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AdvancesinAppliedMathematics
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PublishedOnlineJuly2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.117483
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ThePullbackAttractorsfor
Non-AutonomousKirchhoff-Type
WaveEquationwithStrong
Damping
KaihongTian,XuanWang
∗
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Received:Jun.11
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,2021;accepted:Jul.7
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,2022;published:Jul.14
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,2022
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Abstract
Inthispaper,westudytheasymptoticbehaviorofsolutionsofNon-autonomous
Kirchhoff-type wave equations with strongdamping and nonlinear perturbations.The
existenceoftime-dependentpullbackattractorsisverifiedbyusingcontractionfunc-
tionandasymptoticpriorestimation.
Keywords
ContractionFunction,TheTimeDependentPullbackAttractors,StrongDamping,
KirchhoffWaveEquation
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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DOI:10.12677/aam.2022.1174834566
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DOI:10.12677/aam.2022.1174834567
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DOI:10.12677/aam.2022.1174834568
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DOI:10.12677/aam.2022.1174834569
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t
)
|
ω
t
(
t
)
|
2
+
1
2
|∇
ω
(
t
)
|
2
+
δ
4
(
k∇
u
1
k
2
+
k∇
u
2
k
2
)
|∇
ω
(
t
)
|
2
]d
x.
‰
(3
.
1)
ª
¦
±
e
γt
ω
t
(
t
)
,
k
d
d
t
[
e
γt
K
ω
(
t
)]
−
1
2
ε
0
(
t
)
k
ω
t
(
t
)
k
2
+
h
h
(
u
1
t
)
−
h
(
u
2
t
)
,e
γt
ω
t
(
t
)
i
=
−
δ
2
e
γt
h∇
(
u
1
+
u
2
)
,
∇
ω
(
t
)
ih∇
(
u
1
+
u
2
)
,
∇
ω
t
(
t
)
i
+
γe
γt
K
ω
(
t
)
−
e
γt
k∇
ω
t
(
t
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k
2
−
e
γt
h
f
(
u
1
)
−
f
(
u
2
)
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t
(
t
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i
.
(3.2)
3
[
s,t
]
×
Ω
þ
È
©
,
¿
d
ε
0
(
t
)
<
0
9
h
5
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,
k
e
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ω
(
t
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−
e
γs
K
ω
(
s
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6
−
δ
2
Z
t
s
e
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h∇
(
u
1
+
u
2
)
,
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ω
(
ζ
)
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(
u
1
+
u
2
)
,
∇
ω
t
(
ζ
)
i
d
ζ
+
γ
Z
t
s
e
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K
ω
(
ζ
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ζ
−
Z
t
s
Z
Ω
e
γζ
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ω
t
(
ζ
)
|
2
d
x
d
ζ
−
Z
t
s
Z
Ω
e
γζ
(
f
(
u
1
)
−
f
(
u
2
))
ω
t
(
ζ
)d
x
d
ζ.
(3.3)
DOI:10.12677/aam.2022.1174834570
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X
p
õ
§
à
2
é
(3.3)
ª
'
u
s
3
[
t
−
τ,t
]
þ
È
©
,
k
τe
γt
K
ω
(
t
)
−
Z
t
t
−
τ
e
γs
K
ω
(
s
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s
6
+
γ
Z
t
t
−
τ
Z
t
s
e
γζ
K
ω
(
ζ
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ζ
d
s
−
Z
t
t
−
τ
Z
t
s
Z
Ω
e
γζ
(
f
(
u
1
)
−
f
(
u
2
))
ω
t
(
ζ
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x
d
ζ
d
s
−
δ
2
Z
t
t
−
τ
Z
t
s
e
γζ
h∇
(
u
1
+
u
2
)
,
∇
ω
(
ζ
)
ih∇
(
u
1
+
u
2
)
,
∇
ω
t
(
ζ
)
i
d
ζ
d
s
−
Z
t
t
−
τ
Z
t
s
Z
Ω
e
γζ
k∇
ω
t
(
ζ
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k
2
d
x
d
ζ
d
s.
(3.4)
,
,
‰
(3
.
1)
ª
¦
±
e
γt
ω
(
t
)
,
=
d
d
t
[
e
γt
(
ε
(
t
)
k
ω
t
(
t
)
k
2
+
1
2
k∇
ω
(
t
)
k
2
)]+
δ
2
e
γt
(
k∇
u
1
k
2
+
k∇
u
1
k
2
)
k∇
ω
(
t
)
k
2
=
−
δ
2
e
γt
h∇
(
u
1
+
u
2
)
,
∇
ω
(
t
)
i
2
+
ε
0
(
t
)
ω
t
(
t
)
ω
(
t
)
−h
h
(
u
1
t
)
−
h
(
u
2
t
)
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ω
(
t
)
i
−
e
γt
h
f
(
u
1
)
−
f
(
u
2
)
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(
t
)
i
+
γe
γt
[
ε
(
t
)
k
ω
t
(
t
)
k
2
+
1
2
k∇
ω
(
t
)
k
2
]
.
(3.5)
é
þ
ª
3
[
s,t
]
×
Ω
þ
È
©
,
¿
d
h
5
Ÿ
,
k
δ
2
Z
t
s
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Ω
e
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(
k∇
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1
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2
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k∇
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1
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2
)
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ω
(
ζ
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|
2
d
x
d
ζ
+
Z
t
s
Z
Ω
e
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(
ε
(
t
)
|
ω
t
(
ζ
)
|
2
+
1
2
|∇
ω
(
ζ
)
|
2
)d
x
d
ζ
6
−
Z
Ω
e
γt
(
ε
(
t
)
ω
t
(
t
)
ω
(
t
)+
1
2
|∇
ω
(
t
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|
2
)d
x
+
Z
Ω
e
γs
(
ε
(
s
)
ω
t
(
s
)
ω
(
s
)+
1
2
|∇
ω
(
s
)
|
2
)d
x
−
δ
2
Z
t
s
Z
Ω
e
γζ
h∇
(
u
1
+
u
2
)
,
∇
ω
i
2
d
x
d
ζ
−
Z
t
s
Z
Ω
e
γζ
(
f
(
u
1
)
−
f
(
u
2
))
ω
(
ζ
)d
x
d
ζ
+
γ
Z
t
s
Z
Ω
e
γζ
(
ε
(
t
)
|
ω
t
(
ζ
)
|
2
+
1
2
|∇
ω
(
ζ
)
|
2
)d
x
d
ζ
+
Z
t
s
Z
Ω
ε
0
(
ζ
)
e
γζ
ω
t
(
ζ
)
ω
(
ζ
)d
x
d
ζ.
(3.6)
é
(3.6)
ª
'
u
s
3
[
t
−
τ,t
]
þ
È
©
¿
¦
±
γ
,
k
Z
t
t
−
τ
Z
t
s
Z
Ω
(
γδ
2
e
γζ
(
k∇
u
1
k
2
+
k∇
u
1
k
2
)
|∇
ω
(
ζ
)
|
2
+
γe
γζ
(
ε
(
t
)
|
ω
t
(
ζ
)
|
2
+
1
2
|∇
ω
(
ζ
)
|
2
))d
x
d
ζ
d
s
6
−
δγ
2
Z
t
t
−
τ
Z
t
s
e
γζ
h∇
(
u
1
+
u
2
)
,
∇
ω
i
2
d
ζ
d
s
−
γ
Z
t
t
−
τ
Z
t
s
Z
Ω
e
γζ
(
f
(
u
1
)
−
f
(
u
2
))
ω
(
ζ
)d
x
d
ζ
d
s
−
γτ
Z
Ω
e
γζ
(
ε
(
t
)
ω
t
(
t
)
ω
(
t
)+
1
2
|∇
ω
(
t
)
|
2
)d
x
+
γ
Z
t
t
−
τ
Z
t
s
Z
Ω
ε
0
(
ζ
)
ω
t
(
ζ
)
ω
(
ζ
)d
x
d
ζ
d
s
+
γ
Z
t
t
−
τ
Z
t
s
Z
Ω
e
γζ
(
ε
(
t
)
|
ω
t
(
ζ
)
|
2
+
1
2
|∇
ω
(
ζ
)
|
2
)d
x
d
ζ
d
s
+
γ
Z
t
t
−
τ
Z
Ω
e
γζ
(
ε
(
s
)
ω
t
(
s
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ω
(
s
)+
1
2
|∇
ω
(
s
)
|
2
)d
x
d
s.
(3.7)
DOI:10.12677/aam.2022.1174834571
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X
p
õ
§
à
ò
(3.7)
ª
“
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(3.4)
ª
,
k
e
γt
K
ω
(
t
)
−
Z
t
t
−
τ
e
γs
K
ω
(
s
)d
s
6
−
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4
Z
t
t
−
τ
Z
t
s
e
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h∇
(
u
1
+
u
2
)
,
∇
ω
i
2
d
ζ
d
s
−
Z
t
s
Z
Ω
e
γζ
(
f
(
u
1
)
−
f
(
u
2
))
ω
t
(
ζ
)d
x
d
ζ
−
γτ
2
Z
Ω
e
γζ
(
ε
(
t
)
ω
t
(
t
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ω
(
t
)+
1
2
|∇
ω
(
t
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|
2
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x
+
γ
2
Z
t
t
−
τ
Z
t
s
Z
Ω
ε
0
(
ζ
)
ω
t
(
ζ
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ω
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ζ
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x
d
ζ
d
s
−
Z
t
s
Z
Ω
e
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ω
t
(
ζ
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|
2
d
x
d
ζ
−
Z
t
t
−
τ
Z
t
s
Z
Ω
e
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(
f
(
u
1
)
−
f
(
u
2
))
ω
(
ζ
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x
d
ζ
d
s
−
δ
2
Z
t
s
e
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h∇
(
u
1
+
u
2
)
,
∇
ω
(
ζ
)
ih∇
(
u
1
+
u
2
)
,
∇
ω
t
(
ζ
)
i
d
ζ
+
γ
2
2
Z
t
t
−
τ
Z
t
s
Z
Ω
e
γζ
(
ε
(
t
)
|
ω
t
(
ζ
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2
+
1
2
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ω
(
ζ
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|
2
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x
d
ζ
d
s
+
γ
2
Z
t
t
−
τ
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Ω
e
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(
ε
(
s
)
ω
t
(
s
)
ω
(
s
)+
1
2
|∇
ω
(
s
)
|
2
)d
x
d
s.
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3
[
t
−
τ,t
]
þ
È
©
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ª
,
¿
d
h
5
Ÿ
,
k
δ
2
Z
t
t
−
τ
Z
Ω
e
γs
(
k∇
u
1
k
2
+
k∇
u
2
k
2
)
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ω
(
s
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|
2
d
x
d
s
6
−
δ
2
Z
t
t
−
τ
Z
Ω
e
γs
h∇
(
u
1
+
u
2
)
,
∇
ω
i
2
d
x
d
s
−
Z
t
t
−
τ
Z
Ω
e
γs
(
f
(
u
1
)
−
f
(
u
2
))
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(
s
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x
d
s
+
γ
Z
t
t
−
τ
Z
Ω
e
γs
(
ε
(
s
)
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ω
t
(
s
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2
+
1
2
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ω
(
s
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|
2
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x
d
s
−
Z
t
t
−
τ
Z
Ω
e
γs
ε
(
s
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|
ω
t
(
s
)
|
2
d
x
d
s
−
Z
Ω
e
γt
(
ε
(
t
)
ω
t
(
t
)
ω
(
t
)+
1
2
|∇
ω
(
t
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|
2
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x
+
Z
t
t
−
τ
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ε
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t
(
s
)
ω
(
s
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x
d
s
+
Z
Ω
e
γ
(
t
−
τ
)
(
ε
(
t
−
τ
)
ω
t
(
t
−
τ
)
ω
(
t
−
τ
)+
1
2
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ω
(
t
−
τ
)
|
2
)d
x.
(3.7)
ò
(3.7)
ª
“
\
(3.6)
ª
,
k
e
γt
K
ω
(
t
)+
Z
t
t
−
τ
e
γs
K
ω
(
s
)d
s
6
−
γτ
2
Z
Ω
e
γζ
(
ε
(
t
)
ω
t
(
t
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ω
(
t
)+
1
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ω
(
t
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|
2
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x
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γ
2
Z
t
t
−
τ
Z
t
s
Z
Ω
ε
0
(
ζ
)
ω
t
(
ζ
)
ω
(
ζ
)d
x
d
ζ
d
s
−
δγ
4
Z
t
t
−
τ
Z
t
s
e
γζ
h∇
(
u
1
+
u
2
)
,
∇
ω
i
2
d
ζ
d
s
−
Z
t
s
Z
Ω
e
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(
f
(
u
1
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−
f
(
u
2
))
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t
(
ζ
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x
d
ζ
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δ
2
Z
t
t
−
τ
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e
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1
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2
d
x
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s
−
Z
t
t
−
τ
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Ω
e
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(
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(
u
1
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f
(
u
2
))
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(
s
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x
d
s
+
γ
Z
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t
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τ
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(
s
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t
(
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ω
(
s
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2
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x
d
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Z
t
t
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τ
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Ω
e
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ε
(
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t
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t
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ω
t
(
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2
d
x
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Z
t
t
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τ
Z
t
s
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Ω
e
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(
f
(
u
1
)
−
f
(
u
2
))
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(
ζ
)d
x
d
ζ
d
s
DOI:10.12677/aam.2022.1174834572
A^
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Ð
X
p
õ
§
à
−
Z
Ω
e
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(
ε
(
t
)
ω
t
(
t
)
ω
(
t
)+
1
2
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(
t
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|
2
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+
Z
t
t
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ε
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)
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t
(
s
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ω
(
s
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s
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e
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(
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−
τ
)
(
ε
(
t
−
τ
)
ω
t
(
t
−
τ
)
ω
(
t
−
τ
)+
1
2
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(
t
−
τ
)
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2
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x
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δ
2
Z
t
s
e
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u
1
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u
2
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(
ζ
)
ih∇
(
u
1
+
u
2
)
,
∇
ω
t
(
ζ
)
i
d
ζ
+
γ
2
2
Z
t
t
−
τ
Z
t
s
Z
Ω
e
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(
ε
(
t
)
|
ω
t
(
ζ
)
|
2
+
1
2
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ω
(
ζ
)
|
2
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x
d
ζ
d
s
+
γ
2
Z
t
t
−
τ
Z
Ω
e
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(
ε
(
s
)
ω
t
(
s
)
ω
(
s
)+
1
2
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ω
(
s
)
|
2
)d
x
d
s.
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é
(3.2)
ª
3
[
t
−
τ,t
]
þ
È
©
,
¿
d
ε
(
t
)
,h
(
u
t
)
5
Ÿ
,
Œ
−
Z
t
t
−
τ
e
γs
K
ω
(
s
)d
s
6
−
1
γ
e
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K
ω
(
t
)+
1
γ
e
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(
t
−
τ
)
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ω
(
t
−
τ
)
−
1
γ
Z
t
t
−
τ
Z
Ω
e
γs
(
f
(
u
1
)
−
f
(
u
2
))
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t
(
s
)d
x
d
s
−
δ
2
γ
Z
t
t
−
τ
e
γs
h∇
(
u
1
+
u
2
)
,
∇
ω
(
s
)
ih∇
(
u
1
+
u
2
)
,
∇
ω
t
(
s
)
i
d
s.
(3.9)
ò
(3.9)
ª
‘
\
(3.8)
ª
,
k
τe
γt
K
ω
(
t
)
6
−
γτ
2
Z
Ω
e
γζ
(
ε
(
t
)
ω
t
(
t
)
ω
(
t
)+
1
2
|∇
ω
(
t
)
|
2
)d
x
+
γ
2
Z
t
t
−
τ
Z
t
s
Z
Ω
ε
0
(
ζ
)
ω
t
(
ζ
)
ω
(
ζ
)d
x
d
ζ
d
s
−
δγ
4
Z
t
t
−
τ
Z
t
s
e
γζ
h∇
(
u
1
+
u
2
)
,
∇
ω
i
2
d
ζ
d
s
−
Z
t
s
Z
Ω
e
γζ
(
f
(
u
1
)
−
f
(
u
2
))
ω
t
(
ζ
)d
x
d
ζ
−
δ
2
Z
t
t
−
τ
Z
Ω
e
γs
h∇
(
u
1
+
u
2
)
,
∇
ω
i
2
d
x
d
s
−
Z
t
t
−
τ
Z
Ω
e
γs
(
f
(
u
1
)
−
f
(
u
2
))
ω
(
s
)d
x
d
s
+
γ
Z
t
t
−
τ
Z
Ω
e
γs
(
ε
(
s
)
|
ω
t
(
s
)
|
2
+
1
2
|∇
ω
(
s
)
|
2
)d
x
d
s
−
Z
t
t
−
τ
Z
Ω
e
γs
ε
(
s
)
|
ω
t
(
s
)
|
2
d
x
d
s
−
1
γ
e
γt
K
ω
(
t
)+
1
γ
e
γ
(
t
−
τ
)
K
ω
(
t
−
τ
)
−
1
γ
Z
t
t
−
τ
Z
Ω
e
γs
(
f
(
u
1
)
−
f
(
u
2
))
ω
t
(
s
)d
x
d
s
−
Z
t
s
Z
Ω
e
γζ
|∇
ω
t
(
ζ
)
|
2
d
x
d
ζ
−
Z
t
t
−
τ
Z
t
s
Z
Ω
e
γζ
(
f
(
u
1
)
−
f
(
u
2
))
ω
(
ζ
)d
x
d
ζ
d
s
−
Z
Ω
e
γt
(
ε
(
t
)
ω
t
(
t
)
ω
(
t
)+
1
2
|∇
ω
(
t
)
|
2
)d
x
+
Z
t
t
−
τ
Z
Ω
ε
0
(
s
)
ω
t
(
s
)
ω
(
s
)d
x
d
s
+
Z
Ω
e
γ
(
t
−
τ
)
(
ε
(
t
−
τ
)
ω
t
(
t
−
τ
)
ω
(
t
−
τ
)+
1
2
|∇
ω
(
t
−
τ
)
|
2
)d
x
−
δ
2
γ
Z
t
t
−
τ
e
γs
h∇
(
u
1
+
u
2
)
,
∇
ω
(
s
)
ih∇
(
u
1
+
u
2
)
,
∇
ω
t
(
s
)
i
d
s
−
δ
2
Z
t
s
e
γζ
h∇
(
u
1
+
u
2
)
,
∇
ω
(
ζ
)
ih∇
(
u
1
+
u
2
)
,
∇
ω
t
(
ζ
)
i
d
ζ
+
γ
2
2
Z
t
t
−
τ
Z
t
s
Z
Ω
e
γζ
(
ε
(
t
)
|
ω
t
(
ζ
)
|
2
+
1
2
|∇
ω
(
ζ
)
|
2
)d
x
d
ζ
d
s
+
γ
2
Z
t
t
−
τ
Z
Ω
e
γζ
(
ε
(
s
)
ω
t
(
s
)
ω
(
s
)+
1
2
|∇
ω
(
s
)
|
2
)d
x
d
s.
(3.10)
DOI:10.12677/aam.2022.1174834573
A^
ê
Æ
?
Ð
X
p
õ
§
à
(
Ü
(3.8)
ª
-(3.10)
ª
,
Œ
Φ
t,τ
((
u
1
0
,v
1
0
)
,
(
u
2
0
,v
2
0
))
=
−
(
γ
2
τ
+
1
τ
)
Z
Ω
e
γζ
(
ε
(
t
)
ω
t
(
t
)
ω
(
t
)+
1
2
|∇
ω
(
t
)
|
2
)d
x
+
γ
2
τ
Z
t
t
−
τ
Z
t
s
Z
Ω
ε
0
(
ζ
)
ω
t
(
ζ
)
ω
(
ζ
)d
x
d
ζ
d
s
+
γ
2
τ
Z
t
t
−
τ
Z
Ω
e
γζ
(
ε
(
s
)
ω
t
(
s
)
ω
(
s
)+
1
2
|∇
ω
(
s
)
|
2
)d
x
d
s
+
1
τ
Z
t
t
−
τ
Z
Ω
ε
0
(
s
)
ω
t
(
s
)
ω
(
s
)d
x
d
s
−
δγ
4
τ
Z
t
t
−
τ
Z
t
s
e
γζ
h∇
(
u
1
+
u
2
)
,
∇
ω
i
2
d
ζ
d
s
−
1
τ
Z
t
s
Z
Ω
e
γζ
(
f
(
u
1
)
−
f
(
u
2
))
ω
t
(
ζ
)d
x
d
ζ
+
γ
τ
Z
t
t
−
τ
Z
Ω
e
γs
(
ε
(
s
)
|
ω
t
(
s
)
|
2
+
1
2
|∇
ω
(
s
)
|
2
)d
x
d
s
−
1
τ
Z
t
t
−
τ
Z
Ω
e
γs
ε
(
s
)
|
ω
t
(
s
)
|
2
d
x
d
s
−
1
τ
Z
t
s
Z
Ω
e
γζ
|∇
ω
t
(
ζ
)
|
2
d
x
d
ζ
−
1
τ
Z
t
t
−
τ
Z
t
s
Z
Ω
e
γζ
(
f
(
u
1
)
−
f
(
u
2
))
ω
(
ζ
)d
x
d
ζ
d
s
−
δ
2
τ
Z
t
t
−
τ
e
γs
h∇
(
u
1
+
u
2
)
,
∇
ω
i
2
d
s
−
1
τ
Z
t
t
−
τ
Z
Ω
e
γs
(
f
(
u
1
)
−
f
(
u
2
))
ω
(
s
)d
x
d
s
−
δ
2
γτ
Z
t
t
−
τ
e
γs
h∇
(
u
1
+
u
2
)
,
∇
ω
(
s
)
ih∇
(
u
1
+
u
2
)
,
∇
ω
t
(
s
)
i
d
s
−
δ
2
τ
Z
t
s
e
γζ
h∇
(
u
1
+
u
2
)
,
∇
ω
(
ζ
)
ih∇
(
u
1
+
u
2
)
,
∇
ω
t
(
ζ
)
i
d
ζ
−
1
τγ
e
γt
K
ω
(
t
)
−
1
γτ
Z
t
t
−
τ
Z
Ω
e
γs
(
f
(
u
1
)
−
f
(
u
2
))
ω
t
(
s
)d
x
d
s
+
γ
2
2
τ
Z
t
t
−
τ
Z
t
s
Z
Ω
e
γζ
(
ε
(
t
)
|
ω
t
(
ζ
)
|
2
+
1
2
|∇
ω
(
ζ
)
|
2
)d
x
d
ζ
d
s.
(3.11)
l
K
0
ω
(
t
)
6
C
e
−
βτ
τ
˜
R
2
t
−
τ
+Φ
t,τ
((
u
1
0
)(
v
1
0
)
,
(
u
2
0
)(
v
2
0
))
,
(3.12)
Ù
¥
˜
R
2
t
−
τ
=
R
Ω
e
γ
(
t
−
τ
)
(
ε
(
t
−
τ
)
ω
t
(
t
−
τ
)
ω
(
t
−
τ
)+
1
2
|∇
ω
(
t
−
τ
)
|
2
)d
x
+
1
γ
e
γ
(
t
−
τ
)
K
ω
(
t
−
τ
)
.
-
(
u
n
,u
n
t
)
•
é
A
Ð
©
Š
(
u
i
0
,u
i
1
0
)
∈
˜
D
t
−
τ
0
×
˜
D
t
−
τ
0
)
.
®
•
k∇
u
n
k
2
´
k
.
.
d
(1.2)-(1.3)
ª
•
:
é
u
?
¿
½
t
−
τ
0
,ζ
∈
[
t
−
τ
0
,t
]
,ε
(
ζ
)
k
.
,
ε
(
ζ
)
k
u
n
t
k
2
k
.
.
Ï
d
k∇
u
n
k
2
+
ε
(
ζ
)
k
u
n
t
k
2
k
.
.
Š
â
Alaoglu
½
n
,
k
±
e
(
J
:
u
n
→
u
3
L
∞
(
τ,T
;
H
1
0
(Ω))
¥
f
∗
Â
ñ
,(3.13)
u
n
t
→
u
t
3
L
∞
(
τ,T
;
L
2
(Ω))
¥
f
∗
Â
ñ
,(3.14)
u
n
→
u
3
L
p
+1
(
τ,T
;
L
p
+1
(Ω))
¥
r
Â
ñ
,(3.15)
u
n
→
u
3
L
2
(
τ,T
;
L
2
(Ω))
¥
r
Â
ñ
.(3.16)
e
¡
O
(3
.
11)
ª
¥
z
˜
‘
.
d
(3
.
13)-(3
.
15)
ª
,
k
lim
n
→∞
lim
m
→∞
Z
Ω
e
γζ
(
ε
(
t
)(
∂
t
u
n
−
∂
t
u
m
)(
u
n
−
u
m
)+
1
2
|∇
(
u
n
−
u
m
)
|
2
)d
x
= 0
.
(3
.
17)
DOI:10.12677/aam.2022.1174834574
A^
ê
Æ
?
Ð
X
p
õ
§
à
lim
n
→∞
lim
m
→∞
Z
t
t
−
τ
Z
Ω
e
γζ
L
(
∂
t
u
n
−
∂
t
u
m
)(
u
n
−
u
m
)d
x
d
s
= 0
.
(3
.
18)
lim
n
→∞
lim
m
→∞
Z
t
t
−
τ
Z
t
s
Z
Ω
L
(
∂
t
u
n
−
∂
t
u
m
)(
u
n
−
u
m
)d
x
d
ζ
d
s
= 0
.
(3
.
19)
lim
n
→∞
lim
m
→∞
Z
t
t
−
τ
Z
Ω
e
γζ
(
L
(
∂
t
u
n
−
∂
t
u
m
)(
u
n
−
u
m
)+
1
2
|∇
(
u
n
−
u
m
)
|
2
)d
x
d
s
= 0
.
(3
.
20)
lim
n
→∞
lim
m
→∞
Z
t
t
−
τ
Z
t
s
Z
Ω
e
γζ
(
L
|
∂
t
u
n
−
∂
t
u
m
|
2
+
1
2
|∇
ω
(
u
n
−
u
m
)
|
2
)d
x
d
ζ
d
s
= 0
.
(3
.
21)
lim
n
→∞
lim
m
→∞
Z
t
t
−
τ
e
γs
h∇
(
u
1
+
u
2
)
,
∇
(
u
n
−
u
m
)
i
2
d
s
= 0
.
(3
.
22)
lim
n
→∞
lim
m
→∞
Z
t
t
−
τ
Z
t
s
e
γζ
h∇
(
u
1
+
u
2
)
,
∇
(
u
n
−
u
m
)
i
2
d
ζ
d
s
= 0
.
(3
.
23)
lim
n
→∞
lim
m
→∞
Z
t
s
e
γζ
h∇
(
u
1
+
u
2
)
,
∇
(
u
n
−
u
m
)
ih∇
(
u
1
+
u
2
)
,
∇
(
∂
t
u
n
−
∂
t
u
m
)
i
d
ζ
= 0
.
(3
.
24)
lim
n
→∞
lim
m
→∞
Z
t
t
−
τ
e
γs
h∇
(
u
1
+
u
2
)
,
∇
(
u
n
−
u
m
)
ih∇
(
u
1
+
u
2
)
,
∇
(
∂
t
u
n
−
∂
t
u
m
)
i
d
s
= 0
.
(3
.
25)
lim
n
→∞
lim
m
→∞
Z
t
t
−
τ
Z
Ω
e
γs
(
L
|
(
∂
t
u
n
−
∂
t
u
m
)
|
2
+
1
2
|∇
(
u
n
−
u
m
)
|
2
)d
x
d
s
= 0
.
(3
.
26)
lim
n
→∞
lim
m
→∞
Z
t
t
−
τ
Z
Ω
e
γζ
(
f
(
u
n
)
−
f
(
u
m
))(
u
n
−
u
m
)d
x
d
ζ
= 0
.
(3
.
27)
lim
n
→∞
lim
m
→∞
Z
t
t
−
τ
Z
t
s
Z
Ω
e
βζ
(
f
(
u
n
)
−
f
(
u
m
))(
u
n
−
u
m
)d
x
d
ζ
d
s
= 0
.
(3
.
28)
d
(2.8)
ª
±
9
H¨older
Ø
ª
,
k
|
Z
t
t
−
τ
Z
Ω
e
βζ
(
f
(
u
n
)
−
f
(
u
m
))
·
(
u
n
t
−
u
m
t
)d
x
d
ζ
|
6
Z
t
t
−
τ
Z
Ω
e
βζ
C
(1+
|
u
n
|
p
−
1
+
|
u
m
|
p
−
1
)
·|
u
n
−
u
m
|·|
u
n
t
−
u
m
t
|
d
x
d
ζ
6
Z
t
t
−
τ
e
βζ
(
C
(1+
k
u
n
k
p
−
1
L
2(
p
+1)
(Ω)
+
k
u
m
k
p
−
1
L
2(
p
+1)
(Ω)
)
·k
u
n
−
u
m
k
L
p
+1
(Ω)
·k
u
n
t
−
u
m
t
k
L
2
(Ω)
d
ζ
6
Z
t
t
−
τ
e
βξ
C
(1+
k
u
n
(
s
)
k
p
−
1
V
1
+
k
u
m
(
s
)
k
p
−
1
V
1
)
·k
u
n
(
s
)
−
u
m
(
s
)
k
L
p
+1
(Ω)
·k
u
n
t
−
u
m
t
k
L
2
(Ω)
d
ζ,
lim
n
→∞
lim
m
→∞
Z
t
t
−
τ
Z
Ω
e
βζ
(
f
(
u
n
)
−
f
(
u
m
))(
u
n
t
−
u
m
t
)d
x
d
ζ
= 0
.
(3
.
29)
DOI:10.12677/aam.2022.1174834575
A^
ê
Æ
?
Ð
X
p
õ
§
à
a
q
/
,
é
u
½
t
,
|
R
t
s
R
Ω
e
βζ
(
f
(
u
n
(
ξ
))
−
f
(
u
m
(
ζ
)))(
u
n
t
(
ζ
)
−
u
m
t
(
ζ
))d
x
d
ζ
|
k
.
,
d
V
‚
›
›
Â
ñ
½
n
Œ
lim
n
→∞
lim
m
→∞
Z
t
t
−
τ
Z
t
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