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PureMathematics
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,2023,13(5),1173-1189
PublishedOnlineMay2023inHans.https://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2023.135122
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TheTimePeriodicSolutionsforthe
BoussinesqEquationwithStrong
Damping
HaoXu
SchoolofMathematicalSciences,ChongqingNormalUniversity,Chongqing
Received:Apr.3
rd
,2023;accepted:May5
th
,2023;published:May12
th
,2023
Abstract
In this paper,the problemoftime periodic solutions for theBoussinesqequation with
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,2023,13(5):1173-1189.
DOI:10.12677/pm.2023.135122
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strong damping is studied.When the periodic external force has some small property,
theexistenceanduniquenessofthetime-periodicsolutionsfortheBoussinesqequation
withStrongdampingareproved byspectralanalysisofthesolutionoperatorandthe
principleofcompressionmapping.Moreover,theperiodofthesolutionisthesame
asthatoftheexternalforceterm.
Keywords
Boussinesq Equation, StrongDamping, Periodic Solutions, Existencesand Uniqueness
Copyright
c
2023byauthor(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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.
?
˜
Ú
§
·
‚
„
Œ
k
∂
k
x
(
f
(
u
1
)
−
f
(
u
2
))
k
L
r
6
C
n
(
k
∂
k
x
u
1
k
L
p
+
k
∂
k
x
u
2
k
L
p
)
k
u
1
−
u
2
k
L
q
+(
k
u
1
k
L
q
+
k
u
2
k
L
q
)
k
∂
k
x
(
u
1
−
u
2
)
k
L
p
o
(
k
u
1
k
L
∞
+
k
u
2
k
L
∞
)
α
−
1
.
e
¡
·
‚
Š
â
Duhamel
n
¦
Ñ
•
§
(1.1)
È
©
L
ˆ
ª
§
é
•
§
(1.1)
‰
Fourier
C
†
Œ
ˆ
u
tt
+
|
ξ
|
2
ˆ
u
+
|
ξ
|
4
ˆ
u
+
|
ξ
|
2
ˆ
u
t
=
−|
ξ
|
2
(
ˆ
f
(
u
)+ˆ
ϕ
)
,
(2.1)
Ù
‚
5
Ü
©
é
A
A
•
§
•
λ
2
+
|
ξ
|
2
λ
+
|
ξ
|
2
+
|
ξ
|
4
= 0
,
(2.2)
¦
)
λ
(
|
ξ
|
) =
−
1
2
|
ξ
|
2
±
iw
(
|
ξ
|
)
,w
(
|
ξ
|
) =
|
ξ
|
r
1+
3
4
|
ξ
|
2
.
(2.3)
d
Duhamel
n
Œ
•
u
(
t
) =
G
(
x,t
−
s
)
∗
u
t
(
s
)+
H
(
x,t
−
s
)
∗
u
(
s
)+
Z
t
s
G
(
x,t
−
τ
)
∗
∆(
f
(
u
)+
ϕ
)
dτ,t
>
s,
(2.4)
ù
p
ˆ
G
(
ξ,t
) =
e
λ
+
(
ξ
)
t
−
e
λ
−
(
ξ
)
t
λ
+
(
ξ
)
−
λ
−
(
ξ
)
,
(2.5)
ˆ
H
(
ξ,t
) =
λ
+
(
ξ
)
e
λ
−
(
ξ
)
t
−
λ
−
(
ξ
)
e
λ
+
(
ξ
)
t
λ
+
(
ξ
)
−
λ
−
(
ξ
)
.
(2.6)
ò
(2.5)
Ú
(2.6)
ª
é
ž
m
C
þ
t
¦
§
·
‚
DOI:10.12677/pm.2023.1351221177
n
Ø
ê
Æ
Â
Ó
∂
t
ˆ
G
(
ξ,t
) =
λ
+
(
ξ
)
λ
+
(
ξ
)
−
λ
−
(
ξ
)
e
λ
+
(
ξ
)
t
−
λ
−
(
ξ
)
λ
+
(
ξ
)
−
λ
−
(
ξ
)
e
λ
−
(
ξ
)
t
,
(2.7)
∂
t
ˆ
H
(
ξ,t
) =
λ
+
λ
−
(
ξ
)
λ
+
(
ξ
)
−
λ
−
(
ξ
)
e
λ
−
(
ξ
)
t
−
e
λ
+
(
ξ
)
t
.
(2.8)
3.
)
Ž
f
P
~
O
ù
˜
!
ï
á
)
Ž
f
P
~
O
"
·
‚
Ú
\
˜
‡
1
w
ä
¼
ê
§
-
χ
(
ξ
) =
(
1
,
|
ξ
|
<r,
0
,
|
ξ
|
>
2
r,
Ù
¥
0
<r<
1
´
˜
‡
~
ê
"
·
‚
½
Â
ˆ
f
l
=
χ
(
ξ
)
ˆ
f
(
ξ
)
,
ˆ
f
h
= (1
−
χ
(
ξ
))
ˆ
f
(
ξ
)
.
@
o
f
l
=
χ
(
D
)
f
(
x
)
,f
h
= (1
−
χ
(
D
))
f
(
x
)
,
Ù
¥
Ž
f
χ
(
D
) =
F
−
1
[
χ
(
ξ
)]
.
Ú
n
3.1.
é
u
¼
ê
ˆ
G
(
ξ,t
),
∂
t
ˆ
G
(
ξ,t
),
ˆ
H
(
ξ,t
),
∂
t
ˆ
H
(
ξ,t
),
·
‚
k
X
e
O
|
ˆ
G
l
(
ξ,t
)
|
6
C
1
|
ξ
|
e
−
c
|
ξ
|
2
t
,
|
ˆ
G
h
(
ξ,t
)
|
6
C
1
|
ξ
|
2
e
−
c
|
ξ
|
2
t
,
(3.1)
|
∂
t
ˆ
G
l
(
ξ,t
)
|
6
Ce
−
c
|
ξ
|
2
t
,
|
∂
t
ˆ
G
h
(
ξ,t
)
|
6
Ce
−
c
|
ξ
|
2
t
,
(3.2)
|
ˆ
H
l
(
ξ,t
)
|
6
Ce
−
c
|
ξ
|
2
t
,
|
ˆ
H
h
(
ξ,t
)
|
6
Ce
−
c
|
ξ
|
2
t
,
(3.3)
|
∂
t
ˆ
H
l
(
ξ,t
)
|
6
C
|
ξ
|
e
−
c
|
ξ
|
2
t
,
|
∂
t
ˆ
H
h
(
ξ,t
)
|
6
C
|
ξ
|
2
e
−
c
|
ξ
|
2
t
.
(3.4)
y
²
|
ξ
|
6
r<
1
ž
§
d
(2.3)
ª
±
9
Taylor
Ð
m
ú
ª
§
·
‚
w
(
|
ξ
|
) =
|
ξ
|
r
1+
3
4
|
ξ
|
2
=
|
ξ
|
+
O
(
|
ξ
|
3
)
.
(3.5)
w
(
|
ξ
|
)
−
1
=
|
ξ
|
r
1+
3
4
|
ξ
|
2
!
−
1
=
1
|
ξ
|
−
3
8
|
ξ
|
+
O
(
|
ξ
|
3
)
.
(3.6)
DOI:10.12677/pm.2023.1351221178
n
Ø
ê
Æ
Â
Ó
1
λ
+
(
ξ
)
−
λ
−
(
ξ
)
=
1
2
iw
(
|
ξ
|
)
=
1
2
i
(
1
|
ξ
|
−
3
8
|
ξ
|
+
O
(
|
ξ
|
3
))
.
(3.7)
λ
+
(
ξ
)
λ
+
(
ξ
)
−
λ
−
(
ξ
)
=
−
1
2
|
ξ
|
2
+
iw
(
|
ξ
|
)
2
iw
(
|
ξ
|
)
=
1
2
−
1
4
i
|
ξ
|
2
(
w
(
|
ξ
|
))
−
1
=
1
2
−
1
4
i
|
ξ
|
2
(
1
|
ξ
|
−
3
8
|
ξ
|
+
O
(
|
ξ
|
3
))
=
1
2
−
1
4
i
(
|
ξ
|
+
O
(
|
ξ
|
3
))
.
(3.8)
λ
−
(
ξ
)
λ
+
(
ξ
)
−
λ
−
(
ξ
)
=
−
1
2
|
ξ
|
2
−
iw
(
|
ξ
|
)
2
iw
(
|
ξ
|
)
=
−
1
2
−
1
4
i
|
ξ
|
2
(
w
(
|
ξ
|
))
−
1
=
−
1
2
−
1
4
i
|
ξ
|
2
(
1
|
ξ
|
−
3
8
|
ξ
|
+
O
(
|
ξ
|
3
))
=
−
1
2
−
1
4
i
(
|
ξ
|
+
O
(
|
ξ
|
3
))
.
(3.9)
λ
+
λ
−
(
ξ
)
λ
+
(
ξ
)
−
λ
−
(
ξ
)
=
1
4
|
ξ
|
4
+(
w
(
|
ξ
|
))
2
2
iw
(
|
ξ
|
)
=
1
2
i
w
(
|
ξ
|
))+
1
4
|
ξ
|
4
(
w
(
|
ξ
|
))
−
1
=
1
2
i
(
|
ξ
|
+
O
(
|
ξ
|
3
))+
1
4
|
ξ
|
4
(
1
|
ξ
|
−
3
8
|
ξ
|
+
O
(
|
ξ
|
3
))
=
1
2
i
|
ξ
|
+
O
(
|
ξ
|
3
)
.
(3.10)
|
ξ
|
>
r
ž
§
d
(2.3)
ª
±
9
Taylor
Ð
m
ú
ª
§
·
‚
w
(
|
ξ
|
) =
|
ξ
|
r
1+
3
4
|
ξ
|
2
=
√
3
2
|
ξ
|
2
1+
4
3
1
|
ξ
|
2
1
2
=
√
3
2
|
ξ
|
2
+
√
3
3
+
O
(
1
|
ξ
|
2
)
.
(3.11)
w
(
|
ξ
|
)
−
1
=
√
3
2
|
ξ
|
2
s
1+
4
3
1
|
ξ
|
2
!
−
1
=
2
√
3
3
1
|
ξ
|
2
1+
4
3
1
|
ξ
|
2
−
1
2
=
2
√
3
3
1
|
ξ
|
2
+
O
(
1
|
ξ
|
4
)
.
(3.12)
1
λ
+
(
ξ
)
−
λ
−
(
ξ
)
=
1
2
iw
(
|
ξ
|
)
=
1
2
i
2
√
3
3
1
|
ξ
|
2
+
O
(
1
|
ξ
|
4
)
!
.
(3.13)
λ
+
(
ξ
)
λ
+
(
ξ
)
−
λ
−
(
ξ
)
=
−
1
2
|
ξ
|
2
+
iw
(
|
ξ
|
)
2
iw
(
|
ξ
|
)
=
1
2
−
1
4
i
|
ξ
|
2
(
w
(
|
ξ
|
))
−
1
=
1
2
−
1
4
i
|
ξ
|
2
(
2
√
3
3
1
|
ξ
|
2
+
O
(
1
|
ξ
|
4
))
=
1
2
−
1
4
i
(
2
√
3
3
+
O
(
1
|
ξ
|
2
))
.
(3.14)
DOI:10.12677/pm.2023.1351221179
n
Ø
ê
Æ
Â
Ó
λ
−
(
ξ
)
λ
+
(
ξ
)
−
λ
−
(
ξ
)
=
−
1
2
|
ξ
|
2
−
iw
(
|
ξ
|
)
2
iw
(
|
ξ
|
)
=
−
1
2
−
1
4
i
|
ξ
|
2
(
w
(
|
ξ
|
))
−
1
=
−
1
2
−
1
4
i
|
ξ
|
2
(
2
√
3
3
1
|
ξ
|
2
+
O
(
1
|
ξ
|
4
))
=
−
1
2
−
1
4
i
(
2
√
3
3
+
O
(
1
|
ξ
|
2
))
.
(3.15)
λ
+
λ
−
(
ξ
)
λ
+
(
ξ
)
−
λ
−
(
ξ
)
=
1
4
|
ξ
|
4
+(
w
(
|
ξ
|
))
2
2
iw
(
|
ξ
|
)
=
1
2
i
w
(
|
ξ
|
))+
1
4
|
ξ
|
4
(
w
(
|
ξ
|
))
−
1
=
1
2
i
(
√
3
2
|
ξ
|
2
+
√
3
3
+
O
(
1
|
ξ
|
2
)))
+
1
4
|
ξ
|
4
(
2
√
3
3
1
|
ξ
|
2
−
4
√
3
9
1
|
ξ
|
4
+
O
(
1
|
ξ
|
6
))
=
1
2
i
2
√
3
3
|
ξ
|
2
+
2
√
3
9
+
O
(
1
|
ξ
|
2
)
!
.
(3.16)
Š
â
þ
ã
O
Ž
(
J
§
·
‚
ï
á
Green
¼
ê
Å
:
O
"
|
ξ
|
6
r<
1
ž
§
·
‚
Œ
|
ˆ
G
l
(
ξ,t
)
|
=
1
λ
+
(
ξ
)
−
λ
−
(
ξ
)
(
e
λ
+
(
ξ
)
t
−
e
λ
−
(
ξ
)
t
)
=
1
2
i
1
|
ξ
|
−
3
8
|
ξ
|
+
O
(
|
ξ
|
3
)
e
−
1
2
|
ξ
|
2
t
(
e
itw
(
|
ξ
|
)
−
e
−
itw
(
|
ξ
|
)
)
6
C
1
|
ξ
|
e
−
c
|
ξ
|
2
t
.
(3.17)
|
∂
t
ˆ
G
l
(
ξ,t
)
|
=
λ
+
(
ξ
)
λ
+
(
ξ
)
−
λ
−
(
ξ
)
e
λ
+
(
ξ
)
t
−
λ
−
(
ξ
)
λ
+
(
ξ
)
−
λ
−
(
ξ
)
e
λ
−
(
ξ
)
t
=
1
2
−
1
4
i
(
|
ξ
|
+
O
(
|
ξ
|
3
))
e
−
1
2
|
ξ
|
2
t
e
itw
(
|
ξ
|
)
−
−
1
2
−
1
4
i
(
|
ξ
|
+
O
(
|
ξ
|
3
))
e
−
1
2
|
ξ
|
2
t
e
−
itw
(
|
ξ
|
)
6
Ce
−
c
|
ξ
|
2
t
.
(3.18)
|
ˆ
H
l
(
ξ,t
)
|
=
λ
+
(
ξ
)
λ
+
(
ξ
)
−
λ
−
(
ξ
)
e
λ
−
(
ξ
)
t
−
λ
−
(
ξ
)
λ
+
(
ξ
)
−
λ
−
(
ξ
)
e
λ
+
(
ξ
)
t
=
1
2
−
1
4
i
(
|
ξ
|
+
O
(
|
ξ
|
3
))
e
−
1
2
|
ξ
|
2
t
e
−
itw
(
|
ξ
|
)
−
−
1
2
−
1
4
i
(
|
ξ
|
+
O
(
|
ξ
|
3
))
e
−
1
2
|
ξ
|
2
t
e
itw
(
|
ξ
|
)
6
Ce
−
c
|
ξ
|
2
t
.
(3.19)
DOI:10.12677/pm.2023.1351221180
n
Ø
ê
Æ
Â
Ó
|
∂
t
ˆ
H
l
(
ξ,t
)
|
=
λ
+
(
ξ
)
λ
−
(
ξ
)
λ
+
(
ξ
)
−
λ
−
(
ξ
)
e
λ
−
(
ξ
)
t
−
e
λ
+
(
ξ
)
t
=
1
2
i
|
ξ
|
+
O
(
|
ξ
|
3
)
e
−
1
2
|
ξ
|
2
t
e
itw
(
|
ξ
|
)
−
e
−
itw
(
|
ξ
|
)
6
C
|
ξ
|
e
−
c
|
ξ
|
2
t
.
(3.20)
|
ξ
|
>
r
ž
§
·
‚
Œ
|
ˆ
G
h
(
ξ,t
)
|
=
1
λ
+
(
ξ
)
−
λ
−
(
ξ
)
(
e
λ
+
(
ξ
)
t
−
e
λ
−
(
ξ
)
t
)
=
1
2
i
2
√
3
3
1
|
ξ
|
2
+
O
(
1
|
ξ
|
4
)
!
e
−
1
2
|
ξ
|
2
t
(
e
itw
(
|
ξ
|
)
−
e
−
itw
(
|
ξ
|
)
)
6
C
1
|
ξ
|
2
e
−
c
|
ξ
|
2
t
.
(3.21)
|
∂
t
ˆ
G
h
(
ξ,t
)
|
=
λ
+
(
ξ
)
λ
+
(
ξ
)
−
λ
−
(
ξ
)
e
λ
+
(
ξ
)
t
−
λ
−
(
ξ
)
λ
+
(
ξ
)
−
λ
−
(
ξ
)
e
λ
−
(
ξ
)
t
=
1
2
−
1
4
i
(
2
√
3
3
+
O
(
1
|
ξ
|
2
))
!
e
−
1
2
|
ξ
|
2
t
e
itw
(
|
ξ
|
)
−
−
1
2
−
1
4
i
(
2
√
3
3
+
O
(
1
|
ξ
|
2
))
!
e
−
1
2
|
ξ
|
2
t
e
−
itw
(
|
ξ
|
)
6
Ce
−
c
|
ξ
|
2
t
.
(3.22)
|
ˆ
H
h
(
ξ,t
)
|
=
λ
+
(
ξ
)
λ
+
(
ξ
)
−
λ
−
(
ξ
)
e
λ
−
(
ξ
)
t
−
λ
−
(
ξ
)
λ
+
(
ξ
)
−
λ
−
(
ξ
)
e
λ
+
(
ξ
)
t
=
1
2
−
1
4
i
(
2
√
3
3
+
O
(
1
|
ξ
|
2
))
!
e
−
1
2
|
ξ
|
2
t
e
−
itw
(
|
ξ
|
)
−
−
1
2
−
1
4
i
(
2
√
3
3
+
O
(
1
|
ξ
|
2
))
!
e
−
1
2
|
ξ
|
2
t
e
itw
(
|
ξ
|
)
6
Ce
−
c
|
ξ
|
2
t
.
(3.23)
|
∂
t
ˆ
H
(
ξ,t
)
|
=
λ
+
(
ξ
)
λ
−
(
ξ
)
λ
+
(
ξ
)
−
λ
−
(
ξ
)
e
λ
−
(
ξ
)
t
−
e
λ
+
(
ξ
)
t
=
1
2
i
2
√
3
3
|
ξ
|
2
+
2
√
3
9
+
O
(
1
|
ξ
|
2
)
!
e
−
1
2
|
ξ
|
2
t
e
itw
(
|
ξ
|
)
−
e
−
itw
(
|
ξ
|
)
6
C
|
ξ
|
2
e
−
c
|
ξ
|
2
t
.
(3.24)
n
þ
¤
ã
§
Ú
n
3.1
¤
á
"
DOI:10.12677/pm.2023.1351221181
n
Ø
ê
Æ
Â
Ó
e
¡
·
‚
|
^
Ú
n
3.1
ï
á
)
Ž
f
P
~
O
"
Ú
n
3.2.
1
6
p
6
2,
k,j,l
•
š
K
ê
§
0
6
j
6
k
,
k
+
l
−
2
>
0,
K
k
e
O
k
∂
k
x
G
(
x,t
)
∗
f
k
L
2
6
C
(1+
t
)
−
n
(
1
p
−
1
2
)
−
1
2
−
k
−
j
2
k
∂
j
x
f
k
L
p
+
Ce
−
ct
k
∂
k
+
l
−
2
x
f
k
L
2
.
(3.25)
k
∂
k
x
∂
t
G
(
x,t
)
∗
f
k
L
2
6
C
(1+
t
)
−
n
(
1
p
−
1
2
)
2
−
k
−
j
2
k
∂
j
x
f
k
L
p
+
Ce
−
ct
k
∂
k
+
l
x
f
k
L
2
.
(3.26)
k
∂
k
x
H
(
x,t
)
∗
f
k
L
2
6
C
(1+
t
)
−
n
(
1
p
−
1
2
)
2
−
k
−
j
2
k
∂
j
x
f
k
L
p
+
Ce
−
ct
k
∂
k
+
l
x
f
k
L
2
.
(3.27)
k
∂
k
x
∂
t
H
(
x,t
)
∗
f
k
L
2
6
C
(1+
t
)
−
n
(
1
p
−
1
2
)
2
−
k
+1
−
j
2
k
∂
j
x
f
k
L
p
+
Ce
−
ct
k
∂
k
+2+
l
x
f
k
L
2
.
(3.28)
k
∂
k
x
G
(
x,t
)
∗
∆
f
k
L
2
6
C
(1+
t
)
−
n
(
1
p
−
1
2
)
2
−
k
+1
−
j
2
k
∂
j
x
f
k
L
p
+
Ce
−
ct
k
∂
k
+
l
x
f
k
L
2
.
(3.29)
k
∂
k
x
∂
t
G
(
x,t
)
∗
∆
f
k
L
2
6
C
(1+
t
)
−
n
(
1
p
−
1
2
)
2
−
k
+2
−
j
2
k
∂
j
x
f
k
L
p
+
Ce
−
ct
k
∂
k
+
l
+2
x
f
k
L
2
.
(3.30)
y
²
é
u
∂
k
x
G
(
x,t
)
∗
f
,
d
Plancherel
½
n
Œ
•
k
∂
k
x
G
(
x,t
)
∗
f
k
2
L
2
=
Z
R
n
|
ξ
|
2
k
|
ˆ
G
l
(
ξ,t
)
|
2
|
ˆ
f
|
2
dξ
+
Z
R
n
|
ξ
|
2
k
|
ˆ
G
h
(
ξ,t
)
|
2
|
ˆ
f
|
2
dξ
=
I
1
+
I
2
.
(3.31)
é
u
I
1
,
-
1
6
p
6
2,
1
p
+
1
p
0
= 1,
2
p
0
+
1
q
= 1,
d
(3.1)
§
H¨older
Ø
ª
±
9
Hausdorff-Young
Ø
ª
Œ
•
I
1
6
Z
|
ξ
|
6
2
r
|
ξ
|
2
k
|
ˆ
G
l
(
ξ,t
)
|
2
|
ˆ
f
|
2
dξ
6
C
Z
|
ξ
|
6
2
r
|
ξ
|
2
k
1
|
ξ
|
e
−
c
|
ξ
|
2
t
2
|
ˆ
f
|
2
dξ
6
C
Z
|
ξ
|
6
2
r
|
ξ
|
2(
k
−
1
−
j
)
e
−
c
|
ξ
|
2
t
q
dξ
1
q
Z
|
ξ
|
6
2
r
|
ξ
|
2
j
|
ˆ
f
|
2
p
0
2
dξ
!
2
p
0
6
C
(1+
t
)
−
(
n
(
1
p
−
1
2
)
−
1)
−
(
k
−
j
)
k
∂
j
x
f
k
2
L
p
,
(3.32)
é
u
I
2
,
d
(3.1)
Œ
I
2
6
C
Z
|
ξ
|
>
r
|
ξ
|
2
k
|
ˆ
G
h
(
ξ,t
)
|
2
|
ˆ
f
|
2
dξ
6
C
Z
|
ξ
|
>
r
|
ξ
|
2
k
1
|
ξ
|
2
e
−
c
|
ξ
|
2
t
2
|
ˆ
f
|
2
dξ
6
C
Z
|
ξ
|
>
r
|
ξ
|
2(
k
−
2)
e
−
c
|
ξ
|
2
t
|
ˆ
f
|
2
dξ
6
C
Z
|
ξ
|
>
r
|
ξ
|
2(
k
−
2)
|
ξ
|
2
l
e
−
c
|
ξ
|
2
t
|
ˆ
f
|
2
dξ
6
Ce
−
ct
Z
|
ξ
|
>
r
|
ξ
|
2(
k
+
l
−
2)
|
ˆ
f
|
2
dξ
6
Ce
−
ct
k
∂
k
+
l
−
2
x
f
k
2
L
2
.
(3.33)
DOI:10.12677/pm.2023.1351221182
n
Ø
ê
Æ
Â
Ó
d
(3.32)
Ú
(3.33)
Œ
k
∂
k
x
G
(
x,t
)
∗
f
k
L
2
6
C
(1+
t
)
−
n
(
1
p
−
1
2
)
−
1
2
−
k
−
j
2
k
∂
j
x
f
k
L
p
+
Ce
−
ct
k
∂
k
+
l
−
2
x
f
k
L
2
.
(3.34)
Ù
{
ˆ
ª
y
²
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§
a
q
§
ù
p
Ø
2
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ã
§
Ï
d
Ú
n
3.2
¤
á
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4.
ž
m
±
Ï
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3
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5
½
n
4.1.
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3
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m
>
n
2
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ê
§
ϕ
∈
C
([0
,T
];
L
1
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C
([0
,T
];
H
m
)
´
±
Ï
•
T
ž
m
±
Ï
¼
ê
§
-
E
0
=sup
0
6
t
6
T
(
k
ϕ
k
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1
+
k
ϕ
k
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m
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•
3
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‡
¿
©
~
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0
>
0,
E
0
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0
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3
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ž
m
±
Ï
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u
per
∈
C
([0
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H
m
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1
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v
sup
0
6
t
6
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(
k
u
per
(
t
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k
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u
per
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(
t
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k
H
m
−
2
)
6
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0
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y
²
1
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§
b
•
§
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3
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per
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m
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C
1
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m
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y
²
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per
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m
±
Ï
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‚
½
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e
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µ
u
per
(
t
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G
(
t
−
s
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∗
u
per
t
(
s
)+
H
(
t
−
s
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∗
u
per
(
s
)+
Z
t
s
G
(
t
−
τ
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∗
∆[
f
(
u
per
)+
ϕ
](
τ
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dτ.
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Ù
¥
ϕ
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±
Ï
•
T
ž
m
±
Ï
¼
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"
d
)
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©
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•
§
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•
§
(1.1)
9
Ð
Š
^
‡
t
=
s
:
u
0
=
u
per
(
s
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1
=
u
per
t
(
s
)(4.2)
)
"
-
(4.1)
¥
s
=
−
kT,k
∈
N
,
K
k
u
per
(
t
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G
(
t
+
kT
)
∗
u
per
t
(
−
kT
)+
H
(
t
+
kT
)
∗
u
per
(
−
kT
)
+
Z
t
−
kT
G
(
t
−
τ
)
∗
∆[
f
(
u
per
)+
ϕ
](
τ
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dτ.
(4.3)
n
>
3
ž
§
-
Ú
n
3.2
¥
(3.25)
ª
µ
p
= 1
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= 0
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= 0,
·
‚
Œ
k
G
(
t
+
kT
)
∗
g
k
H
m
6
C
(1+
t
+
kT
)
−
n
−
2
4
(
k
g
k
L
1
+
k
g
k
H
m
)
.
(4.4)
Ï
•
L
2
∩
L
1
3
L
2
¥
È
—
§
k
→∞
ž
§
é
?
¿
g
∈
H
m
,
d
(4.4)
Œ
k
G
(
t
+
kT
)
∗
g
k
H
m
→
0
.
(4.5)
a
q
§
n
>
1
ž
§
-
Ú
n
3.2
¥
(3.27)
ª
µ
p
= 1
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·
‚
Œ
DOI:10.12677/pm.2023.1351221183
n
Ø
ê
Æ
Â
Ó
k
H
(
t
+
kT
)
∗
g
k
H
m
6
C
(1+
t
+
kT
)
−
n
4
(
k
g
k
L
1
+
k
g
k
H
m
)
.
(4.6)
Ï
•
L
2
∩
L
1
3
L
2
¥
È
—
§
é
?
¿
g
∈
H
m
,
k
→∞
ž
§
d
(4.6)
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k
H
(
t
+
kT
)
∗
g
k
H
m
→
0
.
(4.7)
d
(4.5)
Ú
(4.7)
Œ
•
§
k
→∞
ž
§
é
(4.3)
4
•
Œ
u
per
(
t
) =
Z
t
−∞
G
(
t
−
τ
)
∗
∆[
f
(
u
per
)+
ϕ
](
τ
)
dτ.
(4.8)
·
‚
E
X
e
/
ª
N
µ
N
(
u
per
(
t
)) =
Z
t
−∞
G
(
t
−
τ
)
∗
∆[
f
(
u
per
)+
ϕ
](
τ
)
dτ.
(4.9)
b
N
N
•
3
•
˜
Ø
Ä:
u
per
1
(
t
),
K
k
N
(
u
per
1
)(
t
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u
per
1
(
t
),
-
u
per
2
(
t
) =
u
per
1
(
t
+
T
),
d
(4.9)
±
9
ϕ
(
t
+
T
) =
ϕ
(
t
)
Œ
•
u
per
2
(
t
) =
u
per
1
(
t
+
T
) =
N
(
u
per
1
(
t
+
T
))
=
Z
t
+
T
−∞
G
(
t
+
T
−
(
τ
+
T
))
∗
∆[
f
(
u
per
1
)+
ϕ
](
τ
+
T
)
dτ.
=
Z
t
−∞
G
(
t
−
τ
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∗
∆[
f
(
u
per
2
)+
ϕ
](
τ
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dτ.
=
N
(
u
per
2
)(
t
)
.
(4.10)
d
(4.10)
Œ
•
§
u
2
(
t
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•
´
N
N
Ø
Ä:
§
d
Ø
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•
˜
5
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1
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t
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per
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t
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u
per
1
(
t
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T
)
,
(4.11)
Ï
d
u
per
1
(
t
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´
±
Ï
•
T
ž
m
±
Ï
¼
ê
"
1
Ú
§
·
‚
|
^
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N
n
y
²
¹
Ê
5
Boussinesq
•
§
(1.1)
•
3
•
˜
)
u
per
∈
C
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,T
];
H
m
)
∩
C
1
([0
,T
];
H
m
−
2
).
·
‚
k
½
Â
E
0
=sup
0
6
t
6
T
(
k
ϕ
k
L
1
+
k
ϕ
k
H
m
)
.
2
E
˜
‡
T
Ý
þ
˜
m
X
=
{
u
per
∈
C
([0
,T
];
H
m
)
∩
C
1
([0
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];
H
m
−
2
) :
k
u
per
k
X
6
ρ
}
,
Ù
¥
ρ
´
˜
‡
~
ê
§
Ú
?
‰
ê
k·k
X
=sup
0
6
t
6
T
(
k
u
per
k
H
m
+
k
u
per
t
k
H
m
−
2
)
,
DOI:10.12677/pm.2023.1351221184
n
Ø
ê
Æ
Â
Ó
½
Â
Ý
þ
d
=
k
u
per
1
−
u
per
2
k
X
,
d
I
O
z
•{
Œ
•
(
X,d
)
´
Ý
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m
§
ë
©
z
[17]
"
·
‚
‡
y
•
§
(1.1)
)
•
3
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˜
5
§
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I
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²
(4.9)
¥
N
N
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DOI:10.12677/pm.2023.1351221185
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DOI:10.12677/pm.2023.1351221186
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[1]Russell,J.S.(1845)OnWaves,Reportofthe14thMeetingoftheBritishAssociationforthe
AdvancementofScience.JohnMurray,London,311-390.
[2]Boussinesq,J.(1872)Th´eoriedesondesetdesremousquisepropagentlelongd’uncanal
rectangulairehorizontalencommuniquantauliquidecontenudanscecanaldesvitessessen-
DOI:10.12677/pm.2023.1351221187
n
Ø
ê
Æ
Â
Ó
siblementpareillesdelasurfaceaufond.
JournaldeMath´ematiquesPuresetAppliqu´ees
,
17
,
55-108.
[3]Bona,J.L. andSachs, R.L.(1988) GlobalExistence ofSmoothSolutions andStabilityof Soli-
tary Waves fora Generalized Boussinesq Equation.
CommunicationsinMathematicalPhysics
,
118
,15-29.https://doi.org/10.1007/BF01218475
[4]Farah,L.G.(2009)LocalSolutionsinSobolevSpaceswithNegativeIndicesforthe“Good”
BoussinesqEquation.
CommunicationsinPartialDifferentialEquations
,
34
,52-57.
https://doi.org/10.1080/03605300802682283
[5]Kishimoto,N.andTsugawa,K.(2010)LocalWell-PosednessforQuadraticNonlinear
Schr¨odingerEquationsandthe“Good”BoussinesqEquation.
DifferentialandIntegralEqua-
tions
,
23
,463-493.https://doi.org/10.57262/die/1356019307
[6]Linares,F. (1993) GlobalExistenceofSmallSolutionsfora Generalized BoussinesqEquation.
JournalofDifferentialEquations
,
106
,257-293.https://doi.org/10.1006/jdeq.1993.1108
[7]Tsutsumi,M.andMatahashi,T.(1991)OntheCauchyProblemfortheBoussinesqType
Equation.
MathematicaJaponica
,
36
,321-347.
[8]Liu,Y. (1997)Decay andScatteringof SmallSolutionsofa GeneralizedBoussinesqEquation.
JournalofFunctionalAnalysis
,
147
,51-68.https://doi.org/10.1006/jfan.1996.3052
[9]Cho,Y.andOzawa,T.(2007)OnSmallAmplitudeSolutionstotheGeneralizedBoussinesq
Equations.
DiscreteandContinuousDynamicalSystems
,
17
,691-711.
https://doi.org/10.3934/dcds.2007.17.691
[10]Sachs,R.L.(1990)OntheBlow-UpofCertainSolutionsofthe“Good”BoussinesqEquation.
ApplicableAnalysis
,
36
,145-152.https://doi.org/10.1080/00036819008839928
[11]Straughan,B.(1992)GlobalNonexistenceofSolutionstoSomeBoussinesqTypeEquations.
JournalofMathematicalandPhysicalSciences
,
26
,145-152.
[12]Liu,Y.andXu,R.(2008)GlobalExistenceandBlowUpofSolutionsforCauchyProblemof
GeneralizedBoussinesqEquation.
PhysicaD:NonlinearPhenomena
,
237
,721-731.
https://doi.org/10.1016/j.physd.2007.09.028
[13]Yang,Z.andGuo,B.(2008)CauchyProblemfortheMulti-DimensionalBoussinesqType
Equation.
JournalofMathematicalAnalysisandApplications
,
340
,64-80.
https://doi.org/10.1016/j.jmaa.2007.08.017
[14]Varlamov,V.(1996)ExistenceandUniquenessofaSolutiontotheCauchyProblemforthe
DampedBoussinesqEquation.
MathematicalMethodsintheAppliedSciences
,
19
,639-649.
https://doi.org/10.1002/(SICI)1099-1476(19960525)19:8
h
639::AID-MMA786
i
3.0.CO;2-C
[15]Wang,Y.X.(2013)AsymptoticDecayEstimateofSolutionstotheGeneralizedDampedBq
Equation.
JournalofInequalitiesandApplications
,
2013
,ArticleNo.323.
https://doi.org/10.1186/1029-242X-2013-323
DOI:10.12677/pm.2023.1351221188
n
Ø
ê
Æ
Â
Ó
[16]Liu,M.andWang,W.(2014)GlobalExistenceandPointwiseEstimateofSolutionsforthe
MultidimensionalGeneralizedBoussinesqTypeEquation.
CommunicationsonPureandAp-
pliedAnalysis
,
13
,1203-1222.https://doi.org/10.3934/cpaa.2014.13.1203
[17]Liu,G.andWang,W.(2019)Inviscid LimitfortheDampedBoussinesq Equation.
Journalof
DifferentialEquations
,
267
,5521-5542.https://doi.org/10.1016/j.jde.2019.05.037
[18]Liu,G.andWang,W.(2020)DecayEstimatesforaDissipative-DispersiveLinearSemigroup
andApplicationtotheViscousBoussinesqEquation.
JournalofFunctionalAnalysis
,
278
,
Article108413.https://doi.org/10.1016/j.jfa.2019.108413
[19]Xu,R.Z.,Luo,Y.B.,Shen,J.H.andHuang,S.B.(2017)GlobalExistenceandBlowUp
forDampedGeneralizedBoussinesqEquation.
ActaMathematicaeApplicataeSinica,English
Series
,
33
,251-262.https://doi.org/10.1007/s10255-017-0655-4
[20]Wang,Y.andLi, Y.(2018)TimePeriodic Solutionsto theBeamEquationwith WeakDamp-
ing.
JournalofMathematicalPhysics
,
59
,Article111503.https://doi.org/10.1063/1.5046821
[21]
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DOI:10.12677/pm.2023.1351221189
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