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PureMathematicsnØêÆ,2022,12(3),448-457
PublishedOnlineMarch2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.123050
£Ä‚¸eFisher-KPP•§É½1Å)
•35
ëëë
•ânóŒÆêƆÚOÆ§H•â
ÂvFϵ2022c218F¶¹^Fϵ2022c321F¶uÙFϵ2022c328F
Á‡
•Ä£Ä‚¸eFisher-KPP•§3ÙSO•Ç¼êšK^‡eɽ1Å)•35"|^ü
NS“(Üþe)•{E|§y²?¿ð½„Ý£Äeš~ɽ1Å•35"
'…c
£Ä‚¸§Fisher-KPP•§§üNS“
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
©ÙÚ^:ë.£Ä‚¸eFisher-KPP•§É½1Å)•35[J].nØêÆ,2022,12(3):448-457.
DOI:10.12677/pm.2022.123050
ë
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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DOI:10.12677/pm.2022.123050449nØêÆ
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00
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0
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00
(ξ)+U(ξ)[r(−∞)−U(ξ)].(4)
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DOI:10.12677/pm.2022.123050450nØêÆ
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,
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+
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−
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Z
ξ
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e
λ
−
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Z
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λ
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Γ = {U∈BC(R,R)|U≤U≤U}.
w,,8ÜΓ•BC(R,R)¥š˜k.4à8.
é?¿U∈Γ, ½ÂXeŽf
H(U)(ξ) := αU(ξ)+U(ξ)[r(ξ)−U(ξ)].(8)
e5,·‚‰ÑŽfHü‡5Ÿ.
Ún2.1e¡(Øþ¤áµ
DOI:10.12677/pm.2022.123050451nØêÆ
ë
(i)H(Γ) •BC(R,R)¥k.8¶
(ii)ŽfH: Γ →BC(R,R)ëY.
y²UÚU´(2)˜ékSþe).P
D:= max{sup
ξ∈R
|U(ξ)|,sup
ξ∈R
|U(ξ)|}.(9)
˜•¡,é?¿U∈ΓÑk
|H(U)(ξ)|≤(α+r(ξ)+|U(ξ)|)|U(ξ)|≤(α+r(+∞)+D)D.
Ïd
|H(U)|≤(α+r(+∞)+D)D,
H(Γ)•BC(R,R)¥k.8.
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1
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2
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1
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2
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2
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2
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2
2
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2
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=
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1
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2
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1
−U
2
|.
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e5,·‚½ÂNF(U) := ∆
−1
H(U),∀U∈Γ ⊂BC(R,R).·‚k?ØNF˜5Ÿ.
Ún2.2F´˜‡š~Žf.,,eU(ξ) ∈Γš~,KF(U)(ξ) 'uξ•š~.
y²
˜
U,
ˆ
U∈Γ …÷v
˜
U≥
ˆ
U,Ké?Ûξ∈Rk
H(
˜
U)(ξ)−H(
ˆ
U)(ξ) = (α+r(ξ)−
˜
U(ξ)−
ˆ
U(ξ))(
˜
U(ξ)−
ˆ
U(ξ)) ≥0.
l,é∀ξ∈R,kF(
˜
U)(ξ) ≥F(
ˆ
U)(ξ).=,F´˜‡š~Žf.
eU(ξ) ∈Γ´˜‡'uξš~¼ê,Ké∀ζ>0Ú∀s∈R,k
H(U)(s+ζ)−H(U)(s)
= (U(s+ζ)−U(s))[α+r(s+ζ)−U(s+ζ)+U(s)]+(r(s+ζ)−r(s))U(s) ≥0.
éu∀ξ∈R,5¿h∈B
µ
(R,R) ž,k(∆
−1
h(s+ζ))(ξ) = (∆
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h(s))(ξ+ζ),@o
F(U)(ξ+ζ) = [∆
−1
H(U)(s)](ξ+ζ) = [∆
−1
H(U)(s+ζ)](ξ)
≥[∆
−1
H(U)(s)](ξ) = F(U)(ξ).
DOI:10.12677/pm.2022.123050452nØêÆ
ë
y..
Ún2.3F(Γ) ⊂Γ.
y²dÚn2.1´•F(Γ) ⊂BC(R,R).Ïd,·‚•Iyé¤kU∈ΓÑk
U≤F(U) ≤U.
duU,U´˜ékSþe),(Ü©z[18]Ún2.1 ·‚k
F(U) = ∆
−1
H(U) ≥∆
−1
∆(U) ≥U,F(U) = ∆
−1
H(U) ≤∆
−1
∆U≤U.(10)
dÚn2.3•F´˜‡š~Žf.¤±é∀U∈Γk
F(U) ≤F(U) ≤F(U).(11)
(Ü(10)Ú(11)F(Γ) ⊂Γ.y..
•§(2)Œ•
∆U= H(U).(12)
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U= F(U),
KTØÄ:7´(12) ).eTØÄ:„÷v>.^‡(5),K7´•§(1) ɽ1Å.ùÒ´·
‚ïÄ8I.
3.ɽ1Å)•35
½n3.1e(A) ¤á,Ké?¿c>0,•§(1) o•3˜‡÷v>.^‡(5)š~ɽ1
Å.
y²Äk•Äœ/(i):r(−∞) >0 .
U(ξ)=r(+∞) ÚU(ξ)=r(−∞),w,,U(ξ) ÚU(ξ) ÷v½Â2.1 ¤k^‡§=U(ξ)
ÚU(ξ) ´(2) ˜ékSþe).u´·‚˜‡k8:
Γ
1
:= {U∈BC(R,R)|U≤U≤U},
EXeS“S:
U
(1)
= F(U),U
(n+1)
= F(U
(n)
), ∀n≥1.
ÏU(ξ)∈Γ
1
´Rþš~¼ê,(ÜÚn2.2 ÚÚn2.3é¤kn≥1,U
(n)
(ξ)•´Rþ
š~¼ê…÷vØª
U(ξ) ≥U
(1)
(ξ) ≥U
(2)
(ξ) ≥···≥U
(n)
(ξ) ≥U
(n+1)
(ξ) ≥···≥U(ξ).
DOI:10.12677/pm.2022.123050453nØêÆ
ë
l,•3˜‡k .š~¼êU(ξ)¦lim
n→∞
U
(n)
(ξ)= U(ξ).éN´wÑ,é¤kn≥1,ξ∈R
k|H(U
(n)
)(ξ)|≤r(+∞)(r(+∞)+α).l|^Lebesgue’s››Âñ½n
U(ξ) =lim
n→∞
U
(n+1)
(ξ) =lim
n→∞
F(U
(n)
)(ξ)
=lim
n→∞
(∆
−1
H(U
(n)
)(ξ))
=
1
d(λ
+
−λ
−
)
"
Z
ξ
−∞
e
λ
−
(ξ−η)
H(U)(η)dη+
Z
+∞
ξ
e
λ
+
(ξ−η)
H(U)(η)dη
#
=F(U)(ξ).
=U(ξ) ∈Γ
1
´ŽfFØÄ:,•Ò´`U(ξ)´•§(2)).
e5y²U(ξ) ÷v>.^‡(5).ÏU(ξ)´Rþk.š~¼ê,PA
1
:=lim
ξ→−∞
U(ξ)Ú
B
1
:=lim
ξ→+∞
U(ξ).w,k
0 <r(−∞) ≤A
1
≤r(+∞),0 <r(−∞) ≤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{KŒ
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)dη+
Z
+∞
ξ
e
λ
+
(ξ−η)
H(U)(η)dη
#
=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(−∞).Ón,
B
1
=lim
ξ→+∞
U(ξ) =B
1
+
B
1
[r(+∞)−B
1
]
α
.
¤±B
1
= r(+∞).
e¡•Äœ/(ii): r(−∞)=0.P˜r(ξ) ÷v©[18]¥b(H),@o•3ϕ(ξ) ´©[18]¥•
§(1.2)).w,,ϕ(ξ) oØŒu(2) ).-V(ξ) = r(+∞) ÚV(ξ) = ϕ(ξ).Šâ½Â2.1,Ø
JyV(ξ) ÚV(ξ) ´(2) ˜ékSþe).?˜Ú,½ÂXek8
Γ
2
:= {V∈BC(R,R)|V≤V≤V},
DOI:10.12677/pm.2022.123050454nØêÆ
ë
aquœ/(i)y²Œ•µ•3V∈Γ
2
´ŽfFØÄ:.,ξ→−∞ž,Ó¦^L’Hˆopital
{K
lim
ξ→−∞
V(ξ) = 0.
ξ→+∞ž, kV(ξ) →r(+∞).dY%OK,
lim
ξ→+∞
V(ξ) = r(+∞).
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