Proof.
(
1
)
First we prove that
∥
u
ω
(
x
,
t
)
-
u
(
x
,
t
)
∥
L
q
(
0
,
l
;
L
r
)
→
0
as
|
ω
|
→
∞
.
By (34), there must be constants
L
and
M
such that
(36)
sup
|
ω
|
≥
L
∥
D
2
u
ω
∥
L
γ
(
0
,
l
;
L
ρ
(
R
n
)
)
≤
M
.
From (11) and (12), we have
(37)
u
ω
(
x
,
t
)
-
u
(
x
,
t
)
=
i
(
I
1
+
I
2
)
,
where
I
1
=
∫
0
t
θ
(
ω
s
)
W
(
t
-
s
)
[
|
u
ω
(
s
)
|
8
/
(
n
-
4
)
u
ω
(
s
)
-
|
u
(
s
)
|
8
/
(
n
-
4
)
u
(
s
)
]
d
s
and
I
2
=
∫
0
t
[
θ
(
ω
s
)
-
I
(
θ
)
]
W
(
t
-
s
)
|
u
(
s
)
|
8
/
(
n
-
4
)
u
(
s
)
d
s
.
By Lemma 10, we have
(38)
∥
I
2
∥
L
q
(
0
,
l
;
L
r
(
R
n
)
)
∶
=
C
ε
ω
⟶
0
as
|
ω
|
⟶
∞
.
Using Hölder inequality and Sobolev embedding inequality, we obtain
(39)
∥
I
1
∥
L
q
(
0
,
l
;
L
r
(
R
n
)
)
≤
C
∥
|
u
ω
(
s
)
|
8
/
(
n
-
4
)
u
ω
(
s
)
-
|
u
(
s
)
|
8
/
(
n
-
4
)
u
(
s
)
∥
L
γ
′
(
0
,
l
;
L
ρ
′
(
R
n
)
)
≤
C
(
∥
D
2
u
ω
∥
L
γ
(
0
,
l
;
L
ρ
(
R
n
)
)
8
/
(
n
-
4
)
+
∥
D
2
u
∥
L
γ
(
0
,
l
;
L
ρ
(
R
n
)
)
8
/
(
n
-
4
)
)
×
∥
u
ω
-
u
∥
L
γ
(
0
,
l
;
L
ρ
(
R
n
)
)
≤
C
(
M
8
/
(
n
-
4
)
+
N
8
/
(
n
-
4
)
)
∥
u
ω
-
u
∥
L
γ
(
0
,
l
;
L
ρ
(
R
n
)
)
,
where
N
=
∥
u
∥
L
γ
(
0
,
l
;
H
2
,
ρ
(
R
n
)
)
.
Using (38) and (39), we have
(40)
∥
u
ω
(
x
,
t
)
-
u
(
x
,
t
)
∥
L
q
(
0
,
l
;
L
r
(
R
n
)
)
=
∥
I
1
+
I
2
∥
L
q
(
0
,
l
;
L
r
(
R
n
)
)
≤
∥
I
1
∥
L
q
(
0
,
l
;
L
r
(
R
n
)
)
+
∥
I
2
∥
L
q
(
0
,
l
;
L
r
(
R
n
)
)
≤
C
ε
ω
+
C
(
M
8
/
(
n
-
4
)
+
N
8
/
(
n
-
4
)
)
×
∥
u
ω
(
x
,
t
)
-
u
(
x
,
t
)
∥
L
γ
(
0
,
l
;
L
ρ
(
R
n
)
)
.
In the following we will prove that
∥
u
ω
(
x
,
t
)
-
u
(
x
,
t
)
∥
L
γ
(
0
,
l
;
L
ρ
(
R
n
)
)
.
Since
sup
|
ω
|
≥
L
∥
D
2
u
ω
∥
L
γ
(
0
,
l
;
L
ρ
(
R
n
)
)
≤
M
, we can divide the time interval
[
0
,
l
]
into subintervals
[
t
i
,
t
i
+
1
]
,
i
=
0
,
…
,
J
-
1
, where
t
0
=
0
,
t
J
-
1
=
l
, such that in each part
C
(
∥
D
2
u
ω
∥
L
γ
(
t
i
,
t
i
+
1
;
L
ρ
(
R
n
)
)
8
/
(
n
-
4
)
+
∥
D
2
u
∥
L
γ
(
t
i
,
t
i
+
1
;
L
ρ
(
R
n
)
)
8
/
(
n
-
4
)
)
=
1
/
2
.
On
[
t
0
,
t
1
]
, since
u
ω
(
t
0
)
=
u
(
t
0
)
=
φ
, we have
(41)
∥
u
ω
(
x
,
t
)
-
u
(
x
,
t
)
∥
L
q
(
t
0
,
t
1
;
L
r
(
R
n
)
)
≤
C
ε
ω
+
C
(
∥
D
2
u
ω
∥
L
γ
(
t
0
,
t
1
;
L
ρ
(
R
n
)
)
8
/
(
n
-
4
)
m
m
m
m
m
m
+
∥
D
2
u
∥
L
γ
(
t
0
,
t
1
;
L
ρ
(
R
n
)
)
8
/
(
n
-
4
)
)
×
∥
u
ω
(
x
,
t
)
-
u
(
x
,
t
)
∥
L
γ
(
t
0
,
t
1
;
L
ρ
(
R
n
)
)
≤
C
ε
ω
+
1
2
∥
u
ω
(
x
,
t
)
-
u
(
x
,
t
)
∥
L
γ
(
t
0
,
t
1
;
L
ρ
(
R
n
)
)
.
For the case
(
q
,
r
)
=
(
γ
,
ρ
)
, we have
(42)
∥
u
ω
(
x
,
t
)
-
u
(
x
,
t
)
∥
L
γ
(
t
0
,
t
1
;
L
ρ
(
R
n
)
)
≤
2
C
ε
ω
.
For the case
(
q
,
r
)
=
(
∞
,
2
)
, we have
(43)
∥
u
ω
(
x
,
t
)
-
u
(
x
,
t
)
∥
L
∞
(
t
0
,
t
1
;
L
2
(
R
n
)
)
≤
2
C
ε
ω
.
On
[
t
1
,
t
2
]
, we have
(44)
∥
u
ω
(
x
,
t
)
-
u
(
x
,
t
)
∥
L
q
(
t
1
,
t
2
;
L
r
(
R
n
)
)
≤
∥
u
ω
(
t
1
)
-
u
(
t
1
)
∥
L
2
(
R
n
)
+
C
ε
ω
+
1
2
∥
u
ω
(
x
,
t
)
-
u
(
x
,
t
)
∥
L
γ
(
t
1
,
t
2
;
L
ρ
(
R
n
)
)
≤
3
C
ε
ω
+
1
2
∥
u
ω
(
x
,
t
)
-
u
(
x
,
t
)
∥
L
γ
(
t
1
,
t
2
;
L
ρ
(
R
n
)
)
.
Similarly for the case
(
q
,
r
)
=
(
γ
,
ρ
)
, we have
(45)
∥
u
ω
(
x
,
t
)
-
u
(
x
,
t
)
∥
L
γ
(
t
1
,
t
2
;
L
ρ
(
R
n
)
)
≤
6
C
ε
ω
.
For the case
(
q
,
r
)
=
(
∞
,
2
)
, we have
(46)
∥
u
ω
(
x
,
t
)
-
u
(
x
,
t
)
∥
L
∞
(
t
1
,
t
2
;
L
2
(
R
n
)
)
≤
6
C
ε
ω
.
By induction, we have
(47)
∥
u
ω
(
x
,
t
)
-
u
(
x
,
t
)
∥
L
γ
(
t
i
,
t
i
+
1
;
L
ρ
(
R
n
)
)
≤
2
(
2
i
+
1
-
1
)
C
ε
ω
,
∥
u
ω
(
x
,
t
)
-
u
(
x
,
t
)
∥
L
∞
(
t
i
,
t
i
+
1
;
L
2
(
R
n
)
)
≤
2
(
2
i
+
1
-
1
)
C
ε
ω
,
for
i
=
0
,
…
,
J
-
1
.
So we have
(48)
∥
u
ω
(
x
,
t
)
-
u
(
x
,
t
)
∥
L
γ
(
0
,
l
;
L
ρ
(
R
n
)
)
≤
∑
i
=
0
J
-
1
2
(
2
i
+
1
-
1
)
C
ε
ω
=
[
4
(
2
J
-
1
)
-
2
J
]
C
ε
ω
⟶
0
as
|
ω
|
⟶
∞
.
Using the above estimate and (40), we have
(49)
∥
u
ω
(
x
,
t
)
-
u
(
x
,
t
)
∥
L
q
(
0
,
l
;
L
r
(
R
n
)
)
≤
C
ε
ω
+
C
(
M
8
/
(
n
-
4
)
+
N
8
/
(
n
-
4
)
)
×
∥
u
ω
(
x
,
t
)
-
u
(
x
,
t
)
∥
L
γ
(
0
,
l
;
L
ρ
(
R
n
)
)
⟶
0
as
|
ω
|
⟶
∞
.
(
2
)
In the following we discuss the estimate
∥
∇
(
u
ω
-
u
)
∥
L
q
(
0
,
l
;
L
r
(
R
n
)
)
.
By (11) and (12), we have
(50)
∇
(
u
ω
-
u
)
=
i
(
J
1
+
J
2
)
,
where
J
1
=
∫
0
t
θ
(
ω
s
)
W
(
t
-
s
)
∇
(
|
u
ω
|
8
/
(
n
-
4
)
u
ω
-
|
u
|
8
/
(
n
-
4
)
u
)
d
s
and
J
2
=
∫
0
t
[
θ
(
ω
s
)
-
I
(
θ
)
]
W
(
t
-
s
)
∇
(
|
u
|
8
/
(
n
-
4
)
u
)
d
s
.
Using (11)-(12) and Lemma 5, we obtain
(51)
∥
∇
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
≤
C
∥
∇
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
(
R
n
)
+
C
(
∥
J
1
∥
L
q
(
0
,
t
1
;
L
r
(
R
n
)
)
+
∥
J
2
∥
L
q
(
0
,
t
1
;
L
r
(
R
n
)
)
)
≤
C
∥
∇
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
(
R
n
)
+
C
∥
∇
(
|
u
ω
|
8
/
(
n
-
4
)
u
ω
m
m
m
m
l
v
l
-
|
u
|
8
/
(
n
-
4
)
u
)
∥
L
2
n
(
n
+
4
)
/
(
n
2
+
4
n
+
8
)
(
0
,
t
1
;
L
2
n
2
(
n
+
4
)
/
(
n
3
+
8
n
2
+
16
n
-
32
)
(
R
n
)
)
+
C
∥
J
2
∥
L
q
(
0
,
t
1
;
L
r
(
R
n
)
)
.
Noting that
(52)
∥
∇
(
|
u
|
8
/
(
n
-
4
)
u
)
∥
L
γ
′
(
0
,
t
1
;
L
ρ
′
(
R
n
)
)
≤
∥
|
u
|
8
/
(
n
-
4
)
∥
L
n
/
4
(
0
,
t
1
;
L
n
2
/
4
(
n
-
4
)
(
R
n
)
)
∥
∇
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
=
∥
u
∥
L
2
n
/
(
n
-
4
)
(
0
,
t
1
;
L
2
n
2
/
(
n
-
4
)
2
(
R
n
)
)
8
/
(
n
-
4
)
∥
∇
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
≤
C
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
8
/
(
n
-
4
)
∥
∇
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
,
so
∇
(
|
u
|
8
/
(
n
-
4
)
u
)
∈
L
γ
′
(
0
,
t
1
;
L
ρ
′
(
R
n
)
)
.
Using Lemma 5 and Lemma 10, we have
(53)
∥
J
2
∥
L
q
(
0
,
t
1
;
L
r
(
R
n
)
)
∶
=
C
ϵ
ω
⟶
0
as
|
ω
|
⟶
∞
.
Using Hölder inequality, Sobolev embedding, and Lemma 7, we obtain
(54)
∥
∇
(
|
u
ω
|
8
/
(
n
-
4
)
u
ω
m
l
l
l
-
|
u
|
8
/
(
n
-
4
)
u
)
∥
L
2
n
(
n
+
4
)
/
(
n
2
+
4
n
+
8
)
(
0
,
t
1
;
L
2
n
2
(
n
+
4
)
/
(
n
3
+
8
n
2
+
16
n
-
32
)
(
R
n
)
)
≤
C
∥
|
u
ω
|
8
/
(
n
-
4
)
∥
L
n
(
n
+
4
)
/
16
(
0
,
t
1
;
L
n
2
(
n
+
4
)
/
4
(
n
2
+
4
n
-
16
)
(
R
n
)
)
×
∥
∇
(
u
ω
-
u
)
∥
L
2
n
(
n
+
4
)
/
(
n
2
+
4
n
-
24
)
(
0
,
t
1
;
L
2
n
2
(
n
+
4
)
/
(
n
3
-
16
n
+
96
)
(
R
n
)
)
+
C
(
∥
|
u
ω
|
(
12
-
n
)
/
(
n
-
4
)
∥
L
n
(
n
+
4
)
/
2
(
12
-
n
)
(
0
,
t
1
;
L
2
n
(
n
+
4
)
/
(
12
-
n
)
(
n
2
+
4
n
-
16
)
(
R
n
)
)
m
m
+
∥
|
u
|
(
12
-
n
)
/
(
n
-
4
)
∥
L
n
(
n
+
4
)
/
2
(
12
-
n
)
(
0
,
t
1
;
L
2
n
(
n
+
4
)
/
(
12
-
n
)
(
n
2
+
4
n
-
16
)
(
R
n
)
)
)
×
∥
u
ω
-
u
∥
L
n
(
n
+
4
)
/
2
(
n
-
4
)
(
0
,
t
1
;
L
2
n
2
(
n
+
4
)
/
(
n
-
4
)
(
n
2
+
4
n
-
16
)
(
R
n
)
)
×
∥
∇
u
∥
L
2
n
(
n
+
4
)
/
(
n
2
+
4
n
-
24
)
(
0
,
t
1
;
L
2
n
2
(
n
+
4
)
/
(
n
3
-
16
n
+
96
)
(
R
n
)
)
≤
C
∥
u
ω
∥
X
0
(
0
,
t
1
)
8
/
(
n
-
4
)
∥
∇
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
+
C
(
∥
u
ω
∥
X
0
(
0
,
t
1
)
(
12
-
n
)
/
(
n
-
4
)
+
∥
u
∥
X
0
(
0
,
t
1
)
(
12
-
n
)
/
(
n
-
4
)
)
×
∥
u
ω
-
u
∥
X
0
(
0
,
t
1
)
∥
∇
u
∥
S
0
(
0
,
t
1
)
≤
C
∥
u
ω
∥
X
0
(
0
,
t
1
)
8
/
(
n
-
4
)
∥
∇
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
+
C
∥
u
ω
-
u
∥
X
0
(
0
,
t
1
)
8
/
(
n
-
4
)
∥
∇
u
∥
S
0
(
0
,
t
1
)
+
C
∥
u
∥
X
0
(
0
,
t
1
)
(
12
-
n
)
/
(
n
-
4
)
∥
u
ω
-
u
∥
X
0
(
0
,
t
1
)
∥
∇
u
∥
S
0
(
0
,
t
1
)
≤
C
∥
u
ω
-
u
∥
X
0
(
0
,
t
1
)
8
/
(
n
-
4
)
∥
∇
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
+
C
∥
u
∥
X
0
(
0
,
t
1
)
8
/
(
n
-
4
)
∥
∇
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
+
C
∥
u
ω
-
u
∥
X
0
(
0
,
t
1
)
8
/
(
n
-
4
)
∥
∇
u
∥
S
0
(
0
,
t
1
)
+
C
∥
u
∥
X
0
(
0
,
t
1
)
(
12
-
n
)
/
(
n
-
4
)
∥
u
ω
-
u
∥
X
0
(
0
,
t
1
)
∥
∇
u
∥
S
0
(
0
,
t
1
)
≤
C
∥
u
ω
-
u
∥
X
0
(
0
,
t
1
)
8
/
(
n
-
4
)
∥
∇
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
+
C
∥
D
2
u
∥
S
0
(
0
,
t
1
)
8
/
(
n
-
4
)
∥
∇
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
+
C
∥
u
ω
-
u
∥
X
0
(
0
,
t
1
)
8
/
(
n
-
4
)
∥
∇
u
∥
S
0
(
t
0
,
t
1
)
+
C
∥
D
2
u
∥
S
0
(
0
,
t
1
)
(
12
-
n
)
/
(
n
-
4
)
∥
u
ω
-
u
∥
X
0
(
0
,
t
1
)
∥
∇
u
∥
S
0
(
0
,
t
1
)
≤
C
∥
u
ω
-
u
∥
X
(
0
,
t
1
)
8
/
(
n
-
4
)
∥
∇
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
+
C
∥
D
2
u
∥
S
0
(
0
,
t
1
)
8
/
(
n
-
4
)
∥
∇
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
+
C
∥
u
ω
-
u
∥
X
(
0
,
t
1
)
8
/
(
n
-
4
)
∥
∇
u
∥
S
0
(
0
,
t
1
)
+
C
∥
D
2
u
∥
S
0
(
0
,
t
1
)
(
12
-
n
)
/
(
n
-
4
)
∥
u
ω
-
u
∥
X
(
0
,
t
1
)
∥
∇
u
∥
S
0
(
0
,
t
1
)
.
Substituting (53)-(54) into (51), we obtain
(55)
∥
∇
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
≤
C
∥
∇
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
(
R
n
)
+
C
ϵ
ω
+
C
∥
u
ω
-
u
∥
X
(
0
,
t
1
)
8
/
(
n
-
4
)
∥
∇
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
+
C
∥
D
2
u
∥
S
0
(
0
,
t
1
)
8
/
(
n
-
4
)
∥
∇
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
+
C
∥
u
ω
-
u
∥
X
(
0
,
t
1
)
8
/
(
n
-
4
)
∥
∇
u
∥
S
0
(
0
,
t
1
)
+
C
∥
D
2
u
∥
S
0
(
0
,
t
1
)
(
12
-
n
)
/
(
n
-
4
)
∥
u
ω
-
u
∥
X
(
0
,
t
1
)
∥
∇
u
∥
S
0
(
0
,
t
1
)
.
Taking
t
1
such that
C
∥
D
2
u
∥
S
0
(
0
,
t
1
)
8
/
(
n
-
4
)
≤
1
/
2
,
C
∥
∇
u
∥
S
0
(
0
,
t
1
)
≤
1
,
C
∥
D
2
u
∥
S
0
(
0
,
t
1
)
(
12
-
n
)
/
(
n
-
4
)
∥
∇
u
∥
S
0
(
0
,
t
1
)
≤
1
, using Young’s inequality and Lemma 7, we obtain
(56)
∥
∇
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
≤
2
C
∥
∇
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
(
R
n
)
+
2
C
ϵ
ω
+
2
C
∥
u
ω
-
u
∥
X
(
0
,
t
1
)
8
/
(
n
-
4
)
∥
∇
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
+
2
∥
u
ω
-
u
∥
X
(
0
,
t
1
)
8
/
(
n
-
4
)
+
2
∥
u
ω
-
u
∥
X
(
0
,
t
1
)
≤
2
C
∥
∇
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
(
R
n
)
+
2
C
ϵ
ω
+
C
∥
u
ω
-
u
∥
X
(
0
,
t
1
)
(
n
+
4
)
/
(
n
-
4
)
+
C
∥
∇
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
8
/
(
n
-
4
)
+
2
∥
u
ω
-
u
∥
X
(
0
,
t
1
)
8
/
(
n
-
4
)
+
2
∥
u
ω
-
u
∥
X
(
0
,
t
1
)
.
Evidently, in order to get the estimate
∥
∇
(
u
ω
-
u
)
∥
S
0
(
t
0
,
t
1
)
, we have to estimate the norm
∥
u
ω
-
u
∥
X
(
t
0
,
t
1
)
.
By (11)-(12), Lemmas 6 and 9, we obtain
(57)
∥
u
ω
-
u
∥
X
(
0
,
t
1
)
≤
C
∥
W
(
t
)
(
u
ω
(
0
)
-
u
(
0
)
)
∥
X
(
0
,
t
1
)
+
C
∥
|
u
ω
|
8
/
(
n
-
4
)
u
ω
-
|
u
|
8
/
(
n
-
4
)
u
∥
Y
(
0
,
t
1
)
+
∥
∫
0
t
(
θ
(
ω
s
-
I
(
θ
)
)
)
W
(
t
)
(
|
u
|
8
/
(
n
-
4
)
u
)
(
t
)
d
t
∥
X
(
0
,
t
1
)
≤
C
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
+
C
(
∥
u
ω
-
u
∥
X
(
0
,
t
1
)
4
(
n
+
8
)
/
(
n
2
-
16
)
∥
D
2
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
4
n
/
(
n
2
-
16
)
mm
+
∥
u
∥
X
(
0
,
t
1
)
4
(
n
+
8
)
/
(
n
2
-
16
)
∥
D
2
u
∥
S
0
(
0
,
t
1
)
4
n
/
(
n
2
-
16
)
)
∥
u
ω
-
u
∥
X
(
0
,
t
1
)
+
∥
∫
0
t
(
θ
(
ω
s
)
-
I
(
θ
)
)
W
(
t
)
D
2
(
|
u
|
8
/
(
n
-
4
)
u
)
d
t
∥
S
0
(
0
,
t
1
)
.
Let
∥
∫
0
t
(
θ
(
ω
s
-
I
(
θ
)
)
)
W
(
t
)
D
2
(
|
u
|
8
/
(
n
-
4
)
u
)
d
t
∥
S
0
(
0
,
t
1
)
=
C
ξ
ω
. Noting that
(58)
∥
D
2
(
|
u
|
8
/
(
n
-
4
)
u
)
∥
L
γ
′
(
0
,
t
1
;
L
ρ
′
(
R
n
)
)
≤
C
∥
|
u
|
(
12
-
n
)
/
(
n
-
4
)
∥
L
2
n
/
(
12
-
n
)
(
0
,
t
1
;
L
2
n
2
/
(
n
-
4
)
(
12
-
n
)
(
R
n
)
)
m
×
∥
∇
u
∥
L
2
n
/
(
n
-
4
)
(
0
,
t
1
;
L
2
n
2
/
(
n
2
-
6
n
+
16
)
(
R
n
)
)
2
+
C
∥
|
u
|
8
/
(
n
-
4
)
∥
L
γ
/
(
γ
-
2
)
(
0
,
t
1
;
L
ρ
/
(
ρ
-
2
)
(
R
n
)
)
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
≤
C
∥
u
∥
L
2
n
/
(
n
-
4
)
(
0
,
t
1
;
L
2
n
2
/
(
n
-
4
)
2
(
R
n
)
)
(
12
-
n
)
/
(
n
-
4
)
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
2
+
C
∥
|
u
|
8
/
(
n
-
4
)
∥
L
8
γ
/
(
n
-
4
)
(
γ
-
2
)
(
0
,
t
1
;
L
8
ρ
/
(
n
-
4
)
(
ρ
-
2
)
(
R
n
)
)
×
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
≤
C
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
(
n
+
4
)
/
(
n
-
4
)
,
so
D
2
(
|
u
|
8
/
(
n
-
4
)
u
)
∈
L
γ
′
(
0
,
t
1
;
L
ρ
′
(
R
n
)
)
.
Using Lemma 10, we obtain
(59)
C
ξ
ω
⟶
0
as
ω
⟶
∞
.
From (57)–(59), we get
(60)
∥
u
ω
-
u
∥
X
(
0
,
t
1
)
≤
C
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
+
C
(
∥
u
ω
-
u
∥
X
(
0
,
t
1
)
(
n
2
+
4
n
+
16
)
/
(
n
2
-
16
)
∥
D
2
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
4
n
/
(
n
2
-
16
)
mm
l
+
∥
D
2
u
∥
S
0
(
0
,
t
1
)
8
/
(
n
-
4
)
∥
u
ω
-
u
∥
X
(
0
,
t
1
)
)
+
C
ξ
ω
≤
C
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
+
C
(
∥
D
2
u
ω
∥
S
0
(
0
,
t
1
)
4
n
/
(
n
2
-
16
)
+
∥
D
2
u
∥
S
0
(
0
,
t
1
)
4
n
/
(
n
2
-
16
)
)
m
×
∥
u
ω
-
u
∥
X
(
0
,
t
1
)
(
n
2
+
4
n
+
16
)
/
(
n
2
-
16
)
+
C
∥
D
2
u
∥
S
0
(
0
,
t
1
)
8
/
(
n
-
4
)
×
∥
u
ω
-
u
∥
X
(
0
,
t
1
)
+
C
ξ
ω
.
Let
t
1
→
0
such that
∥
D
2
u
∥
S
0
(
0
,
t
1
)
8
/
(
n
-
4
)
≤
1
/
2
, thus we have
(61)
∥
u
ω
-
u
∥
X
(
0
,
t
1
)
≤
2
C
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
+
2
C
(
∥
D
2
u
ω
∥
S
0
(
0
,
t
1
)
4
n
/
(
n
2
-
16
)
+
∥
D
2
u
∥
S
0
(
0
,
t
1
)
4
n
/
(
n
2
-
16
)
)
×
∥
u
ω
-
u
∥
X
(
0
,
t
1
)
(
n
2
+
4
n
+
16
)
/
(
n
2
-
16
)
+
2
C
ξ
ω
≤
2
C
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
+
2
C
(
2
M
1
)
4
n
/
(
n
2
-
16
)
∥
u
ω
-
u
∥
X
(
0
,
t
1
)
(
n
2
+
4
n
+
16
)
/
(
n
2
-
16
)
+
2
C
ξ
ω
,
where
M
1
=
max
(
∥
D
2
u
∥
S
0
(
0
,
l
)
,
∥
D
2
u
ω
∥
S
0
(
0
,
l
)
)
.
If
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
and
ξ
ω
are small such that
(62)
(
4
C
)
(
n
2
+
4
n
+
16
)
/
(
n
2
-
16
)
(
2
M
1
)
4
n
/
(
n
2
-
16
)
×
(
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
+
ξ
ω
)
4
(
n
+
8
)
/
(
n
2
-
16
)
<
1
,
we obtain
(63)
∥
u
ω
-
u
∥
X
(
0
,
t
1
)
≤
4
C
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
+
4
C
ξ
ω
.
Taking (63) into (56), we can get
(64)
∥
∇
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
≤
2
C
∥
∇
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
(
R
n
)
+
2
C
ϵ
ω
+
(
4
C
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
+
4
C
ξ
ω
)
(
n
+
4
)
/
(
n
-
4
)
+
(
4
C
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
+
4
C
ξ
ω
)
8
/
(
n
-
4
)
+
4
C
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
+
4
C
ξ
ω
+
C
∥
∇
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
8
/
(
n
-
4
)
.
If
∥
∇
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
,
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
and
ϵ
ω
,
ξ
ω
are small such that
(65)
C
(
2
C
∥
∇
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
(
R
n
)
+
2
C
ϵ
ω
+
(
4
C
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
+
4
C
ξ
ω
)
(
n
+
4
)
/
(
n
-
4
)
+
(
4
C
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
+
4
C
ξ
ω
)
8
/
(
n
-
4
)
+
4
C
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
+
4
C
ξ
ω
)
(
12
-
n
)
/
(
n
-
4
)
<
1
2
,
then we can obtain
(66)
∥
∇
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
≤
4
C
∥
∇
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
(
R
n
)
+
4
C
ϵ
ω
+
2
(
4
C
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
+
4
C
ξ
ω
)
(
n
+
4
)
/
(
n
-
4
)
+
2
(
4
C
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
+
4
C
ξ
ω
)
8
/
(
n
-
4
)
+
8
C
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
+
8
C
ξ
ω
.
So we have
(67)
∥
∇
(
u
ω
(
x
,
t
)
-
u
(
x
,
t
)
)
∥
L
q
(
0
,
t
1
;
L
r
(
R
n
)
)
⟶
0
as
|
ω
|
⟶
∞
.
Let
T
1
=
sup
{
t
∣
0
<
t
<
l
,
∥
∇
(
u
ω
-
u
)
∥
L
q
(
0
,
t
;
L
r
(
R
n
)
)
→
0
as
|
ω
|
→
∞
}
. By continuous extension method and contradiction method, we can prove that
T
1
=
l
.
(
3
)
At last, we discuss the estimate
∥
D
2
(
u
ω
-
u
)
∥
L
q
(
0
,
l
;
L
r
(
R
n
)
)
.
As in Lemmas 8 and 9, we define
F
(
u
)
=
|
u
|
8
/
(
n
-
4
)
u
.
By (11) and (12), we have
(68)
D
2
(
u
ω
-
u
)
=
i
(
K
1
+
K
2
+
K
3
)
,
where
K
1
=
∫
0
t
θ
(
ω
s
)
W
(
t
-
s
)
A
1
(
u
ω
,
u
)
d
s
,
K
2
=
∫
0
t
θ
(
ω
s
)
W
(
t
-
s
)
A
2
(
u
ω
,
u
)
d
s
,
K
3
=
∫
0
t
[
θ
(
ω
s
)
-
I
(
θ
)
]
W
(
t
-
s
)
A
3
(
u
)
d
s
, and
A
1
(
u
ω
,
u
)
,
A
2
(
u
ω
,
u
)
, and
A
3
(
u
)
are as follows:
(69)
A
1
(
u
ω
,
u
)
=
F
′
(
u
ω
)
D
2
(
u
ω
-
u
)
+
(
D
u
ω
-
D
u
)
⊥
×
F
′′
(
u
ω
)
D
(
u
ω
)
,
A
2
(
u
ω
,
u
)
=
(
D
u
)
⊥
×
[
F
′′
(
u
ω
)
D
u
ω
-
F
′′
(
u
)
D
u
]
+
[
F
′
(
u
ω
)
-
F
′
(
u
)
]
D
2
u
,
A
3
(
u
)
=
(
D
u
)
⊥
×
F
′′
(
u
)
D
u
+
F
′
(
u
)
D
2
u
are all
n
×
n
matrixes.
Using (11)-(12) and Lemma 5, we obtain
(70)
∥
D
2
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
≤
C
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
(
R
n
)
+
C
(
∥
K
1
∥
L
q
(
0
,
t
1
;
L
r
(
R
n
)
)
+
∥
K
2
∥
L
q
(
0
,
t
1
;
L
r
(
R
n
)
)
mm
+
∥
K
3
∥
L
q
(
0
,
t
1
;
L
r
(
R
n
)
)
)
≤
C
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
(
R
n
)
+
C
(
∥
A
1
∥
L
γ
′
(
0
,
t
1
;
L
ρ
′
(
R
n
)
)
+
∥
A
2
∥
L
γ
′
(
0
,
t
1
;
L
ρ
′
(
R
n
)
)
mml
+
∥
A
3
∥
L
γ
′
(
0
,
t
1
;
L
ρ
′
(
R
n
)
)
)
.
For the case
5
≤
n
<
12
, using Hölder inequality and Sobolev embedding, we have
(71)
∥
(
D
u
)
⊥
×
F
′′
(
u
)
D
u
∥
L
γ
′
(
0
,
t
1
;
L
ρ
′
(
R
n
)
)
≤
C
∥
|
u
|
(
12
-
n
)
/
(
n
-
4
)
∥
L
2
n
/
(
12
-
n
)
(
0
,
t
1
;
L
2
n
2
/
(
n
-
4
)
(
12
-
n
)
(
R
n
)
)
×
∥
D
u
∥
L
2
n
/
(
n
-
4
)
(
0
,
t
1
;
L
2
n
2
/
(
n
2
-
6
n
+
16
)
(
R
n
)
)
2
≤
∥
u
∥
L
2
n
/
(
n
-
4
)
(
0
,
t
1
;
L
2
n
2
/
(
n
-
4
)
2
(
R
n
)
)
(
12
-
n
)
/
(
n
-
4
)
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
2
≤
C
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
(
n
+
4
)
/
(
n
-
4
)
,
(72)
∥
F
′
(
u
)
D
2
u
∥
L
γ
′
(
0
,
t
1
;
L
ρ
′
(
R
n
)
)
≤
C
∥
|
u
|
8
/
(
n
-
4
)
∥
L
γ
/
(
γ
-
2
)
(
0
,
t
1
;
L
ρ
/
(
ρ
-
2
)
(
R
n
)
)
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
=
C
∥
u
∥
L
8
γ
/
(
γ
-
2
)
(
n
-
4
)
(
0
,
t
1
;
L
8
ρ
/
(
ρ
-
2
)
(
n
-
4
)
(
R
n
)
)
8
/
(
n
-
4
)
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
≤
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
(
n
+
4
)
/
(
n
-
4
)
.
Since
u
∈
L
γ
(
0
,
l
;
H
2
,
ρ
(
R
n
)
)
, we have
A
3
(
u
)
∈
L
γ
′
(
0
,
t
1
;
L
ρ
′
(
R
n
)
)
; thus by Lemma 10,
(73)
∥
A
3
(
u
)
∥
L
γ
′
(
0
,
t
1
;
L
ρ
′
(
R
n
)
)
∶
=
C
ζ
ω
⟶
0
as
ω
⟶
∞
.
In the following, we analyze the norm
∥
A
1
(
u
ω
,
u
)
∥
L
γ
′
(
0
,
t
1
;
L
ρ
′
(
R
n
)
)
for
5
≤
n
<
12
.
Using Hölder inequality and Sobolev embedding, we obtain
(74)
∥
(
D
u
ω
-
D
u
)
⊥
×
F
′′
(
u
ω
)
D
u
ω
∥
L
γ
′
(
0
,
t
1
;
L
ρ
′
(
R
n
)
)
≤
C
∥
D
u
ω
-
D
u
∥
L
γ
(
0
,
t
1
;
L
ρ
1
(
R
n
)
)
×
∥
|
u
ω
|
(
12
-
n
)
/
(
n
-
4
)
∥
L
a
(
0
,
t
1
;
L
b
(
R
n
)
)
∥
D
u
ω
∥
L
γ
(
0
,
t
1
;
L
ρ
1
(
R
n
)
)
≤
C
∥
D
u
ω
-
D
u
∥
L
γ
(
0
,
t
1
;
L
ρ
1
(
R
n
)
)
∥
u
ω
∥
L
γ
(
0
,
t
1
;
L
(
12
-
n
)
b
/
(
n
-
4
)
(
R
n
)
)
(
12
-
n
)
/
(
n
-
4
)
×
∥
D
u
ω
∥
L
γ
(
0
,
t
1
;
L
ρ
1
(
R
n
)
)
≤
C
∥
D
2
(
u
ω
-
u
)
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
∥
D
2
u
ω
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
(
12
-
n
)
/
(
n
-
4
)
×
∥
D
2
u
ω
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
≤
C
∥
D
2
(
u
ω
-
u
)
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
(
n
+
4
)
/
(
n
-
4
)
+
C
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
8
/
(
n
-
4
)
∥
D
2
(
u
ω
-
u
)
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
,
where
a
=
γ
/
(
γ
-
3
)
,
b
=
n
(
n
-
4
)
ρ
/
(
n
-
2
ρ
)
(
12
-
n
)
,
ρ
1
=
n
ρ
/
(
n
-
ρ
)
.
Similarly, we can get
(75)
∥
F
′
(
u
ω
)
D
2
(
u
ω
-
u
)
∥
L
γ
′
(
0
,
t
1
;
L
ρ
′
(
R
n
)
)
≤
C
∥
D
2
(
u
ω
-
u
)
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
(
n
+
4
)
/
(
n
-
4
)
+
C
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
8
/
(
n
-
4
)
∥
D
2
(
u
ω
-
u
)
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
.
Combing (74) and (75), we have
(76)
∥
A
1
(
u
ω
,
u
)
∥
L
γ
′
(
0
,
t
1
;
L
ρ
′
(
R
n
)
)
≤
C
∥
D
2
(
u
ω
-
u
)
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
(
n
+
4
)
/
(
n
-
4
)
+
C
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
8
/
(
n
-
4
)
∥
D
2
(
u
ω
-
u
)
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
.
At last, we analyze the norm
∥
A
2
(
u
ω
,
u
)
∥
L
γ
′
(
0
,
t
1
;
L
ρ
′
(
R
n
)
)
. We divide it into two cases.
Case I (
5
≤
n
≤
8
). Using Hölder inequality, Sobolev embedding, and (73), we can get
(77)
∥
(
D
u
)
⊥
×
[
F
′′
(
u
ω
)
D
u
ω
-
F
′′
(
u
)
D
u
]
∥
L
γ
′
(
0
,
t
1
;
L
ρ
′
(
R
n
)
)
≤
∥
(
D
u
)
⊥
×
F
′′
(
u
ω
)
D
(
u
ω
-
u
)
∥
L
γ
′
(
0
,
t
1
;
L
ρ
′
(
R
n
)
)
+
∥
(
D
u
)
⊥
×
[
F
′′
(
u
ω
)
-
F
′′
(
u
)
]
D
u
∥
L
γ
′
(
0
,
t
1
;
L
ρ
′
(
R
n
)
)
≤
C
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
∥
D
2
u
ω
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
(
12
-
n
)
/
(
n
-
4
)
×
∥
D
2
(
u
ω
-
u
)
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
+
C
∥
(
D
u
)
⊥
×
(
|
u
ω
|
2
(
8
-
n
)
/
(
n
-
4
)
+
|
u
|
2
(
8
-
n
)
/
(
n
-
4
)
)
mm
l
×
|
u
ω
-
u
|
D
u
(
D
u
)
⊥
×
(
|
u
ω
|
2
(
8
-
n
)
/
(
n
-
4
)
+
|
u
|
2
(
8
-
n
)
/
(
n
-
4
)
)
∥
L
γ
′
(
0
,
t
1
;
L
ρ
′
(
R
n
)
)
≤
C
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
∥
D
2
u
ω
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
(
12
-
n
)
/
(
n
-
4
)
×
∥
D
2
(
u
ω
-
u
)
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
+
C
(
∥
|
u
ω
|
2
(
8
-
n
)
/
(
n
-
4
)
∥
L
8
/
(
8
-
n
)
(
0
,
t
1
;
L
n
2
/
(
n
-
4
)
(
8
-
n
)
(
R
n
)
)
+
∥
|
u
|
2
(
8
-
n
)
/
(
n
-
4
)
∥
L
8
/
(
8
-
n
)
(
0
,
t
1
;
L
n
2
/
(
n
-
4
)
(
8
-
n
)
(
R
n
)
)
∥
|
u
ω
|
2
(
8
-
n
)
/
(
n
-
4
)
∥
L
8
/
(
8
-
n
)
(
0
,
t
1
;
L
n
2
/
(
n
-
4
)
(
8
-
n
)
(
R
n
)
)
)
×
∥
u
ω
-
u
∥
L
2
n
/
(
n
-
4
)
(
0
,
t
1
;
L
2
n
2
/
(
n
2
-
8
n
+
16
)
(
R
n
)
)
×
∥
D
u
∥
L
2
n
/
(
n
-
4
)
(
0
,
t
1
;
L
2
n
2
/
(
n
2
-
6
n
+
16
)
(
R
n
)
)
2
≤
C
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
∥
D
2
u
ω
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
(
12
-
n
)
/
(
n
-
4
)
×
∥
D
2
(
u
ω
-
u
)
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
+
C
(
∥
D
2
u
ω
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
2
(
8
-
n
)
/
(
n
-
4
)
+
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
2
(
8
-
n
)
/
(
n
-
4
)
)
×
∥
D
2
(
u
ω
-
u
)
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
2
≤
C
∥
D
2
(
u
ω
-
u
)
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
8
/
(
n
-
4
)
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
+
C
∥
D
2
(
u
ω
-
u
)
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
8
/
(
n
-
4
)
+
C
∥
D
2
(
u
ω
-
u
)
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
(
12
-
n
)
/
(
n
-
4
)
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
2
.
Similarly, we can get
(78)
∥
[
F
′
(
u
ω
)
-
F
′
(
u
)
]
D
2
u
∥
L
γ
′
(
0
,
t
1
;
L
ρ
′
(
R
n
)
)
≤
C
∥
|
u
ω
-
u
|
(
|
u
ω
|
(
12
-
n
)
/
(
n
-
4
)
+
|
u
|
(
12
-
n
)
/
(
n
-
4
)
)
m
l
×
D
2
u
∥
L
γ
′
(
0
,
t
1
;
L
ρ
′
(
R
n
)
)
≤
C
(
∥
|
u
ω
|
(
12
-
n
)
/
(
n
-
4
)
∥
L
12
n
/
(
12
-
n
)
(
0
,
t
1
;
L
2
n
2
/
(
n
2
-
8
n
+
16
)
(
R
n
)
)
m
m
l
l
+
∥
|
u
|
(
12
-
n
)
/
(
n
-
4
)
∥
L
12
n
/
(
12
-
n
)
(
0
,
t
1
;
L
2
n
2
/
(
n
2
-
8
n
+
16
)
(
R
n
)
)
)
×
∥
u
ω
-
u
∥
L
2
n
/
(
n
-
4
)
(
0
,
t
1
;
L
2
n
2
/
(
n
2
-
8
n
+
16
)
(
R
n
)
)
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
≤
C
(
∥
D
2
u
ω
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
(
12
-
n
)
/
(
n
-
4
)
+
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
(
12
-
n
)
/
(
n
-
4
)
)
×
∥
D
2
(
u
ω
-
u
)
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
≤
C
∥
D
2
(
u
ω
-
u
)
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
8
/
(
n
-
4
)
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
+
C
∥
D
2
(
u
ω
-
u
)
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
8
/
(
n
-
4
)
.
Combing (77) and (78), we can obtain
(79)
∥
A
2
(
u
ω
,
u
)
∥
L
γ
′
(
0
,
t
1
;
L
ρ
′
(
R
n
)
)
≤
C
∥
D
2
(
u
ω
-
u
)
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
8
/
(
n
-
4
)
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
+
C
∥
D
2
(
u
ω
-
u
)
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
8
/
(
n
-
4
)
+
C
∥
D
2
(
u
ω
-
u
)
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
(
12
-
n
)
/
(
n
-
4
)
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
2
.
From (70), (73), (76), and (79), we have
(80)
∥
D
2
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
≤
C
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
(
R
n
)
+
C
ζ
ω
+
C
∥
D
2
(
u
ω
-
u
)
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
8
/
(
n
-
4
)
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
+
C
∥
D
2
(
u
ω
-
u
)
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
8
/
(
n
-
4
)
+
C
∥
D
2
(
u
ω
-
u
)
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
(
n
+
4
)
/
(
n
-
4
)
+
C
∥
D
2
(
u
ω
-
u
)
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
(
12
-
n
)
/
(
n
-
4
)
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
2
.
Taking
C
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
8
/
(
n
-
4
)
<
1
/
2
, we can get
(81)
∥
D
2
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
≤
2
C
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
(
R
n
)
+
2
C
ζ
ω
+
2
C
∥
D
2
(
u
ω
-
u
)
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
8
/
(
n
-
4
)
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
+
2
C
∥
D
2
(
u
ω
-
u
)
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
(
n
+
4
)
/
(
n
-
4
)
+
2
C
∥
D
2
(
u
ω
-
u
)
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
(
12
-
n
)
/
(
n
-
4
)
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
2
.
Furthermore, if
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
(
R
n
)
and
ζ
ω
are small enough such that
(82)
(
2
C
)
(
12
-
n
)
/
8
(
4
C
)
(
12
-
n
)
/
(
n
-
4
)
×
(
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
(
R
n
)
+
ζ
ω
)
(
12
-
n
)
/
(
n
-
4
)
+
(
2
C
)
(
8
-
n
)
/
4
(
4
C
)
2
(
8
-
n
)
/
(
n
-
4
)
×
(
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
(
R
n
)
+
ζ
ω
)
2
(
8
-
n
)
/
(
n
-
4
)
+
2
C
(
4
C
)
8
/
(
n
-
4
)
(
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
(
R
n
)
+
ζ
ω
)
8
/
(
n
-
4
)
<
1
2
,
then we have
(83)
∥
D
2
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
≤
4
C
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
(
R
n
)
+
4
C
ζ
ω
.
Case II (
8
<
n
<
12
). Noting that
(84)
∥
(
D
u
)
⊥
×
[
F
′′
(
u
ω
)
-
F
′′
(
u
)
]
D
u
∥
L
γ
1
′
(
0
,
t
1
;
L
ρ
1
′
(
R
n
)
)
≤
C
∥
|
u
ω
-
u
|
(
12
-
n
)
/
(
n
-
4
)
∥
L
n
(
n
+
4
)
/
2
(
12
-
n
)
(
0
,
t
1
;
L
2
n
2
(
n
+
4
)
/
(
12
-
n
)
(
n
2
+
4
n
-
16
)
(
R
n
)
)
×
∥
D
u
∥
L
q
(
0
,
t
1
;
L
p
(
R
n
)
)
2
≤
C
∥
u
ω
-
u
∥
X
0
(
0
,
t
1
)
(
12
-
n
)
/
(
n
-
4
)
∥
D
2
u
∥
L
q
(
0
,
t
1
;
L
p
1
(
R
n
)
)
2
≤
C
∥
u
ω
-
u
∥
X
(
0
,
t
1
)
(
12
-
n
)
/
(
n
-
4
)
∥
D
2
u
∥
S
0
(
0
,
t
1
)
2
,
where
γ
1
′
=
2
n
(
n
+
4
)
/
(
n
2
+
4
n
+
8
)
,
ρ
1
′
=
2
n
2
(
n
+
4
)
/
(
n
3
+
8
n
2
+
16
n
-
32
)
,
q
=
4
n
(
n
+
4
)
/
(
n
2
+
8
n
-
40
)
,
p
=
2
n
2
(
n
+
4
)
/
(
n
3
-
24
n
+
80
)
, and
p
1
=
2
n
2
(
n
+
4
)
/
(
n
3
+
2
n
2
-
16
n
+
80
)
;
thus we have
(85)
∥
(
D
u
)
⊥
×
[
F
′′
(
u
ω
)
D
u
ω
-
F
′′
(
u
)
D
u
]
∥
L
γ
1
′
(
0
,
t
1
;
L
ρ
1
′
(
R
n
)
)
≤
∥
(
D
u
)
⊥
×
F
′′
(
u
ω
)
D
(
u
ω
-
u
)
∥
L
γ
1
′
(
0
,
t
1
;
L
ρ
1
′
(
R
n
)
)
+
∥
(
D
u
)
⊥
×
[
F
′′
(
u
ω
)
-
F
′′
(
u
)
]
D
u
∥
L
γ
1
′
(
0
,
t
1
;
L
ρ
1
′
(
R
n
)
)
≤
C
∥
D
2
u
∥
S
0
(
0
,
t
1
)
∥
u
ω
∥
X
(
0
,
t
1
)
(
12
-
n
)
/
(
n
-
4
)
∥
D
2
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
+
C
∥
D
2
u
∥
S
0
(
0
,
t
1
)
2
∥
u
ω
-
u
∥
X
(
0
,
t
1
)
(
12
-
n
)
/
(
n
-
4
)
≤
C
∥
D
2
u
∥
S
0
(
0
,
t
1
)
∥
D
2
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
8
/
(
n
-
4
)
+
C
∥
D
2
u
∥
S
0
(
0
,
t
1
)
8
/
(
n
-
4
)
∥
D
2
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
+
C
∥
D
2
u
∥
S
0
(
0
,
t
1
)
2
∥
u
ω
-
u
∥
X
(
0
,
t
1
)
(
12
-
n
)
/
(
n
-
4
)
.
Combing (78) and (85), we can get
(86)
∥
A
2
(
u
ω
,
u
)
∥
L
γ
1
′
(
0
,
t
1
;
L
ρ
1
′
(
R
n
)
)
≤
C
∥
D
2
u
∥
S
0
(
0
,
t
1
)
∥
D
2
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
8
/
(
n
-
4
)
+
C
∥
D
2
u
∥
S
0
(
0
,
t
1
)
8
/
(
n
-
4
)
∥
D
2
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
+
C
∥
D
2
u
∥
S
0
(
0
,
t
1
)
2
∥
u
ω
-
u
∥
X
(
0
,
t
1
)
(
12
-
n
)
/
(
n
-
4
)
.
From (70), (73), (76), and (86), we have
(87)
∥
D
2
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
≤
C
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
(
R
n
)
+
C
∥
D
2
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
(
n
+
4
)
/
(
n
-
4
)
+
C
∥
D
2
u
∥
S
0
(
0
,
t
1
)
∥
D
2
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
8
/
(
n
-
4
)
+
C
∥
D
2
u
∥
S
0
(
0
,
t
1
)
8
/
(
n
-
4
)
∥
D
2
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
+
C
∥
D
2
u
∥
S
0
(
0
,
t
1
)
2
∥
u
ω
-
u
∥
X
(
0
,
t
1
)
(
12
-
n
)
/
(
n
-
4
)
+
C
ζ
ω
.
Taking
C
∥
D
2
u
∥
L
γ
(
0
,
t
1
;
L
ρ
(
R
n
)
)
8
/
(
n
-
4
)
<
1
/
2
,
C
∥
D
2
u
∥
S
0
(
0
,
t
1
)
≤
1
, and
C
∥
D
2
u
∥
S
0
(
0
,
t
1
)
2
≤
1
, we can get
(88)
∥
D
2
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
≤
2
C
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
(
R
n
)
+
2
C
∥
D
2
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
(
n
+
4
)
/
(
n
-
4
)
+
2
∥
D
2
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
8
/
(
n
-
4
)
+
2
∥
u
ω
-
u
∥
X
(
0
,
t
1
)
(
12
-
n
)
/
(
n
-
4
)
+
2
C
ζ
ω
.
Combing (63) and (88), we can obtain
(89)
∥
D
2
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
≤
C
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
(
R
n
)
+
C
∥
D
2
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
(
n
+
4
)
/
(
n
-
4
)
+
C
∥
D
2
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
8
/
(
n
-
4
)
+
C
ζ
ω
+
C
ξ
ω
.
Taking
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
(
R
n
)
,
ζ
ω
,
ξ
ω
small such that
2
C
[
2
C
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
(
R
n
)
+
2
C
ζ
ω
+
2
C
ξ
ω
]
8
/
(
n
-
4
)
+
2
C
[
2
C
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
(
R
n
)
+
2
C
ζ
ω
+
2
C
ξ
ω
]
(
12
-
n
)
/
(
n
-
4
)
<
1
, then we can get
(90)
∥
D
2
(
u
ω
-
u
)
∥
S
0
(
0
,
t
1
)
≤
2
C
∥
D
2
(
u
ω
(
0
)
-
u
(
0
)
)
∥
L
2
(
R
n
)
+
2
C
ζ
ω
+
2
C
ξ
ω
.
From (83) and (90), we have
(91)
∥
D
2
(
u
ω
(
x
,
t
)
-
u
(
x
,
t
)
)
∥
L
q
(
0
,
t
1
;
L
r
(
R
n
)
)
⟶
0
as
|
ω
|
⟶
∞
.
Similarly, let
T
2
=
sup
{
t
∣
0
<
t
<
l
,
∥
D
2
(
u
ω
-
u
)
∥
L
q
(
0
,
t
;
L
r
(
R
n
)
)
→
0
as
|
ω
|
→
∞
}
. By continuous extension method and contradiction method, we can prove that
T
2
=
l
.
From (53), (67) and (91), the desired result holds.