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PureMathematics
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,2021,11(8),1546-1558
PublishedOnlineAugust2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.118173
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TheWell-PosednessofaMemory-Type
EvolutionEquationwithNonclassical
Dissipation
XimengLiu
∗
,DiLiu,JiangweiZhang
SchoolofMathematicsandStatistics,ChangshaUniversityofScienceandTechnology,Changsha
Hunan
Received:Jul.15
th
,2021;accepted:Aug.18
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[J].
n
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,2021,11(8):
1546-1558.DOI:10.12677/pm.2021.118173
4
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Abstract
Inthispaper,wemainlydiscussthewell-posednessproblemofaMemory-typeEvolu-
tionEquationwithnonclassicaldissipation.Theexistenceofweaksolutionisobtained
byusingtheGalerkin’smethodandanalyticaltechniques.Also,weprovetheunique-
nessofthesolutionandthecontinuousdependenceoninitialvalue.
Keywords
EvolutionEquation,Memory,NonclassicalDissipation,Well-Posedness
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
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ψ
j
(
x
)
,
u
mt
(
x,t
) =
∂
t
u
m
(
x,t
) =
m
X
j
=1
∂
t
ϕ
j
(
t
)
ψ
j
(
x
)
,
η
t
m
(
x,t
) =
m
X
j
=1
κ
j
(
t
)
ϑ
j
(
x
)
.
DOI:10.12677/pm.2021.1181731550
n
Ø
ê
Æ
4
Ü
†
Ù
¥
ϕ
j
(
t
) =
h
u,ψ
j
i
,κ
j
=
h
η
t
,ϑ
j
i
M
,
…
z
m
= (
u
m
,u
mt
,η
t
m
)
÷
v
u
mtt
−
M
(
|∇
u
m
|
2
2
)∆
u
m
−
Z
∞
0
µ
(
s
)∆
η
t
m
(
s
)d
s
+
αu
3
mt
−
βu
mt
+
f
(
u
m
) =
g
m
(
x
)
,
η
t
mt
=
−
η
t
ms
+
u
mt
,
(
u
m
,η
t
m
)
|
∂
Ω
×
R
+
= (0
,
0)
,
u
m
(0) =
m
X
j
=1
ψ
j
(
x
)
ξ
j
,u
mt
(0) =
m
X
j
=1
ς
j
ψ
j
,η
t
m
(0) =
m
X
j
=1
η
j
ϑ
j
.
(5)
A
O
/
,
m
→∞
ž
,
k
m
X
j
=1
(
u
0
,ψ
j
)
ψ
j
(
x
) =
m
X
j
=1
ξ
j
ψ
j
(
x
)
→
u
0
∈
H
1
0
(Ω)
,
m
X
j
=1
(
u
1
,ψ
j
)
ψ
j
(
x
) =
m
X
j
=1
ς
j
ψ
j
(
x
)
→
u
1
∈
L
2
(Ω)
,
m
X
j
=1
h
η
0
,ϑ
j
i
M
ϑ
j
(
x
) =
m
X
j
=1
η
j
ϑ
j
(
x
)
→
η
0
∈M
.
Ï
•
{
ψ
j
(
x
)
}
+
∞
j
=1
´
3
þ
ã
˜
m
¥
È
—
5
,
Œ
•
þ
ãÂ
ñ
´
¤
á
.
Ú
n
3.2
b
^
‡
(
H
1
)
-
(
H
2
)
¤
á
,
K
•
3
~
ê
C
T
>
0
,
¦
(5)
ª
)
z
m
(
t
) = (
u
m
(
t
)
,u
mt
(
t
)
,η
t
m
)
,
÷
v
|
u
mt
|
2
2
+
|∇
u
m
|
2
2
+
η
t
m
2
M
+
Z
t
0
k
η
s
m
k
2
M
ds
+
Z
t
0
|
u
mt
(
s
)
|
4
4
ds
≤
C
T
.
(6)
y
²
:
^
u
mt
¦
±
(5)
ª
,
¿
3
Ω
þ
È
©
,
Œ
1
2
d
d
t
|
u
mt
|
2
2
+
1
2
d
d
t
Z
|∇
u
m
|
2
2
0
M
(
s
)d
s
+
1
2
d
d
t
η
t
m
2
M
+
η
t
m
,η
t
ms
M
+
α
|
u
mt
|
4
4
−
β
|
u
mt
|
2
2
+
d
d
t
Z
Ω
b
f
(
u
m
)d
x
−
Z
Ω
g
m
(
x
)
u
mt
d
x
= 0
.
(7)
-
E
(
t
) =
|
u
mt
|
2
2
+
Z
|∇
u
m
|
2
2
0
M
(
s
)d
s
+
η
t
m
2
M
+2
Z
Ω
b
f
(
u
m
)d
x
−
2
Z
Ω
g
m
(
x
)
u
m
d
x.
(8)
¤
±
(7)
ª
Œ
z
•
1
2
d
d
t
E
(
t
)+
α
|
u
mt
|
4
4
−
β
|
u
mt
|
2
2
+
η
t
m
,η
t
ms
M
= 0
.
(9)
Ï
•
β
|
u
mt
|
2
2
=
β
Z
Ω
|
u
mt
|
2
≤
β
(
Z
Ω
|
u
mt
|
4
)
1
2
·
(
Z
Ω
1)
1
2
≤
β
(
|
u
mt
|
2
4
)
·|
Ω
|
1
2
≤
α
2
|
u
mt
|
4
4
+
β
2
2
α
|
Ω
|
.
(10)
DOI:10.12677/pm.2021.1181731551
n
Ø
ê
Æ
4
Ü
†
d
b
^
‡
(
H
2
)
Œ
•
:
η
t
m
,η
t
ms
M
=
Z
∞
0
µ
(
s
)
Z
Ω
∇
η
t
m
(
x,s
)
∇
η
t
ms
(
x,s
)d
x
d
s
=
1
2
Z
∞
0
µ
(
s
)
d
d
s
∇
η
t
m
(
x,s
)
2
d
s
= 0
−
1
2
Z
∞
0
µ
0
(
s
)
∇
η
t
m
(
x,s
)
2
ds
≥
δ
2
k
η
t
m
k
2
M
.
(11)
2
d
M
(
r
)
b
±
9
^
‡
(
H
1
)
Œ
•
:
Z
|∇
u
m
|
2
2
0
M
(
s
)d
s
≥
a
|∇
u
m
|
2
2
,
(12)
Z
Ω
b
f
(
u
m
)d
x
≥
0
,
(13)
,
,
|
^
Young
Ø
ª
Ú
Poinc´
a
re
Ø
ª
,
k
Z
Ω
g
m
(
x
)
u
m
d
x
≤
ε
λ
1
|∇
u
m
|
2
2
+
1
4
ε
|
g
m
|
2
2
.
(14)
Ù
¥
λ
1
´
Ž
f
−
∆
1
˜
A
Š
.
é
(9)
ª
'
u
t
3
[0
,T
]
þ
È
©
,
¿
(
Ü
(10)-(14),
Œ
:
|
u
mt
|
2
2
+
M
0
−
2
ε
λ
1
|∇
u
m
|
2
2
+
η
t
m
2
M
+
δ
Z
t
0
k
η
s
m
k
2
M
ds
+
α
Z
t
0
|
u
mt
(
s
)
|
4
4
ds
≤
β
2
α
|
Ω
|
+
|
u
m
1
|
2
2
+
M
0
−
2
ε
λ
1
|∇
u
m
0
|
2
2
+
k
η
m
0
k
2
M
.
(15)
ε
v
ž
,
-
λ
=
min
{
1
,M
0
−
2
ε
λ
1
,δ,α
}
>
0,
K
•
3
~
ê
C
T
>
0
Œ
¦
e
ª
¤
á
:
|
u
mt
|
2
2
+
|∇
u
m
|
2
2
+
η
t
m
2
M
+
Z
t
0
k
η
s
m
k
2
M
ds
+
Z
t
0
|
u
mt
(
s
)
|
4
4
ds
≤
C
T
.
(16)
y
.
œ
½
n
3.3
b
^
‡
(
H
1
)
–
(
H
2
)
¤
á
,
…
z
0
= (
u
0
,u
1
,η
0
)
∈H
.
K
•
§
•
3
N
f
)
z
= (
u,u
t
,η
t
)
÷
v
u
∈
L
2
([0
,T
];
H
1
0
(Ω))
,
u
t
∈
L
∞
([0
,T
];
L
2
(Ω))
∩
L
4
([0
,T
];
L
4
(Ω))
Ú
η
t
∈
L
2
([0
,T
];
L
2
µ
(
R
+
;
H
1
0
(Ω)))
.
y
²
µ
d
Ú
n
3.2,
Œ
•
u
m
3
L
2
([0
,T
];
H
1
0
(Ω))
¥
˜
—
k
.
,
u
mt
3
L
∞
([0
,T
];
L
2
(Ω))
∩
L
4
([0
,T
];
L
4
(Ω))
¥
˜
—
k
.
,
η
t
m
3
L
2
([0
,T
];
L
2
µ
(
R
+
;
H
1
0
(Ω)))
¥
˜
—
k
.
.
(17)
DOI:10.12677/pm.2021.1181731552
n
Ø
ê
Æ
4
Ü
†
Ï
d
,
•
3
{
u
m
}
f
(
E
P
•
{
u
m
}
),
¦
u
m
→
u
3
L
2
([0
,T
];
H
1
0
(Ω))
¥
f
Â
ñ
,
u
mt
→
u
t
3
L
4
([0
,T
];
L
4
(Ω))
¥
f
Â
ñ
,
u
mt
→
u
t
3
L
∞
([0
,T
];
L
2
(Ω))
¥
f
*
Â
ñ
,
η
t
m
*η
t
3
L
2
([0
,T
];
L
2
µ
(
R
+
;
H
1
0
(Ω)))
¥
f
Â
ñ
.
(18)
du
u
m
(
t
)
3
L
2
([0
,T
];
H
1
0
(Ω))
¥
˜
—
k
.
,
u
mt
(
t
)
3
L
∞
([0
,T
];
L
2
(Ω))
∩
L
4
([0
,T
];
L
4
(Ω))
⊂
L
2
([0
,T
];
L
2
(Ω))
¥
˜
—
k
.
,
Œ
•
u
m
(
t
)
•
H
1
(
Q
T
)
¥
k
.
8
.
q
H
1
(
Q
T
)
;
i
\
L
2
(
Q
T
),
K
•
3
{
u
m
(
t
)
}
f
(
E
P
•
{
u
m
}
),
÷
v
u
m
*u
3
L
2
(
Q
T
)
¥
r
Â
ñ
.
qd
(
H
1
)
Ú
Ú
n
3.2
Œ
Z
T
0
Z
Ω
|
f
(
u
m
)
|
q
dxdt
≤
C
0
Z
T
0
(1+
|
u
m
(
t
)
|
p
p
)
dt
≤
C
Z
T
0
(1+
|
u
m
(
t
)
|
p
2
p
+2
)
dt
≤
C
Z
T
0
(1+
|∇
u
m
(
t
)
|
p
2
)
dt
≤
ρ
0
.
Ù
¥
q
•
p
é
ó
ê
,
ρ
0
>
0
•
~
ê
.
K
•
â
þ
ã
f
(
u
m
)
˜
—
k
.
5
,
Œ
f
(
u
m
)
*χ
3
L
q
([0
,T
];Ω)
¥
f
Â
ñ
.
d
,
•
â
u
m
3
L
2
(
Q
T
)
¥
r
Â
ñ
5
,
Œ
•
u
m
→
u
(
m
→∞
)
3
Q
T
þ
A
? ?
Â
ñ
.
2
d
f
ë
Y5
,
Œ
f
(
u
m
(
x,t
))
*f
(
u
(
x,t
))
3
Q
T
þ
A
??
Â
ñ
.
q
L
q
(
Q
T
)
⊂
H
−
1
(
Q
T
),
K
d
Lebesgue
Å
‘
È
©
½
n
Œ
•
,
•
3
φ
∈
L
2
(0
,T
;
H
1
0
(Ω)),
¦
lim
m
→∞
Z
T
0
Z
Ω
f
(
u
m
)
φdxdt
=
Z
T
0
Z
Ω
f
(
u
)
φdxdt.
Œ
f
(
u
m
)
→
f
(
u
)
3
H
−
1
(
Q
T
)
¥
f
Â
ñ
,
d
f
4
•
•
˜
5
,
:
χ
=
f
(
u
).
φ
∈
C
∞
0
(0
,T
;
C
∞
0
(Ω)),
k
Z
t
0
h
u
mtt
,φ
i
dτ
=
Z
t
0
h
M
(
|∇
u
m
|
2
2
)∆
u
m
+
Z
+
∞
0
µ
(
s
)∆
η
t
m
(
s
)d
s
−
h
(
u
mt
)
−
f
(
u
m
)+
g
m
,φ
i
dτ.
(19)
DOI:10.12677/pm.2021.1181731553
n
Ø
ê
Æ
4
Ü
†
•
g
O
þ
ª
m
>
z
˜
‘
,
¿
(
Ü
Young
Ø
ª
,H¨
o
lder
Ø
ª
,
Ú
Ú
n
3.2,
Œ
Z
t
0
h
M
(
|∇
u
m
|
2
2
)∆
u
m
,φ
i
dτ
≤
Z
t
0
M
(
|∇
u
m
|
2
2
)
|∇
u
m
|
2
|∇
φ
|
2
dτ
≤
C
T
k∇
φ
k
L
2
(0
,T
;
H
1
0
(Ω))
.
(20)
Z
t
0
Z
+
∞
0
µ
(
s
)
h
∆
η
t
m
(
s
)
,φ
i
dsdτ
≤
Z
t
0
Z
+
∞
0
µ
(
s
)
|∇
η
t
m
(
s
)
|
2
|∇
φ
|
2
dsdτ
≤
K
1
2
0
Z
t
0
|∇
φ
|
2
k
η
t
m
k
M
dτ
≤
C
T
k∇
φ
k
L
2
(0
,T
;
H
1
0
(Ω))
.
(21)
Z
t
0
h−
αu
3
mt
,φ
i
dτ
≤
α
Z
t
0
|
u
mt
|
3
4
|
φ
|
4
dτ
≤
C
T
k∇
φ
k
L
4
(0
,T
;
L
4
(Ω))
≤
C
T
k∇
φ
k
L
2
(0
,T
;
L
2
(Ω))
.
(22)
Z
t
0
h
βu
mt
,φ
i
dτ
≤
β
Z
t
0
|
u
mt
|
2
|
φ
|
2
dτ
≤
C
T
k∇
φ
k
L
2
(0
,T
;
L
2
(Ω))
.
(23)
Z
t
0
h
f
(
u
m
)
,φ
i
dτ
≤
Z
t
0
k
f
(
u
m
)
k
H
−
1
k
φ
k
0
dτ
≤
C
T
k
φ
k
L
2
(0
,T
;
H
1
0
(Ω))
.
(24)
Z
t
0
h
g
m
,φ
i
dτ
≤
Z
t
0
|
g
m
|
2
|
φ
|
2
dτ
≤
C
T
k
φ
k
L
2
(0
,T
;
L
2
(Ω))
.
(25)
¤
±
(
Ü
(20)-(25),
Œ
Z
t
0
h
u
mtt
,φ
i
dτ
≤
C
T
(
k
φ
k
L
2
(0
,T
;
H
1
0
(Ω))
+
k
φ
k
L
2
(0
,T
;
L
2
(Ω))
)
.
K
u
mtt
∈
L
2
(0
,T
;
H
−
1
(Ω))
.
(26)
•
â
(17)
ª
Ú
(26)
Œ
•
u
mt
→
u
t
3
L
2
(
Q
T
)
¥
r
Â
ñ
.
(27)
DOI:10.12677/pm.2021.1181731554
n
Ø
ê
Æ
4
Ü
†
u
mt
→
u
t
3
Q
T
¥
A
??
Â
ñ
.
u
´
u
3
mt
→
u
3
t
3
Q
T
¥
A
??
¤
á
.
q
u
3
mt
→
χ
1
3
L
4
3
(
Q
T
)
¥
f
Â
ñ
.
d
f
4
•
•
˜
5
•
:
u
3
t
=
χ
1
.
,
,
du
g
m
(
x
)
∈
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2
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u
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x
∈
Ω,
k
g
m
→
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(
m
→∞
)
´
f
Â
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,
…
g
(
x
)
∈
L
2
(0
,T
;
L
2
(Ω)).
n
þ
¤
ã
,
m
→∞
ž
,
é
?
¿
ω
∈
H
1
0
(Ω)
,ξ
∈M
,
k
:
h
u
tt
,ω
i−h
M
(
|∇
u
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2
2
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u,
∇
ω
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Z
∞
0
µ
(
s
)
h∇
η
t
(
s
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,
∇
ω
i
d
s
+
α
h
u
3
t
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i−
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h
u
t
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i
+
h
f
(
u
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i
=
h
g
(
x
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i
.
h
η
t
t
,ξ
i
M
=
h−
η
t
s
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h
u
t
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i
M
.
e
5
©
O
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= (
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t
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÷
v
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©
^
‡
:
d
u
∈
L
2
(0
,T
;
H
1
0
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Ú
u
t
∈
L
2
(0
,T
;
L
2
(Ω)),
¤
±
k
u
∈
C
(0
,T
;
L
2
(Ω)).
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u
m
(0)
*u
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3
L
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¥
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u
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2
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¤
á
.
ν
(
t
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C
1
[0
,T
],
…
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(
T
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K3
(5)
ª
ü
à
Ó
¦
±
ν
(
t
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¿
'
u
t
3
[0
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þ
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k
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h
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m
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T
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h
g
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(
t
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i
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z
{
Œ
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u
mt
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i
dt
+
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T
0
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(
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m
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m
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t
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dt
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T
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k
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t
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i
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s
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h∇
η
t
(
s
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ν
(
t
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i
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Z
T
0
h
u
3
t
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(
t
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i
dt
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Z
T
0
h
u
t
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(
t
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i
dt
+
Z
T
0
h
f
(
u
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(
t
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i
dt
=
Z
T
0
h
g,ν
(
t
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i
dt.
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DOI:10.12677/pm.2021.1181731555
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q^
ν
(
t
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t
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t
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s
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t
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s
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t
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0
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(
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dt.
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é
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(28)
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(29)
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,
k
h
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1
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i
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u
t
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(0)
i
,
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±
h
(
u
1
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u
t
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(0)
i
= 0,
u
t
(0) =
u
1
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y
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(
t
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;
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2
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;
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T
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2
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(0
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;
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2
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1
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s
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t
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−
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0
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i
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= 0
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=
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t
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1
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(
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t
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t
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2
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0
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v
0
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1
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0
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∈H
ü
‡
)
,
…
ω
=
ω
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u
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v,u
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v
t
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t
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t
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•
ω
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0
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1
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v
•
§
:
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tt
−
a
∆
ω
−
Z
+
∞
0
µ
(
s
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ζ
t
(
s
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s
+
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3
t
−
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3
t
−
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t
+
f
(
u
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−
f
(
v
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b
(
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u
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2
2
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v
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2
2
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,
ζ
t
t
=
−
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t
s
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ω
t
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(30)
d
M
(
r
)=
a
+
br
Œ
•
M
(
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u
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2
2
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a
+
b
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u
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2
2
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t
†
•
§
(30)
3
L
2
(Ω)
þ
S
È
,
2
d
(11)
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1
2
d
d
t
|
ω
t
|
2
2
+
a
2
d
dt
|∇
ω
|
2
2
+
1
2
d
d
t
ζ
t
2
M
+
δ
2
ζ
t
2
M
+
α
((
|
u
t
|
3
−|
v
t
|
3
)
,ω
t
)
≤
β
|
ω
t
|
2
2
+(
f
(
v
)
−
f
(
u
)
,ω
t
)+
b
(
|∇
u
|
2
2
−|∇
v
|
2
2
,ω
t
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.
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e
¡
é
(31)
Ø
ª
m
>
1
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?
1
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n
,
w
,
p
2(
p
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+
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p
+1)
+
p
+1
2(
p
+1)
= 1
¤
á
,
2
Š
â
i
\
DOI:10.12677/pm.2021.1181731556
n
Ø
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4
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½
n
Ú
Young
Ø
ª
,
K
k
:
(
|
f
(
v
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−
f
(
u
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|
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t
)
≤
C
Z
Ω
(1+
|
u
|
p
+
|
v
|
p
)
|
ω
||
ω
t
|
dx
≤
C
[1+
|
u
|
p
2(
p
+1)
+
|
v
|
p
2(
p
+1)
]
·|
ω
|
2(
p
+1)
·|
ω
t
|
2
≤
C
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u
|
p
2
+
|∇
v
|
p
2)
]
·|∇
ω
|
2
·|
ω
t
|
2
≤
C
[
|∇
ω
|
2
2
+
|
ω
t
|
2
2
+
ζ
t
2
M
]
,
(32)
e
¡
é
(31)
Ø
ª
m
>
1
n
‘
?
1
?
n
,
k
:
(
|∇
u
|
2
2
−|∇
v
|
2
2
,ω
t
)
≤≤
(
|∇
u
|
2
+
|∇
v
2
|
)
·|
(
|∇
(
u
−
v
)
|
2
)
|·
(
Z
Ω
|
ω
t
|
2
dx
)
1
2
≤
C
T
|∇
ω
|
2
·|
ω
t
|
2
≤
C
T
[
|∇
ω
|
2
2
+
|
ω
t
|
2
2
+
ζ
t
2
M
]
.
(33)
du
u
3
t
•
ü
N4
O
¼
ê
,
Z
Ω
(
|
u
t
|
3
−|
v
t
|
3
)
·
ω
t
dx
≥
0
.
n
þ
,
(31)
ª
Œ
z
•
d
dt
[
|
ω
t
|
2
2
+
a
|∇
ω
|
2
2
+
k
ζ
t
k
2
M
]
≤
C
[
|
ω
t
|
2
2
+
a
|∇
ω
|
2
2
+
ζ
t
2
M
]
.
e
λ
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,
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k
ω
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k
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≤
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k
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Ï
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z
[1]
Ü
ƒ
w
,
Ü
ï
©
,
°
ÿ
.
˜
a
ä
P
Á
‘
[
‚
5
Å
Ä
•
§
•
§
N
á
Ú
f
[J].
A^
ê
Æ
,2020,
33(4):894-904.
[2]Li,C.,Liang,J.andXiao,T.J.(2021)Long-TermDynamicalBehavioroftheWaveModel
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https://doi.org/10.1016/j.cnsns.2020.105472
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[3]Liao,M.,Guo,B.andZhu,X.(2020)BoundsforBlow-UpTimetoaViscoelasticHyperbolic
Equation ofKirchhoffType withVariable Sources.
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[4]Zhang,Z.Y.,Liu,Z.H.andGan,X.Y.(2013)GlobalExistenceandGeneralDecayfora
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