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PureMathematicsnØêÆ,2023,13(5),1173-1189
PublishedOnlineMay2023inHans.https://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2023.135122
‘r{ZBoussinesq•§žm±Ï)
ÂÂÂÓÓÓ
-Ÿ“‰ŒÆ§êƉÆÆ§-Ÿ
ÂvFϵ2023c43F¶¹^Fϵ2023c55F¶uÙFϵ2023c512F
Á‡
©ïÄ‘r {ZBoussinesq •§žm±Ï)¯K§ÏLé)ŽfÌ©ÛÚØ N
ny²3±Ïåäk,«5ž§‘r{ZBoussinesq •§žm±Ï)•3•˜5§
¿…)±Ï†å‘±ÏƒÓ"
'…c
Boussinesq•§§r{Z§žm±Ï)§•3•˜5
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
©ÙÚ^:ÂÓ.‘r{ZBoussinesq•§žm±Ï)[J].nØêÆ,2023,13(5):1173-1189.
DOI:10.12677/pm.2023.135122
ÂÓ
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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DOI:10.12677/pm.2023.1351221174nØêÆ
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DOI:10.12677/pm.2023.1351221175nØêÆ
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L
p
kvk
L
q
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DOI:10.12677/pm.2023.1351221176nØêÆ
ÂÓ
Ún2.4.[21](H¨olderØª)
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L
p
kvk
L
q
.
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∞
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L
∞
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+
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k∂
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f(u)k
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L
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kuk
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))k
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n
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u
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ˆu
tt
+|ξ|
2
ˆu+|ξ|
4
ˆu+|ξ|
2
ˆu
t
= −|ξ|
2
(
ˆ
f(u)+ˆϕ),(2.1)
Ù‚5Ü©éA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)
dDuhamelnŒ•
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)
ùp
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)ªéžmCþt¦§·‚
DOI:10.12677/pm.2023.1351221177nØêÆ
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∂
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.)ŽfP~O
ù˜!ïá)ŽfP~O"·‚Ú\˜‡1wä¼ê§-
χ(ξ) =
(
1,|ξ|<r,
0,|ξ|>2r,
Ù¥0 <r<1´˜‡~ê"·‚½Â
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¥Žfχ(D) = F
−1
[χ(ξ)].
Ún3.1.éu¼ê
ˆ
G(ξ,t),∂
t
ˆ
G(ξ,t),
ˆ
H(ξ,t),∂
t
ˆ
H(ξ,t),·‚kXeO
|
ˆ
G
l
(ξ,t)|6C
1
|ξ|
e
−c|ξ|
2
t
,|
ˆ
G
h
(ξ,t)|6C
1
|ξ|
2
e
−c|ξ|
2
t
,(3.1)
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t
ˆ
G
l
(ξ,t)|6Ce
−c|ξ|
2
t
,|∂
t
ˆ
G
h
(ξ,t)|6Ce
−c|ξ|
2
t
,(3.2)
|
ˆ
H
l
(ξ,t)|6Ce
−c|ξ|
2
t
,|
ˆ
H
h
(ξ,t)|6Ce
−c|ξ|
2
t
,(3.3)
|∂
t
ˆ
H
l
(ξ,t)|6C|ξ|e
−c|ξ|
2
t
,|∂
t
ˆ
H
h
(ξ,t)|6C|ξ|
2
e
−c|ξ|
2
t
.(3.4)
y²|ξ|6r<1ž§d(2.3)ª±9TaylorÐmúª§·‚
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.2023.1351221178nØêÆ
ÂÓ
1
λ
+
(ξ)−λ
−
(ξ)
=
1
2iw(|ξ|)
=
1
2i
(
1
|ξ|
−
3
8
|ξ|+O(|ξ|
3
)).(3.7)
λ
+
(ξ)
λ
+
(ξ)−λ
−
(ξ)
=
−
1
2
|ξ|
2
+iw(|ξ|)
2iw(|ξ|)
=
1
2
−
1
4i
|ξ|
2
(w(|ξ|))
−1
=
1
2
−
1
4i
|ξ|
2
(
1
|ξ|
−
3
8
|ξ|+O(|ξ|
3
))
=
1
2
−
1
4i
(|ξ|+O(|ξ|
3
)).(3.8)
λ
−
(ξ)
λ
+
(ξ)−λ
−
(ξ)
=
−
1
2
|ξ|
2
−iw(|ξ|)
2iw(|ξ|)
= −
1
2
−
1
4i
|ξ|
2
(w(|ξ|))
−1
= −
1
2
−
1
4i
|ξ|
2
(
1
|ξ|
−
3
8
|ξ|+O(|ξ|
3
))
= −
1
2
−
1
4i
(|ξ|+O(|ξ|
3
)).(3.9)
λ
+
λ
−
(ξ)
λ
+
(ξ)−λ
−
(ξ)
=
1
4
|ξ|
4
+(w(|ξ|))
2
2iw(|ξ|)
=
1
2i

w(|ξ|))+
1
4
|ξ|
4
(w(|ξ|))
−1

=
1
2i

(|ξ|+O(|ξ|
3
))+
1
4
|ξ|
4
(
1
|ξ|
−
3
8
|ξ|+O(|ξ|
3
))

=
1
2i

|ξ|+O(|ξ|
3
)

.(3.10)
|ξ|>rž§d(2.3)ª±9TaylorÐmúª§·‚
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
2iw(|ξ|)
=
1
2i
2
√
3
3
1
|ξ|
2
+O(
1
|ξ|
4
)
!
.(3.13)
λ
+
(ξ)
λ
+
(ξ)−λ
−
(ξ)
=
−
1
2
|ξ|
2
+iw(|ξ|)
2iw(|ξ|)
=
1
2
−
1
4i
|ξ|
2
(w(|ξ|))
−1
=
1
2
−
1
4i
|ξ|
2
(
2
√
3
3
1
|ξ|
2
+O(
1
|ξ|
4
))
=
1
2
−
1
4i
(
2
√
3
3
+O(
1
|ξ|
2
)).(3.14)
DOI:10.12677/pm.2023.1351221179nØêÆ
ÂÓ
λ
−
(ξ)
λ
+
(ξ)−λ
−
(ξ)
=
−
1
2
|ξ|
2
−iw(|ξ|)
2iw(|ξ|)
= −
1
2
−
1
4i
|ξ|
2
(w(|ξ|))
−1
= −
1
2
−
1
4i
|ξ|
2
(
2
√
3
3
1
|ξ|
2
+O(
1
|ξ|
4
))
= −
1
2
−
1
4i
(
2
√
3
3
+O(
1
|ξ|
2
)).(3.15)
λ
+
λ
−
(ξ)
λ
+
(ξ)−λ
−
(ξ)
=
1
4
|ξ|
4
+(w(|ξ|))
2
2iw(|ξ|)
=
1
2i

w(|ξ|))+
1
4
|ξ|
4
(w(|ξ|))
−1

=
1
2i

(
√
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
2i
2
√
3
3
|ξ|
2
+
2
√
3
9
+O(
1
|ξ|
2
)
!
.(3.16)
ŠâþãOŽ(J§·‚ïáGreen ¼êÅ:O"|ξ|6r<1ž§·‚Œ
|
ˆ
G
l
(ξ,t)|=



1
λ
+
(ξ)−λ
−
(ξ)
(e
λ
+
(ξ)t
−e
λ
−
(ξ)t
)



=



1
2i

1
|ξ|
−
3
8
|ξ|+O(|ξ|
3
)

e
−
1
2
|ξ|
2
t
(e
itw(|ξ|)
−e
−itw(|ξ|)
)



6C
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
4i
(|ξ|+O(|ξ|
3
))

e
−
1
2
|ξ|
2
t
e
itw(|ξ|)
−

−
1
2
−
1
4i
(|ξ|+O(|ξ|
3
))

e
−
1
2
|ξ|
2
t
e
−itw(|ξ|)



6Ce
−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
4i
(|ξ|+O(|ξ|
3
))

e
−
1
2
|ξ|
2
t
e
−itw(|ξ|)
−

−
1
2
−
1
4i
(|ξ|+O(|ξ|
3
))

e
−
1
2
|ξ|
2
t
e
itw(|ξ|)



6Ce
−c|ξ|
2
t
.(3.19)
DOI:10.12677/pm.2023.1351221180nØêÆ
ÂÓ
|∂
t
ˆ
H
l
(ξ,t)|=



λ
+
(ξ)λ
−
(ξ)
λ
+
(ξ)−λ
−
(ξ)

e
λ
−
(ξ)t
−e
λ
+
(ξ)t




=



1
2i

|ξ|+O(|ξ|
3
)

e
−
1
2
|ξ|
2
t

e
itw(|ξ|)
−e
−itw(|ξ|)




6C|ξ|e
−c|ξ|
2
t
.(3.20)
|ξ|>rž§·‚Œ
|
ˆ
G
h
(ξ,t)|=



1
λ
+
(ξ)−λ
−
(ξ)
(e
λ
+
(ξ)t
−e
λ
−
(ξ)t
)



=



1
2i
2
√
3
3
1
|ξ|
2
+O(
1
|ξ|
4
)
!
e
−
1
2
|ξ|
2
t
(e
itw(|ξ|)
−e
−itw(|ξ|)
)



6C
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
4i
(
2
√
3
3
+O(
1
|ξ|
2
))
!
e
−
1
2
|ξ|
2
t
e
itw(|ξ|)
−
−
1
2
−
1
4i
(
2
√
3
3
+O(
1
|ξ|
2
))
!
e
−
1
2
|ξ|
2
t
e
−itw(|ξ|)



6Ce
−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
4i
(
2
√
3
3
+O(
1
|ξ|
2
))
!
e
−
1
2
|ξ|
2
t
e
−itw(|ξ|)
−
−
1
2
−
1
4i
(
2
√
3
3
+O(
1
|ξ|
2
))
!
e
−
1
2
|ξ|
2
t
e
itw(|ξ|)



6Ce
−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
2i
2
√
3
3
|ξ|
2
+
2
√
3
9
+O(
1
|ξ|
2
)
!
e
−
1
2
|ξ|
2
t

e
itw(|ξ|)
−e
−itw(|ξ|)




6C|ξ|
2
e
−c|ξ|
2
t
.(3.24)
nþ¤ã§Ún3.1¤á"
DOI:10.12677/pm.2023.1351221181nØêÆ
ÂÓ
e¡·‚|^Ún3.1 ïá)ŽfP~O"
Ún3.2.1 6p62,k,j,l•šKê§0 6j6k,k+l−2 >0,KkeO
k∂
k
x
G(x,t)∗fk
L
2
6C(1+t)
−
n(
1
p
−
1
2
)−1
2
−
k−j
2
k∂
j
x
fk
L
p
+Ce
−ct
k∂
k+l−2
x
fk
L
2
.(3.25)
k∂
k
x
∂
t
G(x,t)∗fk
L
2
6C(1+t)
−
n(
1
p
−
1
2
)
2
−
k−j
2
k∂
j
x
fk
L
p
+Ce
−ct
k∂
k+l
x
fk
L
2
.(3.26)
k∂
k
x
H(x,t)∗fk
L
2
6C(1+t)
−
n(
1
p
−
1
2
)
2
−
k−j
2
k∂
j
x
fk
L
p
+Ce
−ct
k∂
k+l
x
fk
L
2
.(3.27)
k∂
k
x
∂
t
H(x,t)∗fk
L
2
6C(1+t)
−
n(
1
p
−
1
2
)
2
−
k+1−j
2
k∂
j
x
fk
L
p
+Ce
−ct
k∂
k+2+l
x
fk
L
2
.(3.28)
k∂
k
x
G(x,t)∗∆fk
L
2
6C(1+t)
−
n(
1
p
−
1
2
)
2
−
k+1−j
2
k∂
j
x
fk
L
p
+Ce
−ct
k∂
k+l
x
fk
L
2
.(3.29)
k∂
k
x
∂
t
G(x,t)∗∆fk
L
2
6C(1+t)
−
n(
1
p
−
1
2
)
2
−
k+2−j
2
k∂
j
x
fk
L
p
+Ce
−ct
k∂
k+l+2
x
fk
L
2
.(3.30)
y²éu∂
k
x
G(x,t)∗f,dPlancherel½nŒ•
k∂
k
x
G(x,t)∗fk
2
L
2
=
Z
R
n
|ξ|
2k
|
ˆ
G
l
(ξ,t)|
2
|
ˆ
f|
2
dξ+
Z
R
n
|ξ|
2k
|
ˆ
G
h
(ξ,t)|
2
|
ˆ
f|
2
dξ
= I
1
+I
2
.(3.31)
éuI
1
,-1 6p62,
1
p
+
1
p
0
= 1,
2
p
0
+
1
q
= 1,d(3.1)§H¨olderØª±9Hausdorff-Young Øª
Υ
I
1
6
Z
|ξ|62r
|ξ|
2k
|
ˆ
G
l
(ξ,t)|
2
|
ˆ
f|
2
dξ6C
Z
|ξ|62r
|ξ|
2k



1
|ξ|
e
−c|ξ|
2
t



2
|
ˆ
f|
2
dξ
6C

Z
|ξ|62r



|ξ|
2(k−1−j)
e
−c|ξ|
2
t



q
dξ

1
q
Z
|ξ|62r



|ξ|
2j
|
ˆ
f|
2



p
0
2
dξ
!
2
p
0
6C(1+t)
−(n(
1
p
−
1
2
)−1)−(k−j)
k∂
j
x
fk
2
L
p
,(3.32)
éuI
2
,d(3.1)Œ
I
2
6C
Z
|ξ|>r
|ξ|
2k
|
ˆ
G
h
(ξ,t)|
2
|
ˆ
f|
2
dξ6C
Z
|ξ|>r
|ξ|
2k



1
|ξ|
2
e
−c|ξ|
2
t



2
|
ˆ
f|
2
dξ
6C
Z
|ξ|>r
|ξ|
2(k−2)
e
−c|ξ|
2
t
|
ˆ
f|
2
dξ6C
Z
|ξ|>r
|ξ|
2(k−2)
|ξ|
2l
e
−c|ξ|
2
t
|
ˆ
f|
2
dξ
6Ce
−ct
Z
|ξ|>r
|ξ|
2(k+l−2)
|
ˆ
f|
2
dξ6Ce
−ct
k∂
k+l−2
x
fk
2
L
2
.(3.33)
DOI:10.12677/pm.2023.1351221182nØêÆ
ÂÓ
d(3.32)Ú(3.33)Œ
k∂
k
x
G(x,t)∗fk
L
2
6C(1+t)
−
n(
1
p
−
1
2
)−1
2
−
k−j
2
k∂
j
x
fk
L
p
+Ce
−ct
k∂
k+l−2
x
fk
L
2
.(3.34)
Ù{ˆªy²L§aq§ùpØ2•ã§ÏdÚn3.2¤á"

4.žm±Ï)•3•˜5
½n4.1.n>3 Úm>
n
2
Ñ•ê§ϕ∈C([0,T];L
1
)∩C([0,T];H
m
) ´±Ï•Tžm
±Ï¼ê§-
E
0
=sup
06t6T
(kϕk
L
1
+kϕk
H
m
),
K•3˜‡¿©~êδ
0
>0,E
0
<δ
0
ž§•§(1.1)•3•˜žm±Ï)u
per
∈
C([0,T];H
m
)∩C
1
([0,T];H
m−2
),¿…÷v
sup
06t6T
(ku
per
(t)k
H
m
+ku
per
t
(t)k
H
m−2
) 6CE
0
.
y²1˜Ú§b•§(1.1)•3•˜)u
per
∈C([0,T];H
m
)∩C
1
([0,T];H
m−2
),·‚òy²
u
per
Ò´žm±Ï)"·‚½ÂXe/ªÈ©•§µ
u
per
(t) = G(t−s)∗u
per
t
(s)+H(t−s)∗u
per
(s)+
Z
t
s
G(t−τ)∗∆[f(u
per
)+ϕ](τ)dτ.(4.1)
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t= s: u
0
= u
per
(s),u
1
= u
per
t
(s)(4.2)
)"-(4.1)¥s= −kT,k∈N,Kk
u
per
(t) = G(t+kT)∗u
per
t
(−kT)+H(t+kT)∗u
per
(−kT)
+
Z
t
−kT
G(t−τ)∗∆[f(u
per
)+ϕ](τ)dτ.(4.3)
n>3 ž§-Ún3.2¥(3.25)ªµp= 1,j= 0,l= 0,·‚Œ
kG(t+kT)∗gk
H
m
6C(1+t+kT)
−
n−2
4
(kgk
L
1
+kgk
H
m
).(4.4)
Ï•L
2
∩L
1
3L
2
¥È—§k→∞ž§é?¿g∈H
m
,d(4.4)Œ
kG(t+kT)∗gk
H
m
→0.(4.5)
aq§n>1 ž§-Ún3.2¥(3.27)ªµp= 1,j= 0,l= 0,·‚Œ
DOI:10.12677/pm.2023.1351221183nØêÆ
ÂÓ
kH(t+kT)∗gk
H
m
6C(1+t+kT)
−
n
4
(kgk
L
1
+kgk
H
m
).(4.6)
Ï•L
2
∩L
1
3L
2
¥È—§é?¿g∈H
m
,k→∞ž§d(4.6) Œ
kH(t+kT)∗gk
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)
·‚EXe/ªNµ
N(u
per
(t)) =
Z
t
−∞
G(t−τ)∗∆[f(u
per
)+ϕ](τ)dτ.(4.9)
bNN•3•˜ØÄ:u
per
1
(t),KkN(u
per
1
)(t) = u
per
1
(t),-u
per
2
(t) = u
per
1
(t+T),d(4.9)±
9ϕ(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)∗∆[f(u
per
2
)+ϕ](τ)dτ.
= N(u
per
2
)(t).(4.10)
d(4.10)Œ•§u
2
(t)•´NNØÄ:§dØÄ:•˜5Œ•
u
per
1
(t) = u
per
2
(t) = u
per
1
(t+T),(4.11)
Ïdu
per
1
(t)´±Ï•Tžm±Ï¼ê"
1Ú§·‚|^Ø Nny²¹Ê5Boussinesq•§(1.1)•3•˜)u
per
∈C([0,T];H
m
)∩
C
1
([0,T];H
m−2
).·‚k½Â
E
0
=sup
06t6T
(kϕk
L
1
+kϕk
H
m
).
2E˜‡TÝþ˜m
X= {u
per
∈C([0,T];H
m
)∩C
1
([0,T];H
m−2
) : ku
per
k
X
6ρ},
Ù¥ρ´˜‡~ê§Ú?‰ê
k·k
X
=sup
06t6T
(ku
per
k
H
m
+ku
per
t
k
H
m−2
),
DOI:10.12677/pm.2023.1351221184nØêÆ
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½ÂÝþd= ku
per
1
−u
per
2
k
X
,dIOz•{Œ•(X,d) ´Ýþ˜m§ë©z[17]"
·‚‡y•§(1.1))•3•˜5§•Iy²(4.9)¥NN3Ýþ˜mX¥•3•˜ØÄ
:"•N
N(u
per
(t)) =
Z
t
−∞
G(t−τ)∗4[f(u
per
)+ϕ](τ)dτ.(4.12)
·‚Äky²N :X →X.é?¿u
per
∈X,k,m´ê…÷v06k6m,-Ú
n3.2¥(3.29)ªµp=1,j=0,l=0,dÚn2.5±9sobolevi\½nH
m
→L
∞
(m>
n
2
),·
‚Œ
k∂
k
x
N(u
per
(t))k
L
2
6
Z
t
−∞
k∂
k
x
G(t−τ)∗∆[f(u
per
)+ϕ](τ)k
L
2
dτ
6C
Z
t
−∞
(1+t−τ)
−
n
4
−
k+1
2
kf(u
per
)k
L
1
dτ+C
Z
t
−∞
e
−c(t−τ)
k∂
k
x
f(u
per
)k
L
2
dτ
+C
Z
t
−∞
(1+t−τ)
−
n
4
−
k+1
2
kϕk
L
1
dτ+C
Z
t
−∞
e
−c(t−τ)
k∂
k
x
ϕk
L
2
dτ
6C
Z
t
−∞
(1+t−τ)
−
n
4
−
k+1
2

kf(u
per
)k
L
1
+k∂
k
x
f(u
per
)k
L
2

dτ
+C
Z
t
−∞
(1+t−τ)
−
n
4
−
k+1
2

kϕk
L
1
+k∂
k
x
ϕk
L
2

dτ
6C
Z
t
−∞
(1+t−τ)
−
n
4
−
k+1
2

ku
per
k
σ−1
L
∞
ku
per
k
L
2
+ku
per
k
σ−1
L
∞
k∂
k
x
u
per
k
L
2

dτ
+C
Z
t
−∞
(1+t−τ)
−
n
4
−
k+1
2

kϕk
L
1
+k∂
k
x
ϕk
L
2

dτ
6C

ku
per
k
σ
X
+sup
06t6T
(kϕk
L
1
+kϕk
H
m
)

Z
t
−∞
(1+t−τ)
−
n
4
−
k+1
2
dτ
6Cku
per
k
σ
X
+CE
0
.(4.13)
aq§é?¿u
per
∈X,
e
k,m´ê…÷v06
e
k6m−2,-Ún3.2¥(3.30)ªµp=1,j=
0,l= 0,dÚn2.5±9sobolevi\½nH
m
→L
∞
(m>
n
2
),·‚Œ
k∂
e
k
x
∂
t
N(u
per
(t))k
L
2
6
Z
t
−∞
k∂
e
k
x
∂
t
G(t−τ)∗∆[f(u
per
)+ϕ](τ)k
L
2
dτ
6C
Z
t
−∞
(1+t−τ)
−
n
4
−
e
k+2
2
kf(u
per
)k
L
1
dτ+C
Z
t
−∞
e
−c(t−τ)
k∂
e
k+2
x
f(u
per
)k
L
2
dτ
+C
Z
t
−∞
(1+t−τ)
−
n
4
−
e
k+2
2
kϕk
L
1
dτ+C
Z
t
−∞
e
−c(t−τ)
k∂
e
k+2
x
ϕk
L
2
dτ
6C
Z
t
−∞
(1+t−τ)
−
n
4
−
e
k+2
2

kf(u
per
)k
L
1
+k∂
e
k+2
x
f(u
per
)k
L
2

dτ
+C
Z
t
−∞
(1+t−τ)
−
n
4
−
e
k+2
2

kϕk
L
1
+k∂
e
k+2
x
ϕk
L
2

dτ
DOI:10.12677/pm.2023.1351221185nØêÆ
ÂÓ
6C
Z
t
−∞
(1+t−τ)
−
n
4
−
e
k+2
2

ku
per
k
σ−1
L
∞
ku
per
k
L
2
+ku
per
k
σ−1
L
∞
k∂
e
k+2
x
u
per
k
L
2

dτ
+C
Z
t
−∞
(1+t−τ)
−
n
4
−
e
k+2
2

kϕk
L
1
+k∂
e
k+2
x
ϕk
L
2

dτ
6C

ku
per
k
σ
X
+sup
06t6T
(kϕk
L
1
+kϕk
H
m
)

Z
t
−∞
(1+t−τ)
−
n
4
−
˜
k+2
2
dτ
6Cku
per
k
σ
X
+CE
0
.(4.14)
d(4.13)Ú(4.14)Œ
kN(u
per
)k
X
6Cku
per
k
σ
X
+CE
0
.(4.15)
Ï•σ>2,·‚ρ= 4CE
0
,E
0
¿©ž§Kk
kN(u
per
)k
X
6Cku
per
k
σ
X
+CE
0
62CE
0
6ρ.(4.16)
d(4.16)Œ•,N: X→X.
•·‚y²N´Ø N§é?¿u
per
1
,u
per
2
∈X,d(4.9)Œ•
[N(u
per
1
)−N(u
per
2
)](t) =
Z
t
−∞
G(t−τ)∗∆[f(u
per
1
)−f(u
per
2
)](τ)dτ.(4.17)
k,m´ê…÷v06k6m,-Ún3.2¥(3.29)ªµp=1,j=0,l=0,dÚn2.5±9
sobolevi\½nH
m
→L
∞
(m>
n
2
),·‚Œ
k∂
k
x
[N(u
per
1
)−N(u
per
2
)](t)k
L
2
6
Z
t
−∞
k∂
k
x
G(t−τ)∗∆

f(u
per
1
)−f(u
per
2
)

(τ)k
L
2
dτ
6C
Z
t
−∞
(1+t−τ)
−
n
4
−
k+1
2
k

f(u
per
1
)−f(u
per
2
)

(τ)k
L
1
dτ
+C
Z
t
−∞
e
−c(t−τ)
k∂
k
x

f(u
per
1
)−f(u
per
2
)

(τ)k
L
2
dτ
6C
Z
t
−∞
(1+t−τ)
−
n
4
−
k+1
2
n
kf(u
per
1
)−f(u
per
2
)k
L
1
+k∂
k
x

f(u
per
1
)−f(u
per
2
)

(τ)k
L
2
o
dτ
6C
Z
t
−∞
(1+t−τ)
−
n
4
−
k+1
2
n
(ku
per
1
k
L
∞
+ku
per
2
k
L
∞
)
σ−2
(ku
per
1
k
L
2
+ku
per
2
k
L
2
)ku
per
1
−u
per
2
k
L
2
+(ku
per
1
k
L
∞
+ku
per
2
k
L
∞
)
σ−2
h
(k∂
k
x
u
per
1
k
L
2
+k∂
k
x
u
per
2
k
L
2
)ku
per
1
−u
per
2
k
L
∞
+(ku
per
1
k
L
∞
+ku
per
2
k
L
∞
)k∂
k
x
[u
per
1
−u
per
2
]k
L
2
io
dτ
6C(ku
per
1
k
X
+ku
per
2
k
X
)
σ−1
ku
per
1
−u
per
2
k
X
Z
t
−∞
(1+t−τ)
−
n
4
−
k+1
2
dτ
6C(ku
per
1
k
X
+ku
per
2
k
X
)
σ−1
ku
per
1
−u
per
2
k
X
.(4.18)
DOI:10.12677/pm.2023.1351221186nØêÆ
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aq§
e
k,m´ê…÷v06
e
k6m−2,-Ún3.2¥(3.30)ªµp=1,j=0,l=0,dÚ
n2.5±9sobolevi\½nH
m
→L
∞
(m>
n
2
)Υ
k∂
e
k
x
∂
t
[N(u
per
1
)−N(u
per
2
)](t)k
L
2
6
Z
t
−∞
k∂
e
k
x
∂
t
G(t−τ)∗∆

f(u
per
1
)−f(u
per
2
)

(τ)k
L
2
dτ
6C
Z
t
−∞
(1+t−τ)
−
n
4
−
e
k+2
2
k

f(u
per
1
)−f(u
per
2
)

(τ)k
L
1
dτ
+C
Z
t
−∞
e
−c(t−τ)
k∂
e
k+2
x

f(u
per
1
)−f(u
per
2
)

(τ)k
L
2
dτ
6C(ku
per
1
k
X
+ku
per
2
k
X
)
σ−1
ku
per
1
−u
per
2
k
X
.(4.19)
d(4.18)Ú(4.19)Œ
k[N(u
per
1
)−N(u
per
2
)](t)k
X
6C(ku
per
1
k
X
+ku
per
2
k
X
)
σ−1
ku
per
1
−u
per
2
k
X
6Cρ
σ−1
ku
per
1
−u
per
2
k
X
.(4.20)
5¿σ>2,ρ= 4CE
0
,E
0
¿©ž§d(4.20)Œ
k[N(u
per
1
)−N(u
per
2
)](t)k
X
6
1
2
ku
per
1
−u
per
2
k
X
.(4.21)
¤±N´X¥Ø N§dØ NnŒ•§NN•3•˜ØÄ:u
per
∈X,¤±•
§(1.1)•3•˜žm±Ï)u
per
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m
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1
([0,T];H
m−2
).Ïd½n4.1¤á"
5.(Ø
©ïÄ‘r{ZBoussinesq •§3±ÏåŠ^ežm±Ï)•3•˜5§·‚
ïÄ•{̇´|^r{Z‘‘5ÑÑA¼Green ¼êÅ:O,lïá)Žf
3L
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ë•©z
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DOI:10.12677/pm.2023.1351221189nØêÆ

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