / (𝑞 − 1) − 1
𝜆
2
)
𝑟1−𝜆2/(𝑞−1) .
In a similar discussion on the boundary condition (36), for
enough small 𝐴 1 , the following inequality is needed:
In particular, we let
2𝜎 (𝑞 − 1)
𝑝𝛽
−
> 0.
𝑞−2
𝑝−2
(49)
Note that 1 < 𝑝 and 𝑞 < 2; therefore, (48)-(49) are equal to
2 (𝑝 − 1)
𝑝−2
𝑝𝛽
.
<𝜎
<
𝑞𝛼
𝑞 − 2 2 (𝑞 − 1)
(50)
Recall the assumption that 𝛼𝛽 > 4(𝑝 − 1)(𝑞 − 1)/𝑝𝑞, so there
exists a constant 𝜎 > 0, such that (50) holds. Furthermore, we
can choose 𝐴 1 as small as needed.
Therefore, the solution (𝑢, V) of the problem (12)–(14)
blows up in a finite time if (𝑢0 (𝑟), V0 (𝑟)) is large enough such
that
𝑢0 (𝑟) ≥ 𝑢 (𝑟, 0) ,
V0 (𝑟) ≥ V (𝑟, 0) ,
𝑟 > 1.
Proof of Proposition 5. We seek the steady-state solution of
the system (12)–(14):
V (𝑟, 𝑡) = ̃V (𝑟) ,
𝑟 > 1, 𝑡 > 0.
̃ / (𝑝 − 1) − 1
𝜆
1
(52)
= 0,
𝐴2 −
𝐴1
̃ / (𝑞 − 1) − 1
𝜆
2
= 0.
(56)
̃ /(𝑝 − 1) − 1, 𝐶 = 𝜆
̃ /(𝑞 − 1) − 1, and then
Define 𝐶1 = 𝜆
1
2
2
𝐶1 > 0, 𝐶2 > 0. Noticing that 𝛼𝛽 ≠ (𝑝 − 1)(𝑞 − 1), we get
𝛽/(𝑞−1)
𝐴 2 = (𝐶1
(𝑝−1)(𝑞−1)/((𝑝−1)(𝑞−1)−𝛼𝛽)
𝐶2 )
𝛼/(𝑝−1)
𝐴 1 = 𝐶1 𝐴 2
,
(57)
.
Therefore, the bounded positive functions:
𝛼/(𝑝−1)
𝑢̃ (𝑟) =
Proposition 5. If 𝛼𝛽 ≠ (𝑝 − 1)(𝑞 − 1), then every nonnegative
nontrivial solution of the system (12)–(14) with small initial
data exists globally.
𝑢 (𝑟, 𝑡) = 𝑢̃ (𝑟) ,
𝐴1 −
(51)
The proof is completed.
𝛽/(𝑞−1)
𝛼/(𝑝−1)
𝐴2
𝐴2
𝐶1
𝑟−𝐶1 ,
𝛽/(𝑞−1)
̃V (𝑟) =
𝐴1
𝐶2
𝑟−𝐶2
(58)
are just a couple of steady-state solutions of the problem
(12)–(14) with the initial data 𝑢̃(𝑟), ̃V(𝑟). By the comparison
principle, for any initial data 𝑢0 (𝑟), V0 (𝑟) which is small
enough to satisfy
𝑢0 (𝑟) ≤ 𝑢̃ (𝑟) ,
V0 (𝑟) ≤ ̃V (𝑟) ,
𝑟 > 1,
(59)
the solutions of the problem (12)–(14) exist globally in time.
6
Abstract and Applied Analysis
Remark 6. Due to the fact that (𝑝 − 1)(𝑞 − 1) < 4(𝑝 − 1)(𝑞 −
1)/(𝑝𝑞) for 1 < 𝑝, 𝑞 < 2, it is seen from Propositions 3 and
5 that all nonnegative solutions to the system (12)–(14) with
enough small initial data exist globally in time.
Now, we prove the main result for the system (7)–(9), that
is, Theorem 2.
Proof of Theorem 2. Noticing that the functions 𝑢0 (𝑥), V0 (𝑥)
are bounded, we can choose two bounded, radially symmetrical functions denoted by 𝑢1 (𝑥), V1 (𝑥) satisfying that 𝑢1 (𝑥) =
𝑢1 (|𝑥|) ≥ 𝑢0 (𝑥), V1 (𝑥) = V1 (|𝑥|) ≥ V0 (𝑥), respectively. By
using Proposition 3 and the comparison principle, we can
obtain the global existence of solutions to the system (7)–(9).
For the initial data (𝑢0 , V0 ) is large enough such that
𝑢0 (𝑥) ≥ 𝑢(|𝑥|, 0), V0 (𝑥) ≥ V(|𝑥|, 0), here 𝑢(|𝑥|, 0), V(|𝑥|, 0) are
defined in the proof of Proposition 4 if 𝛼𝛽 > 4(𝑝 − 1)(𝑞 −
1)/(𝑝𝑞), then the solutions of the system (7)–(9) with such
(𝑢0 , V0 ) blow up in a finite time by the comparison principle
and Proposition 4.
On the other hand, using the comparison principle again
and combining with Proposition 5, we see that the solution
(𝑢, V) of (7)–(9) exists globally if
𝛼/(𝑝−1)
(𝑝 − 1) 𝐴 2
𝑢0 (𝑥) ≤
|𝑥|−(𝜆 1 +𝑁−𝑝)/(𝑝−1) ,
𝜆1 + 𝑁 − 𝑝
𝛽/(𝑞−1)
(60)
(𝑞 − 1) 𝐴 1
V0 (𝑥) ≤
|𝑥|−(𝜆 2 +𝑁−𝑞)/(𝑞−1) ,
𝜆2 + 𝑁 − 𝑞
where
𝐴 2 = ((
𝜆 1 + 𝑁 − 𝑝 𝛽/(𝑞−1)
)
𝑝−1
×(
𝐴1 =
𝜆 2 + 𝑁 − 𝑞 (𝑝−1)(𝑞−1)/[(𝑝−1)(𝑞−1)−𝛼𝛽]
,
))
𝑞−1
(61)
𝜆 1 + 𝑁 − 𝑝 𝛼/(𝑝−1)
.
𝐴2
𝑝−1
The proof is complete.
Acknowledgments
The authors would like to express their many thanks to the
Editor and Reviewers for their constructive suggestions to
improve the previous version of this paper. This work was
supported by the NNSF.
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