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AdvancesinAppliedMathematics
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,2021,10(7),2442-2456
PublishedOnlineJuly2021inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2021.107256
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LongTimeCharacterizationof
SolutionsofNonautonomous
BoissonadeSystems
ZhenqiongCui
∗
,ChengmingYang
ShanghaiNormalUniversity,Shanghai
Received:Jun.19
th
,2021;accepted:Jul.11
th
,2021;published:Jul.22
nd
,2021
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Abstract
This paper mainly includesthe following twoaspects:Firstly, we prove theuniqueness
ofweaksolutionofautonomousBoissonadesystem.Becausethequadratictermof
theautonomousBoissonadesystemis
uv
insteadof
u
2
,itisdifferentfromthegeneral
methodwhenprovingtheuniquenessofweaksolution.Therefore,thispapergivesa
specificmethodtoprovetheuniquenessofweaksolution.Finally,accordingtothe
sufficientandnecessary conditionsfortheexistenceofuniform attractor,the existence
ofuniformattractorin
E
ofnon-autonomousBoissonadesystemisproved.
Keywords
Non-AutonomousDynamicalSystems,UniformAttractor,ContinuousProcess,
GlobalAttractor,TranslationBounded
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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u
2
)
2
+
u
2
(
v
1
−
v
2
)(
u
1
−
u
2
)]
dx
≤
γ
Z
Ω
v
4
1
dx
1
4
Z
Ω
(
u
1
−
u
2
)
2
dx
1
2
Z
Ω
(
u
1
−
u
2
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4
dx
1
4
+
γ
Z
Ω
u
2
2
dx
1
2
Z
Ω
(
v
1
−
v
2
)
4
dx
1
4
(
Z
Ω
(
u
1
−
u
2
)
4
dx
)
1
4
≤
γ
k
v
1
k
L
4
k
u
1
−
u
2
kk
u
1
−
u
2
k
L
4
+
γ
k
u
2
kk
v
1
−
v
2
k
L
4
k
u
1
−
u
2
k
L
4
≤
C
2
γ
k
v
1
k
H
1
0
k
u
1
−
u
2
kk
u
1
−
u
2
k
H
1
0
+
C
2
γ
k
u
2
kk
v
1
−
v
2
k
H
1
0
k
u
1
−
u
2
k
H
1
0
≤
C
4
γ
2
d
1
k
v
1
k
2
H
1
0
k
u
1
−
u
2
k
2
+
d
1
4
k
u
1
−
u
2
k
2
H
1
0
+
C
4
M
2
γ
2
d
1
k
v
1
−
v
2
k
2
H
1
0
+
d
1
4
k
u
1
−
u
2
k
2
H
1
0
=
C
4
γ
2
d
1
k
v
1
k
2
H
1
0
k
u
1
−
u
2
k
2
+
C
4
M
2
γ
2
d
1
k
v
1
−
v
2
k
2
H
1
0
+
d
1
2
k
u
1
−
u
2
k
2
H
1
0
,
(3.4)
DOI:10.12677/aam.2021.1072562446
A^
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(3.2)
ª
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1
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Z
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u
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u
2
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2
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v
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2
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2
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2
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u
2
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(3.5)
r
(3.4)
“
\
(3.5)
2
¦
±
2
d
d
t
k
u
1
−
u
2
k
2
+
d
1
k
u
1
−
u
2
k
H
1
0
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2
k
u
1
−
u
2
k
2
+
α
k
u
1
−
u
2
k
2
+
α
k
v
1
−
v
2
k
2
+
2
C
4
γ
2
d
1
k
v
1
k
2
H
1
0
k
v
1
−
v
2
k
2
+
2
C
4
M
2
γ
2
d
1
k
v
1
−
v
2
k
2
.
(3.6)
(3.3)
ª
•
Œ
±
¤
d
d
t
k
v
1
−
v
2
k
2
+2
d
2
k
v
1
−
v
2
k
2
H
1
0
=
Z
Ω
[(
u
1
−
u
2
)(
v
1
−
v
2
)
−
β
(
v
1
−
v
2
)(
v
1
−
v
2
)]
dx
≤k
u
1
−
u
2
k
2
+
k
v
1
−
v
2
k
2
+2
β
k
v
1
−
v
2
k
2
.
(3.7)
(3.7)
ª
¦
±
3
C
4
M
2
γ
2
2
d
1
d
2
3
C
4
M
2
γ
2
2
d
1
d
2
d
d
t
k
v
1
−
v
2
k
2
+
3
C
4
M
2
γ
2
d
1
k
v
1
−
v
2
k
2
H
1
0
≤
3
C
4
M
2
γ
2
2
d
1
d
2
k
u
1
−
u
2
k
2
+
3
C
4
M
2
γ
2
2
d
1
d
2
k
v
1
−
v
2
k
2
+
3
C
4
M
2
γ
2
β
d
1
d
2
k
v
1
−
v
2
k
2
.
(3.8)
DOI:10.12677/aam.2021.1072562447
A^
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Ð
w
§
¤
²
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(3
.
6)
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(3
.
8)
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ª
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d
d
t
k
u
1
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2
k
2
+
3
C
4
M
2
γ
2
2
d
1
d
2
k
v
1
−
v
2
k
2
+
d
1
k
u
1
−
u
2
k
H
1
0
+
C
4
M
2
γ
2
d
1
k
v
1
−
v
2
k
2
H
1
0
≤
2+
α
+
3
C
4
M
2
γ
2
2
d
1
d
2
+
2
C
4
γ
2
d
1
k
v
1
k
H
1
0
k
u
1
−
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2
k
2
+
α
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3
C
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M
2
γ
2
β
d
1
d
2
2
d
1
d
2
3
C
4
M
2
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2
3
C
4
M
2
γ
2
2
d
1
d
2
k
v
1
−
v
2
k
2
≤
2+
α
+
3
C
4
M
2
γ
2
2
d
1
d
2
+
2
C
4
γ
2
d
1
k
v
1
k
H
1
0
+
α
+1+
3
C
4
M
2
γ
2
β
d
1
d
2
2
d
1
d
2
3
C
4
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2
γ
2
k
u
1
−
u
2
k
2
+
3
C
4
M
2
γ
2
2
d
1
d
2
k
v
1
−
v
2
k
2
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(3.9)
|
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.
s
Ø
ª
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·
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k
k
u
1
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2
k
2
+
3
C
4
M
2
γ
2
2
d
1
d
2
k
v
1
−
v
2
k
2
≤
e
R
t
0
[2+
α
+
3
C
4
M
2
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2
2
d
1
d
2
+
2
C
4
γ
2
d
1
k
v
1
(
s
)
k
H
1
0
+(
α
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3
C
4
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2
γ
2
β
d
1
d
2
)
2
d
1
d
2
3
C
4
M
2
γ
2
]
ds
(
k
u
10
−
u
20
k
2
+
3
C
4
M
2
γ
2
2
d
1
d
2
k
v
10
−
v
20
k
2
)
= 0
.
(3.10)
Ï
d
,
·
‚
u
1
=
u
2
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1
=
v
2
,
ù
L
²
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¯
K
(3
.
1)
f
)
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l
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.
10)
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‚
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±
)
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ë
Y
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6
5
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n
3.2
é
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¿
τ,T
∈
R
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,
X
J
u
τ
∈
E
,
K
Ð
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¯
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(3
.
1)
•
3
•
˜
r
)
g
(
t
)=
(
u
(
t
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(
t
))
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∈
(
τ,T
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÷
v
g
∈
C
([
τ,T
];
E
)
∩
L
2
(
τ,T
;
D
(
A
))
.
½
n
3.1
é
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¿
τ,T
∈
R
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g
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Boissonade
u
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.
1)
•
3
•
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r
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(
t
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u
(
t
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(
t
))
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∈
(
τ,
∞
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d
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n
3.1
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g
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u
Ð
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(3
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1)
N
r
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½
Â
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L
§
:
{
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:
U
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t,τ
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g
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=
g
(
t
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u
(
t
)
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(
t
))
.
DOI:10.12677/aam.2021.1072562448
A^
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§
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²
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n
3.3
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h
(
t
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∈
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Loc
(
R
τ
;
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£
k
.
§
K
•
3
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1
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0
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4
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1
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{∈
V
:
k
g
k
2
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≤
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t,τ
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g
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Σ
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k
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k
2
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y
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[9]
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k
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L
p
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y
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©
z
[10]
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4.
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3
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n
4.1
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R
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,
X
J
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≥
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•
3
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ê
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0
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sup
t
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t
+
η
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k
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2
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á
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L
2
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2
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(
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2
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n
4.2
X
J
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¼
ê
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0
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L
2
n
(
R
;
E
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@
o
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τ
∈
R
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k
lim
γ
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sup
t
≥
τ
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t
τ
e
−
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(
t
−
s
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k
ϕ
(
s
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k
2
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Ú
n
4.3
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h
1
(
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8
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τ
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τ
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3
M
≤
0
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0
≤
0
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≤
0
¦
Z
t
+
η
t
Z
Ω(
u
≥
M
)
|
u
|
3
|
(
u
−
M
)
+
|
3
dxds
≤
ε,t
≥
t
0
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∈
Σ
,
(4.1)
Z
t
+
η
t
Z
Ω(
u
≤−
M
)
|
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|
3
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(
u
+
M
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−
|
3
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≤
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t
0
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Z
t
+
η
t
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Ω(
v
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−
M
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dxds
≤
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≥
t
0
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Σ
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(4.3)
DOI:10.12677/aam.2021.1072562449
A^
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Ð
w
§
¤
²
Z
t
+
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+
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t
0
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¤
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−
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m
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1
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1
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h
1
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h
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t
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2
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2
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1
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v
2
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2
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1
0
)+
d
1
k
∆
u
2
k
2
+
d
2
k
∆
v
2
k
2
= (
f
1
(
u,v
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,
∆
u
2
)
−
(
f
h
1
(
t
)
,
∆
u
2
)
+(
f
2
(
u,v
)
,
∆
v
2
)
−
(
f
h
2
(
t
)
,
∆
v
2
)
.
(4.6)
d
(1.3)
Œ
•
|
f
1
(
u,v
)
|≤
C
1
(
|
u
|
3
+1)
.
DOI:10.12677/aam.2021.1072562450
A^
ê
Æ
?
Ð
w
§
¤
²
d
Young’s
Ø
ª
,
·
‚
|
(
f
1
(
u,v
)
,
∆
u
2
)
|≤k
f
1
(
u,v
)
kk
∆
u
2
k
≤
d
1
4
k
∆
u
2
k
2
+
1
d
1
k
f
1
(
u,v
)
k
2
≤
d
1
4
k
∆
u
2
k
2
+
1
d
1
Z
Ω
C
2
1
(
|
u
|
3
+1)
2
dx
≤
d
1
4
k
∆
u
2
k
2
+
2
C
2
1
d
1
|
Ω
|
+
2
C
2
1
d
1
Z
Ω
|
u
|
6
dx
≤
d
1
4
k
∆
u
2
k
2
+
2
C
2
1
d
1
|
Ω
|
+
2
C
2
1
d
1
Z
Ω(
|
u
|≥
M
)
|
u
|
6
dx
+
2
C
2
1
d
1
Z
Ω(
|
u
|≤
M
)
|
u
|
6
dx
≤
d
1
4
k
∆
u
2
k
2
+
2
C
2
1
d
1
|
Ω
|
+
2
C
2
1
M
6
d
1
|
Ω
|
+
2
C
2
1
d
1
Z
Ω(
|
u
|≥
M
)
|
u
|
6
dx,
(4.7)
±
9
|
(
f
h
1
(
t
)
,
−
∆
u
2
)
|
=
|
Z
Ω
f
h
1
(
t
)(
−
∆
u
2
)
dx
|
≤k
f
h
1
(
t
)
k
2
k
∆
u
2
k
2
≤
d
1
4
k
∆
u
2
k
2
+
1
d
1
k
f
h
1
(
t
)
k
2
.
(4.8)
d
(1
.
4)
Œ
•
|
f
2
(
u,v
)
|≤
C
2
(
|
u
|
+1)
.
d
Young’s
Ø
ª
,
·
‚
|
f
2
(
u,v
)
,
∆
v
2
|
=
|
Z
Ω
f
2
(
u,v
)∆
v
2
dx
|
≤k
f
2
(
u,v
)
kk
∆
v
2
k
≤
d
2
4
k
∆
v
2
k
2
+
1
d
2
k
f
2
(
u,v
)
k
2
≤
d
2
4
k
∆
v
2
k
2
+
1
d
2
Z
Ω
C
2
2
(
|
v
|
+1)
2
dx
≤
d
2
4
k
∆
v
2
k
2
+
2
C
2
2
d
2
|
Ω
|
+
2
C
2
2
d
2
Z
Ω
|
v
|
2
dx
≤
d
2
4
k
∆
v
2
k
2
+
2
C
2
2
d
2
|
Ω
|
+
2
C
2
2
d
2
Z
Ω(
|
v
|≥
M
)
|
v
|
2
dx
+
2
C
2
2
d
2
Z
Ω(
|
v
|≤
M
)
|
v
|
2
dx
≤
d
2
4
k
∆
v
2
k
2
+
2
C
2
2
d
2
|
Ω
|
+
2
C
2
2
M
2
d
2
|
Ω
|
+
2
C
2
2
d
2
Z
Ω(
|
v
|≥
M
)
|
v
|
2
dx,
(4.9)
DOI:10.12677/aam.2021.1072562451
A^
ê
Æ
?
Ð
w
§
¤
²
±
9
|
(
f
h
2
(
t
)
,
−
∆
v
2
)
|≤k
f
h
2
(
t
)
k
2
k
∆
v
2
k
2
≤
d
2
4
k
∆
v
2
k
2
+
1
d
2
k
f
h
2
(
t
)
k
2
.
(4.10)
ò
(4
.
7)
−
(4
.
10)
‘
\
(4
.
6),
d
d
t
(
k
u
2
k
2
H
1
0
+
k
v
2
k
2
H
1
0
)+
d
1
k
∆
u
2
k
2
+
d
2
k
∆
v
2
k
2
≤
4
C
2
1
d
1
|
Ω
|
+
4
C
2
1
M
6
d
1
|
Ω
|
+
4
C
2
2
d
2
|
Ω
|
+
4
C
2
2
M
2
d
2
|
Ω
|
+
4
C
2
1
d
1
Z
Ω(
|
u
|≥
M
)
|
u
|
6
dx
+
4
C
2
2
d
2
Z
Ω(
|
v
|≥
M
)
|
v
|
2
dx
+
2
d
1
k
f
h
1
(
t
)
k
2
+
2
d
2
k
f
h
2
(
t
)
k
2
.
Š
â
poincar´
e
Ø
ª
k
ϕ
k≤
1
p
λ
m
+1
k
Du
k
2
,
∀
ϕ
∈
H
1
0
,
-
d
=
min
{
d
1
,d
2
}
,
Ò
k
d
d
t
(
k
u
2
k
2
H
1
0
+
k
v
2
k
2
H
1
0
)+
λ
m
+1
d
(
k
u
2
k
2
H
1
0
+
k
v
2
k
2
H
1
0
)
≤
4
C
2
1
d
1
|
Ω
|
+
4
C
2
1
M
6
d
1
|
Ω
|
+
4
C
2
2
d
2
|
Ω
|
+
4
C
2
2
M
2
d
2
|
Ω
|
+
4
C
2
1
d
1
Z
Ω(
|
u
|≥
M
)
|
u
|
6
dx
+
4
C
2
2
d
2
Z
Ω(
|
v
|≥
M
)
|
v
|
2
dx
+
2
d
1
k
f
h
1
(
t
)
k
2
+
2
d
2
k
f
h
2
(
t
)
k
2
.
(4.11)
-
k>max
{
T
B
,t
B
}
,(4
.
11)
ª
^
Gronwall’s
Ø
ª
,
k
u
2
k
2
H
1
0
+
k
v
2
k
2
H
1
0
≤
e
−
λ
m
+1
d
(
t
−
k
)
(
k
u
2
(
τ
)
k
2
H
1
0
+
k
v
2
(
τ
)
k
2
H
1
0
)
+
Z
t
k
e
−
λ
m
+1
d
(
t
−
s
)
(
4
C
2
1
d
1
|
Ω
|
+
4
C
2
1
M
6
d
1
|
Ω
|
+
4
C
2
2
d
2
|
Ω
|
+
4
C
2
2
M
2
d
2
|
Ω
|
)
ds
+
4
C
2
1
d
1
Z
t
k
e
−
λ
m
+1
d
(
t
−
s
)
Z
Ω(
|
u
|≥
M
)
|
u
(
s
)
|
6
dxds
+
4
C
2
2
d
2
Z
t
k
e
−
λ
m
+1
d
(
t
−
s
)
Z
Ω(
|
v
|≥
M
)
|
v
(
s
)
|
2
dxds
+
2
d
1
Z
t
k
e
−
λ
m
+1
d
(
t
−
s
)
k
f
h
1
(
s
)
k
2
+
2
d
2
Z
t
k
e
−
λ
m
+1
d
(
t
−
s
)
f
h
2
(
s
)
k
2
ds
≤
ρ
2
V
e
−
λ
m
+1
d
(
t
−
k
)
+
1
dλ
m
+1
4
C
2
1
d
1
|
Ω
|
+
4
C
2
1
M
6
d
1
|
Ω
|
+
4
C
2
2
d
2
|
Ω
|
+
4
C
2
2
M
2
d
2
|
Ω
|
+
4
C
2
1
d
1
Z
t
k
e
−
λ
m
+1
d
(
t
−
s
)
Z
Ω(
|
u
|≥
M
)
|
u
(
s
)
|
6
dxds
+
4
C
2
2
d
2
Z
t
k
e
−
λ
m
+1
d
(
t
−
s
)
Z
Ω(
|
v
|≥
M
)
|
v
(
s
)
|
2
dxds
+
2
d
1
Z
t
k
e
−
λ
m
+1
d
(
t
−
s
)
k
f
h
1
(
s
)
k
2
+
2
d
2
Z
t
k
e
−
λ
m
+1
d
(
t
−
s
)
f
h
2
(
s
)
k
2
ds,
(4.12)
DOI:10.12677/aam.2021.1072562452
A^
ê
Æ
?
Ð
w
§
¤
²
Ù
¥
T
B
´
Ú
n
3.3
¥
,
t
B
´
Ú
n
3.4
¥
.
,
˜
•
¡
§
·
‚
k
Z
Ω(
|
u
|≥
M
)
|
u
|
6
dx
=
Z
Ω(
u
≥
M
)
|
u
|
6
dx
+
Z
Ω(
u
≤−
M
)
|
u
|
6
dx
=
Z
Ω(
u
≥
M
)
|
u
|
3
|
u
|
3
dx
+
Z
Ω(
u
≤−
M
)
|
u
|
3
|
u
|
3
dx
=
Z
Ω(
u
≥
M
)
|
u
|
3
|
u
−
M
+
M
|
3
dx
+
Z
Ω(
u
≤−
M
)
|
u
|
3
|
u
+
M
−
M
|
3
dx
≤
4[
Z
Ω(
u
≥
M
)
|
u
|
3
(
|
(
u
−
M
)
+
|
3
+
|
M
|
3
)
dx
+
Z
Ω(
u
≤−
M
)
|
u
|
3
(
|
(
u
+
M
)
−
|
3
+
|
M
|
3
)
dx
]
≤
4[
Z
Ω(
u
≥
M
)
|
u
|
3
|
(
u
−
M
)
+
|
3
dx
+
Z
Ω(
u
≤−
M
)
|
u
|
3
|
(
u
+
M
)
−
|
3
dx
]+4
M
3
Z
Ω
(
|
u
|≥
M
)
|
u
|
3
dx
≤
4[
Z
Ω(
u
≥
M
)
|
u
|
3
|
(
u
−
M
)
+
|
3
dx
+
Z
Ω(
u
≤−
M
)
|
u
|
3
|
(
u
+
M
)
−
|
3
dx
]+4
M
2
Z
Ω(
|
u
|≥
M
)
|
u
|
4
dx,
(4.13)
±
9
Z
Ω(
|
v
|≥
M
)
|
v
|
2
dx
=
Z
Ω(
v
≥
M
)
|
v
|
2
dx
+
Z
Ω(
v
≤−
M
)
|
v
|
2
dx
=
Z
Ω(
v
≥
M
)
|
v
||
v
|
dx
+
Z
Ω(
v
≤−
M
)
|
v
||
v
|
dx
=
Z
Ω(
v
≥
M
)
|
v
||
v
−
M
+
M
|
dx
+
Z
Ω(
v
≤−
M
)
|
v
||
v
+
M
−
M
|
dx
≤
Z
Ω(
v
≥
M
)
(
|
v
||
v
−
M
|
+
|
M
|
)
dx
+
Z
Ω(
v
≤−
M
)
|
v
|
(
|
v
+
M
|
+
|
M
|
)
dx
≤
Z
Ω(
v
≥
M
)
|
v
||
(
v
−
M
)
+
|
dx
+
Z
Ω(
v
≤−
M
)
|
v
||
(
v
+
M
)
−
|
dx
+
M
Z
Ω(
|
v
|≥
M
)
|
v
|
dx
≤
Z
Ω(
v
≥
M
)
|
v
||
(
v
−
M
)
+
|
dx
+
Z
Ω(
v
≤−
M
)
|
v
||
(
v
+
M
)
−
|
dx
+
Z
Ω(
|
v
|≥
M
)
|
v
|
2
dx.
(4.14)
d
Ú
n
5.1,
·
‚
•
é
?
¿
k
.
8
e
B
⊂
L
2
(Ω),
•
3
M¿0,
t
0
>
0,
η>
0
¦
Z
t
+
η
t
Z
Ω(
u
≥
M
)
|
u
|
3
|
(
u
−
M
)
+
|
3
dxds
≤
d
1
ε
320
C
2
1
,t
≥
t
0
,σ
∈
Σ
,u
τ
∈
e
B,
(4.15)
Z
t
+
η
t
Z
Ω(
u
≤−
M
)
|
u
|
3
|
(
u
+
M
)
−
|
3
dxds
≤
d
1
ε
320
C
2
1
,t
≥
t
0
,σ
∈
Σ
,u
τ
∈
e
B,
(4.16)
Z
t
+
η
t
Z
Ω(
v
≥
M
)
|
v
||
(
v
−
M
)
+
|
dxds
≤
d
2
ε
80
C
2
2
,t
≥
t
0
,σ
∈
Σ
,u
τ
∈
e
B.
(4.17)
Z
t
+
η
t
Z
Ω(
v
≤−
M
)
|
v
||
(
v
+
M
)
−
|
dxds
≤
d
2
ε
80
C
2
2
,t
≥
t
0
,σ
∈
Σ
,u
τ
∈
e
B.
(4.18)
DOI:10.12677/aam.2021.1072562453
A^
ê
Æ
?
Ð
w
§
¤
²
5
¿
Z
t
+
η
t
e
−
λ
m
+1
d
(
t
−
s
)
Z
Ω(
u
≥
M
)
|
u
|
3
|
(
u
−
M
)
+
|
3
dxds
=
Z
t
t
−
η
+
Z
t
−
η
t
−
2
η
+
Z
t
−
2
η
t
−
3
η
+
···
e
−
λ
m
+1
d
(
t
−
s
)
Z
Ω(
u
≥
M
)
|
u
|
3
|
(
u
−
M
)
+
|
3
dxds
≤
Z
t
t
−
η
Z
Ω(
u
≥
M
)
|
u
|
3
|
(
u
−
M
)
+
|
3
dxds
+
e
−
λ
m
+1
dη
Z
t
−
η
t
−
2
η
Z
Ω(
u
≥
M
)
|
u
|
3
|
(
u
−
M
)
+
|
3
dxds
+
e
−
2
λ
m
+1
dη
Z
t
−
2
η
t
−
3
η
Z
Ω(
u
≥
M
)
|
u
|
3
|
(
u
−
M
)
+
|
3
dxds
+
···
≤
1
1
−
e
−
dλ
m
+1
η
d
1
ε
320
C
2
1
,fork>t
0
.
(4.19)
Ó
§
·
‚
k
Z
t
+
η
t
e
−
λ
m
+1
d
(
t
−
s
)
Z
Ω(
u
≤−
M
)
|
u
|
3
|
(
u
−
M
)
+
|
3
dxds
≤
1
1
−
e
−
dλ
m
+1
η
d
1
ε
320
C
2
1
,fork>t
0
,
(4.20)
Z
t
+
η
t
e
−
λ
m
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t
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s
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Z
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v
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|
v
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(
v
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M
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1
1
−
e
−
dλ
m
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η
d
2
ε
80
C
2
2
,fork>t
0
,
(4.21)
Z
t
+
η
t
e
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λ
m
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d
(
t
−
s
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Z
Ω(
v
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M
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|
v
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(
v
−
M
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+
|
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1
1
−
e
−
dλ
m
+1
η
d
2
ε
80
C
2
2
,fork>t
0
.
(4.22)
d
(4
.
5),
·
‚
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±
m
+1
v
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,
¦
16
C
2
1
d
1
Z
t
k
e
−
λ
m
+1
d
(
t
−
s
)
Z
Ω(
u
≥
M
)
|
u
|
3
|
(
u
−
M
)
+
|
3
dxds
≤
ε
10
,
(4.23)
16
C
2
1
d
1
Z
t
k
e
−
λ
m
+1
d
(
t
−
s
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Z
Ω(
u
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M
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|
u
|
3
|
(
u
+
M
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−
|
3
dxds
≤
ε
10
,
(4.24)
4
C
2
2
d
2
Z
t
k
e
−
λ
m
+1
d
(
t
−
s
)
Z
Ω(
v
≥
M
)
|
v
||
(
v
−
M
)
+
|
dxds
≤
ε
10
,
(4.25)
4
C
2
2
d
2
Z
t
k
e
−
λ
m
+1
d
(
t
−
s
)
Z
Ω(
v
≤−
M
)
|
v
||
(
v
+
M
)
−
|
dxds
≤
ε
10
,
(4.26)
4
C
2
1
d
1
Z
t
k
e
−
λ
m
+1
d
(
t
−
s
)
Z
Ω(
|
u
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M
)
4
M
2
|
u
(
s
)
|
4
|
dxds
≤
ε
10
,
(4.27)
4
C
2
2
d
2
Z
t
k
e
−
λ
m
+1
d
(
t
−
s
)
Z
Ω(
|
v
|≥
M
)
|
v
(
s
)
|
2
|
dxds
≤
ε
10
,
(4.28)
1
dλ
m
+1
4
C
2
1
d
1
|
Ω
|
+
4
C
2
1
M
6
d
1
|
Ω
|
+
4
C
2
2
d
2
|
Ω
|
+
4
C
2
2
M
2
d
2
|
Ω
|
≤
ε
10
.
(4.29)
DOI:10.12677/aam.2021.1072562454
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t
1
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1
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ln
6
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2
V
ε
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k
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K
t
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t
1
ž
k
ρ
2
V
e
−
λ
m
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(
t
−
k
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ε
10
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d
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0,
Œ
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m
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v
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Z
t
k
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t
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f
h
1
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s
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k
2
+
f
h
2
(
s
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k
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ds
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2
d
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2
d
2
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∀
σ
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Σ
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(4.31)
ò
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13)
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31)
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t
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k
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k
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t
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z
[1]Robinson,J.C.andPierre,C.(2003)Infinite-DimensionalDynamicalSystems:AnIntroduc-
tiontoDissipativeParabolicPDEsandtheTheoryofGlobalAttractors.CambridgeTextsin
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AppliedMechanicsReviews
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[2]Haraux,A. (1991)Syst`emesdynamiquesdissipatifsetapplications.In:
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[3]Chepyzhov,V.V.andVishik,M.I.(2002)AttractorsforEquationsofMathematicalPhysics.
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[4]Ma, Q.F.,Wang, S.H. and Zhong,C.K. (2002) Necessary and Sufficient Conditions for the Ex-
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IndianaUniversityMathematics
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[6]Sell,G.R.andYou,Y.(2002)DynamicsofEvolutionaryEquations.Springer,NewYork.
[7]Tu,J.Y.(2015)GlobalAttractorsandRobustnessoftheBoissonadeSystem.
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[8]Song, H.T.,Ma, S. andZhong, C.K. (2009)Attractors of Non-Autonomous Reaction-Diffusion
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[9]
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2017,47(12):1891-1906.https://doi.org/10.1360/N012017-00062
DOI:10.12677/aam.2021.1072562455
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[10]Song,H.T.andZhong,C.K.(2008)AttractorsofNon-AutonomousReaction-DiffusionEqua-
tionsin
L
p
.
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,1890-1897.
https://doi.org/10.1016/j.na.2007.01.059
DOI:10.12677/aam.2021.1072562456
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