Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 326527, 16 pages
doi:10.1155/2012/326527
Research Article
Roles of Weight Functions to a Nonlocal
Porous Medium Equation with Inner Absorption
and Nonlocal Boundary Condition
Zhong Bo Fang,1 Jianyun Zhang,1 and Su-Cheol Yi2
1
2
School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China
Department of Mathematics, Changwon National University, Changwon 641-773, Republic of Korea
Correspondence should be addressed to Zhong Bo Fang, fangzb7777@hotmail.com
Received 22 August 2012; Revised 24 October 2012; Accepted 7 November 2012
Academic Editor: Dragoş-Pătru Covei
Copyright q 2012 Zhong Bo Fang 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 work is concerned with an initial boundary value problem for a nonlocal porous medium
equation with inner absorption and weighted nonlocal boundary condition. We obtain the roles of
weight function on whether determining the blowup of nonnegative solutions or not and establish
the precise blow-up rate estimates under some suitable condition.
1. Introduction
Our main interest lies in the following nonlocal porous medium equation with inner
absorption term:
ut Δu u
m
p
Ω
uq y, t dy − kur ,
x, t ∈ Ω × 0, ∞,
1.1
subjected to weighted linear nonlocal boundary and initial conditions,
ux, t Ω
f x, y u y, t dy,
ux, 0 u0 x,
x, t ∈ ∂Ω × 0, ∞,
1.2
x ∈ Ω,
1.3
where m > 1, p ≥ 0, q > 0, p q ≥ 1, r ≥ 1, k > 0, and Ω ⊂ RN N ≥ 1 is a bounded
domain with smooth boundary. The weight function fx, y /
≡ 0 is a nonnegative continuous
2
Abstract and Applied Analysis
function defined on ∂Ω × Ω, and Ω fx, ydy > 0 on ∂Ω. The initial value u0 x ∈ C2α Ω
with 0 < α < 1 is a nonnegative continuous function satisfying the compatibility condition on
∂Ω.
Many natural phenomena have been formulated as nonlocal diffusive equation 1.1,
such as the model of non-Newton flux in the mechanics of fluid, the model of population,
biological species, and filtration we refer to 1, 2 and the references therein. For instance,
in the diffusion system of some biological species with human-controlled distribution,
ux, t, Δum , up Ω uq y, tdy, and −k represent the density of the species, the mutation, the
human-controlled distribution, and the decrement rate of biological species at location x and
time t, respectively. Due to the effect of spatial inhomogeneity, the arising of nonlocal term
denotes that the evolution of the species at a point of space depends not only on the density
of species in partial region but also on the total region we refer to 3–5 . However, there
are some important phenomena formulated as parabolic equations which are coupled with
weighted nonlocal boundary conditions in mathematical models, such as thermoelasticity
theory. In this case, the solution ux, t describes entropy per volume of the material we
refer to 6, 7 .
To motivate our work, let us recall some results of global and blow-up solutions to
the initial boundary value problems with nonlocal terms or with nonlocal terms in boundary
conditions we refer to 8–16 . For the study of the initial boundary value problems for the
parabolic equations with local terms which subject to the weighted nonlocal linear boundary
condition 1.2, one can see 8–10 . For example, Friedman 8 studied the linear parabolic
ut − Au 0,
x, t ∈ Ω × 0, T ,
1.4
subjected to the nonlocal Dirichlet boundary condition 1.2, where A is an elliptic operator,
A
n
ai,j x
i,j1
n
∂2
∂
bi x
cx,
∂xi ∂xj i1
∂xi
cx ≤ 0.
1.5
He proved that when Ω fx, ydy ≤ ρ < 1, the solution tends to 0 monotonously and
exponentially as t → ∞. With regard to more general discussions on initial boundary value
problem for linear parabolic equation with nonlocal Neumann boundary condition, one can
see 9 by Pao where the following problem was considered:
ut − Lu gx, u, x ∈ Ω, t > 0,
f x, y u y, t dy, x ∈ ∂Ω, t > 0,
Bu Ω
ux, 0 u0 x,
1.6
x ∈ Ω,
where
Lu n
aij xuxi xj i,j1
n
bj xuxj ,
j1
Bu α0
∂u
u.
∂n
1.7
Abstract and Applied Analysis
3
He studied the asymptotic behavior of solutions and found the influence of weight function
on the existence of global and blow-up solutions. Wang et al. 10 studied porous medium
equation with power form source term
ut Δum up ,
x, t ∈ Ω × 0, ∞,
1.8
subjected to nonlocal boundary condition 1.2. By virtue of the method of upper-lower
solutions, they obtained global existence, blow-up properties, and blow-up rate of solutions.
For the study of the initial boundary value problems for the parabolic equations with
nonlocal terms which subjected to the weighted nonlocal linear boundary condition 1.2, we
refer to 11–16 . Lin and Liu 11 considered the semilinear parabolic equation
ut Δu Ω
gudx,
x, t ∈ Ω × 0, ∞,
1.9
with nonlocal boundary condition 1.2. They established local existence, global existence,
and blow-up properties of solutions. Moreover, they derived the uniform blow-up estimates
for special gu under suitable assumption; Cui and Yang 12 discussed the nonlocal slow
diffusion equation
ut Δum aup
Ω
uq y, t dy,
x, t ∈ Ω × 0, ∞,
1.10
and they built global existence, blow-up properties, and blow-up rate of solutions. For the
system of equations, we refer readers to 13 and the references therein.
Recently, Wang et al. 14 studied the following semilinear parabolic equation with
nonlocal sources and interior absorption term:
ut Δu Ω
uq dx − αur ,
x, t ∈ Ω × 0, ∞,
1.11
with weighted linear nonlocal boundary condition 1.2 and initial condition 1.3, where
q ≥ 1, r ≥ 1, and α > 0. By using comparison principle and the method of upper-lower
solutions, they got the following results.
a If 1 ≤ q < r, then the solution of the problem exists globally.
b If q > r ≥ 1, the problem has solutions blowing up in finite time as well as global
solutions. That is,
i if Ω fx, ydy ≤ 1, and u0 x ≤ α/|Ω|1/r−q , then the solution exists globally;
ii if Ω fx, ydy > 1, and u0 x > α/|Ω| − α1/r |Ω| > α, then the solution
blows up in finite time;
iii for any fx, y ≥ 0, there exists a2 > 0 such that the solution blows up in
finite time provided that u0 x > a2 φx, where φx is the corresponding
normalized eigenfunction
of −Δ with homogeneous Dirichlet boundary
condition, and Ω φxdx 1.
4
Abstract and Applied Analysis
c If q r > 1.
i The solution blows up in finite time for any fx, y ≥ 0 and large enough u0 .
ii If Ω fx, ydy < 1, the solution exists globally for u0 x ≤ a1 Φx for some
a1 > 0, where Φx solves the following problem:
−ΔΦx δ0 , x ∈ Ω,
f x, y dy, x ∈ ∂Ω,
Φx 1.12
Ω
here δ0 is a positive constant such that 0 ≤ Φx ≤ 1.
In addition, for the initial boundary value problem of 1.11 with weighted nonlinear
boundary condition and Dirichlet boundary condition, we refer to 15, 16 and references
therein, respectively.
The aim of this paper is to obtain the sufficient condition of global and blow-up
solutions to problem 1.1–1.3 and to extend the results of the semilinear equation 1.11
to the quasilinear ones. The difficulty lies in finding the roles of weighted function in the
boundary condition and the competitive relationship of nonlocal source and inner absorption
on whether determining the blowup of solutions or not. Our detailed results are as follows.
Theorem 1.1. Suppose that p q > r. If Ω fx, ydy ≥ 1 for x ∈ ∂Ω, and the initial data u0 x >
k/|Ω| − k1/pq |Ω| > k, then the solution of problem 1.1–1.3 blows up in finite time.
Remark 1.2. There may exist a global solution of problem 1.1–1.3 for small enough initial
data under the
condition of Theorem 1.1. Unfortunately, since the weight function satisfies
the condition Ω fx, ydy ≥ 1 on the boundary, we cannot construct a suitable supersolution
of problem 1.1–1.3.
Theorem 1.3. Suppose that p q > r, if Ω fx, ydy ≤ 1 for x ∈ ∂Ω, then the solution of problem
1.1–1.3 exists globally for the initial data u0 x < k/|Ω|1/pq−r . If p q ≥ max{m, r}, then
the solution of problem 1.1–1.3 blows up in finite time for large enough initial data and arbitrary
fx, y > 0.
Theorem 1.4. Suppose that p q < r,then the solution of problem 1.1–1.3 exists globally for
arbitrary fx, y > 0.
Theorem 1.5. Suppose that p q r. Ω fx, y ≤ ρ < 1 for x ∈ ∂Ω, where ρ is a positive constant
and ρ < 1.
1 If m > p q, then every nonnegative solution of problem 1.1–1.3 exists globally.
2 If m p q, then for |Ω| < k δ/M1 1 − ρm , the solution of problem 1.1–1.3 exists
globally, where δ, M1 > 0 are as defined in 3.13.
3 If m < p q, the solution of problem 1.1–1.3 exists globally for sufficient small initial
data while it blows up in finite time for large enough initial data.
In order to show blow-up rate estimate of the blow-up solution, we need the following
assumptions on the initial data u0 x:
p
q
r
C1 Δum
0 u0 Ω u0 y, tdy − ku0 > 0, for x ∈ Ω;
Abstract and Applied Analysis
5
C2 there exists a constant δ > 0, such that
Δum
0
p
u0
Ω
q
pq
u0 y, t dy − kur0 − δ u0 ≥ 0,
1.13
where δ will be determined later.
Theorem 1.6. Suppose that p q > max{m, r},
satisfies the conditions C1 -C2 , then
Ω
fx, ydy ≤ 1 for x ∈ ∂Ω, and the initial data
cT − t−1/pq−1 ≤ ux, t ≤ CT − t−1/pq−1 ,
where c |Ω|p q − 1
−1/pq−1
, C δp q − 1/m
−1/pq−1
1.14
.
The rest of our paper is organized as follows. In Section 2, with the definitions of weak
upper and lower solutions, we will give the comparison principle of problem 1.1–1.3,
which is an important tool in our research. The proofs of results of global existence and blowup of solutions will be given in Section 3. And in Section 4, we will give the blow-up rate
estimate of the blow-up solutions.
2. Comparison Principle and Local Existence
In this section, we establish a suitable comparison principle for problem 1.1–1.3. Let QT Ω × 0, T , QT Ω × 0, T , and ST ∂Ω × 0, T . Firstly, we start with the precise definitions
of upper solution and lower solution of problem 1.1–1.3.
Definition 2.1. Suppose that ux, t ∈ C2,1 QT ∩ CQT is nonnegative and satisfies
ut ≤ Δum up
ux, t ≤
Ω
uq dy − kur ,
x, t ∈ QT ,
2.1
f x, y u y, t dy,
x, t ∈ ST ,
2.2
Ω
ux, 0 ≤ u0 x,
x ∈ Ω,
2.3
then we call ux, t is the lower solution of problem 1.1–1.3, in QT .
Similarly, a nonnegative function ux, t ∈ C2,1 QT ∩ CQT is an upper solution if it
satisfies 2.1–2.3 in the reverse order. We say ux, t is a solution of problem 1.1–1.3 in
QT if it is both an upper solution and a lower solution of problem 1.1–1.3 in QT which is
called classical solution.
The following comparison principle plays a crucial role in our proofs which can be
obtained by establishing suitable test function and Gronwall’s inequality.
Proposition 2.2 comparison principle. Suppose that ux, t and ux, t are the nonnegative subsolution and supersolution of problem 1.1–1.3, respectively. Moreover ux, 0 ≤ ux, 0, ux, 0 ≥
0, and ux, 0 ≥ δ > 0 in Ω, then ux, t ≤ ux, t in QT .
6
Abstract and Applied Analysis
Proof. Let ψx, t ∈ C2,1 QT be a nonnegative function with ψ 0 on ST . Multiplying the
inequality in 2.1 by ψx, t and integrating it on QT , we get
QT
ut ψdx dt ≤
uψt u Δψ ψ u
m
−
0
∂Ω
∂ψ
∂n
Ω
u dy − ku
q
Ω
QT
t p
r
dx dt
2.4
m
f x, y udy
dS dt,
and similarly, the upper solution satisfies the reversed inequality,
ut ψdx dt ≤
m
p
uψt u Δψ ψ u
QT
QT
−
t 0
∂Ω
∂ψ
∂n
Ω
Ω
uq dy − kur dx dt
2.5
m
f x, y udy
dS dt.
Let ωx, t ux, t − ux, t, we have
ψt Φ1 x, tΔψ Φ2 x, t
uq dy − kΦ3 x, t ψ ωdx dt
ωt ψdx dt ≤
QT
Ω
QT
p
u ψ
QT
Ω
Φ4 ωdy dx dt −
t 0
∂Ω
∂ψ m−1
mξ
∂n
Ω
f x, y ωdy dS dt,
2.6
where
Φ1 x, t 1
m−1
m θux, t 1 − θux, t
dθ,
0
Φ2 x, t 1
p−1
p θux, t 1 − θux, t
dθ,
0
Φ3 x, t 1
r−1
r θux, t 1 − θux, t
dθ,
2.7
0
Φ4 x, t 1
q−1
q θux, t 1 − θux, t
dθ,
0
and ξ is a function between Ω fx, yudy and Ω fx, yudy. Noticing that ux, t and ux, t
are bounded functions, it follows from m > 1, p ≥ 1, q ≥ 1, and r ≥ 1 that Φi i 1, 2, 3, 4 are
bounded nonnegative functions. If 0 ≤ p < 1 or 0 < q < 1, we have Φ2 ≤ δp−1 and Φ4 ≤ δq−1 by
the condition that ux, 0 ≥ 0 or ux, 0 ≥ δ > 0. Thus, we may choose an appropriate ψx, t
as in 17, p. 118–123 to obtain
Ω
ω dx ≤ C1
Ω
ωx, 0 dx C2
ωx, tdx dt,
QT
2.8
Abstract and Applied Analysis
7
where ω max{ω, 0} and C1 , C2 > 0. It follows from ux, 0 ≤ ux, 0 that
Ω
ω dx ≤ C2
ωx, tdx dt.
QT
2.9
By Gronwall’s inequality, we know that u − u ≤ 0, that is, ux, t ≤ ux, t in QT .
For x ∈ ∂Ω, y ∈ Ω, t > 0,
u−u≤
Ω
f x, y u − u dy ≤ 0.
2.10
This completes the proof.
Next, we state the local existence and uniqueness theorem without proof.
Theorem 2.3 local existence and uniqueness. Suppose that the nonnegative initial data u0 x ∈
C2α Ω ∩ CΩ 0 < α < 1 satisfies the compatibility condition. Then, there exists a constant
T ∗ > 0 such that the problem 1.1–1.3 admits nonnegative solution ux, t ∈ C2,1 QT ∩ CΩT for each T < T ∗ . Furthermore, either T ∗ ∞ or
lim sup ux, t
t → T∗
∞
∞.
2.11
Remark 2.4. The existence of local nonnegative solutions in time to problem 1.1–1.3 can be
obtained by using the fixed point theorem see 18 or the regular theory to get the suitable
estimate in a standard limiting process see 19, 20 . By the previous comparison principle,
we can get the uniqueness of solution to the problem 1.1–1.3 in the case of p q ≥ 1, r ≥ 1.
3. Global Existence and Blowup of Solutions
Comparing problems with the general homogeneous Dirichlet boundary condition, the existence of weight function on the boundary has a great influence on the global and nonglobal
existence of solutions.
Proof of Theorem 1.1. Consider the following problem:
v t |Ω|vpq − kvr ,
v0 v0 .
3.1
As p q > r, we know that vpq 1 > vr , and |Ω|vpq − kvr ≥ |Ω| − kvpq − k. Therefore, the
solution of 3.1 is an upper solution of the following problem:
v t |Ω| − kvpq − k,
v0 v0 .
3.2
When |Ω| > k and p q > 1, it is known that the solution to the problem 3.2 blows up in
finite time if v0 > k/|Ω| − k1/pq .
8
Abstract and Applied Analysis
It is obvious
that the solution of problem 3.1 is a lower solution of problem 1.1–
1.3 when Ω fx, ydy ≥ 1 and u0 x > v0 . By Proposition 2.2, ux, t is a blow-up solution
of problem 1.1–1.3.
Proof of Theorem 1.3. 1 The case of p q > r. Let u k/|Ω|1/pq−r . It is easy to show
that if Ω fx, ydy < 1 and u0 x < k/|Ω|1/pq−r , ux, t is the upper solution of problem
1.1–1.3, then we can draw the conclusion.
2 The case of p q ≥ max{m, r}. We need to establish a self-similar blow-up solution
in order to prove the blow-up result. We first suppose that ω ∈ C1 Ω, ωx ≥ 0, and ωx is
not identically zero, and ωx|∂Ω 0. Without loss of generality, we assume that 0 ∈ Ω and
ω0 > 0.
Let ux, t T − t−γ V 1/m ξ, V ξ 1 − ξ2 /2A , and ξ |x|T − t−μ , where A >
1, 0 < T < 1, γ and μ > 0. We know that
supp u ·, t B 0, RT − tμ ⊂ B0, RT μ ⊂ Ω,
for sufficiently small T > 0 and R ut AA 2. Calculating the derivative of u, we obtain
mγV 1/m ξ μξV ξV 1−m/m
mT − tγ1
N
Δum −
p
u
Ω
u y, t dy q
≥
where M B0,R
3.3
V
AT − tmγ2μ
p/m
T − tpqγ
,
B0,RT −tμ M
T − tpqγ−Nμ
,
V
q/m
|x|
dx
T − tμ
3.4
,
V q/m |ξ|dξ > 0.
It is easy to see that V ξ ≤ 0 and V < 1, and
ut − Δu − u
m
≤
mγV
1/m
Ω
uq y, t dy kur
ξ μξV ξV 1−m/m
mT − t
−
≤
p
M
T − tpqγ−Nμ
γ
γ1
T − t
γ1
kT − t
N
AT − t
mγ2μ
−rγ
−
V
N
AT − tmγ2μ
r/m
ξ
M
T − t
3.5
pqγ−Nμ
k
.
T − trγ
Abstract and Applied Analysis
9
Since p q ≥ m > 1, choosing γ > 0 such that m < 1 γ/γ < p q, and μ is sufficiently small
such that
μ < min
pq−1 γ −1 pq−r γ pq−m
,
,
.
N
N
2N
3.6
Then, for small enough T > 0, we have
ut − Δu − u
m
p
Ω
uq y, t dy kur ≤ 0.
3.7
If x ∈ ∂Ω, ω0 > 0, and ω is continuous, it is known that there exist positive ε and ρ
suchthat ω ≥ ε for x ∈ B0, ρ. We can get B0, RT σ ⊂ B0, ρ ⊂ Ω if T is small enough. Then,
u ≤ Ω fx, yudy on ∂Ω × 0, T . It follows from 3.3 that ux, 0 ≤ K0 ωx for sufficiently
large K0 . Therefore, one can observe that the solution to 1.1–1.3 exists no later than t T
provided that u0 ≥ K0 ωx. This implies that the solution blows up in finite time for large
enough initial data.
Proof of Theorem 1.4. Suppose that λ1 > 0 is the first eigenvalue of −Δ with homogeneous
Dirichlet boundary condition, and φx is the corresponding eigenfunction. Let
M Ω 1/φy εdy ≤ 1 for some 0 < ε < 1, where M maxx∈∂Ω,y∈Ω fx, y.
Now, we assume that um v, then 1.1–1.3 becomes that
vn t − Δv vnp
vx, t Ω
Ω
vqn dy − kvnr ,
n
x ∈ Ω, t > 0,
m
f x, y v y, t dy
vx, 0 v0 x um
0 x,
x ∈ ∂Ω, t > 0,
,
x ∈ Ω,
∗
∗∗
∗ ∗ ∗
where n 1/m.
Set vx, t Ceγt/n /φx ε, where C is determined later, then it follows that
n
Cn γeγt
v t
n ,
φx ε
−Ceγt/n ∇φx
∇φ
2 − φx ε v,
φx ε
φ ε −vΔφ − ∇φ · ∇v v∇φ · ∇φ
Δv 2
φε
2 2
∇φ
2∇φ
λ1 φv − ∇φ · ∇v
λ1 φ
2 v φ ε 2 v.
φε
φε
φε
∇v 3.8
10
Abstract and Applied Analysis
And, we can get
vn
− Δv − vpn
t
Cn γeγt
φx ε
Ω
vqn dy − kvnr
n −
2 2∇φ
λ1 φ
v
φ ε φ ε2
npq nr Ceγt/n
nr
1
φε
− k Ceγt/n
np
nq
dy
φε
Ω φε
2 n−1
2∇φ
λ1 φ
C
γ−γ/nt
γe
−
v
φε
φ ε φ ε2
3.9
npq Ceγt/n
nr
1
− kCnr eγrt φ ε .
np
nq
dy
φε
Ω φε
Choosing C > max{supΩ φ εnr−p /k
the proper λ1 such that
C
γ
φε
Ω
n−1
1/φ εnq dy
1/nr−p−q
, supφ εu0 x} and
2
2∇φ
λ1 φ
>
,
φ ε φ ε2
3.10
then
n
v t − Δv − vpn
Ω
vqn dy − kvnr ≥ 0.
3.11
For x ∈ ∂Ω, t > 0,
vx, t Ceγt/n
≥
ε
Ω
Ceγt/n M
dy ≥
φ y ε
Ω
f x, y vdy,
3.12
so we can obtain that vx, t is the upper solution of problem ∗–∗ ∗ ∗. From Proposition 2.2,
we know that there exists global solution of problem ∗–∗ ∗ ∗. Since the problem 1.1–1.3
has the same solution with problem ∗–∗ ∗ ∗. We know that there exists a global solution of
problem 1.1–1.3. This completed the proof.
Proof of Theorem 1.5. Suppose that Φx solves the following problem:
−ΔΦ δ,
x ∈ Ω,
Φx 0,
x ∈ ∂Ω,
where δ is a positive constant such that 0 ≤ Φx ≤ 1. Then, let M1 maxx∈Ω Φx ≤ 1.
3.13
Abstract and Applied Analysis
11
Set ωx, t Aρm /1 − ρm Φx/M1 1/m , here A > 0 will be determined later and
0 < ρ < 1,
m
p
ωt − Δω − ω
ωq y, t dy kωr
Ω
m
m
ρ
ρ
Am δ
Φx p/m
Φx q/m
pq
−A
dx
M1
1 − ρm
M1
1 − ρm
M1
Ω
m
ρ
Φx pq/m
kApq
1 − ρm
M1
m
m
m
ρ
ρ
A δ
Φx pq/m
Φx pq/m
pq
≥
− Apq |Ω|
kA
M1
1 − ρm
M1
1 − ρm
M1
pq/m
m
m
ρ
A δ
Φx
Apq k − |Ω|
.
M1
1 − ρm
M1
3.14
It is obvious that the global existence result holds for k ≥ |Ω|. For k < |Ω|, since Φx/M1 ≤ 1,
we know that ρm /1 − ρm Φx/M1 ≤ ρm /1 − ρm 1 1/1 − ρm , then
ωt − Δωm − ωp
pq/m
Am δ
1
ωq y, t dy kωr ≥
− Apq |Ω| − k
.
M1
1 − ρm
Ω
3.15
1 If m > p q, choosing A max{max |u0 x|, |Ω| − kM/δ1/1 − ρm pq/m }, we
have
m
p
ωt − Δω − ω
ωq y, t dy kωr ≥ 0.
3.16
Ω
2 If m p q, selecting |Ω| < k δ/M1 1 − ρm , and A max u0 x, we get
ωt − Δωm − ωp
Ω
ωq y, t dy kωr ≥ 0.
3.17
3 If m < p q, we choose max u0 x ≤ A ≤ |Ω| − kM1 /δ1/1 − ρm −pq/m such
that
ωq y, t dy kωr ≥ 0.
ωt − Δωm − ωp
3.18
Ω
For x ∈ ∂Ω,
1/m
ρm
1
ρ
1 − ρm
m
ρ
Φx 1/m
≥A
f x, y dy
m
1−ρ
M1
Ω
ω y, t f x, y dy.
ωx, t A
ρm
1 − ρm
1/m
A
Ω
Through the previous discussion, we know that the global existence results hold.
3.19
12
Abstract and Applied Analysis
For the blow-up case of m < p q, it holds clearly from the second part of the proof of
Theorem 1.3.
4. Blow-Up Rate Estimates
Next, we will get the following precise blow-up rate estimates for slow diffusion case under
some suitable conditions.
Let v um , we just need to consider the problem ∗–∗ ∗ ∗, and let pn p1 , qn q1 ,
and rn r1 , then ∗ becomes
vn t Δv vp1
Ω
vq1 y, t dy − kvr1 ,
x ∈ Ω, t > 0.
4.1
Suppose that vx, t is the blow-up solution of problem ∗–∗ ∗ ∗ in finite time T , and
set V t maxx∈Ω vx, t.
Proof of Theorem 1.6. 1 We can easily know that V t is Lip continuous and differential
almost everywhere,
V n t ≤ V p1
Ω
V q1 dy − kV r1 ≤ V p1 q1 |Ω| − kV r1 ≤ V p1 q1 |Ω|.
4.2
Then, it follows that
Vt ≤
1
|Ω|V p1 q1 1−n .
n
4.3
Integrating it over t, T , we get
V t ≥
1 |Ω| p1 q1 − n
n
−1/p1 q1 −n
T − t−1/p1 q1 −n ,
4.4
then
ux, t ≥ cT − t−1/pq−1 ,
where c |Ω|p q − 1 −1/pq−1 .
2 Next, we set J vt − δvp1 q1 1−n , where δ > 0, then
Jt vtt − δ p1 q1 1 − n vp1 q1 −n vt
1 − n −n n
1 1−n
p1
q1
r1
v v t vt v
v dy − kv
− δ p1 q1 1 − n vp1 q1 −n vt
Δv v
n
n
Ω
t
p
q
1
1
1
1 − nv−1 vt 2 v1−n Δvt vp1 −n vt
vq1 dy vp1 1−n
vq1 −1 vt dy
n
n
n
Ω
Ω
−
kr1 r1 −n
v
vt − δ p1 q1 1 − n vp1 q1 −n vt ,
n
4.5
Abstract and Applied Analysis
Jt −
13
p1 vp1 −n
v1−n
ΔJ − 2δ1 − nvp1 q1 −n n
n
Ω
vq1 dy −
q1 vp1 1−n
kr1 r1 −n
v
J−
n
n
Ω
vq1 −1 Jdy
2p1 q1 −2n1
p1 δ δ −1 2
2
2p1 2q1 −2n1
−
vq1 dy
1 − nv J δ 1 − nv
p1 q1 1 − n v
n
n
Ω
kr1 δ p1 q1 r1 −2n1 δ kδ p1 q1 1 − n −
p1 q1 − n 1 p1 q1 − n |∇v|2
v
n
n
n
q1 δ p1 1−n
v
vp1 2q1 −n dy
n
Ω
≥ δ2 1 − nv2p1 2q1 −2n1
δ
kδ vq1 dy q1 1 − n v2p1 q1 −2n1
p1 q1 1 − n − r1 vp1 q1 r1 −2n1
n
n
Ω
q1 δ p1 1−n
v
vp1 2q1 −n dy
n
Ω
p1 q1 −n
δ p1 −n1
2
2p1 2q1 −2n1
p1 2q1 −n
v
v
dy − q1 1 − n v
vq1 dy
q1
δ 1 − nv
n
Ω
Ω
kδ p1 q1 1 − n − r1 vp1 q1 r1 −2n1 .
n
4.6
Since q1 /p1 2q1 − n p1 q1 − n/p1 2q1 − n 1, by Hölder’s inequality, we know that
Ω
vq1 dy ≤
Ω
vp1 2q1 −n
q1 /p1 2q1 −n
4.7
,
and by Young inequality, we have
v
p1 q1 −n
Ω
vp1 2q1 −n
q1 /p1 2q1 −n
p1 q1 − n p1 q1 −n p1 2q1 −n/p1 q1 −n
q1
≤
v
p1 2q1 − n
p1 2q1 − n
p1 q1 − n p1 2q1 −n
q1
v
vp1 2q1 −n .
p1 2q1 − n
p1 2q1 − n Ω
Ω
vp1 2q1 −n
4.8
Since p1 q1 > r1 , we get p1 2q1 − n > q1 1 − n and p1 q1 − r1 > 0, then by 4.7-4.8,
Jt −
p1 vp1 −n
v1−n
ΔJ − 2δ1 − nvp1 q1 −n n
n
≥ δ2 1 − nv2p1 2q1 −2n1 ≥ δ1 − n −
Ω
vq1 dy −
q1 vp1 1−n
kr1 r1 −n
v
J−
n
n
Ω
vq1 −1 Jdy
δp1 q1 − n 2p1 2q1 −2n1
kδ v
p1 q1 1 − n − r1 vp1 q1 r1 −2n1 −
n
n
p1 q1 − n
δv2p1 2q1 −2n1 .
n
4.9
14
Abstract and Applied Analysis
Choosing δ ≥ p1 q1 − n/n1 − n, such that
p1 vp1 −n
v1−n
ΔJ − 2δ1 − nvp1 q1 −n Jt −
n
n
q1 vp1 1−n
kr1 r1 −n
v
v dy −
J−
n
n
Ω
q1
Ω
vq1 −1 Jdy
≥ 0.
4.10
For x, t ∈ ST , because of vt J δvp1 q1 1−n , we then have
J vt − δvp1 q1 1−n
m−1 mp1 q1 1−n
m
f x, y v1/m dy
f x, y v1/m dy − δ
f x, y v1/m dy
Ω
Ω
Ω
f x, y v1/m dy
f x, y v1/m dy
t
Ω
m−1
m
Ω
m−1 Ω
f x, y
Ω
v1/m dy − δ
t
Ω
f x, y v1/m dy
pq
f x, y v1−m/m Jdy
δ
Ω
f x, y vpq/m dy −
Ω
f x, y v1/m dy
pq .
4.11
Noticing that 0 < Fx Ω fx, ydy ≤ 1 for x ∈ ∂Ω, p q > max{m, r}, and applying
Jensen’s inequality to the last part in the previous inequality, we can get
f x, y vpq/m dy −
Ω
≥ Fx
Ω
f x, y v
1/m
Ω
f x, y v1/m dy
dy
Fx
pq
−
pq
Ω
f x, y v
1/m
4.12
pq
dy
.
Hence, we can get
J≥
Ω
f x, y v1/m dy
m−1 Ω
f x, y v1−m/m Jdy ≥ 0.
4.13
Since u0 x satisfies the conditions C1 -C2 and v um ,
Jx, t mum−1 ut − δump1 q1 1−n
m−1
m
p
q
r
pq
Δu u
,
mu
u y, t dy − ku − δ u
4.14
Ω
where δ δ/m, then Jx, 0 ≥ 0. Combining 4.10-4.13, we can know that Jx, t ≥ 0 for
x, t ∈ QT , that is, vt ≥ δvp1 q1 1−n .
Abstract and Applied Analysis
15
Integrating it over t, T , we have
−1/p1 q1 −n
v ≤ δ p1 q1 − n
T − t−1/p1 q1 −n ,
4.15
ux, t ≤ CT − t−1/pq−1 ,
4.16
then it follows that
where C δp q − 1/m
−1/pq−1
. This completed the proof.
Acknowledgments
This work is supported by the Natural Science Foundation of Shandong Province of China
ZR2012AM018. The authors would like to deeply thank all the reviewers for their insightful
and constructive comments.
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