# Blow-Up of Solutions for a Class of Sixth Order Nonlinear Strongly Damped Wave Equation

Authors: Huafei Di; Yadong Shang
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

We consider the blow-up phenomenon of sixth order nonlinear strongly damped wave equation. By using the concavity method, we prove a finite time blow-up result under assumptions on the nonlinear term and the initial data.


---

## Body

## 1. Introduction

It is well known that nonlinear strongly damped wave equation is proposed to describe all kinds of viscous vibration system. The global well-posedness of third order nonlinear strongly damped wave equation(1)
u
t
t
-
α
Δ
u
t
-
Δ
u
=
f
(
u
)
,
α
>
0
,
x
∈
Ω
,
t
>
0
,
was studied by Webb [1] firstly. He gave the existence and asymptotic behavior of strong solutions for the problem (1). Then this result was improved by Y. C. Liu and D. C. Liu [2]. The existence and uniqueness of strong solutions were proved under the hypothesis of the weaker conditions. For two classes of strongly damped nonlinear wave equation, the finite time blow-up of solutions was proved by Shang [3]. A number of authors (Chen et al. [4], Zhou [5], and Al’shin et al. [6]) have shown the existence of the global weak solutions and the global attractors for third order nonlinear strongly damped wave equation.For the fourth order nonlinear strongly damped wave equation, there are also some results about initial boundary value problem or Cauchy problem [7–9]. In [7], Shang studied the initial boundary value problem of the following equation:
(2)
u
t
t
-
Δ
u
-
Δ
u
t
-
Δ
u
t
t
=
f
(
u
)
,
x
∈
Ω
,
t
>
0
.
Under some assumptions on f and n
=
1
,
2
,
3, he investigated the existence, uniqueness, asymptotic behavior, and blow-up phenomenon of the solutions.In [8], Xu et al. considered the initial boundary value problem of fourth order wave equation with viscous damping term
(3)
u
t
t
-
α
u
x
x
t
+
u
x
x
x
x
=
f
(
u
x
)
x
,
x
∈
Ω
,
t
>
0
.
They proved the global existence and nonexistence of the solution by argument related to the potential well-convexity method.In order to investigate the water wave problem with surface tension, Schneider and Wayne [10] studied a class of Boussinesq equation as follows:
(4)
u
t
t
=
u
x
x
+
u
x
x
t
t
+
μ
u
x
x
x
x
-
u
x
x
x
x
t
t
+
(
u
2
)
x
x
,
where x
,
t
,
μ
∈
R. This type of equations can be formally derived from the 2D water wave problem and models the water wave problem with surface tension. They proved that the long wave limit can be described approximately by two decoupled Kawahara equations. A more natural model seems to be an extension from the classical Boussinesq equation as follows (see [11]):
(5)
u
t
t
=
u
x
x
+
(
μ
+
1
)
u
x
x
x
x
-
u
x
x
x
x
x
x
+
(
u
2
)
x
x
.
Wang and Mu [12] studied the Cauchy problem of the equation
(6)
u
t
t
=
u
x
x
+
u
x
x
t
t
-
u
x
x
x
x
-
u
x
x
x
x
t
t
+
f
(
u
)
x
x
.
They obtained the existence and uniqueness of the local solutions and proved the blow-up of solutions to the problem (6).  Esfahani et al. [13] studied the solutions of
(7)
u
t
t
=
u
x
x
+
β
u
x
x
x
x
+
u
x
x
x
x
x
x
+
(
|
u
|
α
u
)
x
x
,
where β
=
±
1 and α
>
0. They proved the local well-posedness in L
2
(
R
) and H
1
(
R
) and gave finite time blow-up results to the problem (7).For the sixth order nonlinear wave equation with strong damping term(8)
u
t
t
-
Δ
u
-
ν
Δ
u
t
+
Δ
2
u
+
Δ
2
u
t
t
=
Δ
f
(
u
)
, H. W. Wang and S. B. Wang [14] established a global existence result of small amplitude solutions of the Cauchy problem (8) for all space dimensions n
>
1. When ν
=
0,  H. W. Wang and S. B. Wang [15] studied the long-time behavior of small solutions of the Cauchy problem for a Rosenau equation. The decay and scattering for small amplitude solution are established.In this paper, we study a class of sixth order nonlinear strongly damped wave equation:(9)
u
t
t
-
Δ
u
t
=
d
i
v
σ
→
(
∇
u
)
+
Δ
u
+
μ
Δ
2
u
+
Δ
3
u
,
x
∈
Ω
,
t
>
0
,
u
(
x
,
0
)
=
u
0
(
x
)
,
u
t
(
x
,
0
)
=
u
1
(
x
)
,
x
∈
Ω
,
(
B
i
u
)
(
x
,
t
)
=
0
,
x
∈
∂
Ω
,
t
>
0
,
where μ
>
0, Ω is a bounded domain of R
n
(
n
≥
1
) with a smooth boundary ∂
Ω, and B
i, i
=
0
,
1
,
2, are homogeneous boundary condition:
(10)
D
=
{
u
∈
C
6
(
Ω
¯
)
∣
B
i
u
=
∇
i
u
≡
0
,
i
=
0,1
,
2
,
x
∈
∂
Ω
}
.
By using the ideas of the concavity theory introduced by Levine [17], we prove the finite time blow-up results under assumption on the nonlinear term d
i
v
σ
→
(
∇
u
) and the initial data u
0
,
u
1.


## 2. Preliminaries and Main Results

In this section, we introduce some notations, basic ideas, and important lemmas which will be needed in the course of the paper.LetH
=
L
2
(
Ω
) be a Hilbert space which is equipped with the scalar product (
u
,
v
)
=
∫
Ω
‍
u
(
x
)
v
(
x
)
d
x.Now, we define(11)
A
u
=
-
Δ
u
-
μ
Δ
2
u
-
Δ
3
u
,
where A
:
D
→
H is a symmetric linear operator and satisfies (
u
,
A
v
)
=
(
A
u
,
v
) for all u
,
v
∈
D
⊆
H.For the nonlinear termd
i
v
σ
→
(
∇
u
) of the problem (9), σ
→
=
(
σ
1
,
σ
2
,
…
,
σ
n
)
:
D
n
→
H
n is a vector function which satisfies the following conditions.(a)
Assume that the Fréchet derivativeσ
→
x is a symmetric, bounded, linear operator on H
n and that x
→
σ
→
x is a continuous map from D
n to L
(
H
)
n.
(b)
The scalar valued functionG
:
D
→
R is defined by
(12)
G
(
u
)
=
∫
0
1
(
d
i
v
σ
→
(
ρ
∇
u
)
,
u
)
d
ρ
=
∫
Ω
‍
(
∫
0
1
‍
d
i
v
σ
→
(
ρ
∇
u
)
u
d
ρ
)
d
x
=
∫
Ω
‍
(
∫
0
1
‍
d
d
ρ
(
∫
0
ρ
u
d
i
v
σ
→
(
∇
z
)
d
z
)
d
ρ
)
d
x
=
∫
Ω
‍
(
∫
0
u
d
i
v
σ
→
(
∇
z
)
d
z
)
d
x
,
where G
(
u
) denotes the potential associated with d
i
v
σ
→
(
∇
u
). The Fréchet derivative of G
(
u
) is G
u which can be shown to act as follows:
(13)
G
u
v
=
(
d
i
v
σ
→
(
∇
u
)
,
v
)
,
for all u, v
∈
D.
(c)
Assume that for someα
>
0
(14)
(
d
i
v
σ
→
(
∇
u
)
,
u
)
≥
2
(
2
α
+
1
)
G
(
u
)
,
for all u
∈
D.To obtain the finite time blow-up result, we need the following interpolation inequality of Evance [16] for function in W
0
k
,
p
(
Ω
).Lemma 1.
For allu
∈
W
0
k
,
p
(
Ω
), if 1
≤
p
<
∞, j and k are integers, and 1
≤
j
≤
k, (
j
/
k
)
≤
α
≤
1, then
(15)
∥
D
j
u
∥
L
q
(
Ω
)
≤
C
∥
D
k
u
∥
L
p
(
Ω
)
α
∥
D
u
∥
L
r
(
Ω
)
1
-
α
,
∀
u
∈
W
0
k
,
p
(
Ω
)
,
where (
1
/
q
)
-
(
j
/
n
)
=
α
(
(
1
/
p
)
-
(
k
/
n
)
)
+
(
1
-
α
)
(
(
1
/
r
)
-
(
1
/
n
)
), with the constant C depending only on (
p, r, j, k, α, n
) and Ω.
In particular, ifj
=
2, k
=
3, and p
=
q
=
r
=
2, one has
(16)
∥
D
2
u
∥
L
2
(
Ω
)
≤
C
∥
D
3
u
∥
L
2
(
Ω
)
α
∥
D
u
∥
L
2
(
Ω
)
1
-
α
,
∀
u
∈
W
0
3,2
(
Ω
)
.Lemma 2.
Assume thatu
∈
W
0
3,2
(
Ω
) and 0
<
μ
≤
(
1
/
C
1
)(with C
1 depending on the constant C of Sobolev’s interpolation inequality); then (
A
u
,
u
)
≥
0.Proof.
By Lemma1, we see that
(17)
∥
D
2
u
∥
L
2
(
Ω
)
≤
C
∥
D
3
u
∥
L
2
(
Ω
)
α
∥
D
u
∥
L
2
(
Ω
)
1
-
α
,
∀
u
∈
W
0
3,2
(
Ω
)
.
Using Young’s inequality, we have
(18)
∥
D
2
u
∥
L
2
(
Ω
)
2
≤
C
∥
D
3
u
∥
L
2
(
Ω
)
2
α
∥
D
u
∥
L
2
(
Ω
)
2
(
1
-
α
)
≤
C
[
α
∥
D
3
u
∥
L
2
(
Ω
)
2
+
(
1
-
α
)
∥
D
u
∥
L
2
(
Ω
)
2
]
≤
C
1
[
∥
D
3
u
∥
L
2
(
Ω
)
2
+
∥
D
u
∥
L
2
(
Ω
)
2
]
.
For the operator A, using integration of parts, we have
(19)
(
A
u
,
u
)
=
(
-
Δ
u
-
μ
Δ
2
u
-
Δ
3
u
,
u
)
=
∥
∇
u
∥
L
2
(
Ω
)
2
+
∥
∇
3
u
∥
L
2
(
Ω
)
2
-
μ
∥
∇
2
u
∥
L
2
(
Ω
)
2
≥
1
C
1
∥
∇
2
u
∥
L
2
(
Ω
)
2
-
μ
∥
∇
2
u
∥
L
2
(
Ω
)
2
=
(
1
C
1
-
μ
)
∥
∇
2
u
∥
L
2
(
Ω
)
2
,
where 0
<
μ
≤
(
1
/
C
1
).
The verification of the action ofG
x can be proved from the definition. The details, not being germane to this paper, are omitted here. But a formula will be useful in the sequel as follows.Lemma 3.
LetG
(
u
)
=
∫
0
1
‍
(
d
i
v
σ
(
∇
(
ρ
u
)
)
,
u
)
d
ρ; then one has
(20)
G
(
u
(
t
)
)
-
G
(
u
(
0
)
)
=
∫
0
t
(
d
i
v
σ
(
∇
u
(
η
)
)
,
u
η
(
η
)
)
d
η
,
for u
:
[
0
,
T
)
→
D with a strongly continuous derivative u
t.Proof.
By the chain rule and the action ofG
x, we have
(21)
d
d
t
G
(
u
)
=
d
d
t
∫
0
1
(
d
i
v
σ
→
(
∇
(
ρ
u
)
)
,
u
)
d
ρ
=
-
d
d
t
∫
0
1
(
σ
→
(
∇
(
ρ
u
)
)
,
∇
u
)
d
ρ
=
-
∫
0
1
ρ
(
σ
→
ρ
∇
u
·
∇
u
t
,
∇
u
)
+
(
σ
→
(
ρ
∇
u
)
,
∇
u
t
)
d
ρ
=
-
∫
0
1
ρ
(
σ
→
ρ
∇
u
·
∇
u
,
∇
u
t
)
+
(
σ
→
(
ρ
∇
u
)
,
∇
u
t
)
d
ρ
=
-
∫
0
1
ρ
d
d
ρ
(
σ
→
(
ρ
∇
u
)
,
∇
u
t
)
+
(
σ
→
(
ρ
∇
u
)
,
∇
u
t
)
d
ρ
=
-
∫
0
1
d
d
ρ
[
ρ
(
σ
→
(
ρ
∇
u
)
,
∇
u
t
)
]
d
ρ
=
(
d
i
v
σ
→
(
∇
u
)
,
u
t
)
,
where we have used the symmetry of σ
→
x in the fourth line.The following lemma contributing to the result of this paper is analogous to Corollary 1.1 of [17] with slight modification.Lemma 4.
Assume thatσ
→ is homogenous of degree 1
+
γ for some γ
>
0 (i.e., σ
→
(
s
h
)
=
s
1
+
γ
σ
→
(
h
) for all s
>
0 and for all h
∈
D). Let (
h
0
,
d
i
v
σ
→
(
∇
h
0
)
)
>
0 for some h
0
∈
D. Then there are infinitely many vectors u
0 such that
(22)
G
(
u
0
)
>
1
2
(
u
0
,
A
u
0
)
=
1
2
[
∥
∇
3
u
0
∥
L
2
(
Ω
)
2
+
∥
∇
u
0
∥
L
2
(
Ω
)
2
-
μ
∥
∇
2
u
0
∥
L
2
(
Ω
)
2
]
.Proof.
Letu
0
=
s
h
0, where s is large enough so that
(23)
s
γ
G
(
h
0
)
=
s
γ
∫
0
1
(
d
i
v
σ
→
(
ρ
∇
h
0
)
,
h
0
)
d
ρ
>
1
2
(
h
0
,
A
h
0
)
=
1
2
[
∥
∇
3
h
0
∥
L
2
(
Ω
)
2
+
∥
∇
h
0
∥
L
2
(
Ω
)
2
-
μ
∥
∇
2
h
0
∥
L
2
(
Ω
)
2
]
.
Then for all
(24)
s
>
s
0
=
(
2
+
γ
)
[
∥
∇
3
h
0
∥
L
2
(
Ω
)
2

+
∥
∇
h
0
∥
L
2
(
Ω
)
2

-
μ
∥
∇
2
h
0
∥
L
2
(
Ω
)
2
]
2
(
d
i
v
σ
→
(
∇
h
0
)
,
h
0
)
γ

,
we have
(25)
G
(
u
0
)
=
∫
0
1
(
d
i
v
σ
→
(
ρ
s
∇
h
0
)
,
s
h
0
)
d
ρ
=
-
∫
0
1
(
σ
→
(
ρ
s
∇
h
0
)
,
s
∇
h
0
)
d
ρ
=
s
2
+
γ
G
(
h
0
)
>
1
2
(
u
0
,
A
h
0
)
.
The local existence of solution for the problem (9) can be obtained by the standard Faedo-Galerkin approximation methods. The interested reader is referred to Lions [18] or Robinson [19] for details.Next, we are ready to state the blow-up result of this paper.Theorem 5.
Letu
:
[
0
,
T
)
→
D be a strongly continuously differentiable solution of (9) in the D norm. Suppose that 0
<
μ
≤
(
1
/
C
1
) with (C
1 depending on the constant C of Sobolev’s interpolation inequality) (
d
i
v
σ
→
(
∇
u
)
,
u
)
≥
2
(
2
α
+
1
)
G
(
u
), where (
G
(
u
)
=
∫
0
1
‍
(
d
i
v
σ
→
(
ρ
∇
u
)
,
u
)
d
ρ
). Finally let u
0 satisfy
(26)
G
(
u
0
)
>
1
2
(
u
0
,
A
u
0
)
=
1
2
[
∥
∇
3
u
0
∥
L
2
(
Ω
)
2
+
∥
∇
u
0
∥
L
2
(
Ω
)
2
-
μ
∥
∇
2
u
0
∥
L
2
(
Ω
)
2
]
.
Then the solution u can only exist on a bounded interval [
0
,
T
), and in fact
(27)
T
≤
(
+
[
(
∇
u
0
,
∇
u
0
)
-
2
α
(
u
0
,
u
1
)
]
2
+
4
α
2
β
(
u
0
,
u
0
)
]
2
4
α
2
β
(
u
0
,
u
0
)
+
[
[
(
∇
u
0
,
∇
u
0
)
-
2
α
(
u
0
,
u
1
)
]
2
+
4
α
2
β
(
u
0
,
u
0
)
(
∇
u
0
,
∇
u
0
)
-
2
α
(
u
0
,
u
1
)
+
[
(
∇
u
0
,
∇
u
0
)
-
2
α
(
u
0
,
u
1
)
]
2
+
4
α
2
β
(
u
0
,
u
0
)
]
2
)
T
≤
×
(
+
4
α
2
β
(
u
0
,
u
0
)
)
1
/
2
)
-
1
4
α
2
β
(
[
(
∇
u
0
,
∇
u
0
)
-
2
α
(
u
0
,
u
1
)
]
2
+
4
α
2
β
(
u
0
,
u
0
)
)
1
/
2
)
-
1
,
while also
(28)
lim
⁡
t
→
T
-
[
∥
u
∥
L
2
(
Ω
)
2
+
∫
0
t
∥
∇
u
∥
L
2
(
Ω
)
2
d
η
]
=
+
∞
,
and consequently
(29)
lim
⁡
⁡
sup
⁡
t
→
T
-
∥
u
∥
H
1
2
=
+
∞
.Proof.
For arbitraryT
0, β, τ
>
0 and t
∈
[
0
,
T
0
), let
(30)
F
(
t
)
=
(
u
,
u
)
+
∫
0
t
(
u
,
-
Δ
u
)
d
η
+
(
T
0
-
t
)
(
u
0
,
-
Δ
u
0
)
+
β
(
t
+
τ
)
2
.
A direct computation yields
(31)
F
′
(
t
)
=
2
(
u
,
u
t
)
+
(
u
,
-
Δ
u
)
-
(
u
0
,
-
Δ
u
0
)
+
2
β
(
t
+
τ
)
=
2
(
u
,
u
t
)
+
(
∇
u
,
∇
u
)
-
(
∇
u
0
,
∇
u
0
)
+
2
β
(
t
+
τ
)
=
2
(
u
,
u
t
)
+
2
∫
0
t
(
∇
u
,
∇
u
η
)
d
η
+
2
β
(
t
+
τ
)
.
Suppressing the argument t, we see that
(32)
F
′′
(
t
)
=
2
(
u
,
u
t
t
)
+
2
(
u
t
,
u
t
)
+
2
∫
0
t
(
∇
u
,
∇
u
η
)
η
d
η
+
2
(
∇
u
1
,
∇
u
0
)
+
2
β
.
Hence, from (30), (31), and (32), we find after some algebra that
(33)
F
′′
F
-
(
α
+
1
)
(
F
′
(
t
)
)
2
=
4
(
α
+
1
)
S
2
+
2
F
[
(
u
,
u
t
t
)
-
(
2
α
+
1
)
(
u
t
,
u
t
)
]
+
2
F
[
∫
0
t
(
∇
u
,
∇
u
η
)
η
d
η
-
2
(
α
+
1
)
∫
0
t
(
∇
u
η
,
∇
u
η
)
d
η
]
+
4
(
α
+
1
)
(
T
0
-
t
)
(
∇
u
0
,
∇
u
0
)
×
[
(
u
t
,
u
t
)
+
∫
0
t
(
∇
u
η
,
∇
u
η
)
d
η
+
β
]
+
2
F
[
(
∇
u
1
,
∇
u
0
)
-
(
2
α
+
1
)
β
]
,
where
(34)
S
2
=
-
[
(
u
,
u
t
)
+
∫
0
t
(
∇
u
,
∇
u
η
)
d
η
+
β
(
t
+
τ
)
]
2
+
[
(
u
,
u
)
+
∫
0
t
(
u
,
-
Δ
u
)
d
η
+
β
(
t
+
τ
)
2
]
×
[
(
u
t
,
u
t
)
+
∫
0
t
(
∇
u
η
,
∇
u
η
)
d
η
+
β
]
.
Using Schwarz’s inequality, we have
(35)
(
u
,
u
)
(
u
t
,
u
t
)
≥
(
u
,
u
t
)
2
,
∫
0
t
(
∇
u
,
∇
u
)
d
η
∫
0
t
(
∇
u
η
,
∇
u
η
)
d
η
≥
[
∫
0
t
(
∇
u
,
∇
u
η
)
d
η
]
2
,
(
u
,
u
)
∫
0
t
(
∇
u
η
,
∇
u
η
)
d
η
+
(
u
,
u
t
)
∫
0
t
(
∇
u
,
∇
u
)
d
η
≥
2
(
u
,
u
)
(
u
t
,
u
t
)
∫
0
t
(
∇
u
,
∇
u
)
d
η
∫
0
t
(
∇
u
η
,
∇
u
η
)
d
η
≥
2
(
u
,
u
t
)
∫
0
t
(
∇
u
,
∇
u
η
)
d
η
,
(
u
,
u
)
β
+
(
u
t
,
u
t
)
β
(
t
+
τ
)
2
≥
2
(
u
,
u
)
(
u
t
,
u
t
)
β
2
(
t
+
τ
)
2
≥
2
(
u
,
u
t
)
β
(
t
+
τ
)
.
By (35), we have S
2
≥
0. Let
(36)
H
(
t
)
=
(
u
,
u
t
t
)
-
(
2
α
+
1
)
(
u
t
,
u
t
)
+
(
∇
u
1
,
∇
u
0
)
-
(
2
α
+
1
)
β
-
∫
0
t
(
Δ
u
η
,
u
)
η
d
η
+
2
(
α
+
1
)
∫
0
t
(
Δ
u
η
,
u
η
)
d
η
=
(
u
,
u
t
t
)
-
(
2
α
+
1
)
(
u
t
,
u
t
)
+
(
∇
u
1
,
∇
u
0
)
-
(
2
α
+
1
)
β
-
∫
0
t
(
u
η
η
,
u
)
η
d
η
+
2
(
α
+
1
)
∫
0
t
(
u
η
η
,
u
η
)
d
η
-
∫
0
t
[
(
A
u
,
u
)
η
-
2
(
α
+
1
)
(
A
u
,
u
η
)
]
d
η
+
∫
0
t
[
(
d
i
v
σ
→
(
∇
u
)
,
u
)
η
-
2
(
α
+
1
)
(
d
i
v
σ
→
(
∇
u
)
,
u
η
)
]
d
η
.
Thus
(37)
H
′
(
t
)
=
-
2
α
(
u
t
t
,
u
t
)
+
2
α
(
u
t
,
A
u
)
+
(
d
i
v
σ
→
(
∇
u
)
,
u
)
t
-
2
(
α
+
1
)
d
d
t
G
(
u
)
=
-
2
α
(
Δ
u
t
,
u
t
)
-
2
α
(
d
i
v
σ
→
(
∇
u
)
,
u
t
)
+
4
α
(
u
t
,
A
u
)
+
(
d
i
v
σ
→
(
∇
u
)
,
u
)
t
-
2
(
α
+
1
)
d
d
t
G
(
u
)
=
2
α
(
∇
u
t
,
∇
u
t
)
+
4
α
(
u
t
,
A
u
)
+
(
d
i
v
σ
→
(
∇
u
)
,
u
)
t
-
(
4
α
+
2
)
d
d
t
G
(
u
)
.
Using the positive semidefiniteness of A, Lemma 3, and (14), we have
(38)
H
′
(
t
)
=
H
(
0
)
+
2
α
∫
0
t
(
∇
u
η
,
∇
u
η
)
d
η
+
4
α
∫
0
t
(
u
η
,
A
u
)
d
η
+
∫
0
t
‍
[
(
d
i
v
σ
→
(
∇
u
)
,
u
)
η
-
(
4
α
+
2
)
d
d
η
G
(
u
)
]
d
η
=
(
4
α
+
2
)
G
(
u
0
)
-
(
2
α
+
1
)
(
u
0
,
A
u
0
)
+
(
u
1
,
u
1
)
+
(
∇
u
1
,
∇
u
1
)
-
(
2
α
+
1
)
β
+
2
α
∫
0
t
(
∇
u
η
,
∇
u
η
)
d
η
+
2
α
(
u
,
A
u
)
+
[
(
d
i
v
σ
→
(
∇
u
)
,
u
)
-
(
4
α
+
2
)
G
(
u
)
]
≥
(
4
α
+
2
)
G
(
u
0
)
-
(
2
α
+
1
)
(
u
0
,
A
u
0
)
-
(
2
α
+
1
)
β
=
2
(
2
α
+
1
)
[
G
(
u
0
)
-
1
2
(
A
u
0
,
u
0
)
-
β
2
]
.
Thus, from what has been discussed above, we have
(39)
F
F
′′
-
(
α
+
1
)
(
F
′
)
2
≥
4
(
2
α
+
1
)
F
[
G
(
u
0
)
-
1
2
(
A
u
0
,
u
0
)
-
β
2
]
.
Therefore, for any β
>
0 such that
(40)
β
≤
2
G
(
u
0
)
-
(
A
u
0
,
u
0
)
=
2
∫
0
1
(
d
i
v
σ
→
(
ρ
∇
u
0
)
,
u
0
)
d
ρ
-
[
∥
∇
3
u
0
∥
L
2
(
Ω
)
2
+
∥
∇
u
0
∥
L
2
(
Ω
)
2
-
μ
∥
∇
2
u
0
∥
L
2
(
Ω
)
2
]
,
F
F
′′
-
(
α
+
1
)
(
F
′
)
2
≥
0 and (
F
-
α
(
t
)
)
′′
≤
0. We see that F
(
t
)
>
0, for all t
∈
[
0
,
T
0
) and F
′
(
0
)
=
2
(
u
0
,
u
1
)
+
2
β
τ
>
0, if τ is sufficiently large. Since a concave function must always lie below any tangent line, so we have
(41)
F
-
α
(
t
)
≤
F
-
α
(
0
)
+
[
F
-
α
(
0
)
]
′
t
or
(42)
F
(
t
)
≥
F
1
+
(
1
/
α
)
(
0
)
[
F
(
0
)
-
α
t
F
′
(
0
)
]
-
1
/
α
;
we may choose T
0 such that T
0
≥
(
F
(
0
)
/
α
(
F
′
(
0
)
)
)
≡
T
β
τ. Thus, we see that the interval of existence of u must be contained in [
0
,
F
(
0
)
/
α
(
F
′
(
0
)
)
) and that the finite time blow-up of solution of (9) is proved. Let
(43)
β
0
=
2
G
(
u
0
)
-
(
A
u
0
,
u
0
)
=
2
∫
0
1
(
d
i
v
σ
→
(
ρ
∇
u
0
)
,
u
0
)
d
ρ
-
[
∥
∇
3
u
0
∥
L
2
(
Ω
)
2
+
∥
∇
u
0
∥
L
2
(
Ω
)
2
-
μ
∥
∇
2
u
0
∥
L
2
(
Ω
)
2
]
,
T
β
τ
=
F
(
0
)
α
(
F
′
(
0
)
)
=
(
u
0
,
u
0
)
+
T
0
(
∇
u
0
,
∇
u
0
)
+
β
τ
2
2
α
(
u
0
,
u
1
)
+
2
α
β
τ
.
Since T
0
≥
T
β
τ, we have (
F
-
α
(
t
)
)
′′
≤
0. Even if we take T
0
=
T
β
τ, we have
(44)
T
β
τ
=
[
(
u
0
,
u
0
)
+
β
τ
2
]
[
2
α
(
u
0
,
u
1
)
+
2
α
β
τ
-
(
∇
u
0
,
∇
u
0
)
]
-
1
;
thus we must choose τ so large such that 2
α
β
τ
>
(
∇
u
0
,
∇
u
0
)
-
2
α
(
u
0
,
u
1
). As a function, T
β
τ has a minimum at
(45)
τ
=
τ
(
β
)
=
(
+
[
(
∇
u
0
,
∇
u
0
)
-
2
α
(
u
0
,
u
1
)
]
2
+
4
α
2
β
(
u
0
,
u
0
)
(
∇
u
0
,
∇
u
0
)
-
2
α
(
u
0
,
u
1
)
+
[
(
∇
u
0
,
∇
u
0
)
-
2
α
(
u
0
,
u
1
)
]
2
+
4
α
2
β
(
u
0
,
u
0
)
)
×
(
2
α
β
)
-
1
,
and this minimum is
(46)
T
β
τ
(
β
)
=
(
[
(
∇
u
0
,
∇
u
0
)

-
2
α
(
u
0
,
u
1
)
]
2

+
4
α
2
β
(
u
0
,
u
0
)
]
2
4
α
2
β
(
u
0
,
u
0
)
+
[
[
(
∇
u
0
,
∇
u
0
)
-
2
α
(
u
0
,
u
1
)
]
2
+
4
α
2
β
(
u
0
,
u
0
)
(
∇
u
0
,
∇
u
0
)
-
2
α
(
u
0
,
u
1
)
+
[
(
∇
u
0
,
∇
u
0
)

-
2
α
(
u
0
,
u
1
)
]
2

+
4
α
2
β
(
u
0
,
u
0
)
]
2
)
×
(
+
4
α
2
β
(
u
0
,
u
0
)
)
1
/
2
4
α
2
β
(
[
(
∇
u
0
,
∇
u
0
)
-
2
α
(
u
0
,
u
1
)
]
2
+
4
α
2
β
(
u
0
,
u
0
)
)
1
/
2
)
-
1
.
Since β is restricted to [
0
,
β
0
], we see that T
β
τ
(
β
) attains its minimum at β
=
β
0. Thus T cannot exceed T
β
τ
(
β
0
).



---

*Source: 310297-2014-05-15.xml*