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PureMathematics
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,2022,12(3),448-457
PublishedOnlineMarch2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.123050
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ExistenceofForcedTravelingWaves
forFisher-KPPEquationundera
ShiftingHabitat
LinlinZhao
SchoolofMathematicsandStatistics,ChangshaUniversityofScienceandTechnology,Changsha
Hunan
Received:Feb.18
th
,2022;accepted:Mar.21
st
,2022;published:Mar.28
th
,2022
Abstract
Inthispaper,weareconcernedwiththeexistenceofforcedtravelingwavesolutions
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n
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,2022,12(3):448-457.
DOI:10.12677/pm.2022.123050
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forFisher-KPPequationinthehabitatshiftingundertheconditionthatitsintrin-
sicgrowthratefunctionisnonnegative.Usingthetechniqueofmonotoneiteration
combinedwiththeupperandlowersolutionmethod,theexistenceofnon-decreasing
forcedtravelingwavesunderarbitrarypositiveconstantshiftingspeedisproved.
Keywords
ShiftingHabitat,Fisher-KPPEquation,MonotoneIterative
Copyright
c
2022byauthor(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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±
E
k
8
Γ:
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{
U
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BC
(
R
,
R
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|
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≤
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≤
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}
.
w
,
,
8
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à
8
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é
?
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Γ,
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X
e
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ξ
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(
ξ
)+
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(
ξ
)[
r
(
ξ
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−
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(
ξ
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.
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e
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f
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.
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n
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DOI:10.12677/pm.2022.123050451
n
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k
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U
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ξ
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K
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∀
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0
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∀
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k
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5
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DOI:10.12677/pm.2022.123050452
n
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y
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Ú
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2.3
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2.3
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k
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1
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X
e
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n
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∀
n
≥
1
.
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U
(
ξ
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∈
Γ
1
´
R
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š
~
¼
ê
,
(
Ü
Ú
n
2.2
Ú
Ú
n
2.3
é
¤
k
n
≥
1,
U
(
n
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(
ξ
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¼
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v
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n
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ξ
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ξ
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≥···≥
U
(
ξ
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.
DOI:10.12677/pm.2022.123050453
n
Ø
ê
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ë
l
,
•
3
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k
.
š
~
¼
ê
U
(
ξ
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lim
n
→∞
U
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n
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(
ξ
)=
U
(
ξ
).
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N
´
w
Ñ
,
é
¤
k
n
≥
1,
ξ
∈
R
k
|
H
(
U
(
n
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)(
ξ
)
|≤
r
(+
∞
)(
r
(+
∞
)+
α
).
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|
^
Lebesgue’s
›
›
Â
ñ
½
n
U
(
ξ
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n
→∞
U
(
n
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(
ξ
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n
→∞
F
(
U
(
n
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)(
ξ
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=lim
n
→∞
(∆
−
1
H
(
U
(
n
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)(
ξ
))
=
1
d
(
λ
+
−
λ
−
)
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Z
ξ
−∞
e
λ
−
(
ξ
−
η
)
H
(
U
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η
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η
+
Z
+
∞
ξ
e
λ
+
(
ξ
−
η
)
H
(
U
)(
η
)d
η
#
=
F
(
U
)(
ξ
)
.
=
U
(
ξ
)
∈
Γ
1
´
Ž
f
F
Ø
Ä:
,
•
Ò
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U
(
ξ
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§
(2)
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e
5
y
²
U
(
ξ
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÷
v
>
.
^
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(5).
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U
(
ξ
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R
þ
k
.
š
~
¼
ê
,
P
A
1
:=lim
ξ
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U
(
ξ
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Ú
B
1
:=lim
ξ
→
+
∞
U
(
ξ
).
w
,
k
0
<r
(
−∞
)
≤
A
1
≤
r
(+
∞
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,
0
<r
(
−∞
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≤
B
1
≤
r
(+
∞
)
.
d
lim
ξ
→−∞
H
(
U
)(
ξ
) =
A
1
(
α
+
r
(
−∞
)
−
A
1
)
,
lim
ξ
→
+
∞
H
(
U
)(
ξ
) =
B
1
(
α
+
r
(+
∞
)
−
B
1
)
,
|
^
L’Hˆopital
{
K
Œ
A
1
=lim
ξ
→−∞
U
(
ξ
) =lim
ξ
→−∞
(∆
−
1
H
(
U
)(
ξ
))
=lim
ξ
→−∞
1
d
(
λ
+
−
λ
−
)
"
Z
ξ
−∞
e
λ
−
(
ξ
−
η
)
H
(
U
)(
η
)d
η
+
Z
+
∞
ξ
e
λ
+
(
ξ
−
η
)
H
(
U
)(
η
)d
η
#
=lim
ξ
→−∞
1
d
(
λ
+
−
λ
−
)
H
(
U
)(
ξ
)
−
λ
−
+
H
(
U
)(
ξ
)
λ
+
=
A
1
+
A
1
[
r
(
−∞
)
−
A
1
]
α
.
u
´
A
1
=
r
(
−∞
).
Ó
n
,
B
1
=lim
ξ
→
+
∞
U
(
ξ
) =
B
1
+
B
1
[
r
(+
∞
)
−
B
1
]
α
.
¤
±
B
1
=
r
(+
∞
).
e
¡
•
Ä
œ
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(ii):
r
(
−∞
)=0.
P
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r
(
ξ
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÷
v
©
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¥
b
(H),
@
o
•
3
ϕ
(
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w
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ϕ
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-
V
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r
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V
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ξ
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ϕ
(
ξ
).
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â
½
Â
2.1,
Ø
J
y
V
(
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Ú
V
(
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k
S
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)
.
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Ú
,
½
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X
ek
8
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2
:=
{
V
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BC
(
R
,
R
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|
V
≤
V
≤
V
}
,
DOI:10.12677/pm.2022.123050454
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Ø
ê
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ë
a
q
u
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/
(i)
y
²
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µ
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3
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2
´
Ž
f
F
Ø
Ä:
.
,
ξ
→−∞
ž
,
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¦
^
L’Hˆopital
{
K
lim
ξ
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V
(
ξ
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.
ξ
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+
∞
ž
,
k
V
(
ξ
)
→
r
(+
∞
).
d
Y
%
O
K
,
lim
ξ
→
+
∞
V
(
ξ
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r
(+
∞
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.
y
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.
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7
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8
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Ž
g
,
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©
z
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