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AdvancesinAppliedMathematicsA^êÆ?Ð,2023,12(5),2480-2492
PublishedOnlineMay2023inHans.https://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2023.125250
˜a •¼‡©•§ÿÝ
–.)93)ÔêÆ.¥
A^
•••
úô•…’EâÆÄ:ܧúôɲ
ÂvFϵ2023c428F¶¹^Fϵ2023c521F¶uÙFϵ2023c531F
Á‡
©ïÄ˜aäkÕž¢α-. •¼‡©•§ÿÝ–.)•35Ú•˜5"©¥b‚
5ŽfÜ©3Banach˜mXþ)¤;)ÛŒ+§,|^©ê˜ŽfnØÚŽfŒ+nØy
²•§)•3•˜5§¿òÙA^u˜a)ÔêÆ.¥§y(Ø(5"
'…c
•¼‡©•§§Ã¡ž¢§ÿÝا©ê˜Žf
A Class of Measure Pseudo Type Solution
ofPartialFunctionalDifferential
EquationsandApplicationsin
BiomathematicalModel
FangWang
DepartmentofBasicCourses,ZhejiangChangzhengVocationalTechnicalCollege,
HangzhouZhejiang
Received:Apr.28
th
,2023;accepted:May21
st
,2023;published:May31
st
,2023
©ÙÚ^:•.˜a •¼‡©•§ÿÝ–.)93)ÔêÆ.¥A^[J].A^êÆ?Ð,2023,12(5):
2480-2492.DOI:10.12677/aam.2023.125250
•
Abstract
Inthispaper, theexistence, uniquenessofmeasure pseudotype solutionsintheα-form
forpartialfunctionaldifferentialequationswithinfinitedelayareinvestigated.Here
weassumethatthelinearpartgeneratesacompactanalyticsemigrouponaBanach
spaceX,thedelayedpartisassumedtobecontinuouswithrespecttothefractional
power of thegenerator.Finally,anexampleofbiomathematicalmodelispresented to
illustratethemainfindings.
Keywords
PartialFunctionalDifferentialEquations,InfiniteDelay,MeasureTheory,Fractional
PowerofOperators
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
3©z[1]¥,ïÄXeäkអ •¼‡©•§({P•PFDEs)V±Ï)•35,





d
dt
u(t) = −Au(t)+L(u
t
)+f(t),t≥σ,
u
σ
= ϕ∈B.
(1.1)
Ù¥−A÷vHille-Yosida^‡§¼êu
t
∈B½Â•u
t
(θ)=u(t+θ),θ∈(−∞,0],B•l(−∞,0]
NX¼ê¤Banach˜m,…÷v©ò‡0˜ún5^‡.L•lBNX
k.‚5Žf,f•V±Ï¼ê.d,?˜ÚXJ−A3Xþ)¤)ÛŒ+(T(t))
t≥0
,KŠâ©
꘎fnØ,©z[2]ïÄ‡©•§AgÅ5.
ÏäkអPFDEs3)Ô!zÆÚÔn+•Ñk2•A^,8c®²¤•ÄåXÚ
˜‡-‡ïÄ+•.'uអPFDEs©zš~õ,~X3[3]¥ïÄ±Ï5,3[4–6]
¥ïÄV±Ï5,[7,8]?ØAgÅ5,[9,10]ïÄ•§ÿÝ–V±Ï5ÚÿÝ–Ag
Å5,)K5Ú-½5K©O3[11–13]Ú[12,14,15]¥k¤&?.3ÄåÆ5ŸïÄ¥§
DOI:10.12677/aam.2023.1252502481A^êÆ?Ð
•
±Ï5§V±Ï5§AgÅ5§–V±Ï5§ –AgÅ5´É'5ïÄé–§é
k©zéù5Ÿ?1Ú˜/ïÄ",˜•¡§3± ©z¥§ŒÜ©©z−A)¤´ëYŒ
+§é)¤)ÛŒ +ïÄ¿Øõ„"©3c¡©zÄ:þ§|^©ê˜ŽfnØÚŽfŒ
+•{§ïÄ.(1.1)ÿÝ–.)•35Ú•˜5,Ù¥−A)¤)ÛŒ+.ƒéuk•ž
¢,អw,´•E,,Ï•)äN5Ÿ†ƒ˜mBÀJ—ƒƒ'§3©¥§·‚
ƒ˜mB÷vHaleÚKato‰ÑÄún5^‡[16].
2.ý•£
P(X,|·|)•˜‡EBanach˜m,N,R,R
+
ÚC©OL«g,ê8,¢ê8,šK¢ê8ÚE
ê8.A´½Â3Xþ‚5Žf,D(A)Úσ(A)L«A½Â•ÚÌ.
½Â2.1.¡f∈C(R,X)•‡±Ï¼ê(½±Ï¼ê),e•3ω∈R−{0}¦éu¤kt∈R÷
vf(t+ω) = −f(t)(½f(t+ω) = f(t)).òùa¼ê8ÜP•P
ωap
(R,X)(½P
ω
(R,X)).
52.1.ef∈P
ωap
(R,X),Kkf∈P
2ω
(R,X).
½Â2.2.[17]éu‰½ω,k∈R,¡f∈C(R,X)•(ω,k)-±Ï¼ê(½Bloch-±Ï¼ê),eéu
¤kt∈R÷vf(t+ω) = e
ikω
f(t).òùa¼ê8ÜP•P
ω,k
(R,X).
52.2.¡k•fFloquet•ê.ef´(ω,k)-±Ï¼ê…ωk=2π,Kf´ω-±Ï¼ê;eωk=π,
Kf´ω-‡±Ï¼ê.
½Â2.3.[18]éu‰½ρ∈C(R,X),¡f∈C(R,X)•\±Ï¼ê,eéu¤kt∈R÷
vf(t+ω) = ρ(t)f(t).òùa\±Ï¼ê8ÜP•P
ω,ρ(t)
(R,X).
½Â2.4.[19]¡f∈C(R,X)•V±Ï¼ê,eéu?¿ε>0,•3l(ε)>0,¦3z‡•Ý
•l(ε)«mIþ–•3˜‡τ¦Ù÷vkf(t+τ)−f(t)k<ε阃t∈R¤á.ùaV±Ï¼
ê8ÜP•AP(R,X).
½Â2.5.[20] ¡f∈C(R,X) •Agżê, eéuz‡¢ê(s
0
n
)
n∈N
§•3˜‡fê(s
n
)
n∈N
¦g(t) :=lim
n→∞
f(t+s
n
)éu?¿t∈RÑk½Â,…lim
n→∞
g(t−s
n
) = f(t)é?¿t∈R¤á.ù
a¼ê8ÜP•AA(R,X).
e¡,·‚|^ÿÝnØ•{§ÏLH{˜mVg§Ú\ÿÝ–.¼êVg,=ÿÝ–
‡±Ï¼ê,ÿÝ–±Ï¼ê,ÿÝ–Bloch-±Ï¼ê,ÿÝ–\±Ï¼ê,ÿÝ–V±Ï¼êÚ
ÿÝ–Agżê.
LL«RþLebesgueσ•,ML«3Lþ¤k÷vXe^‡ÿݵ8ܵéu¤k
a,b∈R(a≤b),…µ(R) = ∞Úµ([a,b]) <∞.
½Â2.6.µ,ν∈M.e÷v
lim
T→+∞
1
µ([−T,T])
Z
[−T,T]
|f(t)|dν(t) = 0,
K¡¼êf∈C(R,X)•(µ,ν)-H{.ùa¼ê8ÜP•E(R,X,µ,ν).
½Â2.7.µ,ν∈M.¡f∈C(R,X)•ÿÝ–‡±Ï¼ê(½ÿÝ–±Ï¼ê,ÿÝ–Bloch-±
ϼê,ÿÝ–\±Ï¼ê,ÿÝ–V±Ï¼êÚÿÝ–Agżê),e¼êŒ©)•f=
g+ ϕ,Ù¥g∈P
ωap
(R,X)(½P
ω
(R,X),P
ω,k
(R,X),P
ω,ρ(t)
(R,X),AP(R,X),AA(R,X))…ϕ∈
DOI:10.12677/aam.2023.1252502482A^êÆ?Ð
•
E(R,X,µ,ν).ùa¼ê8ÜP•MPP
ωap
(R,X,µ,ν)(½MPP
ω
(R,X,µ,ν),MPP
ω,k
(R,X,µ,ν),
MPP
ω,ρ(t)
(R,X,µ,ν),MPAP(R,X,µ,ν),MPAA(R,X,µ,ν)).
52.3.dþ¡½Â§ØJwÑXe•¹'X¤áµ
(i)MPP
ωap
(R,X,µ,ν) ⊂MPP
ω
(R,X,µ,ν) ⊂MPP
ω,k
(R,X,µ,ν) ⊂MPP
ω,ρ(t)
(R,X,µ,ν).
(ii)MPP
ωap
(R,X,µ,ν) ⊂MPAP(R,X,µ,ν) ⊂MPAA(R,X,µ,ν).
(iii)MPP
ω
(R,X,µ,ν) ⊂MPAP(R,X,µ,ν),MPP
ω
(R,X,µ,ν) ⊂MPAA(R,X,µ,ν).
©O½ÂXe8ܵ
A(R,X) = {P
ωap
(R,X),P
ω
(R,X),P
ω,k
(R,X),P
ω,ρ(t)
(R,X),AP(R,X),AA(R,X)},
P(R,X,µ,ν) = {MPP
ωap
(R,X,µ,ν),MPP
ω
(R,X,µ,ν),MPP
ω,k
(R,X,µ,ν),MPP
ω,ρ(t)
(R,X,µ,ν),
MPAP(R,X,µ,ν),MPAA(R,X,µ,ν)},
Kf∈P(R,X,µ,ν)…=÷vfŒ±©)•f= g+ϕ,Ù¥g∈A(R,X),ϕ∈E(R,X,µ,ν).·
‚•Ú˜¡P(R,X,µ,ν)p¡¼ê•ÿÝ–.¼ê.
e5‰ÑP(R,X,µ,ν)˜5Ÿ,ÄkŠXeb:
(M
1
)éu¤kτ∈R,•3β>0Úk.«mI¦
eA∈L…A∩I= ∅,Kkµ({a+τ,a∈A}) ≤βµ(A).
(M
2
)µ,ν∈M…limsup
T→∞
µ([−T,T])
ν([−T,T])
<∞
†[21]y²aq,·‚Œ±
Ún2.1.µ,ν∈M÷v(M
1
),KE(R,X,µ,ν)ÚP(R,X,µ,ν)´²£ØC,…ÿÝ–.¼ê
©)´•˜.
Ún2.2.µ,ν∈M÷v(M
1
)Ú(M
2
),KP(R,X,µ,ν)3þ(.‰ê|·|
∞
e´Banach˜m.
Ún2.3.µ,ν∈M…÷v(M
1
),(M
2
),ef∈E(R,X,µ,ν),G∈L
1
(R,B(X)),KE(R,X,µ,ν)
þòÈf∗G
(f∗G)(t) =
Z
+∞
−∞
G(s)f(t−s)ds∈E(R,X,µ,ν),t∈R.
y².dÚn2.1ØJwÑ,éu¤ks∈Rkf(·−s) ∈E(R,X,µ,ν),ŠâFubini½n,k
1
ν([−T,T])
Z
[−T,T]
|(f∗G)(t)|dµ(t)
≤
1
ν([−T,T])
Z
[−T,T]
Z
+∞
−∞
|G(s)||f(t−s)|dsdµ(t)
≤
Z
+∞
−∞
|G(s)|
ν([−T,T])
Z
[−T,T]
|f(t−s)|dµ(t)ds.
DOI:10.12677/aam.2023.1252502483A^êÆ?Ð
•
2ŠâG∈L
1
(R,B(X))±9
0 ≤
|G(s)|
ν([−T,T])
Z
[−T,T]
|f(t−s)|dµ(t) ≤sup
T→∞
µ([−T,T])
ν([−T,T])
·|G(s)|·|f|s∈R,
dLebesgue››Âñ½nÚf(·−s) ∈E(R,X,µ,ν),
lim
T→+∞
Z
∞
−∞
|G(s)|
ν([−T,T])
Z
[−T,T]
|f(t−s)|dµ(t)ds= 0.
¤±f∗G∈E(R,X,µ,ν).
3.̇(Ø
ù˜Ü©,·‚òïÄäkអPFDEsP(R,X,µ,ν))•35Ú•˜5¯K:





d
dt
u(t) = −Au(t)+L(u
t
)+f(t),t≥σ,
u
σ
= ϕ∈B.
(3.1)
Ù¥f•[σ,+∞)NXëY¼ê§−A3Xþ)¤)ÛŒ+(T(t))
t≥0
§éu0<α<1,A
α
L«A©ê˜Žf.L•½Â3f˜mB
α
Š3X¥k.‚5Žf,Ù¥B
α
½Â•
B
α
= {ϕ∈B: ϕ(θ) ∈D(A
α
),θ≤0…A
α
ϕ∈B},
éuϕ∈B
α
½Â‰êkϕk
α
= kA
α
ϕk,A
α
ϕ½Â•
(A
α
ϕ)(θ) = A
α
(ϕ(θ)),θ≤0.
¼êu
t
∈B
α
½Â•u
t
(θ) = u(t+θ),Ù¥θ≤0.
•ïÄ(3.1),ÄkÚ\Xeb:
(H
1
)−A: D(A) →X´Banach˜mXþ)ÛŒ+(T(t))
t≥0
á),¦
|T(t)x|≤Me
ωt
|x|,t≥0,x∈X,
Ù¥M≥1…ω∈R.Ø”˜„5,·‚Œ±b0∈ρ(A),ÄK,Œò−A+ δIO†•−A
¦0 ∈ρ(−A+δI).
(H
2
)éϕ∈BkA
−α
ϕ∈B,Ù¥A
−α
ϕ½Â•(A
−α
ϕ)(θ) = A
−α
(ϕ(θ)),é¤kθ≤0¤á.
(H
3
)T(t)3Xþ´;,é¤kt>0¤á.
DOI:10.12677/aam.2023.1252502484A^êÆ?Ð
•
w,,÷vb(H
1
)…0 <α<1ž,·‚Œ±½ÂXþ‚5ŽfA
−α
,
A
−α
=
1
Γ(α)
Z
∞
0
t
α−1
T(t)dt,
Ù¥ΓL«Gamma¼ê
Γ(α) =
Z
∞
0
t
α−1
e
−αt
dt.
éu©ê˜ŽfA
α
ÚÙ_ŽfA
−α
§Ù¥0 <α<1,·‚kXe5Ÿµ
Ún3.1.[22]0 <α<1…(H
1
)¤á,K
(i)3‰ê|x|
α
=|A
α
x|,x∈D(A
α
)e§D(A
α
)•Banach˜m.e©¥,·‚^X
α
L«Banach
˜m(D(A
α
),|·|
α
).
(ii)T(t) : X→D(A
α
)é¤kt>0¤á.
(iii)é¤kx∈D(A
α
)…t≥0,kT(t)A
α
x= A
α
T(t)x.
(iv)é¤kt>0,A
α
T(t)3Xþk.,…
|A
α
T(t)|≤M
α
t
−α
e
ωt
,x∈X,t>0,(3.2)
Ù¥M
α
>0´~ê,ω∈R…d(H
1
)‰Ñ.
(v)A
−α
´Xþk.‚5Žf,…D(A
α
)=Im(A
−α
).
e5,·‚0[16]¥ÄgÚ\ƒ˜mBúnz½Â.bB´¼êl(−∞,0]NX
D‰˜m…÷vXeÄún:
(A)•3~êN,[0,+∞)þÛÜk.¼êM(·)±93[0,+∞)þëY¼êK(·),¦x:
(−∞,a] →X3[σ,a]ëY…x
σ
∈B,σ<a¤á,Kéu¤kt∈[σ,a],k:
(i)x
t
∈B,
(ii)3«m[σ,a]þ,t→x
t
'uk·këY,
(iii)N|x|≤kx
t
k≤K(t−σ)sup
σ≤s≤t
|x(s)|+M(t−σ)kx
σ
k.
(B)B´Banach˜m.
l·‚Œ±
Ún3.2.[23]C
00
´äk;| òëY¼êl(−∞,0]NX˜m,C
a
00
•[−a,0]¥
äk¼ê| f˜m…±‰êÿÀ˜—§KkC
a
00
→B.
·K3.1.[2]b(H
1
),(H
2
)¤á,eB÷vún(A)-(B),KB
α
÷vXeún(
e
A)-(
e
B):
(
e
A)ex: (−∞,a] →X
α
3[σ,a]þëY…x
σ
∈B
α
,σ<a,Kéu¤kt∈[σ,a],k:
(i)x
t
∈B
α
,
(ii)3«m[σ,a]þ,t→x
t
'uk·k
α
ëY,
DOI:10.12677/aam.2023.1252502485A^êÆ?Ð
•
(iii)N|x|
α
≤kx
t
k
α
≤K(t−σ)sup
σ≤s≤t
|x(s)|
α
+M(t−σ)kx
σ
k
α
.
(
e
B)B
α
´Banach˜m.
½Â3.1.[2]¡¼êu: (−∞,a] →X
α
•(3.1)),eÙ÷v
(i)u3[σ,a]þëY.
(ii)u(t) = T(t−σ)ϕ(0)+
R
t
σ
T(t−s)[L(u
s
)+f(s)]dsét∈[σ,a]¤á.
(iii)u
σ
= ϕ.
½n3.1.[15]b(H
1
),(H
2
)¤á,Kéuϕ∈B
α
,3t≥0ž,(3.1)•3•˜).
éut≥0,·‚3B
α
þ½ÂŽfU(t):
U(t)(ϕ) = u
t
(·,σ,ϕ,0),
Ù¥u(·,σ,ϕ,0)´(3.1)¥f= 0,σ= 0ž).
·K3.2.[15](U(t))
t≥0
x´B
α
þrëYŽfŒ+,Ùá)P•A
U
.
éuφ∈B,t≥0…θ≤0,·‚½Â‚5ŽfW(t):
[W(t)φ](θ) =



φ(0),XJt+θ≥0,
φ(t+θ)XJt+θ<0,
KW(t)´÷veª)Œ+





d
dt
u(t) = 0,
u
0
= ϕ.
W
0
(t) = W(t)|
e
B
,Ù¥
e
B:= {φ∈B: φ(0) = 0},2bB÷vún:
(C)e(ϕ
n
)
n≥0
´BSCauchy¿…3(−∞,0]þ;Âñϕ,Kϕ∈B…kϕ
n
−ϕk→0.
½Â3.2.eB÷vún(A),(B),(C)…t→+∞žkW
0
(t)k→0,K¡Ù•P~PÁ˜m.
½Â3.3.eσ(A
U
)∩iR= ∅,K¡Œ+(U(t))
t≥0
´V-.
½n3.2.[2]b(H
1
)-(H
3
)¤á.2bB•P~PÁ˜m…Œ+(U(t))
t≥0
´V-,K˜ mB
α
Œ±©)•ü‡U-ØC4f˜mSÚU†Ú,¦3Uþ•›Œ+´˜‡+…•3~
êN
0
,÷v
kU(t)ϕk
α
≤N
0
e
−%t
kϕk
α
,t≥0,ϕ∈S,
kU(t)ϕk
α
≤N
0
e
%t
kϕk
α
,t≤0,ϕ∈U,
Ù¥SÚU©O¡Š´-½˜mÚØ-½˜m.
DOI:10.12677/aam.2023.1252502486A^êÆ?Ð
•
éun>ω,x∈X,½ÂXedXNB‚5ŽfΘ
n
,
(Θ
n
x)(θ) =







n(nθ+1)R(n,A)x,−
1
n
≤θ≤0,
0,θ<−
1
n
,
Ù¥R(n,−A)=(nI+A)
−1
.éux∈X,¼êΘ
n
x∈C
00
((−∞,0],X
α
)3[−1,0]Sk| ,
KΘ
n
x∈B
α
.d,éux∈X,k
kΘ
n
xk
α
≤
n
(n−ω)
1−α
K(1)M
α
Γ(1−α)|x|.(3.3)
äNy²Œë„©z[2].
dV-5^‡,·‚Xe½nµ
½n3.3.[2]b(H
1
)-(H
3
)¤á.2bB•P~PÁ˜m… Œ+(U(t))
t≥0
´V-.ef3R
þk.,K(3.1)3Rþk•˜k.)u,¿kXeLˆª:
u
t
=lim
n→+∞
Z
t
−∞
U
s
(t−s)Π
s
(Θ
n
f(s))ds−lim
n→+∞
Z
+∞
t
U
u
(t−s)Π
u
(Θ
n
f(s))dst∈R,(3.4)
Ù¥U
s
(t),U
u
(t)©O•U(t)3SÚUþ•›,Π
s
,Π
u
©O•B
α
SÚUþÝK.
½n3.4.b(M
1
),(M
2
),(H
1
)-(H
3
)¤á¿…f∈P(R,X,µ,ν).2bB•P~PÁ˜m…Œ
+(U(t))
t≥0
´V-.K(3.1)k•˜)u∈P(R,X
α
,µ,ν),…d(3.4)ª‰Ñ.
y².Šâ½n3.3,(3.1)k•˜k.)…uLˆªd(3.4)‰Ñ,
u
t
= (Γ
s
f)(t)−(Γ
u
f)(t),
Ù¥
(Γ
s
f)(t) =lim
n→+∞
Z
t
−∞
U
s
(t−s)Π
s
(Θ
n
f(s))ds,
(Γ
u
f)(t) =lim
n→+∞
Z
+∞
t
U
u
(t−s)Π
u
(Θ
n
f(s))ds.
Ï•f∈P(R,X,µ,ν),lŒf=g+ ϕ,Ù¥g∈A(R,X),ϕ∈E(R,X,µ,ν),K(Γ
s
f)(t)=
F
1
(t)+F
2
(t),Ù¥
F
1
(t) =lim
n→+∞
Z
t
−∞
U
s
(t−s)Π
s
(Θ
n
g(s))ds,
F
2
(t) =lim
n→+∞
Z
t
−∞
U
s
(t−s)Π
s
(Θ
n
ϕ(s))ds.
(i)F
1
∈A(R,B
α
).
DOI:10.12677/aam.2023.1252502487A^êÆ?Ð
•
éug∈P
ω,ρ(t)
(R,X),ØJy²
F
1
(t+ω) =lim
n→+∞
Z
t
−∞
U
s
(t−s)Π
s
(Θ
n
g(s+ω))ds= ρ(t)F
1
(t),
KF
1
∈P
ω,ρ(t)
(R,B
α
).aq,eg∈P
ωap
(R,X),g∈P
ω
(R,X),g∈P
ω,k
(R,X),©OŒ±
F
1
∈P
ωap
(R,B
α
),F
1
∈P
ω
(R,B
α
),F
1
∈P
ω,k
(R,B
α
).
éug∈AP(R,X),Šâ½n3.2,•3~ê
f
M>0÷v
kF
1
(t+τ)−F
1
(τ)k
α
≤lim
n→+∞
Z
t
−∞
N
0
e
−%(t−s)
kΠ
s
(Θ
n
(g(s+τ)−g(s)))k
α
ds≤
f
Mε,
KF
1
∈AP(R,B
α
).aq(ØéuAA(R,X)•¤á.
Ïd,eg∈A((R,X),ŒF
1
∈A(R,B
α
).
(ii)F
2
∈E(R,B
α
,µ,ν).
d(H
1
)Ú(3.3)Œ,•3~ê
e
K>0÷v
kF
2
(t)k
α
≤
e
K
Z
t
−∞
e
−%(t−s)
|ϕ(s)|ds.(3.5)
½Â©ã¼êG:t≥0žG(t) = e
−%t
,t<0žG(t) = 0,¤±
Z
t
−∞
e
−%(t−s)
|ϕ(s)|ds=
Z
+∞
0
e
−%s
|ϕ(t−s)|ds=
Z
+∞
−∞
G(s)|ϕ(t−s)|ds.
dut→|ϕ(t)|∈E(R,R,µ,ν),ŠâÚn2.3 Œt→
R
t
−∞
e
−%(t−s)
|ϕ(s)|ds∈E(R,R,µ,ν),Šâ(3.5)
ŒF
2
∈E(R,B
α
,µ,ν),ÏdΓ
s
f∈P(R,B
α
,µ,ν).aq/,·‚kΓ
u
f∈P(R,B
α
,µ,ν).½ny.
4.~f
•Ä˜a2•ïÄ)ÔêÆ.[2,3,10]§=Xeäkអ ‡©•§:



















∂
∂t
u(t,x) =
n
X
i,j=1
∂
∂x
i

a
ij
(x)
∂
∂x
j
u(t,x)

−a
0
u(t,x)+ε
n
X
i=1
∂
∂x
i
u(t−r,x)
+
R
0
−∞
β(θ)u(t+θ,x)dθ+Θ(t,x)t≥σ,x∈Ω,
u(t,x) = 0t≥σ,x∈∂Ω,
u(σ+θ,x) = ϕ
0
(θ,x)−∞<θ≤0,x∈Ω,
(4.1)
Ù¥σ∈R,a
0
,rÚε•~ê,Ω•R
n
Sk.m8…äk1w>.∂Ω,β:(−∞,0]→R•
¼ê,Θ:[σ,+∞) ×Ω→R•ëY¼ê,a
ij
∈L
∞
(Ω)´é¡…•3η>0éx∈Ω,ξ∈R
n
DOI:10.12677/aam.2023.1252502488A^êÆ?Ð
•
k
n
P
i,j=1
a
ij
ξ
i
ξ
j
≥η|ξ|
2
.
X= L
2
(Ω),2½Â‚5ŽfA: D(A) ⊂X→X:







D(A) = H
2
(Ω)∩H
1
0
(Ω)
A= −
n
P
i,j=1
∂
∂x
i

a
ij
(x)
∂
∂x
j

.
Ún4.1.[22]−A´Xþ;)ÛŒ+(T(t))
≥0
á)¤.d,Ìσ(−A)´lÑÌ
…σ(−A) = {λ
n
: n∈N},Ù¥···<λ
n+1
<λ
n
<···<λ
0
<0.
Ún4.2.[24]|∇|Úk·k
1
2
3D(A
1
2
)þd.d,·‚k
√
η|∇ψ|≤kψk
1
2
≤
q
nmax
1≤i,j≤n
|a
ij
|
L
∞
|∇ψ|,
Ù¥∇L«FÝ•þ.
éuγ>0,½Â
B= C
γ
= {ϕ∈C((−∞,0],X) :lim
θ→−∞
e
γθ
ϕ(θ)3Xþ•3}
éuϕ∈C
γ
,½Â‰ê|ϕ|= sup
θ≤0
e
γθ
ϕ(θ).
Ún4.3.[23]B÷v(A),(B),(C)…•P~PÁ˜m.
½Â‚5ŽfL: B
1
2
→X:
L(φ) = −a
0
φ(0)+ε
n
X
i=1
∂
∂x
i
φ(−r)+
Z
0
−∞
β(θ)φ(θ)dθ,
3B
1
2
þ½Â‰êkϕk
1
2
= sup
θ≤0
e
γθ
|A
1
2
ϕ(θ)|.
e¡·‚b:
(A
1
)e
−2γ
β∈L
2
(R
−
).
(A
2
)
R
0
−∞
|β(θ)|dθ<a
0
.
Ún4.4.[2](A
1
)¤á,KL´dB
1
2
NXk.‚5Žf.
½Âf: R→X•f(t)(x) = Θ(t,x),Ù¥t∈R,x∈Ω…





u(t)(x) = u(t,x)t≥σ,x∈Ω,
ϕ(θ)(x) = ϕ
0
(θ,x)θ≤0,x∈Ω,
DOI:10.12677/aam.2023.1252502489A^êÆ?Ð
•
K(4.1)Œ-#Lˆ•XeÄ–‡©•§/ªµ





d
dt
u(t) = −Au(t)+L(u
t
)+f(t),t≥0,
u
0
= ϕ.
(4.2)
ϕ∈B
1
2
,Šâ½n3.1,K(4.2)3(−∞,+∞)þ•3•˜)u.
(U(t))
t≥0
´(4.2)3B
1
2
þ)Œ+,…A
U
´Ùá)¤.Šâ(U(t))
t≥0
V-5,·‚
Œ±
Ún4.5.[2]e(A
1
),(A
2
)¤á,Kéuε<
1
nM
1
2
q
−λ
0
η
π
,kσ(A
U
) ⊂{λ∈C: Re(λ) <0}.
µ= ν¿bÙRadon-Nikodymê•
ρ(t) =



e
t
,t≤0,
1,t>0.
d[25]Œ•,µ,ν∈M÷v(M
1
)Ú(M
2
),2Šâ½n3.4,Œ±
½n4.1.3±þbe,…ef∈P(R,X,µ,ν),K(4.1)k•˜k.)u∈P(R,X
1
2
,µ,ν).
54.1.©|^©ê˜ŽfnØÚŽfŒ+nØy²PFDEsÿÝ–.)•3•˜5§ÏLÚ
\ÿݧ½Â˜aÿÝ–.¼ê§y²Ù¤˜mäk5§é‡±Ï§±Ï§V±Ï§
Agŧ–V±Ï§–AgÅÄåÆ5Ÿ?1Ú˜/ïħl/ªÚ(Ø•\{'"Ó
ž,˜•¡§·‚•5¿§3^‡(H
3
)¥,‡¦)ÛŒ+T(t)3Xþ´;,ù´˜‡'
r^‡§äNA^¥•ØBuy§XÛK½ö~f^‡(H
3
)´Š&?˜‡¯K§•´
·‚8ïÄ••"
ë•©z
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