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AdvancesinAppliedMathematics
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,2023,12(5),2340-2363
PublishedOnlineMay2023inHans.https://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2023.125238
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TheTime-DependentGlobal
AttractorsforanExtensible
BeamEquationwithStructural
Damping
RuiGuo,XuanWang
∗
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Received:Apr.28
th
,2023;accepted:May21
st
,2023;published:May29
th
,2023
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,2023,12(5):2340-2363.
DOI:10.12677/aam.2023.125238
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Abstract
Thispaperstudieslongtimebehaviorofsolutionsforanextensiblebeamequation
withstructuraldampinginthe
H
t
space.Inthefirstinstance,we showthattheglobal
well-posednessofsolutionsfortheequationwithnonlinearity
f
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undersubcritical
condition.Moreover,theasymptoticcompactnessofthesolutionprocess
{
U
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t,τ
)
}
isprovedbyusingthemethodofcontractionfunction.Finally,theexistenceofthe
time-dependentglobalattractorisgained.
Keywords
StructuralDamping,ContractionFunction,TheTime-DependentAttractors,
ExtensibleBeamEquation
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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´
k
.
,
K
B
3
f
ÿ
À
˜
m
X
¥
ƒ
é
;
.
½
Â
2.2
[12]
{H
t
}
t
∈
R
´
˜
x
D
‰
˜
m
,
V
ë
ê
Ž
fx
{
U
(
t,τ
):
H
τ
→H
t
,t
>
τ,τ
∈
R
}
÷
v
±
e
5
Ÿ
:
1)
é
?
¿
τ
∈
R
,U
(
τ,τ
) = Id
´
H
τ
þ
ð
Ž
f
;
2)
é
z
‡
σ
∈
R
Ú
?
¿
t
>
τ
>
σ,U
(
t,τ
)
U
(
τ,σ
) =
U
(
t,σ
)
.
K
¡
U
(
t,τ
)
´
˜
‡
L
§
.
½
Â
2.3
[12]
X
J
é
?
¿
t
∈
R
,
•
3
˜
‡
~
ê
R>
0,
¦
C
t
⊂{
z
∈H
t
:
k
z
k
H
t
6
R
}
=
DOI:10.12677/aam.2023.1252382344
A^
ê
Æ
?
Ð
H
a
§
à
B
t
(
R
),
K
¡
k
.
8
C
t
⊂H
t
8
Ü
x
C
=
{
C
t
}
t
∈
R
´
˜
—
k
.
.
½
Â
2.4
[12]
X
J
é
?
¿
R>
0,
•
3
~
ê
t
0
(
t,R
)
6
t
,
¦
τ
6
t
−
t
0
⇒
U
(
t,τ
)
B
τ
(
R
)
⊂
B
t
,
K
¡
˜
—
k
.
8
x
B
=
{
B
t
}
t
∈
R
´
L
§
U
(
t,τ
)
ž
m
•
6
á
Â
8
.
½
Â
2.5
[12]
X
J
B
=
{
B
t
}
t
∈
R
˜
—
k
.
,
…
é
?
¿
R>
0,
•
3
~
ê
t
0
=
t
0
(
t,R
)
6
t
,
¦
τ
6
t
0
⇒
U
(
t,τ
)
B
τ
(
R
)
⊂
B
t
,
K
¡
B
´
.
£
á
Â
.
½
Â
2.6
[12]
L
§
U
(
t,τ
)
ž
m
•
6
á
Ú
f
´
÷
v
X
e
5
Ÿ
•
x
A
=
{
A
t
}
t
∈
R
:
1)
z
‡
A
t
3
H
t
¥
´
;
;
2)
A
´
.
£
á
Ú
,
=
é
z
‡
˜
—
k
.
x
C
=
{
C
t
}
t
∈
R
,
4
•
lim
τ
→−∞
δ
t
(
U
(
t,τ
)
C
τ
,A
t
) = 0
¤
á
,
Ù
¥
δ
t
(
B,C
)=sup
x
∈
B
inf
y
∈
C
k
x
−
y
k
H
t
L
«
8
Ü
B
Ú
C
Hausdorff
Œ
å
l
.
½
n
2.3
[12]
X
JL
§
U
(
t,τ
)
ì
C;
,
=8
Ü
K
=
{K
=
{
K
t
}
t
∈
R
:
K
t
⊂H
t
•
;
8
,
K
•
.
£
á
Ú
}
´
š
˜
;
,
K
ž
m
•
6
á
Ú
f
A
•
3
…
•
˜
.
½
Â
2.7
[12]
X
J
∀
t
>
τ,U
(
t,τ
)
A
τ
=
A
t
,
K
¡
ž
m
•
6
á
Ú
f
A
=
{
A
t
}
t
∈
R
´
ØC
.
½
n
2.4
[12]
U
(
·
,
·
)
•
Š
^u
Banach
˜
m
x
{H
t
}
t
∈
R
L
§
,
K
U
(
·
,
·
)
k
ž
m
•
6
Û
á
Ú
f
U
∗
=
{
A
∗
t
}
t
∈
R
÷
v
A
∗
t
=
\
s
6
t
[
τ
6
s
U
(
t,τ
)
B
τ
,
…
=
(i)
U
(
·
,
·
)
•
3
ž
m
•
6
á
Â
8
x
B
=
{
B
t
}
t
∈
R
;
(ii)
U
(
·
,
·
)
´
ì
C;
.
3.
·
½
5
½
n
3.1
e
^
‡
(C
1
)-(C
3
)
¤
á
,
K
¯
K
(2
.
1)
k
˜
‡
f
)
u
,
…
(
u,∂
t
u
)
∈
L
∞
([
τ,T
];
H
t
)
,∂
t
u
∈
L
2
([
τ,T
];
V
θ
),
…
k
e
ª
¤
á
k
(
u,∂
t
u
)
k
2
H
t
+
Z
t
τ
k
∂
t
u
(
s
)
k
2
V
θ
d
s
6
C
0
,
∀
t
>
τ,
(3
.
1)
Ù
¥
C
0
=
C
(
R,δ,
k
g
k
)
.
d
,
1
6
p<p
θ
=
N
+2
θ
N
−
4
ž
)
„
ä
k
±
e
5
Ÿ
:
(i)
é
?
¿
t
>
s
>
τ
Ú
(
u,∂
t
u
)
∈H
t
k
e
U
þ
ð
ª
¤
á
DOI:10.12677/aam.2023.1252382345
A^
ê
Æ
?
Ð
H
a
§
à
E
(
u
(
s
)
,∂
t
u
(
s
))+
1
2
Z
t
s
ε
0
(
r
)
k
∂
t
u
(
r
)
k
2
d
r
=
E
(
u
(
t
)
,∂
t
u
(
t
))+
γ
Z
t
s
k
∂
t
u
(
r
)
k
2
V
θ
d
r,
(3
.
2)
Ù
¥
E
(
u
(
t
)
,∂
t
u
(
t
)) =
1
2
(
k
u
k
2
V
2
+
ε
(
t
)
k
∂
t
u
k
2
)+
Z
Ω
F
(
u
)d
x
−h
g,u
i
.
(ii)
z
=
u
−
v
•¯
K
(2
.
1)
é
Au
Ð
Š
(
u
0
,u
1
)
,
(
v
0
,v
1
)
)
,
K
T
)
3
˜
m
V
2
−
θ
×
V
−
θ
¥
Lipschitz
ë
Y
,
=
é
τ
6
t
6
T
,
k
k
(
z,∂
t
z
)(
t
))
k
2
V
2
−
θ
×
V
−
θ
6
C
3
k
(
z,∂
t
z
)(
τ
)
k
2
V
2
−
θ
×
V
−
θ
,
(3
.
3)
Ù
¥
C
3
= max
{
C
1
,C
2
}
= max
{{
2
γ
+
bδL
+
δ,L
+
bL
}
,
{
17
δ
8
+
δL
2
+
δbL,bδγ,b
}}
,
ù
p
δ,b
¿
©
,
¿
…
z
„
÷
v
e
ª
k
z
(
t
)
k
2
V
2
−
θ
+
ε
(
t
)
k
∂
t
z
(
t
)
k
2
V
−
θ
6
e
−
bt
(
k
z
(
τ
)
k
2
V
2
−
θ
+
ε
(
τ
)
k
∂
t
z
(
τ
)
k
2
V
−
θ
)
+
C
3
Z
t
τ
e
−
b
(
t
−
s
)
(
k
∂
t
z
(
s
)
k
2
+
k
z
(
s
)
k
2
V
2
)d
s.
(3.4)
(iii)
é
?
¿
τ<ka<a
6
t
6
T,
(
u,∂
t
u,∂
2
t
u
)
∈
L
∞
([
a,T
];
V
2+
θ
×
V
2
−
θ
×
V
−
θ
)
∩
L
2
([
a,T
];
V
4
×
V
2
×
L
2
(Ω)),
k
e
ª
¤
á
k
u
k
2
V
2+
θ
+
k
∂
t
u
k
2
V
2
−
θ
+
ε
(
t
)
k
∂
2
t
u
k
2
V
−
θ
+
Z
t
+1
t
(
k
u
(
s
)
k
2
V
4
+
k
∂
t
u
(
s
)
k
2
V
2
+
k
∂
2
t
u
(
s
)
k
2
)d
s
6
(1+
30
7
ν
)
e
C
0
(
t
−
τ
)
t
2
θ
+(
2
γ
+
16
7
)
C
0
(
t
−
ka
)
1
−
θ
2
−
θ
+(
2
νγ
+
16
7
ν
)
h
(
t
)+
8
7
C
4
,
(3.5)
Ù
¥
k>
0
,ν
= min
{
9
8
δ,γ
−
δL
+
δ
}
,
h
(
t
) =
e
C
0
(
ka
−
τ
)
e
C
0
(
t
−
ka
)
(
ka
)
2
θ
(
t
−
ka
)
1
2
−
θ
,C
4
=
C
(
R,δ,
k
g
k
,L
) .
y
²
é
•
§
(2
.
1)
¦
±
∂
t
u
k
d
d
t
E
(
u,∂
t
u
)+
γ
k
∂
t
u
k
2
V
θ
−
1
2
ε
0
(
t
)
k
∂
t
u
k
2
= 0
,
(3
.
6)
Ù
¥
E
(
u,∂
t
u
) =
1
2
(
k
u
k
2
V
2
+
ε
(
t
)
k
∂
t
u
k
2
)+
Z
Ω
F
(
u
)d
x
−h
g,u
i
.
ò
(3.6)
3
[
s,t
]
þ
È
©
Œ
(3.2).
d
5
1, H¨older
Ø
ª
,Young
Ø
ª
±
9
Poincar´e
Ø
ª
Œ
Z
Ω
F
(
u
)d
x
>
−
η
2
k
u
k
2
−
C
>
−
η
2
λ
1
k
u
k
2
V
2
−
C
(
η
)
.
(3
.
7)
h
g,u
i
6
1
4
δ
2
k
g
k
2
+
δ
2
λ
1
k
u
k
2
V
2
.
(3
.
8)
DOI:10.12677/aam.2023.1252382346
A^
ê
Æ
?
Ð
H
a
§
à
¤
±
k
E
(
u,∂
t
u
) =
1
2
(
ε
(
t
)
k
∂
t
u
k
2
+
k
u
k
2
V
2
)+
Z
Ω
F
(
u
)d
x
−h
g,u
i
>
1
2
(
ε
(
t
)
k
∂
t
u
k
2
+(1
−
η
λ
1
)
k
u
k
2
V
2
)
−h
g,u
i−
C
(
η
)
>
d
k
(
u,∂
t
u
)
k
2
H
t
−
C
(
δ,
k
g
k
)
,
(3.9)
Ù
¥
d
= min
{
1
2
,
1
2
−
η
+2
δ
2
2
λ
1
}
.
é
(3.6)
l
τ
t
È
©
Œ
E
(
u
(
t
)
,∂
t
u
(
t
))+
γ
Z
t
τ
k
∂
t
u
(
s
)
k
2
V
θ
d
s
−
1
2
Z
t
τ
ε
0
(
s
)
k
∂
t
u
(
s
)
k
2
d
s
=
E
(
u
(
τ
)
,∂
t
u
(
τ
))
.
(3
.
10)
d
ε
(
t
)
4
~
5
Œ
•
E
(
u
(
t
)
,∂
t
u
(
t
))
6
E
(
u
(
τ
)
,∂
t
u
(
τ
))
.
K
k
k
(
u,∂
t
u
)
k
2
X
t
6
C
(
R,δ,
k
g
k
)
.
(3
.
11)
Ï
•
γ>
0
,
2
(
Ü
(3.11),
k
Z
t
τ
k
∂
t
u
(
s
)
k
2
V
θ
d
s
6
C
(
R,δ,
k
g
k
)
.
(3
.
12)
P
C
0
=
C
(
R,δ,
k
g
k
)
,
2
(
Ü
(3.11)
Ú
(3.12)
Œ
(3.1).
(ii)
u,v
•
•
§
(2.1)
é
Au
Ð
Š
(
u
0
,u
1
)
,
(
v
0
,v
1
)
)
,
K
z
=
u
−
v
÷
v
e
ª
ε
(
t
)
∂
2
t
z
+
Az
+
γA
θ
2
∂
t
z
+
f
(
u
)
−
f
(
v
) = 0
.
(3
.
13)
ò
(3.13)
†
2
A
−
θ
2
∂
t
z
+2
δz
‰
S
È
k
d
d
t
H
1
(
z,∂
t
z
)+2
δ
k
z
k
2
V
2
+[2
γ
−
2
δε
(
t
)]
k
∂
t
z
k
2
−
ε
0
(
t
)
k
∂
t
z
k
2
V
−
θ
−
2
δε
0
(
t
)
h
∂
t
z,z
i
=
−h
f
(
u
)
−
f
(
v
)
,
2
A
−
θ
2
∂
t
z
+2
δz
i
,
(3.14)
Ù
¥
H
1
(
z,∂
t
z
) =
ε
(
t
)
k
∂
t
z
k
2
V
−
θ
+2
δε
(
t
)
h
∂
t
z,z
i
+
k
z
k
2
V
2
−
θ
+
γδ
k
z
k
2
V
θ
.
Ï
•
V
2
−
θ
→
V
θ
,
¤
±
|
ε
(
t
)
h
∂
t
z,z
i|
6
ε
(
t
)
k
A
−
θ
4
∂
t
z
k·k
A
θ
4
z
k
6
ε
(
t
)
2
k
∂
t
z
k
2
V
−
θ
+
L
2
k
z
k
2
V
2
−
θ
,
δ
¿
©
ž
,
H
1
(
z,∂
t
z
)
∼
ε
(
t
)
k
∂
t
z
k
2
V
−
θ
+
k
z
k
2
V
2
−
θ
.
(3
.
15)
DOI:10.12677/aam.2023.1252382347
A^
ê
Æ
?
Ð
H
a
§
à
N
=4
ž
,
é
1
<r<
∞
,
k
V
2
→
L
r
;
N
>
5
ž
,
é
0
<δ
1,
k
2
N
N
−
2(2
−
δ
)
<
2
N
N
−
4
,
,
k
V
2
→
L
2
N
N
−
4
;
Ï
•
1
6
p<p
θ
=
N
+2
θ
N
−
4
,
¤
±
k
N
(
p
−
1)
θ
+2
−
δ
<
2
N
N
−
4
,
?
k
V
2
→
L
N
(
p
−
1)
θ
+2
−
δ
.
d
H¨older
Ø
ª
,Young
Ø
ª
±
9
¥Š
½
n
Œ
|h
f
(
u
)
−
f
(
v
)
,
2
A
−
θ
2
∂
t
z
+2
δz
i|
6
C
Z
Ω
(1+
|
u
|
p
−
1
+
|
v
|
p
−
1
)
|
z
|
(
|
A
−
θ
2
∂
t
z
|
+
δ
|
z
|
)d
x
6
C
(1+
k
u
k
p
−
1
N
(
p
−
1)
θ
+2
−
δ
+
k
v
k
p
−
1
N
(
p
−
1)
θ
+2
−
δ
)
k
z
k
2
N
N
−
2(2
−
δ
)
(
k
A
−
θ
2
∂
t
z
k
2
N
N
−
2
θ
+
δ
k
z
k
2
N
N
−
2
θ
)
6
C
(1+
k
u
k
p
−
1
V
2
+
k
v
k
p
−
1
V
2
)
k
z
k
V
2
−
δ
(
k
A
−
θ
2
∂
t
z
k
V
θ
+
δ
k
z
k
V
θ
)
6
C
0
k
z
k
V
2
(
k
∂
t
z
k
+
δ
k
z
k
V
2
)
6
δ
k
∂
t
z
k
2
+
δ
8
k
z
k
2
V
2
+
C
0
,
(3.16)
Ù
¥
^
θ
+2
−
δ
N
+
N
−
2(2
−
δ
)
2
N
+
N
−
2
θ
2
N
= 1
,
2
−
δ<
2
,θ<
2
.
¤
±
é
¿
©
b>
0,
ò
(3.16)
“
\
(3.14)
Œ
d
d
t
H
1
(
z,∂
t
z
)+
bH
1
(
z,∂
t
z
)
6
[2
−
2
δε
(
t
)]
k
∂
t
z
k
2
+[
ε
0
(
t
)+
bε
(
t
)]
k
∂
t
z
k
2
V
−
θ
+
δ
k
∂
t
z
k
2
+
b
k
z
k
2
V
2
−
θ
+2
δ
k
z
k
2
V
2
+
bδ
k
z
k
2
V
θ
+2
δε
0
(
t
)
h
∂
t
z,z
i
+2
δbε
(
t
)
h
∂
t
z,z
i
+
δ
8
k
z
k
2
V
2
+
C
0
.
(3.17)
|
2
δε
0
(
t
)
h
∂
t
z,z
i|
6
2
δL
k
∂
t
z
kk
z
k
6
2
δL
k
∂
t
z
k
2
+
δL
2
k
z
k
2
V
2
.
(3.18)
|
2
δbε
(
t
)
h
∂
t
z,z
i|
6
δbL
(
k
∂
t
z
k
2
+
k
z
k
2
V
2
)
.
(3
.
19)
(
Ü
(3.17)-(3.19),
C
1
= max
{
2
γ
+
bδL
+
δ,L
+
bL
}
,C
2
= max
{
17
δ
8
+
δL
2
+
δbL,bδγ,b
}
,
K
k
|
(2
γ
−
2
δε
(
t
))
|k
∂
t
z
k
2
+
|
(
ε
0
(
t
)+
bε
(
t
))
|k
∂
t
z
k
2
V
−
θ
+(2
δL
+
δbL
)
k
∂
t
z
k
2
+
δ
k
∂
t
z
k
2
6
(2
γ
+
bδL
+
δ
)
k
∂
t
z
k
2
+(
L
+
bL
)
k
∂
t
z
k
2
V
−
θ
6
C
1
k
∂
t
z
k
2
.
(3.20)
2
δ
k
z
k
2
V
2
+
bδγ
k
z
k
2
V
θ
+
b
k
z
k
2
V
2
−
θ
+
δL
2
k
z
k
2
V
2
+
δbL
k
z
k
2
V
2
+
δ
8
k
z
k
2
V
2
6
(
17
δ
8
+
δL
2
+
δbL
)
k
z
k
2
V
2
+
bδγ
k
z
k
2
V
θ
+
b
k
z
k
2
V
2
−
θ
6
C
2
k
z
k
2
V
2
.
(3.21)
DOI:10.12677/aam.2023.1252382348
A^
ê
Æ
?
Ð
H
a
§
à
ò
(3.20)-(3.21)
“
\
(3.17)
d
d
t
H
1
(
z,∂
t
z
)+
bH
1
(
z,∂
t
z
)
6
C
3
(
k
∂
t
z
k
2
+
k
z
k
2
V
2
)
,
(3
.
22)
Ù
¥
C
3
= max
{
C
1
,C
2
}
.
é
(3.22)
3
[
τ,t
]
þ
$
^
Gronwall
Ú
n
Œ
(3.4).
(iii)
ò
•
§
(2.1)
'
u
t
¦
,
Œ
v
=
∂
t
u
÷
v
•
§
ε
(
t
)
∂
2
t
v
+
ε
0
(
t
)
∂
t
v
+
Av
+
γA
θ
2
∂
t
v
+
f
0
(
u
)
v
= 0
.
(3
.
23)
^
A
−
θ
2
∂
t
v
+
δv
†
(3.23)
Š
^
Œ
d
d
t
H
2
(
v,∂
t
v
)+
δ
k
v
k
2
V
2
+
1
2
ε
0
(
t
)
k
∂
t
v
k
2
V
−
θ
+[
γ
−
δε
(
t
)]
k
∂
t
v
k
2
+
h
f
0
(
u
)
v,A
−
θ
2
∂
t
v
+
δv
i
= 0
,
(3.24)
Ù
¥
H
2
(
v,∂
t
v
) =
1
2
[
ε
(
t
)
k
∂
t
v
k
2
V
−
θ
+
k
v
k
2
V
2
−
θ
]+
δε
(
t
)
h
∂
t
v,v
i
+
1
2
δγ
k
v
k
2
V
θ
.
Ï
•
V
2
−
θ
→
V
θ
,
¤
±
|
ε
(
t
)
h
∂
t
v,v
i|
6
ε
(
t
)
k
A
−
θ
4
∂
t
v
kk
A
θ
4
v
k
6
ε
(
t
)
2
k
∂
t
v
k
2
V
−
θ
+
L
2
k
v
k
2
V
2
−
θ
,
δ
¿
©
ž
,
H
2
(
v,∂
t
v
)
∼
ε
(
t
)
k
∂
t
v
k
2
V
−
θ
+
k
v
k
2
V
2
−
θ
.
(3
.
25)
†
(3.16)
O
a
q
,
k
|h
f
0
(
u
)
v,A
−
θ
2
∂
t
v
+
δv
i|
6
C
Z
Ω
(1+
|
u
|
p
−
1
)
|
v
|
(
|
A
−
θ
2
∂
t
v
|
+
δ
|
v
|
)d
x
6
C
(1+
k
u
k
p
−
1
N
(
p
−
1)
θ
+2
−
δ
)
k
v
k
2
N
N
−
2(2
−
δ
)
(
k
A
−
θ
2
∂
t
v
k
2
N
N
−
2
θ
+
δ
k
v
k
2
N
N
−
2
θ
)
6
C
(1+
k
u
k
p
−
1
V
2
)
k
v
k
V
2
−
δ
(
k
A
−
θ
2
∂
t
v
k
V
θ
+
δ
k
v
k
V
θ
)
6
C
0
k
v
k
V
2
(
k
∂
t
v
k
+
δ
k
v
k
V
2
)
6
δ
k
∂
t
v
k
2
+
δ
8
k
v
k
2
V
2
+
C
0
.
(3.26)
DOI:10.12677/aam.2023.1252382349
A^
ê
Æ
?
Ð
H
a
§
à
ò
(3.26)
“
\
(3.25)
d
d
t
H
2
(
v,∂
t
v
)+
9
8
δ
k
v
k
2
V
2
+[
γ
−
δε
(
t
)+
δ
]
k
∂
t
v
k
2
6
C
0
k
v
k
2
V
2
−
θ
−
1
2
ε
0
(
t
)
k
∂
t
v
k
2
V
−
θ
+
C
0
.
d
ε
(
t
)
4
~
5
Œ
,
é
¿
©
δ
Ú
ν
= min
{
9
8
δ,γ
−
δL
+
δ
}
,
k
d
d
t
H
2
(
v,∂
t
v
)+
ν
(
k
v
k
2
V
2
+
k
∂
t
v
k
2
)
6
C
0
H
2
(
v,∂
t
v
)+
C
0
.
(3
.
27)
^
t
2
θ
¦
±
(3.27)
Œ
d
d
t
(
t
2
θ
H
2
(
v,∂
t
v
))+
νt
2
θ
(
k
v
k
2
V
2
+
k
∂
t
v
k
2
)
6
C
0
t
2
θ
H
2
(
v,∂
t
v
)+
C
0
t
2
θ
+
Ct
2
−
θ
θ
(
ε
(
t
)
k
∂
t
v
k
2
V
−
θ
+
k
v
k
2
V
2
−
θ
)
, τ
∈
R
.
(3.28)
d
V
2
→
→
L
2
→
V
−
2
Ú
(3.1)
Œ
k
f
(
u
)
k
V
−
2
6
C
k
f
(
u
)
k
6
C
(
k
u
k
+
k
u
k
p
V
2
)
6
C
0
.
(3
.
29)
ε
(
t
)
k
∂
t
v
k
V
−
2
=
ε
(
t
)
k
∂
2
t
v
k
V
−
2
6
k
Au
k
V
−
2
+
γ
k
A
θ
2
∂
t
u
k
V
−
2
+
k
f
(
u
)
k
V
−
2
+
k
g
k
V
−
2
6
k
u
k
V
2
+
γ
k
∂
t
u
k
V
2
θ
−
2
+
k
f
(
u
)
k
V
−
2
+
k
g
k
6
C
0
.
(3.30)
(
Ü
(3.30)
Ú
Š
½
n
Œ
t
2
−
θ
θ
ε
(
t
)
k
∂
t
v
k
2
V
−
θ
6
t
2
−
θ
θ
k
∂
t
v
k
ε
(
t
)
k
∂
t
v
k
θ
V
−
2
6
ν
2
t
2
θ
k
∂
t
v
k
2
+
C
0
,
t
2
−
θ
θ
k
v
k
2
V
2
−
θ
6
t
2
−
θ
θ
k
v
k
2
−
θ
V
2
k
v
k
6
ν
2
t
2
θ
k
v
k
2
V
2
+
C
0
,
Ù
¥
^
2
θ
−
1
<
2
θ
.
ò
±
þ
(
J
“
\
(3.28),
¿
|
^
V
2
→
→
L
2
d
d
t
(
t
2
θ
H
2
(
v,∂
t
v
))+
ν
2
t
2
θ
(
k
v
k
2
V
2
+
k
∂
t
v
k
2
)
6
C
0
t
2
θ
H
2
(
v,∂
t
v
)+
C
0
t
2
θ
.
(3
.
31)
^
e
−
C
0
(
t
−
τ
)
¦
±
(3.31),
¿
3
[
τ,t
]
þ
È
©
Œ
ε
(
t
)
k
∂
t
v
k
2
V
−
θ
+
k
v
k
2
V
2
−
θ
6
1
t
2
θ
e
C
0
(
t
−
τ
)
.
(3.32)
DOI:10.12677/aam.2023.1252382350
A^
ê
Æ
?
Ð
H
a
§
à
é
?
¿
τ<a
6
t
,
‰
(3.31)
¦
±
e
−
C
0
(
t
−
a
)
,
¿
3
[
a,t
]
þ
È
©
Œ
Z
t
a
(
k
v
k
2
V
2
+
k
∂
t
v
k
2
)d
s
6
2
ν
e
C
0
(
a
−
τ
)
e
C
0
(
t
−
a
)
a
2
θ
.
^
e
−
C
0
(
t
−
τ
)
¦
±
(3.31)
¿
3
[
t,t
+1]
þ
È
©
k
H
2
(
v
(
t
+1)
,∂
t
v
(
t
+1))e
−
C
0
(
t
+1
−
τ
)
+
ν
2
Z
t
+1
t
e
−
C
0
(
s
−
τ
)
(
k
v
k
2
V
2
+
k
∂
t
v
k
2
)d
s
6
H
2
(
v
(
t
)
,∂
t
v
(
t
))e
−
C
0
(
t
−
τ
)
.
(3.33)
Ï
d
k
Z
t
+1
t
(
k
v
k
2
V
2
+
k
∂
t
v
k
2
)d
s
6
2
ν
e
C
0
(
t
−
τ
)
t
2
θ
.
(3
.
34)
(
Ü
(3.32)
Ú
(3.34)
k
ε
(
t
)
k
∂
t
v
k
2
V
−
θ
+
k
v
k
2
V
2
−
θ
+
Z
t
+1
t
(
k
v
k
2
V
2
+
k
∂
t
v
k
2
)d
s
6
(1+
2
ν
)
1
t
2
θ
e
C
0
(
t
−
τ
)
.
(3.35)
^
Au
†
(2.1)
‰
S
È
,
Œ
γ
2
d
d
t
k
u
k
2
V
2+
θ
+
k
u
k
2
V
4
=
h
g
−
f
(
u
)
−
ε
(
t
)
∂
2
t
u,Au
i
.
(3.36)
Ï
•
p
+1
<
2
N
+(2
θ
−
4)
N
−
4
,
¤
±
k
V
2
→
L
p
+1
,
2
|
^
Young
Ø
ª
,H¨older
Ø
ª
Ú
Š
½
n
Œ
|h
f
(
u
)
,Au
i|
6
k
f
(
u
)
kk
Au
k
6
C
(1+
k
u
k
p
−
1
p
+1
)
k
u
k
p
+1
k
u
k
V
4
6
C
0
k
u
k
V
2
k
u
k
V
4
6
C
0
+
1
4
k
u
k
2
V
4
(3.37)
|h
g
−
ε
(
t
)
∂
2
t
u,Au
i|
6
(
k
g
k
+
ε
(
t
)
k
∂
2
t
u
k
)
k
u
k
V
4
6
C
(
k
g
k
2
+
L
2
k
∂
2
t
u
k
2
)+
1
4
k
u
k
2
V
4
(3.38)
ò
(3.37)-(3.38)
“
\
(3.36)
Œ
γ
d
d
t
k
u
k
2
V
2+
θ
+
k
u
k
2
V
4
6
2
C
0
(
k
g
k
2
+
L
2
k
∂
2
t
u
k
2
)
.
(3
.
39)
DOI:10.12677/aam.2023.1252382351
A^
ê
Æ
?
Ð
H
a
§
à
a>
0
,
0
<k<
1;
½
a<
0
,k>
1
ž
,
é
?
¿
τ<ka<a
6
t
,
é
(3.39)
¦
±
(
t
−
ka
)
1
2
−
θ
Œ
γ
d
d
t
[(
t
−
ka
)
1
2
−
θ
k
u
k
2
V
2+
θ
]+
1
2
(
t
−
ka
)
1
2
−
θ
k
u
k
2
V
4
6
C
0
(
t
−
ka
)
1
2
−
θ
(
k
g
k
2
+
L
2
k
∂
2
t
u
k
2
)+
1
2
−
θ
(
t
−
ka
)
θ
−
1
2
−
θ
k
u
k
2
V
2+
θ
.
(3.40)
d
Š
½
n
Ú
Young
Ø
ª
k
1
2
−
θ
(
t
−
ka
)
θ
−
1
2
−
θ
k
u
k
2
V
2+
θ
6
C
1
2
−
θ
(
t
−
ka
)
θ
−
1
2
−
θ
k
u
k
2
−
θ
V
2
k
u
k
θ
V
4
6
1
8
(
t
−
ka
)
1
2
−
θ
k
u
k
2
V
4
+
C
k
u
k
2
V
2
.
(3.41)
¤
±
k
γ
d
d
t
[(
t
−
ka
)
1
2
−
θ
k
u
k
2
V
2+
θ
]
6
2
C
0
(
t
−
ka
)
1
2
−
θ
(
k
g
k
2
+
k
u
k
2
V
2
+
L
2
k
∂
2
t
u
k
2
)
.
(3
.
42)
é
(3.42)
3
[
ka,t
]
È
©
,
2
|
^
c
¡
®
(
J
k
γ
k
u
k
2
V
2+
θ
6
Z
t
ka
(
k
u
(
s
)
k
2
V
2
+
L
2
k
∂
2
t
u
(
s
)
k
2
)d
s
6
2
C
0
(
t
−
ka
)
1
−
θ
2
−
θ
+
2
ν
h
(
t
)
,
(3.43)
Ù
¥
h
(
t
) =
e
C
0
(
ka
−
τ
)
e
C
0
(
t
−
ka
)
(
ka
)
2
θ
(
t
−
ka
)
1
2
−
θ
.
é
(3.39)
3
[a,t]
þ
È
©
Œ
Z
t
a
k
u
k
2
V
4
d
s
6
8
γ
7
k
u
(
a
)
k
2
V
2+
θ
+
8
7
Z
t
a
(
k
u
(
s
)
k
2
V
2
+
L
2
k
∂
2
t
u
(
s
)
k
2
)d
s
6
16
C
0
7
(
a
−
ka
)
1
−
θ
2
−
θ
+
8
C
4
7
(
t
−
a
)+
16
7
ν
h
(
a
)+
16
7
ν
e
C
0
(
a
−
τ
)
e
C
0
(
t
−
a
)
a
2
θ
,
(3.44)
Ù
¥
C
4
=
C
(
R,δ,
k
g
k
,L
)
.
é
(3
.
39)
3
[
t,t
+1]
È
©
Œ
Z
t
+1
t
k
u
(
s
)
k
2
V
4
d
s
6
16
C
0
7
(
t
−
ka
)
1
−
θ
2
−
θ
+
8
C
4
7
+
16
7
ν
h
(
t
)+
16
7
ν
e
C
0
(
t
−
τ
)
t
2
θ
(3
.
45)
(
Ü
(3.43)-(3.45)
Œ
k
u
k
2
V
2+
θ
+
Z
t
+1
t
k
u
(
s
)
k
2
V
4
d
s
6
(
2
γ
+
16
7
)
C
0
(
t
−
ka
)
1
−
θ
2
−
θ
+(
2
νγ
+
16
7
ν
)
h
(
t
)+
16
7
ν
e
C
0
(
t
−
τ
)
t
2
θ
+
8
7
C
4
.
(3.46)
(
Ü
(3.35)
Ú
(3.46)
k
(3.5)
¤
á
.
DOI:10.12677/aam.2023.1252382352
A^
ê
Æ
?
Ð
H
a
§
à
-
z
n
=(
u
n
,∂
t
u
n
)
´
¯
K
(2.1)
é
A
C
q
•
§
)
,
´
•
(3.1)
é
Galerkin
C
q
S
z
n
•
´
¤
á
.
Ï
d
,
•
3
(
u,∂
t
u
)
∈
L
∞
([
τ,T
];
H
t
)
,∂
t
u
∈
L
2
([
τ,T
];
V
θ
)
,
¦
(
u
n
,∂
t
u
n
)
f
∗
Â
ñ
u
(
u,∂
t
u
)
3
L
∞
([
τ,T
];
H
t
)
¥
,
∂
t
u
n
f
Â
ñ
u
∂
t
u
3
L
2
([
τ,T
];
V
θ
)
¥
.
A^
Ú
n
2.1
Œ
u
n
→
u
3
L
2
([
τ,T
];
V
2
),
∂
t
u
n
→
∂
t
u
3
L
2
([
τ,T
];
L
2
(Ω)),
(
u
n
,∂
t
u
n
)
→
(
u,∂
t
u
)
3
C
([
τ,T
];
V
2
−
δ
×
V
−
δ
)
,
(3.47)
f
(
u
n
(
t
))
→
f
(
u
(
t
))
3
L
2
¥
f
Â
ñ
,
t
∈
[
τ,T
]
,
u
n
(
x,t
)
3
Ω
×
[
τ,T
]
¥
A
??
Â
ñ
u
u
(
x,t
)
.
d
V
2
→
L
p
+1
Ú
(2.5)
Œ
,
é
?
¿
ζ
∈
C
∞
0
(Ω)
Z
T
τ
h
f
(
u
n
)
−
f
(
u
)
,ζ
i
d
t
6
C
Z
T
τ
Z
Ω
(1+
|
u
n
|
p
−
1
+
|
u
|
p
−
1
)
|
u
n
−
u
||
ζ
|
d
x
d
t
6
C
Z
T
τ
(1+
k
u
n
k
p
−
1
p
+1
+
k
u
k
p
−
1
p
+1
)
k
u
n
−
u
k
p
+1
|
ζ
|
p
+1
d
t
6
C
Z
T
τ
(1+
k
u
n
k
p
−
1
V
2
+
k
u
k
p
−
1
V
2
)
k
u
n
−
u
k
V
2
k
ζ
k
V
2
d
t
6
C
0
k
u
n
−
u
k
L
2
([
τ,T
];
V
2
)
→
0
.
¤
±
z
= (
u,∂
t
u
)
´
÷
v
(3.1)
¯
K
(2.1)
f
)
.
é
?
¿
t
∈
[
τ,T
],
d
(3.2)
Ú
(3.47)
Œ
lim
s
→
t
E
(
u
(
s
)
,∂
t
u
(
s
)) =
E
(
u
(
t
)
,∂
t
u
(
t
))
,
x
∈
Ω
ž
,
u
(
x,s
)
→
u
(
x,t
)(
s
→
t
)
A
??
,
(
u,∂
t
u
)
∈
C
([
τ,T
];
V
2
−
δ
×
V
−
δ
)
∩
L
∞
([
τ,T
];
H
t
).
d
Fatou
Ú
n
Ú
5
1
Œ
lim
s
→
t
h
g,u
(
s
)
i
=
h
g,u
(
t
)
i
,
k
(
u
(
t
))
,∂
t
u
(
t
)
k
2
H
t
6
liminf
s
→
t
k
(
u
(
s
))
,∂
t
u
(
s
)
k
2
H
t
,
DOI:10.12677/aam.2023.1252382353
A^
ê
Æ
?
Ð
H
a
§
à
Z
Ω
(
F
(
u
(
t
))+
η
2
|
u
(
t
)
|
2
+
C
)d
x
6
liminf
s
→
t
Z
Ω
(
F
(
u
(
s
))+
η
2
|
u
(
s
)
|
2
+
C
)d
x
6
liminf
s
→
t
Z
Ω
F
(
u
(
s
))d
x
+
η
2
k
u
(
t
)
k
2
+
C
|
Ω
|
.
¤
±
k
Z
Ω
F
(
u
(
t
))d
x
6
liminf
s
→
t
Z
Ω
F
(
u
(
s
))d
x.
(3
.
48)
d
±
þ
O
ª
Œ
liminf
s
→
t
(
1
2
ε
(
s
)
k
∂
t
u
(
s
)
k
2
+
1
2
k
u
(
s
)
k
2
V
2
)+liminf
s
→
t
Z
Ω
F
(
u
(
s
))d
x
6
lim
s
→
t
(
1
2
ε
(
s
)
k
∂
t
u
(
s
)
k
2
+
1
2
k
u
(
s
)
k
2
V
2
+
Z
Ω
F
(
u
(
s
))d
x
)
=
1
2
ε
(
t
)
k
∂
t
u
(
t
)
k
2
+
1
2
k
u
(
t
)
k
2
V
2
+
Z
Ω
F
(
u
(
t
))d
x
6
liminf
s
→
t
(
1
2
ε
(
s
)
k
∂
t
u
(
s
)
k
2
+
1
2
k
u
(
s
)
k
2
V
2
)+liminf
s
→
t
Z
Ω
F
(
u
(
s
))d
x.
k
ε
(
t
)
k
∂
t
u
(
t
)
k
2
= liminf
s
→
t
ε
(
s
)
k
∂
t
u
(
s
)
k
2
,
(3
.
49)
k
u
(
t
)
k
2
V
2
= liminf
s
→
t
k
u
(
s
)
k
2
V
2
.
(3
.
50)
(
Ü
(3.49),(3.50),
˜
m
H
t
˜
—
à
5
Ú
(
u,∂
t
u
)
∈
C
w
([
τ,T
];
H
t
),
Œ
(
u,∂
t
u
)
∈
C
([
τ,T
];
H
t
)
.
4.
ž
m
•
6
á
Â
8
½
n
4.1
^
‡
(C
1
)-(C
3
)
¤
á
,
é
?
¿
τ<T,
¯
K
(2.1)
)
ë
Y
•
6
Ð
Š
.
=
X
J
z
1
= (
u
1
,∂
t
u
1
)
,z
2
= (
u
2
,∂
t
u
2
)
´
¯
K
(2
.
1)
'
u
Ð
Š
k
z
1
(
τ
)
k
H
τ
6
R,
k
z
2
(
τ
)
k
H
τ
6
R
)
,
ù
p
R>
0
´
˜
‡
~
ê
,
K
k
z
1
(
t
)
−
z
2
(
t
)
k
2
H
t
6
e
C
0
(
t
−
τ
)
k
z
1
(
τ
)
−k
z
2
(
τ
)
k
2
H
τ
,
∀
t
∈
[
τ,T
]
.
(4
.
1)
y
²
¯
z
(
t
) =
{
¯
u
(
t
)
,∂
t
¯
u
(
t
)
}
=
z
1
(
t
)
−
z
2
(
t
)
,
K
¯
z
(
t
)
÷
v
•
§
ε
(
t
)
∂
2
t
¯
u
(
t
)+
A
¯
u
+
γA
θ
2
∂
t
¯
u
(
t
)+
f
(
u
1
)
−
f
(
u
2
) = 0
.
(4
.
2)
^
2
∂
t
¯
u
(
t
)
†
(4
.
2)
‰
S
È
,
Œ
DOI:10.12677/aam.2023.1252382354
A^
ê
Æ
?
Ð
H
a
§
à
d
d
t
H
3
(¯
u,∂
t
¯
u
(
t
))+2
γ
k
∂
t
¯
u
(
t
)
k
2
V
θ
=
−
2
h
f
(
u
1
)
−
f
(
u
2
)
,∂
t
¯
u
(
t
)
i
+
ε
0
(
t
)
k
∂
t
¯
u
(
t
)
k
2
,
(4
.
3)
Ù
¥
H
3
(¯
u,∂
t
¯
u
(
t
)) =
ε
(
t
)
k
∂
t
¯
u
(
t
)
k
2
+
k
¯
u
k
2
V
2
.
d
ε
(
t
)
4
~
5
Œ
d
d
t
H
3
(¯
u,∂
t
¯
u
(
t
))+2
γ
k
∂
t
¯
u
(
t
)
k
2
V
θ
6
−
2
h
f
(
u
1
)
−
f
(
u
2
)
,∂
t
¯
u
(
t
)
i
.
(4.4)
|−
2
h
f
(
u
1
)
−
f
(
u
2
)
,∂
t
¯
u
(
t
)
i|
6
C
Z
Ω
(1+
|
u
1
|
p
−
1
+
|
u
2
|
p
−
1
)
|
¯
u
||
∂
t
¯
u
(
t
)
|
d
x
6
C
(1+
k
u
1
k
p
−
1
N
(
p
−
1)
θ
+2
−
δ
+
k
u
2
k
p
−
1
N
(
p
−
1)
θ
+2
−
δ
)
k
¯
u
k
2
N
N
−
2(2
−
δ
)
k
∂
t
¯
u
(
t
)
k
2
N
N
−
2
θ
6
C
(1+
k
u
1
k
p
−
1
V
2
+
k
u
2
k
p
−
1
V
2
)
k
¯
u
k
V
2
−
δ
k
∂
t
¯
u
(
t
)
k
V
θ
6
C
0
k
¯
u
k
V
2
−
δ
k
∂
t
¯
u
(
t
)
k
V
θ
6
C
0
k
¯
u
k
2
V
2
+
γ
k
∂
t
¯
u
(
t
)
k
2
V
θ
.
(4.5)
ò
(4.5)
“
\
(4.4)
Œ
d
d
t
H
3
(¯
u,∂
t
¯
u
(
t
))+
γ
k
∂
t
¯
u
(
t
)
k
2
V
θ
6
C
0
(
k
¯
u
k
2
V
2
+
ε
(
t
)
k
∂
t
¯
u
(
t
)
k
2
)
?
k
d
d
t
H
3
(¯
u,∂
t
¯
u
(
t
))
6
C
0
H
3
(¯
u,∂
t
¯
u
(
t
))
.
(4
.
6)
‰
(4.6)
¦
±
e
−
C
0
(
t
−
τ
)
,
¿
3
[
τ,t
]
þ
È
©
Œ
k
z
1
(
t
)
−
z
2
(
t
)
k
2
H
t
6
e
C
0
(
t
−
τ
)
k
z
1
(
τ
)
−k
z
2
(
τ
)
k
2
H
τ
.
d
½
n
3.1,4.1
Œ
•
,
1
6
p<p
θ
ž
,
·
‚
Œ
±
½
Â
¯
K
(2.1)
L
§
U
(
t,τ
) :
H
τ
→H
t
,
U
(
t,τ
)(
u
0
,u
1
) = (
u
(
t
)
,∂
t
u
(
t
))
, t
>
τ,
Ù
¥
u
(
t
)
´
¯
K
(2.1)
ƒ
é
u
Ð
Š
(
u
0
,u
1
)
∈H
τ
•
˜
)
,
¿
…
T
L
§
d
H
τ
N
\
H
t
´
ë
Y
.
½
n
4.2
X
J
^
‡
(C
1
)-(C
3
)
¤
á
,
P
B
t
(
R
) =
{
z
∈H
t
:
k
z
k
H
t
6
R
}
,
@
o
•
3
R
0
>
0,
¦
B
=
{
B
t
(
R
0
)
}
t
∈
R
´
L
§
{
U
(
t,τ
)
}
ž
m
•
6
á
Â
8
,
…
é
S
0
>
R
0
,
÷
v
sup
z
τ
∈
B
τ
(
R
0
)
{k
U
(
t,τ
)
z
τ
k
H
t
+
Z
+
∞
τ
k
∂
t
u
(
y
)
k
d
y
}
6
S
0
.
(4
.
7)
y
²
d
½
n
4.1
9
©
[7],[13],
´
y
(4.7)
¤
á
.
DOI:10.12677/aam.2023.1252382355
A^
ê
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?
Ð
H
a
§
à
5.
ž
m
•
6
á
Ú
f
5.1.
k
O
•
¯
K
(2.1)
L
§
U
(
t,τ
)
ì
C;
5
,
·
‚
k
?
1
±
e
O
.
(
u
i
(
t
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t
u
i
(
t
))
,i
= 1
,
2
´
•
§
(2
.
1)
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u
Ð
Š
(
u
i
(
τ
)
,∂
t
u
i
(
τ
))
∈
B
τ
(
R
0
)
)
,
P
w
(
t
) =
u
1
(
t
)
−
u
2
(
t
).
K
w
(
t
)
÷
v
ε
(
t
)
∂
2
t
w
+
Aw
+
γA
θ
2
∂
t
w
+
f
(
u
1
)
−
f
(
u
2
) = 0
,x
∈
Ω
,t
>
τ,
w
(
x,t
) = 0
,x
∈
∂
Ω
,t
>
τ,
w
(
x,τ
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u
1
(
τ
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u
2
(
τ
)
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t
w
(
x,τ
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∂
t
u
1
(
τ
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∂
t
u
2
(
τ
)
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∈
Ω
.
(5
.
1)
‰
(5.1)
†
m
ü
>
¦
±
w
,
¿
3
[
τ,T
]
×
Ω
þ
È
©
Œ
Z
Ω
ε
(
T
)
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t
w
(
T
)
w
(
T
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x
−
Z
Ω
ε
(
τ
)
∂
t
w
(
τ
)
w
(
τ
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x
−
Z
T
τ
Z
Ω
ε
0
(
s
)
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t
w
(
s
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s
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x
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s
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Z
T
τ
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Ω
ε
(
s
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|
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t
w
(
s
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x
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s
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T
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1
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(
s
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x
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s
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A
θ
4
w
(
T
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2
d
x
+
Z
T
τ
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Ω
(
f
(
u
1
)
−
f
(
u
2
))
w
(
s
)d
x
d
s
−
γ
2
Z
Ω
|
A
θ
4
w
(
τ
)
|
2
d
x
= 0
.
(5.2)
^
w
t
¦
±
(5.1),
¿
3
[
s,T
]
×
Ω
þ
È
©
,
k
1
2
Z
Ω
ε
(
T
)
|
∂
t
w
(
T
)
|
2
d
x
−
1
2
Z
Ω
ε
(
s
)
|
∂
t
w
(
s
)
|
2
d
x
+
1
2
Z
Ω
|
A
1
2
w
(
T
)
|
d
x
−
1
2
Z
Ω
|
A
1
2
w
(
s
)
|
d
x
+
γ
Z
T
s
Z
Ω
|
A
θ
4
∂
t
w
(
t
)
|
2
d
x
d
t
−
1
2
Z
T
s
Z
Ω
ε
0
(
t
)
|
∂
t
w
(
t
)
|
2
d
x
d
t
+
Z
T
s
Z
Ω
(
f
(
u
1
)
−
f
(
u
2
))
∂
t
w
(
t
)d
x
d
t
= 0
.
(5.3)
-
G
w
(
t
) =
1
2
R
Ω
(
ε
(
t
)
|
∂
t
w
(
t
)
|
2
+
|
A
1
2
w
(
t
)
|
)d
x
,
K
d
(5.3)
Œ
G
w
(
T
)
−
G
w
(
s
)+
Z
T
s
Z
Ω
(
f
(
u
1
)
−
f
(
u
2
))
∂
t
w
(
t
)d
x
d
s
−
1
2
Z
T
s
Z
Ω
ε
0
(
t
)
|
∂
t
w
(
t
)
|
2
d
x
d
t
+
γ
Z
T
s
Z
Ω
|
A
θ
4
∂
t
w
(
t
)
|
2
d
x
d
t
= 0
.
(5.4)
d
ε
(
t
)
•
4
~
¼
ê
Œ
ε
0
(
t
)
6
0,
?
k
1
2
Z
T
s
Z
Ω
ε
0
(
t
)
|
∂
t
w
(
t
)
|
2
d
x
d
t
6
G
w
(
s
)
−
Z
T
s
Z
Ω
(
f
(
u
1
)
−
f
(
u
2
))
∂
t
w
(
t
)d
x
d
t
−
γ
Z
T
s
Z
Ω
|
A
θ
4
∂
t
w
(
t
)
|
2
d
x
d
t.
DOI:10.12677/aam.2023.1252382356
A^
ê
Æ
?
Ð
H
a
§
à
ε
(
t
)
|
∂
t
w
(
t
)
|
2
6
1
2
Lε
0
(
t
)
|
∂
t
w
(
t
)
|
2
,
¤
±
k
Z
T
s
Z
Ω
ε
(
t
)
|
∂
t
w
(
t
)
|
2
d
x
d
t
6
LG
w
(
s
)
−
L
Z
T
s
Z
Ω
(
f
(
u
1
)
−
f
(
u
2
))
∂
t
w
(
t
)d
x
d
t
−
Lγ
Z
T
s
Z
Ω
|
A
θ
4
∂
t
w
(
t
)
|
2
d
x
d
t.
(5.5)
é
(5.4)
'
u
s
3
[
τ,T
]
×
Ω
þ
È
©
,
Œ
G
w
(
T
)(
T
−
τ
)
−
1
2
Z
T
τ
Z
T
s
Z
Ω
ε
0
(
t
)
|
∂
t
w
(
t
)
|
2
d
x
d
t
d
s
=
Z
T
τ
G
w
(
s
)d
s
−
Z
T
τ
Z
T
s
Z
Ω
(
f
(
u
1
)
−
f
(
u
2
))
∂
t
w
(
t
)d
x
d
t
d
s
−
γ
Z
T
τ
Z
T
s
Z
Ω
|
A
θ
4
∂
t
w
(
t
)
|
2
d
x
d
t
d
s.
(5.6)
(
Ü
(5.5)
Ú
(5.2)
k
Z
T
τ
G
w
(
s
)d
s
=
1
2
Z
T
τ
Z
Ω
ε
(
s
)
|
∂
t
w
(
s
)
|
2
d
x
d
s
+
1
2
Z
T
τ
Z
Ω
|
A
1
2
w
(
s
)
|
2
d
x
d
s
=
Z
T
τ
Z
Ω
ε
(
s
)
|
∂
t
w
(
s
)
|
2
d
x
d
s
+
1
2
Z
Ω
ε
(
τ
)
∂
t
w
(
τ
)
w
(
τ
)d
x
+
γ
4
Z
Ω
|
A
θ
4
w
(
τ
)
|
2
d
x
−
1
2
Z
Ω
ε
(
T
)
∂
t
w
(
T
)
w
(
T
)d
x
+
1
2
Z
T
τ
Z
Ω
ε
0
(
s
)
∂
t
w
(
s
)
w
(
s
)d
x
d
s
−
1
2
Z
T
τ
Z
Ω
(
f
(
u
1
)
−
f
(
u
2
))
w
(
s
)d
x
d
s
−
γ
4
Z
Ω
|
A
θ
4
w
(
T
)
|
2
d
x
6
LG
w
(
τ
)
−
L
Z
T
τ
Z
Ω
(
f
(
u
1
)
−
f
(
u
2
))
∂
t
w
(
s
)d
x
d
s
−
Lγ
Z
T
τ
Z
Ω
|
A
θ
4
∂
t
w
(
s
)
|
2
d
x
d
s
+
1
2
Z
Ω
ε
(
τ
)
∂
t
w
(
τ
)
w
(
τ
)d
x
−
γ
4
Z
Ω
|
A
θ
4
w
(
T
)
|
2
d
x
+
γ
4
Z
Ω
|
A
θ
4
w
(
τ
)
|
2
d
x
−
1
2
Z
T
τ
Z
Ω
(
f
(
u
1
)
−
f
(
u
2
))
w
(
s
)d
x
d
s
+
1
2
Z
T
τ
Z
Ω
L∂
t
w
(
s
)
w
(
s
)d
x
d
s
−
1
2
Z
Ω
ε
(
T
)
∂
t
w
(
T
)
w
(
T
)d
x.
(5.7)
ò
(5.7)
“
\
(5.6),
2
d
ε
(
t
)
4
~
5
Œ
DOI:10.12677/aam.2023.1252382357
A^
ê
Æ
?
Ð
H
a
§
à
G
w
(
T
)
6
L
T
−
τ
G
w
(
τ
)+
1
2(
T
−
τ
)
Z
Ω
ε
(
τ
)
∂
t
w
(
τ
)
w
(
τ
)d
x
+
γ
4(
T
−
τ
)
Z
Ω
|
A
θ
4
w
(
τ
)
|
2
d
x
−
L
(
T
−
τ
)
Z
T
τ
Z
Ω
(
f
(
u
1
)
−
f
(
u
2
))
∂
t
w
(
s
)d
x
d
s
−
γL
(
T
−
τ
)
Z
T
τ
Z
Ω
|
A
θ
4
∂
t
w
(
s
)
|
2
d
x
d
s
−
1
(
T
−
τ
)
Z
T
τ
Z
T
s
Z
Ω
(
f
(
u
1
)
−
f
(
u
2
))
∂
t
w
(
t
)d
x
d
t
d
s
−
γ
4(
T
−
τ
)
Z
Ω
|
A
θ
4
w
(
T
)
|
2
d
x
+
γ
2(
T
−
τ
)
Z
T
τ
Z
Ω
L∂
t
w
(
s
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w
(
s
)d
x
d
s
−
γ
(
T
−
τ
)
Z
T
τ
Z
T
s
Z
Ω
|
A
θ
4
∂
t
w
(
t
)
|
2
d
x
d
t
d
s
−
1
2(
T
−
τ
)
Z
Ω
ε
(
T
)
∂
t
w
(
T
)
w
(
T
)d
x
−
1
2(
T
−
τ
)
Z
T
τ
Z
Ω
(
f
(
u
1
)
−
f
(
u
2
))
w
(
s
)d
x
d
s
-
C
M
=
LG
w
(
τ
)+
1
2
Z
Ω
ε
(
τ
)
∂
t
w
(
τ
)
w
(
τ
)d
x
+
γ
4
Z
Ω
|
A
θ
4
w
(
τ
)
|
2
d
x.
(5
.
8)
Φ
T
τ
((
u
1
(
τ
)
,∂
t
u
1
(
τ
))
,
(
u
2
(
τ
)
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t
u
2
(
τ
))) =
3
X
i
=1
I
i
.
(5
.
9)
Ù
¥
I
1
=
−
1
2(
T
−
τ
)
Z
Ω
ε
(
T
)
∂
t
w
(
T
)
w
(
T
)d
x
+
1
2(
T
−
τ
)
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T
τ
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Ω
L∂
t
w
(
s
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w
(
s
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x
d
s,
I
2
=
1
(
T
−
τ
)
[
−
L
Z
T
τ
Z
Ω
(
f
(
u
1
)
−
f
(
u
2
))
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t
w
(
s
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x
d
s
−
1
2
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T
τ
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Ω
(
f
(
u
1
)
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f
(
u
2
))
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(
s
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x
d
s
−
Z
T
τ
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T
s
Z
Ω
(
f
(
u
1
)
−
f
(
u
2
))
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t
w
(
t
)d
x
d
t
d
s
]
,
I
3
=
−
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(
T
−
τ
)
Z
T
τ
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T
s
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θ
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t
w
(
t
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|
2
d
x
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t
d
s
−
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4(
T
−
τ
)
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Ω
|
A
θ
4
w
(
T
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|
2
d
x
−
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(
T
−
τ
)
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T
τ
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Ω
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θ
4
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t
w
(
s
)
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2
d
x
d
s.
d
(5
.
8)
,
(5
.
9)
Œ
G
w
(
T
)
6
1
(
T
−
τ
)
C
M
+Φ
T
τ
((
u
1
(
τ
)
,∂
t
u
1
(
τ
))
,
(
u
2
(
τ
)
,∂
t
u
2
(
τ
)))
5.2.
ì
C;
5
e
¡
·
‚
ò
|
^
Â
¼
ê
•{
5
y
²
•
§
(2
.
1)
é
A
L
§
´
ì
C;
.
DOI:10.12677/aam.2023.1252382358
A^
ê
Æ
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H
a
§
à
½
n
5.1
X
J
^
‡
(C
1
)-(C
3
)
¤
á
,
é
?
¿
½
t
∈
R
,
k
.
S
{
x
n
}
∞
n
=1
∈H
τ
n
,
†
?
¿
S
{
τ
n
}
∞
n
=1
∈
[
−∞
,t
)(
n
→∞
,τ
n
→−∞
)
ž
,
K
S
{
U
(
t,τ
n
)
x
n
}
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n
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.
y
²
(
u
n
(
t
)
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t
u
n
(
t
))
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K
(2.1)
'
u
Ð
Š
(
u
i
(
τ
)
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t
u
i
(
τ
))
)
,
d
½
n
3.1
Œ
•
k
u
n
k
2
V
2
+
ε
(
ξ
)
k
∂
t
u
n
k
2
´
k
.
,
…
k
u
n
k
2
V
2
´
k
.
;
d
^
‡
(C
1
)
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•
,
é
ξ
∈
[
τ,T
]
,ε
(
ξ
)
´
k
.
,
¤
±
k
∂
t
u
n
k
2
´
k
.
.
Š
â
Banach
−
Alaoglu
½
n
,
½
n
3.1,
Ú
n
2.1
k
±
e
(
J
:
u
n
→
u
3
L
∞
([
τ,T
];
V
2
)
¥
f
∗
Â
ñ
,(5.10)
∂
t
u
n
→
u
t
3
L
2
([
τ,T
];
V
θ
)
¥
f
Â
ñ
,(5.11)
∂
t
u
n
→
u
t
3
L
∞
([
τ,T
];
L
2
)
¥
f
∗
Â
ñ
,(5.12)
u
n
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u
3
L
2
([
τ,T
];
V
2
),(5.13)
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t
u
n
→
∂
t
u
3
L
2
([
τ,T
];
L
2
),(5.14)
u
n
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3
L
2
([
τ,T
];
V
2
θ
),(5.15)
Ù
¥
^
θ<
2
θ<
2.
é
?
¿
>
0
Ú
½
T>τ,
¦
T
−
τ
v
Œ
,
K
k
1
T
−
τ
C
M
<.
d
½
n
2.1
Œ
•
,
·
‚
•
I
y
²
é
?
¿
½
T
,Φ
T
τ
∈C
(
B
τ
(
R
0
))
¤
á
=
Œ
.
•
d
,
·
‚
Å
‘
?
n
(5
.
9).
Ä
k
,
d
½
n
4.2
Ú
(5.10)-(5.13)
Œ
lim
n
→∞
lim
m
→∞
Z
Ω
|
A
θ
4
(
u
n
−
u
m
)
|
2
d
x
= 0
.
(5
.
16)
lim
n
→∞
lim
m
→∞
Z
T
τ
Z
Ω
|
A
θ
4
(
∂
t
u
n
−
∂
t
u
m
)
|
2
d
x
d
s
= 0
.
(5
.
17)
Ï
•
V
2
→
L
p
+1
,
2
(
Ü
(5.10)
Ú
(5.13)
Œ
lim
n
→∞
lim
m
→∞
Z
Ω
ε
(
t
)(
∂
t
u
n
−
∂
t
u
m
)(
u
n
−
u
m
)d
x
6
C
lim
n
→∞
lim
m
→∞
L
k
∂
t
u
n
−
∂
t
u
m
k
V
2
k
u
n
−
u
m
k
V
2
6
C
lim
n
→∞
lim
m
→∞
L
k
∂
t
u
n
+
∂
t
u
m
k
V
2
k
u
n
−
u
m
k
V
2
= 0
.
(5.18)
lim
n
→∞
lim
m
→∞
Z
T
τ
Z
Ω
L
(
∂
t
u
n
−
∂
t
u
m
)(
u
n
−
u
m
)d
x
d
s
6
C
lim
n
→∞
lim
m
→∞
(
Z
T
τ
k
∂
t
u
n
−
∂
t
u
m
k
2
d
s
)
1
2
(
Z
T
τ
k
u
n
−
u
m
k
2
d
s
)
1
2
= 0
.
(5.19)
DOI:10.12677/aam.2023.1252382359
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H
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§
à
lim
n
→∞
lim
m
→∞
Z
T
τ
Z
Ω
(
f
(
u
n
)
−
f
(
u
m
))(
u
n
−
u
m
)d
x
d
s
6
C
lim
n
→∞
lim
m
→∞
Z
T
τ
Z
Ω
(1+
|
u
n
|
p
−
1
+
|
u
m
|
p
−
1
)
|
u
n
−
u
m
|
2
d
x
d
s
6
C
lim
n
→∞
lim
m
→∞
Z
T
τ
(1+
k
u
n
k
p
−
1
p
+1
+
k
u
m
k
p
−
1
p
+1
)
k
u
n
−
u
m
k
2
p
+1
d
s
6
C
lim
n
→∞
lim
m
→∞
Z
T
τ
(1+
k
u
n
k
p
−
1
V
2
+
k
u
m
k
p
−
1
V
2
)
k
u
n
−
u
m
k
2
V
2
d
s
6
C
0
lim
n
→∞
lim
m
→∞
Z
T
τ
k
u
n
−
u
m
k
2
V
2
d
s
= 0
.
(5.20)
(
Ü
(5.16)-(5.17),
±
9
(5.18)-(5.19),
k
lim
n
→∞
lim
m
→∞
I
1
= 0
,
(5
.
21)
lim
n
→∞
lim
m
→∞
I
3
= 0
.
(5
.
22)
Ù
g
,
Ï
•
Z
T
τ
Z
Ω
(
f
(
u
n
)
−
f
(
u
m
))(
∂
t
u
n
−
∂
t
u
m
)d
x
d
s
=
Z
T
τ
Z
Ω
f
(
u
n
)
∂
t
u
n
d
x
d
s
−
Z
T
τ
Z
Ω
f
(
u
n
)
∂
t
u
m
d
x
d
s
−
Z
T
τ
Z
Ω
f
(
u
m
)
∂
t
u
n
d
x
d
s
+
Z
T
τ
Z
Ω
f
(
u
m
)
∂
t
u
m
d
x
d
s
=
Z
Ω
F
(
u
n
(
T
))d
x
−
Z
Ω
F
(
u
n
(
τ
))d
x
−
Z
T
τ
Z
Ω
f
(
u
n
)
∂
t
u
m
d
x
d
s
−
Z
T
τ
Z
Ω
f
(
u
m
)
∂
t
u
n
d
x
d
s
+
Z
Ω
F
(
u
m
(
T
))d
x
−
Z
Ω
F
(
u
m
(
τ
))d
x.
(5.23)
d
(2.5)
Ú
V
2
→
L
p
+1
,
Œ
|
Z
Ω
(
F
(
u
n
(
t
))
−
F
(
u
(
t
)))d
x
|
6
Z
Ω
|
(
F
(
u
n
(
t
))
−
F
(
u
(
t
)))
|
d
x
6
Z
Ω
|
f
(
u
(
t
)+
λ
(
u
n
(
t
)
−
u
(
t
)))
||
u
n
(
t
)
−
u
(
t
)
|
d
x
6
C
Z
Ω
(1+
|
u
n
(
t
)
|
p
−
1
+
|
u
(
t
)
|
p
−
1
)
|
u
n
(
t
)
−
u
(
t
)
|
2
d
x
6
C
(1+
k
u
n
(
t
)
k
p
−
1
p
+1
+
k
u
(
t
)
k
p
−
1
p
+1
)
k
u
n
(
t
)
−
u
(
t
)
k
2
p
+1
6
C
(1+
k
u
n
(
t
)
k
p
−
1
V
2
+
k
u
(
t
)
k
p
−
1
V
2
)
k
u
n
(
t
)
−
u
(
t
)
k
2
V
2
6
C
0
.
(5.24)
DOI:10.12677/aam.2023.1252382360
A^
ê
Æ
?
Ð
H
a
§
à
d
(5.14)
Œ
,
n
→∞
,m
→∞
ž
k
lim
n
→∞
lim
m
→∞
Z
T
τ
h
f
(
u
n
)
,∂
t
u
m
i
d
s
=lim
n
→∞
Z
T
τ
h
f
(
u
n
)
,∂
t
u
i
d
s
=
Z
T
τ
h
f
(
u
)
,∂
t
u
i
d
s
=
Z
Ω
F
(
u
(
T
))d
x
−
Z
Ω
F
(
u
(
τ
))d
x.
(5.25)
a
q
/
,
k
lim
n
→∞
lim
m
→∞
Z
T
τ
h
f
(
u
m
)
,∂
t
u
n
i
d
s
=
Z
Ω
F
(
u
(
T
))d
x
−
Z
Ω
F
(
u
(
τ
))d
x.
(5
.
26)
(
Ü
(5.23)-(5.26)
Œ
lim
n
→∞
lim
m
→∞
Z
T
τ
Z
Ω
(
f
(
u
n
)
−
f
(
u
m
))(
∂
t
u
n
−
∂
t
u
m
)d
x
d
s
= 0(5
.
27)
é
?
¿
½
T
,
|
R
T
s
R
Ω
(
f
(
u
n
)
−
f
(
u
m
))(
∂
t
u
n
−
∂
t
u
m
)d
x
d
t
|
´
k
.
,
?
d
Lebesgue
›
›
Â
ñ
½
n
Œ
lim
n
→∞
lim
m
→∞
Z
T
τ
Z
T
s
Z
Ω
(
f
(
u
n
)
−
f
(
u
m
))(
∂
t
u
n
−
∂
t
u
m
)d
x
d
t
d
s
=
Z
T
τ
( lim
n
→∞
lim
m
→∞
Z
T
s
Z
Ω
(
f
(
u
n
)
−
f
(
u
m
))(
∂
t
u
n
−
∂
t
u
m
)d
x
d
t
)d
s
=
Z
T
τ
0d
s
= 0
.
(5.28)
(
Ü
(5.27)
Ú
(5.28)
Œ
lim
n
→∞
lim
m
→∞
I
2
= 0
.
(5
.
29)
n
þ
Œ
Φ
T
τ
∈C
(
B
τ
(
R
0
))
.
½
n
5.2
X
J
^
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(C
1
)-(C
3
)
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1)
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(
t,τ
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H
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DOI:10.12677/aam.2023.1252382361
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z
[1]Woinowsky-Krieger,S.(1950)TheEffectofanAxialForceontheVibrationofHingedBars.
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ù
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å
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