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AdvancesinAppliedMathematics
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,2023,12(6),3011-3020
PublishedOnlineJune2023inHans.https://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2023.126302
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NonuniformContinuityofaClass
ofShallowWaterWaveEquation
SenlinCai
SchoolofMathematicalSciences,ChongqingNormalUniversity,Chongqing
Received:May28
th
,2023;accepted:Jun.23
rd
,2023;published:Jun.30
th
,2023
Abstract
Inthispaper,theperiodicCauchyproblemofshallowwaterwavemodelwithmoderate
amplitudeisstudied,wefirstconstructtwosolutionsequences,theyareboundedin
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DOI:10.12677/aam.2023.126302
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,andconvergetozeroattheinitialtimeinterval,buttheboundofthe
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thatthesolutionmappingoftheequationisnon-uniformlycontinuousinSobolev
space.
Keywords
CauchyProblem,NonuniformContinuity,ShallowWaterWave
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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σ
.
n
−
1
+
n
−
s
+
σ
,n
1
.
(3.5)
Proof.
ù
‡
Ú
n
y
²
Ú
©
z
[17]
¥
y
²
a
q
§
©
ò
Ø
2
?
1
y
²
"
e
¡
y
²
(3.3)
H
σ
§
¦
^
Ú
n
3.1
§
·
‚
Œ
±
k−
1
/
2
a
1
λ
−
2
s
+1
sin
2
A
−
a
1
ωλ
−
2
s
sin
2
A
+1
/
2
a
2
λ
−
3
s
sin
2
Acos
2
A
k
H
σ
.
λ
−
2
s
+1+
σ
+
λ
−
2
s
+
σ
+
λ
−
3
s
+
σ
.
(3.6)
e
5
y
²
f
(
u,u
x
)
H
σ
§
Ù
¥
·
‚
I
‡
^
Ú
n
2.1(ii)
!
Cauchy-Schwarz
Ø
ª
Ú
Ú
n
3.1,
k
f
(
u,u
x
)
k
H
σ
=
k
(1
−
∂
x
)
−
1
∂
x
(
b
1
u
2
+
b
2
u
3
+
b
2
u
4
+
b
3
u
4
+
b
4
u
5
+
b
5
u
6
+
b
6
u
7
+
b
7
u
2
x
+
b
8
uu
2
x
)
k
H
σ
+
k
b
9
(1
−
∂
x
)
−
1
u
3
x
k
H
σ
.
λ
−
2
s
+1+
σ
+
λ
−
s
−
1+
σ
.
(3.7)
Š
â
±
þ
O
§
·
‚
Œ
±
±
e
·
K
"
·
K
3.1.
é
u
s>
3
/
2
,
1
/
2
<σ
≤
1
,n
1
,
·
‚
k
k
F
k
H
σ
.
n
−
r
s
,
(3.8)
Ù
¥
r
s
>
0
…
r
s
=
(
2
s
−
1
−
σ,
3
/
2
<s<
2
,
s
+1
−
σ,s
≥
2
.
(3.9)
4.
C
q
)
†
ý
¢
)
ƒ
m
Ø
-
u
ω,n
(
t,x
)
´
•
§
(2.1)
±
Ï
)
§
K
u
ω,n
(
t,x
)
÷
v
(
∂
x
u
ω,n
+(
a
1
u
ω,n
+
a
2
u
2
ω,n
)
∂
x
u
ω,n
−
f
(
u
ω,n
,∂
x
u
ω,n
) = 0
,
u
ω,n
(0
,x
) =
u
ω,n
(0
,x
) =
ωn
−
1
−
n
−
s
cos
(
nx
)
.
(4.1)
•
O
C
q
)
†
ý
¢
)
Ø
§
·
‚
v
=
u
ω,n
−
u
ω,n
.
DOI:10.12677/aam.2023.1263023015
A^
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Ð
é
Ü
w
,
§
v
÷
v
±
e
ª
∂
t
v
=
F
−
a
1
u
ω,n
∂
x
v
−
a
1
v∂
x
u
ω,n
−
a
2
(
u
ω,n
)
2
∂
x
u
ω,n
+
a
2
(
u
ω,n
)
2
∂
x
u
ω,n
+
f
(
u
ω,n
,∂
x
u
ω,n
)
−
f
(
u
ω,n
,∂
x
u
ω,n
)
,
V
(0
,x
) = 0
.
(4.2)
Ù
¥
F
(3.2)
½
Â
…
÷
v
Ø
ª
(3.8).
·
K
4.1.
e
n
1
,s>
3
/
2
,
1
/
2
<σ
≤
min
1
,s
−
1
,
K
k
v
(
t
)
k
H
σ
.
=
k
u
ω,n
(
t
)
−
u
ω,n
(
t
)
k
H
σ
.
n
−
r
s
,n
1
,
0
≤
t
≤
T,
(4.3)
Ù
¥
r
s
>
0
…
d
(3.9)
‰
Ñ
"
Proof.
^
Ž
f
Λ
σ
¦
±
•
§
(4.2)
ü
>
§
,
^
Λ
σ
v
¦
±
(
J
§
2
é
±
Ï
˜
m
S
x
?
1
È
©
·
‚
Œ
±
1
2
d
dt
k
v
(
t
)
k
2
H
σ
=
Z
T
Λ
σ
F
Λ
σ
vdx
−
a
1
Z
T
Λ
σ
u
ω,n
∂
x
v
Λ
σ
vdx
−
a
1
Z
T
Λ
σ
v∂
x
u
ω,n
Λ
σ
vdx
−
a
2
Z
T
Λ
σ
(
u
ω,n
)
2
∂
x
u
ω,n
Λ
σ
vdx
+
a
2
Z
T
Λ
σ
(
u
ω,n
)
2
∂
x
u
ω,n
Λ
σ
vdx
+
Z
T
Λ
σ
(
f
(
u
ω,n
,∂
x
u
ω,n
−
f
(
u
ω,n
,∂
x
u
ω,n
))Λ
σ
vdx
=
E
1
+
E
2
++
E
3
+
E
4
+
E
5
+
E
6
.
(4.4)
e
5
·
‚
é
Ø
F
H
ρ
?
1
O
§
Ä
k
1
˜
‘
O
X
e
µ
|
E
1
|
=
Z
T
Λ
σ
F
Λ
σ
vdx
≤k
F
k
H
σ
+
k
v
k
H
σ
.
n
−
r
s
||
v
(
t
)
||
H
σ
(4.5)
1
‘
O
X
e
µ
|
E
2
|
.
Z
T
Λ
σ
∂
x
(
u
ω,n
,v
)Λ
σ
vdx
+
Z
T
Λ
σ
v∂
x
u
ω,n
Λ
σ
vdx
≤||
[Λ
σ
∂
x
,u
ω,n
]
v
||
L
2
||
v
||
H
σ
+
1
2
Z
T
∂
x
u
ω,n
(Λ
σ
v
)
2
dx
+
||
u
ω,n
||
H
s
||
v
||
2
H
σ
.
||
u
ω,n
||
H
s
||
v
||
2
H
σ
,
(4.6)
Ù
¥
^
©
Ü
È
©
Ú
Ú
n
2.2(ii).
e
5
·
‚
O
ª
(4.4)
1
n
‘
X
e
µ
|
E
3
|
=
Z
T
Λ
σ
v∂
x
u
ω,n
Λ
σ
vdx
≤||
v∂
x
u
ω,n
||
H
σ
||
v
||
H
σ
≤
C
(
||
v
||
H
σ
||
∂
x
u
ω,n
||
L
∞
+
||
∂
x
u
ω,n
||
H
σ
||
v
||
H
σ
)
||
v
||
H
σ
.
||
u
ω,n
||
H
s
||
v
||
2
H
σ
,
(4.7)
DOI:10.12677/aam.2023.1263023016
A^
ê
Æ
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é
Ü
Ù
¥
$
^
Ú
n
2.1(i)
Ú
Cauchy-Schwar
Ø
ª
"
a
q
c
¡
y
²
L
§
§
·
‚
U
•
{
Ø
H
σ
O
§
¿
ò
(
J
(
Ü
å
5
Œ
±
1
2
||
v
||
2
H
σ
.
B
||
v
||
2
H
σ
+
n
−
r
s
||
v
||
H
σ
.
(4.8)
=
d
dt
||
v
||
H
σ
.
B
||
v
||
H
σ
+
n
−
r
s
,
(4.9)
Ù
¥
B
´•
{
‘
N
"
·
‚
A^
Ø
ª
||
u
ω,n
(
t
)
||
H
s
.
||
u
ω,n
(0)
||
H
s
=
||
u
ω,n
(0)
||
H
s
Œ
±
||
v
(
t
)
||
H
σ
.
n
−
r
s
,n
10
≤
t
≤
T,
(4.10)
ù
Ò
y
²
·
K
4.1.
e
5
·
‚
y
²
½
n
1.1
Proof.
4
·
‚
Ä
k
b
s>
3
/
2,
¿
…
u
1
,n
(
x,t
)
Ú
u
−
i,n
(
x,t
)
©
O
´
•
§
(2.1)
3
Ð
Š
^
‡
•
u
1
,n
(
x,
0)
Ú
u
−
1
,n
(
x,
0)
)
,
…
ù
ü
‡
)
á
u
C
([0
,T
];
H
s
(
T
)).
d
ª
(3.1)
Ú
||
u
(
t
)
||
H
s
≤
2
||
u
(0)
||
H
s
,
·
‚
U
||
u
1
,n
(
x,t
)
||
H
s
+
||
u
−
1
,n
(
x,t
)
||
H
s
≤
2
C
||
u
1
,n
(0)
||
H
s
+
||
u
−
1
,n
(0)
||
H
s
.
1
.
(4.11)
t
=
o
…
λ
−→∞
ž
k
||
u
1
,n
(0)
−
u
−
1
,n
(0)
||
H
s
−→
0
.
(4.12)
t<
0
ž
§
$
^
n
Ø
ª
Œ
±
||
u
1
,n
(
x,t
)
||
H
s
+
||
u
−
1
,n
(
x,t
)
||
H
s
≤
2
C
||
u
1
,n
(0)
||
H
s
+
||
u
−
1
,n
(0)
||
H
s
.
1
.
(4.13)
t
=
o
…
λ
−→∞
ž
k
||
u
1
,n
(
t
)
−
u
−
1
,n
(
t
)
||
H
s
≥||
u
1
,n
(
t
)
−
u
−
1
,n
(
t
)
||
H
s
−||
u
1
,n
(
t
)
−
u
1
,n
(
t
)
||
H
s
−||
u
−
1
,n
(
t
)
−
u
1
,n
(
t
)
||
H
s
.
(4.14)
$
^
cosα
−
cosβ
=
−
2
sin
α
+
β
2
sin
α
−
β
2
Œ
||
u
1
,n
(
t
)
−
u
−
1
,n
(
t
)
||
H
s
=
||
2
n
−
1
−
2
n
−
s
sin
(
nx
−
a
1
ωt
−
a
2
ω
2
λ
−
1
t
)
sin
(
a
1
ωt
+
a
2
ω
2
λ
−
1
t
)
||
H
s
.
(4.15)
DOI:10.12677/aam.2023.1263023017
A^
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Ð
é
Ü
k
= [
s
]+2,
$
^
Ú
n
3.1
Ú
||
u
(
t
)
||
H
s
≤
2
||
u
(0)
||
H
s
Œ
||
u
ω,n
(
t
)
−
u
ω,n
(
t
)
||
H
k
≤||
u
ω,n
(
t
)
||
H
k
+2
||
u
ω,n
(0)
||
H
k
.
n
k
−
s
,
0
<t
≤
T.
(4.16)
¦
^
Ú
n
2.3
…
-
s
1
=
σ,s
2
= [
s
]+2
k
||
u
ω,n
(
t
)
−
u
ω,n
(
t
)
||
H
s
≤||
u
ω,n
(
t
)
−
u
ω,n
(
t
)
||
k
−
s
k
−
σ
H
σ
||
u
ω,n
(
t
)
−
u
ω,n
(
t
)
||
s
−
σ
k
−
σ
H
k
.
n
−
r
s
(
k
−
s
)
k
−
σ
n
(
k
−
s
)(
s
−
σ
)
k
−
σ
.
n
−
(
r
s
−
s
+
σ
)
k
−
σ
,
(4.17)
ò
ƒ
c
r
s
“
\þ
ª
Œ
||
u
ω,n
(
t
)
−
u
ω,n
(
t
)
||
H
s
.
n
−
ε
s
,
Ù
¥
ε
s
=
(
(
s
−
1)(
k
−
s
)
k
−
σ
,
3
/
2
<s<
2
,
(
k
−
s
)
k
−
σ
,s
≥
2
,
(4.18)
ù
p
ε
s
Œ
u
"
§
·
‚
(4.14)
ü
>
4
•
e
(
.
k
liminf
n
→∞
k
u
1
,n
(
t
)
−
u
−
1
,n
(
t
)
k
H
s
≥
liminf
n
→∞
(
k
u
1
,n
(
t
)
−
u
−
1
,n
(
t
)
k
H
s
−||
u
1
,n
(
t
)
−
u
1
,n
(
t
)
k
H
s
−||
u
−
1
,n
(
t
)
−
u
−
1
,n
(
t
)
k
H
s
)
&
sin
a
1
t
+
a
2
λ
−
1
t
2
>
0
,
Ù
¥
0
<t<
min
n
T,
2
π
|
a
1
+
a
2
λ
−
1
|
o
.
ù
Ò
y
²
½
n
1.1.
5.
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Ø
9
Ð
"
©
ï
Ä
¥
Ì
f
Y
Å
.
±
Ï
…
ܯ
K
§
·
‚
Ä
k
E
ü
‡
)
S
§
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‚
3
H
s
(
T
),
s>
3
/
2
¥
´
k
.
§
¿
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3
Ð
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ž
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m
Â
ñ
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§
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ù
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m
å
l
e
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3
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N
3
Sobolev
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m
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ë
Y
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š
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k
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z
[1]Quirchmayr,R.(2016)ANewHighlyNonlinearShallowWaterWaveEquation.
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