# Blow-Up Analysis for a Quasilinear Parabolic Equation with Inner Absorption and Nonlinear Neumann Boundary Condition

Authors: Zhong Bo Fang; Yan Chai
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We investigate an initial-boundary value problem for a quasilinear parabolic equation with inner absorption and nonlinear Neumann boundary condition. We establish, respectively, the
conditions on nonlinearity to guarantee thatu
(
x
,
t
) exists globally or blows up at some finite time t
*. Moreover, an upper bound for t
* is derived. Under somewhat more restrictive conditions, a lower bound for t
* is also obtained.


---

## Body

## 1. Introduction

We are concerned with the global existence and blow-up phenomenon for a quasilinear parabolic equation with nonlinear inner absorption term(1)
u
t
=
[
(
|
∇
u
|
p
+
1
)
u
,
i
]
,
i
-
f
(
u
)
,
(
x
,
t
)
∈
Ω
×
(
0
,
t
*
)
,
subjected to the nonlinear Neumann boundary and initial conditions
(2)
(
|
∇
u
|
p
+
1
)
∂
u
∂
ν
=
g
(
u
)
,
(
x
,
t
)
∈
∂
Ω
×
(
0
,
t
*
)
,
(3)
u
(
x
,
0
)
=
u
0
(
0
)
≥
0
,
x
∈
Ω
¯
,
where Ω is a bounded star-shaped region of R
N
(
N
≥
2
) with smooth boundary ∂
Ω, ν is the unit outward normal vector on ∂
Ω, p
≥
0, t
* is the blow-up time if blow-up occurs, or else t
*
=
+
∞, the symbol, i denotes partial differentiation with respect to x
i, i
=
1,2
,
…
,
N, the repeated index indicates summation over the index, and ∇ is gradient operator.Many physical phenomena and biological species theories, such as the concentration of diffusion of some non-Newton fluid through porous medium, the density of some biological species, and heat conduction phenomena, have been formulated as parabolic equation (1) (see [1–3]). The nonlinear Neumann boundary condition (2) can be physically interpreted as the nonlinear radial law (see [4, 5]).In the past decades, there have been many works dealing with existence and nonexistence of global solutions, blow-up of solutions, bounds of blow-up time, blow-up rates, blow-up sets, and asymptotic behavior of solutions to nonlinear parabolic equations; see the books [6–8] and the survey papers [9–11]. Specially, we would like to know whether the solution blows up and at which time when blow-up occurs. A variety of methods have been used to study the problem above (see [12]), and in many cases, these methods, used to show that solutions blow up, often provide an upper bound for the blow-up time. However, lower bounds for blow-up time may be harder to be determined. For the study of the initial boundary value problem of a parabolic equation with homogeneous Dirichlet boundary condition, see [13, 14]. Payne et al. [13] considered the following quasilinear parabolic equation:
(4)
u
t
=
div
⁡
(
ρ
|
∇
u
|
2
∇
u
)
+
f
(
u
)
,
(
x
,
t
)
∈
Ω
×
(
0
,
t
*
)
,
where Ω is a bounded domain in R
3 with smooth boundary ∂
Ω. To get the lower bound for the blow-up time, the authors assumed that ρ is a positive C
1 function which satisfies
(5)
ρ
(
s
)
+
s
ρ
′
(
s
)
>
0
,
s
>
0
.
The lower bound for the blow-up time of solution to (4) with Robin boundary condition was obtained in [15], where ρ is also assumed to satisfy the condition (5). However, under this boundary condition, the best constant of Sobolev inequality used in [13] is no longer applicable. They imposed suitable conditions on f and ρ and determined a lower bound for the blow-up time if blow-up occurs and determined when blow-up cannot occur. Marras and Vernier Piro [14] studied the nonlinear parabolic problem with time dependent coefficients
(6)
k
1
(
t
)
div
⁡
(
g
(
|
∇
u
|
2
∇
u
)
)
+
k
2
(
t
)
f
(
u
)
=
k
3
(
t
)
u
t
,
gdfgfdgdfgfdgfdgfdgfdgfdgf
(
x
,
t
)
∈
Ω
×
(
0
,
t
*
)
,
where Ω is a bounded domain in R
N with smooth boundary ∂
Ω. Under some conditions on the data and geometry of the spatial domain, they obtained upper and lower bounds of the blow-up time. Moreover, the sufficient conditions for global existence of the solution were derived.For the study of the initial boundary value problem of a parabolic equation with Robin boundary condition, we refer to [15–19]. Li et al. [16] investigated the problem of the nonlinear parabolic equation
(7)
u
t
=
[
(
|
∇
u
|
p
+
1
)
u
,
i
]
,
i
+
f
(
u
)
,
(
x
,
t
)
∈
Ω
×
(
0
,
t
*
)
,
where Ω is a bounded domain in R
3 with smooth boundary ∂
Ω. They derived the lower bound for the blow-up time when the blow-up occurs. Clearly, |
∇
u
|
p
+
1 does not satisfy the condition (5). Enache [17] discussed the quasilinear parabolic problem
(8)
u
t
=
(
g
(
u
)
u
,
i
)
,
i
+
f
(
u
)
,
where Ω is a bounded domain in R
N
(
N
≥
2
) with smooth boundary ∂
Ω. By virtue of a first-order differential inequality technique, they showed the sufficient conditions to guarantee that the solution u
(
x
,
t
) exists globally or blows up. In addition, a lower bound for the blow-up time when blow-up occurs was also obtained. Ding [18] studied the nonlinear parabolic problem
(9)
(
b
(
u
)
)
t
=
∇
·
(
g
(
u
)
∇
u
)
+
f
(
u
)
,
(
x
,
t
)
∈
Ω
×
(
0
,
t
*
)
,
where Ω is a bounded domain in R
3 with smooth boundary ∂
Ω. They derived conditions on the data which guarantee the blow-up or the global existence of the solution. A lower bound on blow-up time when blow-up occurs was also obtained. For the problem of the nonlinear nonlocal porous medium equation, we read the paper of Liu [19].Recently, for the problems with nonlinear Neumann boundary conditions, Payne et al. [20] studied the semilinear heat equation with inner absorption term
(10)
u
t
=
Δ
u
-
f
(
u
)
,
(
x
,
t
)
∈
Ω
×
(
0
,
t
*
)
.
They established conditions on nonlinearity to guarantee that the solution u
(
x
,
t
) exists for all time t
>
0 or blows up at some finite time t
*. Moreover, an upper bound for t
* was derived. Under somewhat more restrictive conditions, a lower bound for t
* was derived. Thereafter, they considered the quasilinear parabolic equation
(11)
u
t
=
∇
·
(
|
∇
u
|
p
∇
u
)
,
(
x
,
t
)
∈
Ω
×
(
0
,
t
*
)
,
and they showed that blow-up occurs at some finite time under certain conditions on the nonlinearities and the data; upper and lower bounds for the blow-up time were derived when blow-up occurs; see [21]. Liu et al. The authors [22, 23] studied the reaction diffusion problem with nonlocal source and inner absorption terms or with local source and gradient absorption terms. Very recently, Fang et al. [24] considered lower bounds estimate for the blow-up time to nonlocal problem with homogeneous Dirichlet or Neumann boundary condition.Motivated by the above work, we intend to study the global existence and the blow-up phenomena of problem (1)–(3), and the results of the semilinear equations are extended to the quasilinear equations. Unfortunately, the techniques used for semilinear equation to analysis of blow-up phenomena are no longer applicable to our problem. As a consequence, by using the suitable techniques of differential inequalities, we establish, respectively, the conditions on the nonlinearities f and g to guarantee that u
(
x
,
t
) exists globally or blows up at some finite time. If blow-up occurs, we derive upper and lower bounds of the blow-up time.The rest of our paper is organized as follows. In Section2, we establish conditions on the nonlinearities to guarantee that u
(
x
,
t
) exists globally. In Section 3, we show the conditions on data forcing the solution u
(
x
,
t
) to blow up at some finite time t
* and obtain an upper bound for t
*. A lower bound of blow-up time under some assumptions is derived in Section 4.


## 2. The Global Existence

In this section, we establish the conditions on the nonlinearityf and nonlinearity g to guarantee that u
(
x
,
t
) exists globally. We state our result as follows.Theorem 1.
Assume that the nonnegative functionsf and g satisfy
(12)
f
(
ξ
)
≥
k
1
ξ
q
,
ξ
≥
0
,
g
(
ξ
)
≤
k
2
ξ
s
,
ξ
≥
0
,
where k
1
>
0, k
2
≥
0, s
>
1, 2
s
<
q
+
1, and s
-
1
<
p
<
q
-
1. Then the (nonnegative) solution u
(
x
,
t
) of problem (1)-(3) does not blow up; that is, u
(
x
,
t
) exists for all time t
>
0.Proof.
Set(13)
Ψ
(
t
)
=
∫
Ω
u
2
d
x
.
Similar to Theorem 2.1 in [20], we get
(14)
Ψ
′
(
t
)
≤
{
2
δ
2
∫
Ω
u
2
s
d
x
-
k
1
∫
Ω
u
q
+
1
d
x
}
+
{
2
k
2
N
ρ
0
∫
Ω
u
s
+
1
d
x
f
d
f
d
d
-
2
∫
Ω
|
∇
u
|
p
+
2
d
x
-
k
1
∫
Ω
u
q
+
1
d
x
2
k
2
N
ρ
0
}
=
I
1
+
I
2
,
where δ
=
k
2
(
s
+
1
)
d
/
2
ρ
0, ρ
0
=
min
⁡
x
∈
∂
Ω
(
x
·
ν
), d
=
max
⁡
x
∈
∂
Ω
|
x
|, and
(15)
I
1
≤
(
∫
Ω
u
q
+
1
d
x
)
(
s
+
1
)
/
(
q
+
1
)
×
{
A
1
~
|
Ω
|
(
q
-
s
)
/
(
q
+
1
)
-
A
2
~
(
∫
Ω
u
q
+
1
d
x
)
(
q
-
s
)
/
(
q
+
1
)
}
,
where A
1
~
=
2
δ
2
α
ε
(
α
-
1
)
/
α, A
2
~
=
k
1
-
2
δ
2
(
1
-
α
)
ɛ, α
=
(
q
+
1
-
2
s
)
/
(
q
-
s
)
<
1, ɛ
>
0.
Next, we estimateI
2
=
(
2
k
2
N
/
ρ
0
)
∫
Ω
‍
u
s
+
1
d
x
-
2
∫
Ω
‍
|
∇
u
|
p
+
2
d
x
-
k
1
∫
Ω
‍
u
q
+
1
d
x. Since
(16)
|
∇
u
(
p
/
2
)
+
1
|
2
=
(
p
2
+
1
)
2
u
p
|
∇
u
|
2
,
it follows from Hölder inequality that
(17)
∫
Ω
|
∇
u
(
p
/
2
)
+
1
|
2
d
x
≤
(
p
2
+
1
)
2
(
∫
Ω
|
∇
u
|
p
+
2
d
x
)
2
/
(
p
+
2
)
×
(
∫
Ω
u
p
+
2
d
x
)
p
/
(
p
+
2
)
.
Furthermore, we have
(18)
∫
Ω
u
p
+
2
d
x
≤
[
(
p
+
2
)
2
4
λ
1
]
(
p
/
2
)
+
1
∫
Ω
|
∇
u
|
p
+
2
d
x
,
which follows from (17) and membrane inequality
(19)
λ
1
∫
Ω
ω
2
d
x
≤
∫
Ω
|
∇
ω
|
2
d
x
,
where λ
1 is the first eigenvalue in the fixed membrane problem
(20)
Δ
ω
+
λ
ω
=
0
,
ω
>
0
in
Ω
,
ω
=
0
on
⁡
∂
Ω
.
Combining I
2 and (18), we have
(21)
I
2
≤
2
k
2
N
ρ
0
∫
Ω
u
s
+
1
d
x
-
2
[
4
λ
1
(
p
+
2
)
2
]
(
p
/
2
)
+
1
×
∫
Ω
u
p
+
2
d
x
-
k
1
∫
Ω
u
q
+
1
d
x
.
=
{
2
k
2
N
ρ
0
∫
Ω
u
s
+
1
d
x
-
3
[
4
λ
1
(
p
+
2
)
2
]
(
p
/
2
)
+
1
∫
Ω
u
p
+
2
d
x
}
+
{
[
4
λ
1
(
p
+
2
)
2
]
(
p
/
2
)
+
1
∫
Ω
u
p
+
2
d
x
-
k
1
∫
Ω
u
q
+
1
d
x
}
=
I
21
+
I
22
.
Making use of Hölder inequality, we obtain
(22)
∫
Ω
u
s
+
1
d
x
≤
(
∫
Ω
u
p
+
2
d
x
)
(
s
+
1
)
/
(
p
+
2
)
|
Ω
|
(
p
-
s
+
1
)
/
(
p
+
2
)
,
(23)
Ψ
(
t
)
=
∫
Ω
u
2
d
x
≤
(
∫
Ω
u
s
+
1
d
x
)
2
/
(
s
+
1
)
|
Ω
|
(
s
-
1
)
/
(
s
+
1
)
.
Combining (21), (22) with (23), we get
(24)
I
21
≤
(
∫
Ω
u
s
+
1
d
x
)
{
B
1
-
B
2
Ψ
(
p
-
s
+
1
)
/
2
}
,
with
(25)
B
1
=
2
k
2
N
ρ
0
,
B
2
=
3
[
4
λ
1
(
p
+
2
)
2
]
(
p
/
2
)
+
1
|
Ω
|
-
(
p
-
s
+
1
)
/
2
.
Applying Hölder inequality, we obtain
(26)
∫
Ω
u
p
+
2
d
x
≤
(
∫
Ω
u
q
+
1
d
x
)
(
p
+
2
)
/
(
q
+
1
)
|
Ω
|
(
q
-
p
-
1
)
/
(
q
+
1
)
,
Ψ
(
t
)
=
∫
Ω
u
2
d
x
≤
(
∫
Ω
u
q
+
1
d
x
)
2
/
(
q
+
1
)
|
Ω
|
(
q
-
1
)
/
(
q
+
1
)
.
It follows from (26) that
(27)
I
22
≤
(
∫
Ω
u
q
+
1
d
x
)
(
p
+
2
)
/
(
q
+
1
)
{
C
1
-
C
2
Ψ
(
q
-
p
-
1
)
/
2
}
,
where
(28)
C
1
=
[
4
λ
1
(
p
+
2
)
2
]
(
p
/
2
)
+
1
|
Ω
|
(
q
-
p
-
1
)
/
(
q
+
1
)
,
C
2
=
k
1
|
Ω
|
(
1
-
q
)
(
q
-
p
-
1
)
/
2
(
q
+
1
)
.
Combining (14), (15), (21), and (24) with (27), we obtain
(29)
Ψ
′
(
t
)
≤
(
∫
Ω
u
q
+
1
d
x
)
(
s
+
1
)
/
(
q
+
1
)
{
A
1
-
A
2
Ψ
(
t
)
(
q
-
s
)
/
2
}
+
(
∫
Ω
u
s
+
1
d
x
)
{
B
1
-
B
2
Ψ
(
p
-
s
+
1
)
/
2
}
+
(
∫
Ω
u
q
+
1
d
x
)
(
p
+
2
)
/
(
q
+
1
)
{
C
1
-
C
2
Ψ
(
q
-
p
-
1
)
/
2
}
,
with
(30)
A
1
=
A
1
~
|
Ω
|
(
q
-
s
)
/
(
q
+
1
)
,
A
2
=
A
2
~
|
Ω
|
(
1
-
q
)
(
q
-
s
)
/
2
(
q
+
1
)
.
We conclude from (29) that Ψ
(
t
) is decreasing in each time interval on which we obtain
(31)
Ψ
(
t
)
≥
max
⁡
{
(
A
1
A
2
)
2
/
(
q
-
s
)
,
(
B
1
B
2
)
2
/
(
p
-
s
+
1
)
,
(
C
1
C
2
)
2
/
(
q
-
p
-
1
)
}
,
so that Ψ
(
t
) remains bounded for all time under the conditions in Theorem 1. This completes the proof of Theorem 1.


## 3. Blow-Up and Upper Bound oft
*

In this section,Ω needs not to be star-shaped. We establish the conditions to assure that the solution of (1)–(3) blows up at finite time t
* and derive an upper bound for t
*. More precisely we establish the following result.Theorem 2.
Letu
(
x
,
t
) be the classical solution of problem (1)-(3). Assume that the nonnegative and integrable functions f and g satisfy
(32)
ξ
f
(
ξ
)
≤
2
(
1
+
α
)
F
(
ξ
)
,
ξ
≥
0
,
ξ
g
(
ξ
)
≥
2
(
1
+
β
)
G
(
ξ
)
,
ξ
≥
0
,
with
(33)
F
(
ξ
)
=
∫
0
ξ
f
(
η
)
d
η
,
G
(
ξ
)
=
∫
0
ξ
g
(
η
)
d
η
,
where α
≥
0,
(34)
β
≥
max
⁡
(
p
2
,
α
)
.
Moreover assume that Φ
(
0
)
≥
0 with
(35)
Φ
(
t
)
=
2
∫
∂
Ω
G
(
u
)
d
S
-
∫
Ω
|
∇
u
|
2
(
1
+
2
p
+
2
|
∇
u
|
p
)
d
x
-
2
∫
Ω
F
(
u
)
d
x
.
Then the solution u
(
x
,
t
) of problem (1)-(3) blows up at some finite time t
*
<
T with
(36)
T
=
Ψ
(
0
)
2
β
(
1
+
β
)
Φ
(
0
)
,
β
>
0
,
where Ψ
(
t
) is defined in (13). If β
=
0, we have T
=
∞.Proof.
We compute(37)
Ψ
′
(
t
)
=
2
∫
Ω
u
u
t
d
x
=
2
∫
Ω
u
[
(
(
|
∇
u
|
p
+
1
)
u
,
i
)
,
i
-
f
(
u
)
]
d
x
=
2
∫
∂
Ω
u
(
|
∇
u
|
p
+
1
)
∂
u
∂
ν
d
S
-
2
∫
Ω
(
|
∇
u
|
p
+
1
)
|
∇
u
|
2
d
x
-
2
∫
Ω
u
f
(
u
)
d
x
=
2
∫
∂
Ω
u
g
(
u
)
d
S
-
2
∫
Ω
(
|
∇
u
|
p
+
1
)
|
∇
u
|
2
d
x
-
2
∫
Ω
u
f
(
u
)
d
x
.
Making use of the hypotheses stated in Theorem 2, we have
(38)
Ψ
′
(
t
)
≥
2
(
1
+
β
)
Φ
(
t
)
.
Differentiating (35), we derive
(39)
Φ
′
(
t
)
=
2
∫
∂
Ω
g
(
u
)
u
t
d
S
-
∫
Ω
(
|
∇
u
|
p
+
1
)
(
|
∇
u
|
2
)
t
d
x
-
2
∫
Ω
f
(
u
)
u
t
d
x
.
Integrating the identity ∇
·
(
u
t
(
|
∇
u
|
p
+
1
)
∇
u
)
=
u
t
∇
·
(
(
|
∇
u
|
p
+
1
)
∇
u
)
+
(
1
/
2
)
(
|
∇
u
|
p
+
1
)
(
|
∇
u
|
2
)
t over Ω, we get
(40)
∫
Ω
(
|
∇
u
|
p
+
1
)
(
|
∇
u
|
2
)
t
d
x
=
2
∫
Ω
∇
·
(
u
t
(
|
∇
u
|
p
+
1
)
∇
u
)
d
x
-
2
∫
Ω
u
t
∇
·
(
(
|
∇
u
|
p
+
1
)
∇
u
)
d
x
=
2
∫
∂
Ω
u
t
(
|
∇
u
|
p
+
1
)
∇
u
·
ν
d
S
-
2
∫
Ω
u
t
∇
·
(
(
|
∇
u
|
p
+
1
)
∇
u
)
d
x
=
2
∫
∂
Ω
‍
u
t
(
|
∇
u
|
p
+
1
)
∂
u
∂
ν
d
S
-
2
∫
Ω
u
t
∇
·
(
(
|
∇
u
|
p
+
1
)
∇
u
)
d
x
.
Substituting (40) into (39), we have
(41)
Φ
′
(
t
)
=
2
∫
Ω
u
t
2
d
x
>
0
,
which with Φ
(
0
)
>
0 imply Φ
(
t
)
>
0 for all t
∈
(
0
,
t
*
). Making use of the Schwarz inequality, we obtain
(42)
2
(
1
+
β
)
Ψ
′
Φ
≤
(
Ψ
′
(
t
)
)
2
=
4
(
∫
Ω
u
u
t
d
x
)
2
≤
2
Ψ
(
t
)
Φ
′
(
t
)
.
Multiplying the above inequality by Ψ
-
2
-
β, we deduce
(43)
(
Φ
Ψ
-
(
1
+
β
)
)
′
≥
0
.
Arguing as in Theorem 3.1 in [20], we find
(44)
t
*
≤
T
=
1
2
β
(
1
+
β
)
(
Ψ
(
0
)
)
-
β
=
Ψ
(
0
)
2
β
(
1
+
β
)
Φ
(
0
)
valid for β
>
0. If β
=
0, we have
(45)
Ψ
(
t
)
≥
Ψ
(
0
)
e
2
M
t
valid for t
>
0, implying that t
*
=
∞. This completes the proof of Theorem 2.


## 4. Lower Bounds fort
*

In this section, under the assumption thatΩ is a star shaped domain in R
3, convex in two orthogonal directions, we seek a lower bound for the blow-up time t
*. Now we state the result as follows.Theorem 3.
Letu
(
x
,
t
) be the nonnegative solution of problem (1)-(3) and u
(
x
,
t
) blows up at t
*; moreover, the nonnegative functions f and g satisfy
(46)
f
(
ξ
)
≥
k
1
ξ
q
,
ξ
≥
0
,
g
(
ξ
)
≤
k
2
ξ
s
,
ξ
≥
0
,
with k
1
>
0, k
2
>
0, q
>
1, s
>
1, q
<
s. Define
(47)
φ
(
t
)
=
∫
Ω
u
n
(
s
-
1
)
d
x
,
where n is a parameter restricted by the condition
(48)
n
>
max
⁡
{
4
,
2
s
-
1
}
.
Then φ
(
t
) satisfies inequality
(49)
φ
′
(
t
)
≤
Γ
(
φ
)
,
for some computable function Γ
(
φ
). It follows that t
* is bounded from below. We have
(50)
t
*
≥
∫
φ
(
0
)
∞
d
η
Γ
(
η
)
d
η
.Proof.
Differentiating (47) and making use of the boundary condition (2) together with the conditions (46), we have
(51)
φ
′
(
t
)
=
n
(
s
-
1
)
∫
Ω
u
n
(
s
-
1
)
-
1
u
t
d
x
=
n
(
s
-
1
)
∫
Ω
u
n
(
s
-
1
)
-
1
f
d
s
f
d
f
d
f
d
d
d
×
[
(
(
|
∇
u
|
p
+
1
)
u
,
i
)
,
i
-
f
(
u
)
]
d
x
=
n
(
s
-
1
)
∫
∂
Ω
u
n
(
s
-
1
)
-
1
(
|
∇
u
|
p
+
1
)
∂
u
∂
ν
d
S
-
n
(
s
-
1
)
[
n
(
s
-
1
)
-
1
]
×
∫
Ω
u
n
(
s
-
1
)
-
2
|
∇
u
|
p
+
2
d
x
-
n
(
s
-
1
)
[
n
(
s
-
1
)
-
1
]
×
∫
Ω
u
n
(
s
-
1
)
-
2
|
∇
u
|
2
d
x
-
n
(
s
-
1
)
∫
Ω
u
n
(
s
-
1
)
-
1
f
(
u
)
d
x
≤
k
2
n
(
s
-
1
)
∫
∂
Ω
u
(
n
+
1
)
(
s
-
1
)
d
S
-
n
(
s
-
1
)
[
n
(
s
-
1
)
-
1
]
×
∫
Ω
u
n
(
s
-
1
)
-
2
|
∇
u
|
p
+
2
d
x
-
n
(
s
-
1
)
[
n
(
s
-
1
)
-
1
]
×
∫
Ω
u
n
(
s
-
1
)
-
2
|
∇
u
|
2
d
x
-
k
1
n
(
s
-
1
)
∫
Ω
u
n
(
s
-
1
)
+
q
-
1
d
x
.
Applying inequality (
2.7
) in [20] to the first term on the right hand side of (51), we have
(52)
∫
∂
Ω
u
(
n
+
1
)
(
s
-
1
)
d
S
≤
3
ρ
0
∫
Ω
u
(
n
+
1
)
(
s
-
1
)
d
x
+
(
n
+
1
)
(
s
-
1
)
d
ρ
0
∫
Ω
u
(
n
+
1
)
(
s
-
1
)
-
1
|
∇
u
|
d
x
.
Substituting (52) into (51), we obtain
(53)
φ
′
(
t
)
≤
3
k
2
n
(
s
-
1
)
ρ
0
∫
Ω
u
(
n
+
1
)
(
s
-
1
)
d
x
+
k
2
n
(
n
+
1
)
(
s
-
1
)
2
d
ρ
0
∫
Ω
u
(
n
+
1
)
(
s
-
1
)
-
1
|
∇
u
|
d
x
-
n
(
s
-
1
)
[
n
(
s
-
1
)
-
1
]
∫
Ω
u
n
(
s
-
1
)
-
2
|
∇
u
|
p
+
2
d
x
-
n
(
s
-
1
)
[
n
(
s
-
1
)
-
1
]
∫
Ω
u
n
(
s
-
1
)
-
2
|
∇
u
|
2
d
x
-
k
1
n
(
s
-
1
)
∫
Ω
u
n
(
s
-
1
)
+
q
-
1
d
x
.
Making use of arithmetic-geometric mean inequality, we derive
(54)
∫
Ω
u
(
n
+
1
)
(
s
-
1
)
-
1
|
∇
u
|
d
x
≤
μ
2
∫
Ω
u
n
(
s
-
1
)
-
2
|
∇
u
|
2
d
x
+
1
2
μ
∫
Ω
u
(
n
+
2
)
(
s
-
1
)
d
x
,
for all μ
>
0. Choose μ
>
0 such that
(55)
k
2
n
(
n
+
1
)
(
s
-
1
)
2
d
μ
2
ρ
0
-
n
(
s
-
1
)
[
n
(
s
-
1
)
-
1
]
=
0
.
We rewrite (53) as
(56)
φ
′
(
t
)
≤
3
k
2
n
(
s
-
1
)
ρ
0
∫
Ω
u
(
n
+
1
)
(
s
-
1
)
d
x
+
k
2
n
(
n
+
1
)
(
s
-
1
)
2
d
2
μ
ρ
0
∫
Ω
u
(
n
+
2
)
(
s
-
1
)
d
x
-
n
(
s
-
1
)
[
n
(
s
-
1
)
-
1
]
∫
Ω
u
n
(
s
-
1
)
-
2
|
∇
u
|
p
+
2
d
x
-
k
1
n
(
s
-
1
)
∫
Ω
u
n
(
s
-
1
)
+
q
-
1
d
x
.
Using Hölder inequality, we get
(57)
∫
Ω
u
n
(
s
-
1
)
d
x
≤
(
∫
Ω
u
n
(
s
-
1
)
+
q
-
1
d
x
)
n
(
s
-
1
)
/
(
n
(
s
-
1
)
+
q
-
1
)
×
|
Ω
|
(
q
-
1
)
/
(
n
(
s
-
1
)
+
q
-
1
)
.
Combining (56) with (57), we obtain
(58)
φ
′
(
t
)
≤
3
k
2
n
(
s
-
1
)
ρ
0
∫
Ω
u
(
n
+
1
)
(
s
-
1
)
d
x
+
k
2
n
(
n
+
1
)
(
s
-
1
)
2
d
2
μ
ρ
0
∫
Ω
u
(
n
+
2
)
(
s
-
1
)
d
x
-
n
(
s
-
1
)
[
n
(
s
-
1
)
-
1
]
×
∫
Ω
u
n
(
s
-
1
)
-
2
|
∇
u
|
p
+
2
d
x
-
k
1
n
(
s
-
1
)
|
Ω
|
(
1
-
q
)
/
n
(
s
-
1
)
φ
(
n
(
s
-
1
)
+
q
-
1
)
/
n
(
s
-
1
)
=
3
k
2
n
(
s
-
1
)
ρ
0
J
1
(
t
)
+
k
2
n
(
n
+
1
)
(
s
-
1
)
2
d
2
μ
ρ
0
J
2
(
t
)
-
n
(
s
-
1
)
[
n
(
s
-
1
)
-
1
]
ω
(
t
)
-
k
1
n
(
s
-
1
)
|
Ω
|
(
1
-
q
)
/
n
(
s
-
1
)
φ
(
n
(
s
-
1
)
+
q
-
1
)
/
n
(
s
-
1
)
,
where
(59)
J
1
(
t
)
=
∫
Ω
u
(
n
+
1
)
(
s
-
1
)
d
x
,
J
2
(
t
)
=
∫
Ω
u
(
n
+
2
)
(
s
-
1
)
d
x
,
ω
(
t
)
=
∫
Ω
u
n
(
s
-
1
)
-
2
|
∇
u
|
p
+
2
d
x
.
Using Sobolev type inequality (
A
.
5
) derived by Payne et al. [21], we obtain
(60)
J
1
(
t
)
=
∫
Ω
u
(
n
+
1
)
(
s
-
1
)
d
x
≤
{
3
ρ
0
∫
Ω
u
(
2
/
3
)
(
n
+
1
)
(
s
-
1
)
d
x
+
(
n
+
1
)
(
s
-
1
)
3
f
d
f
×
(
1
+
d
ρ
0
)
∫
Ω
u
(
2
/
3
)
(
n
+
1
)
(
s
-
1
)
-
1
|
∇
u
|
d
x
}
3
/
2
.
We now make use of Hölder inequality to bound the second integral on the right hand side of (60) as follows:
(61)
∫
Ω
u
(
2
/
3
)
(
n
+
1
)
(
s
-
1
)
-
1
|
∇
u
|
d
x
≤
(
∫
Ω
u
(
2
/
3
)
(
n
+
1
)
(
s
-
1
)
(
1
-
δ
1
)
d
x
)
(
p
+
1
)
/
(
p
+
2
)
ω
1
/
(
p
+
2
)
,
with
(62)
δ
1
=
(
n
-
2
)
(
s
-
1
)
+
3
p
2
(
n
+
1
)
(
s
-
1
)
(
p
+
1
)
.
We note that δ
1
<
1 for n
>
(
3
p
-
2
(
s
-
1
)
(
p
+
2
)
)
/
(
s
-
1
)
(
2
p
+
1
), an inequality satisfied in view of (48). Using again Hölder's inequality, we obtain
(63)
∫
Ω
u
(
2
/
3
)
(
n
+
1
)
(
s
-
1
)
(
1
-
δ
1
)
d
x
≤
φ
2
(
n
+
1
)
(
1
-
δ
1
)
/
3
n
|
Ω
|
1
-
(
2
(
n
+
1
)
(
1
-
δ
1
)
/
3
n
)
,
∫
Ω
u
(
2
/
3
)
(
n
+
1
)
(
s
-
1
)
d
x
≤
φ
2
(
n
+
1
)
/
3
n
|
Ω
|
1
-
(
2
(
n
+
1
)
/
3
n
)
,
where |
Ω
|
=
∫
Ω
‍
d
x is the volume of Ω. Substituting (61) and (63) in (60), we obtain the following inequality:
(64)
J
1
(
t
)
≤
{
c
1
~
φ
2
(
n
+
1
)
/
3
n
+
c
2
~
φ
(
2
(
n
+
1
)
(
1
-
δ
1
)
/
3
n
)
(
(
p
+
1
)
/
(
p
+
2
)
)
f
d
s
×
ω
1
/
(
p
+
2
)
}
3
/
2
≤
c
1
φ
(
n
+
1
)
/
n
+
c
2
φ
(
(
n
+
1
)
(
1
-
δ
1
)
/
n
)
(
(
p
+
1
)
/
(
p
+
2
)
)
ω
3
/
2
(
p
+
2
)
,
where c
1
,
c
2 are computable positive constants. Note that the last inequality in (64) follows from Hölder inequality under the particular form (
a
+
b
)
3
/
2
≤
2
(
a
3
/
2
+
b
3
/
2
). Similarly, we can bound J
2 and get
(65)
J
2
(
t
)
≤
c
3
φ
(
n
+
2
)
/
n
+
c
4
φ
(
(
n
+
2
)
(
1
-
δ
2
)
/
n
)
(
(
p
+
1
)
/
(
p
+
2
)
)
ω
3
/
2
(
p
+
2
)
,
where c
3
,
c
4 are computable positive constants,
(66)
δ
2
=
(
n
-
4
)
(
s
-
1
)
+
3
p
2
(
n
+
1
)
(
s
-
1
)
(
p
+
1
)
.
We note that δ
2
<
1 for n
>
(
3
p
-
4
(
s
-
1
)
(
p
+
2
)
)
/
(
s
-
1
)
(
2
p
+
1
), an inequality satisfied in view of (48). Inserting (64) and (65) in (58), we arrive at
(67)
φ
′
(
t
)
≤
d
1
~
φ
(
n
+
1
)
/
n
+
d
2
~
φ
(
(
n
+
1
)
(
1
-
δ
1
)
/
n
)
λ
ω
3
/
2
(
p
+
2
)
+
d
3
φ
(
n
+
2
)
/
n
+
d
4
~
φ
(
(
n
+
2
)
(
1
-
δ
2
)
/
n
)
λ
ω
3
/
2
(
p
+
2
)
-
n
(
s
-
1
)
[
n
(
s
-
1
)
-
1
]
ω
(
t
)
-
k
1
n
(
s
-
1
)
|
Ω
|
(
1
-
q
)
/
n
(
s
-
1
)
φ
(
n
(
s
-
1
)
+
q
-
1
)
/
n
(
s
-
1
)
,
where λ
=
(
p
+
1
)
/
(
p
+
2
), d
3 and d
j
~
(
j
=
1,2
,
4
) are computable positive constants. Next, we want to eliminate the quantity ω
(
t
) in inequality (67). By using the following inequality:
(68)
φ
α
ω
β
=
(
γ
ω
)
β
{
φ
α
/
(
1
-
β
)
γ
β
/
(
1
-
β
)
}
1
-
β
≤
γ
β
ω
+
(
1
-
β
)
γ
β
/
(
β
-
1
)
+
(
1
-
β
)
γ
β
/
(
β
-
1
)
φ
α
/
(
1
-
β
)
,
valid for 0
<
β
<
1, where γ is an arbitrary positive constant, then we have
(69)
d
2
~
φ
(
(
n
+
1
)
(
1
-
δ
1
)
/
n
)
λ
ω
3
/
2
(
p
+
2
)
≤
γ
1
ω
(
t
)
+
d
2
φ
(
2
(
n
+
1
)
(
1
-
δ
1
)
(
p
+
2
)
/
n
(
2
p
+
1
)
)
λ
,
d
4
~
φ
(
(
n
+
2
)
(
1
-
δ
2
)
/
n
)
λ
ω
3
/
2
(
p
+
2
)
≤
γ
2
ω
(
t
)
+
d
4
φ
(
2
(
n
+
2
)
(
1
-
δ
2
)
(
p
+
2
)
/
n
(
2
p
+
1
)
)
λ
,
with arbitrary positive constants γ
1, γ
2 and computable positive constants d
2, d
4. Substitute (69) in (67) and choose the arbitrary (positive) constants γ
1, γ
2 such that γ
1
+
γ
2
-
n
(
s
-
1
)
[
n
(
s
-
1
)
-
1
]
=
0. We obtain
(70)
φ
′
(
t
)
≤
d
1
~
φ
(
n
+
1
)
/
n
+
d
2
φ
(
2
(
n
+
1
)
(
1
-
δ
1
)
(
p
+
2
)
/
n
(
2
p
+
1
)
)
λ
+
d
3
φ
(
n
+
2
)
/
n
+
d
4
φ
(
2
(
n
+
2
)
(
1
-
δ
2
)
(
p
+
2
)
/
n
(
2
p
+
1
)
)
λ
-
k
1
n
(
s
-
1
)
|
Ω
|
(
1
-
q
)
/
n
(
s
-
1
)
φ
(
n
(
s
-
1
)
+
q
-
1
)
/
n
(
s
-
1
)
.
We eliminate the last term in (70), by using the following inequality:
(71)
φ
(
n
+
1
)
/
n
=
{
m
φ
(
n
(
s
-
1
)
+
q
-
1
)
/
n
(
s
-
1
)
}
(
2
n
-
1
)
(
s
-
1
)
/
(
(
2
n
-
1
)
(
s
-
1
)
+
s
-
q
)
×
{
m
(
2
n
-
1
)
(
1
-
s
)
/
(
s
-
q
)
φ
3
}
(
s
-
q
)
/
(
(
2
n
-
1
)
(
s
-
1
)
+
s
-
q
)
≤
(
2
n
-
1
)
(
s
-
1
)
(
2
n
-
1
)
(
s
-
1
)
+
s
-
q
m
φ
(
n
(
s
-
1
)
+
q
-
1
)
/
n
(
s
-
1
)
+
s
-
q
(
2
n
-
1
)
(
s
-
1
)
+
s
-
q
m
(
2
n
-
1
)
(
1
-
s
)
/
(
s
-
q
)
φ
3
,
valid for q
<
s and arbitrary m
>
0, and choose m such that
(72)
(
2
n
-
1
)
(
s
-
1
)
(
2
n
-
1
)
(
s
-
1
)
+
s
-
q
d
1
~
m
-
k
1
n
(
s
-
1
)
|
Ω
|
(
1
-
q
)
/
n
(
s
-
1
)
=
0
.
Then (70) can be rewritten as
(73)
φ
′
(
t
)
≤
d
1
φ
3
+
d
2
φ
(
2
(
n
+
1
)
(
1
-
δ
1
)
(
p
+
2
)
/
n
(
2
p
+
1
)
)
λ
+
d
3
φ
(
n
+
2
)
/
n
+
d
4
φ
(
2
(
n
+
2
)
(
1
-
δ
2
)
(
p
+
2
)
/
n
(
2
p
+
1
)
)
λ
.
Integrating (73) over [
0
,
t
], we conclude
(74)
t
*
≥
∫
φ
(
0
)
∞
d
η
f
d
f
d
d
d
×
(
d
1
η
3
+
d
2
η
(
2
(
n
+
1
)
(
1
-
δ
1
)
(
p
+
2
)
/
n
(
2
p
+
1
)
)
λ
d
s
d
s
d
s
d
s
s
s
+
d
3
η
(
n
+
2
)
/
n
+
d
4
η
(
2
(
n
+
2
)
(
1
-
δ
2
)
(
p
+
2
)
/
n
(
2
p
+
1
)
)
λ
)
-
1
.
This completes the proof of Theorem 3.



---

*Source: 289245-2014-04-30.xml*