时间依赖记忆型经典反应扩散方程的吸引子 Attractors for the Classical Reaction Diffusion Equation with Time-DependentMemory Kernel

doi:10.12677/PM.2023.135150

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PureMathematicsnØêÆ,2023,13(5),1456-1482

PublishedOnlineMay2023inHans.https://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2023.135150

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AttractorsfortheClassicalReaction

DiﬀusionEquationwithTime-Dependent

MemoryKernel

YunaLi,XuanWang

CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu

Received:Apr.23

,2023;accepted:May24

,2023;published:May31

,2023

Abstract

Inthispaper,weconsiderthelong-timedynamicalbehaviorofsolutionsfortheclas-

©ÙÚ^:oŒA,à.žm•6PÁ.²;‡A*Ñ•§áÚf[J].nØêÆ,2023,13(5):1456-1482.

DOI:10.12677/pm.2023.135150

oŒA§à

sicalreactiondiﬀusionequationwithtime-dependentmemorykernelwhennonlinear

term adheresto subcritical growthin thetime-dependent space H

(Ω)×L

(Ω)).

Underthenewtheoricalframework,thewell-posednessandtheregularityoftheso-

lution,theexistenceofthetime-dependentglobalattractorsareprovedbyusingthe

delicateintegralestimationmethodanddecompositiontechnique.

Keywords

ClassicalReactionDiﬀusionEquation,QuadTime-DependentMemoryKernel,

Well-Posed-Ness,Time-DependentGlobalAttractors,Existence

2023byauthor(s)andHansPublishersInc.

This work is licensed undertheCreative Commons Attribution InternationalLicense (CCBY4.0).

http://creativecommons.org/licenses/by/4.0/

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DOI:10.12677/pm.2023.1351501459nØêÆ

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DOI:10.12677/pm.2023.1351501461nØêÆ

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DOI:10.12677/pm.2023.1351501463nØêÆ

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)∩D(T

).@o,éu¤kτ6a6b6T,

DOI:10.12677/pm.2023.1351501464nØêÆ

oŒA§à

keãØª¤á:

kη

−

∞

(∂

(s)+∂

(s))kη

(s)k

dsdt6kη

hu(t),η

.(3.6)

Ún3.6éu¤kτ6a6b6T,eãO¤á

kη

+δ

κ(t)kη

(s)k

dsdt6kη

−

∞

(∂

(s)+∂

(s))kη

(s)k

dsdt

6kη

hu(t),η

.(3.7)

½Â3.7éu?¿T>τ∈R , g∈L

(Ω),…z

= (u

,η

) ∈H

,XJ

(i)u(t) ∈L

∞

([τ,T];V

),η

∈L

∞

([τ,T];M

);

(ii)¼êη

÷vª(2.1) ;

(iii)éuz‡φ∈V

…a.e.t∈[τ,T],

h∂

u,φi+hu,φi

∞

(s)hη

(s),φi

ds+hf(u),φi= hg,φi.

…¡z(t) = (u(t),η

)´¯K(2.2), (2.3)3«m[τ,T] þf).

½n3.8(·½5ÚK5)(1.3), (1.4)ª¤á, …g∈L

(Ω). (H

)-(H

) ¤áž,éz‡

T>τ∈R, ÐŠz(τ)∈H

…kz(τ)k

, K3«m[τ,T] ¯K(2.2), (2.3) •3•˜f)

z(t) = (u(t),η

),÷v

sup

t>τ

kz(t)k

ku(r)k

dr+

κ(r)kη

dr+

k∂

u(r)k

dr6Q,

?˜Ú,(i) ÐŠz(τ) ∈H

…kz(τ)k

,K3«m[τ,T]þ, •3r),k

sup

t>τ

kz(t)k

ku(r)k

dr+

κ(r)kη

dr+

k∂

u(r)k

dr6

¤á.(ii) •3S{z

(τ)}∈H

, ¦z

(τ)→z(τ)∈H

, ÐŠz(τ)∈H

…kz(τ)k

K3«m[τ,T] þ,k

sup

t>τ

kz(t)k

ku(r)k

dr+

κ(r)kη

dr+

k∂

u(r)k

dr6Q,

ùpR

ÚR

þ•~ê,Q= max{Q

Q= max{

}.d,

(t)−z

(t)k

6Ce

C(R,λ

)(t−τ)

(τ)−z

(τ)k

, t∈[τ,T],

ùpz

(t),z

(t)´¯K(2.2), (2.3)ü‡),ÐŠz

= (u

,η

),z

= (u

,η

DOI:10.12677/pm.2023.1351501465nØêÆ

oŒA§à

y²ò•§(2.2) ¦±u, ·‚k

(kuk

)+2kuk

+2hu,η

+2hf(u),ui−2hg,ui= 0.

(3.8)

Šâ(1.4) ª,·‚k

−2hf(u),ui62(1−θ)kuk

+4c

ùp,θ∈(0,1).N´

2hg,ui6θkuk

kgk

·‚½Â

N(t) = kuk

N(t)+θkuk

+2hu,η

kgk

+4c

:= Q

(3.9)

é(3.9) ª3[τ,t] þÈ©,·‚k

N(t)+θ

ku(r)k

dr+2

hu,η

dr6N(τ)+Q

(t−τ), ∀t>τ.

(3.10)

A^½n3.6, ·‚k

N(t)+kη

+θ

ku(r)k

dr−

∞

(∂

(s)+∂

(s))kη

(s)k

dsdr

6N(τ)+kη

(t−τ), ∀t>τ.

½Â

N(t) = N(t)+kη

kz(t)k

6N(t) 6(1+

)kz(t)k

(3.11)

Ïd,

N(t)+θ

ku(r)k

dr−

∞

(∂

(s)+∂

(s))kη

(s)k

dsdr6N(τ)+Q

(t−τ).

(3.12)

¿›X,

sup

t>τ

kz(t)k

ku(r)k

dr+

κ(r)kη

dr6C(R,T,kgk,θ,δ,λ

) := Q

.(3.13)

aq/

sup

t>τ

kz(t)k

ku(r)k

dr+

κ(r)kη

dr6C(R,T,kgk,θ,δ,λ

) :=

.(3.14)

DOI:10.12677/pm.2023.1351501466nØêÆ

oŒA§à

ÐŠz(τ) ∈H

ž,ò•§(2.2) ¦±−∆u,·‚k

(kuk

)+2kuk

+2hu,η

+2hf(u),−∆ui−2hg,−∆ui= 0.

(3.15)

d(1.3) ª,·‚k

−2hf(u),−∆ui= −2

Ω

(u)|∇u|

dx62C

kuk

(3.16)

w,,

2hg,−∆ui6kuk

+kgk

½Â

(t) = kuk

(t)+kuk

+2hu,η

62C

kuk

+kgk

(3.17)

é(3.17) ª3[τ,t] þÈ©,·‚k

(t)+

kuk

dr+2

hu,η

dr6N

(τ)+2C

ku(r)k

dr+kgk

(t−τ).

(3.18)

du½n3.6, ·‚

(t)+

kuk

dr+kη

+δ

κ(r)kη

(τ)+kη

+2C

ku(r)k

dr+kgk

(t−τ), ∀t>τ.

(3.19)

½Â

(t) = N

(t)+kη

kz(t)k

(t) 6(1+

)kz(t)k

Ïd,

(t)+

kuk

dr+δ

κ(r)kη

dr6N

(τ)+2C

ku(s)k

ds+kgk

(t−τ), ∀t>τ.

A^Gronwall Øª,·‚íäÑ

sup

t>τ

kz(t)k

kuk

dr+

κ(r)kη

6C(kz(τ)k

,T,kgk,θ,δ,λ

) := Q

(3.20)

ò•§(2.2) ¦±∂

DOI:10.12677/pm.2023.1351501467nØêÆ

oŒA§à

k∂

= −hu,∂

ui−

∞

(s)h∆η

(s),∂

uids−hf(u),∂

ui+hg,∂

ui.

dª(1.3), ·‚k

|hf(u),∂

ui|6kf(u)kk∂

uk6C(1+ku(t)k

)k∂

uk.

d(H

)Œ



−

∞

(s)h∆η

(s),∂

uids



6k∂

∞

(s)k∆η

(s)kds

6k∂



∞

(s)ds





∞

(s)kη

(s)k



6k∂

κ(t)kη

k∂

6C(ku(t)k

+1+ku(t)k

κ(t)kη

+kgk)k∂

6C(1+Q

κ(t)kη

+kgk)k∂

k∂

+C(R,T,c

,kgk,θ,δ,λ

)(1+κ(t)kη

)

k∂

(1+κ(t)kη

),∀t∈[τ,T].

(3.21)

Ïd,

k∂

u(s)k

ds62Q

(1+

κ(s)kη

ds) 6Q

(3.22)

{w

}´L

(Ω)IOÄ,•3V

¥IO,¿…−∆w

= λ

,j= 1,2,···.{ζ

}

´L

) IOÄ, •3L

) ¥IO, ¿…−∆ζ

=λ

,j=1,2,···. éu

z‡n∈N, k•‘f˜m½ÂXe:

= span{w

,···,w

}⊂V

= span{ζ

,···,ζ

}⊂L

: V

→H

L«3H

þÝK,Q

: L

) →M

L«3M

þÝK.

Ð©^‡z

= (u

,η

)Cqu˜‡S{z

= (u

,η

)}⊂H

,Ù¥

= P

→u

inV

(3.23)

= Q

→η

inM

(3.24)

éz‡n∈N, z

= (u

,η

)•¯K(2.2), (2.3)%C).Ù¥u

= Σ

j=1

(t)w

∈C

([τ,T])¿…η

=Σ

j=1

(t)ζ

, Λ

∈C

([τ,T]).¤±éz‡Á¼êψ∈H

, ¿…z‡

t∈[τ,T],z

= (u

,η

))ûeã¯K:

h∂

,ψi+hu

,ψi

∞

(s)hη

(s),ψi

ds+hf(u

),ψi= hg,ψi,

(3.25)

DOI:10.12677/pm.2023.1351501468nØêÆ

oŒA§à

¿…

(s) =







(t−r)dr,0 <s6t−τ,

(s−t+τ)+

t−τ

(t−r)dr,s>t−τ.

(3.26)

bψ∈H

´½.@oéz‡n>m,·‚k(3.25)ª¤á.‰(3.25)ª¦±ϕ∈

∞

([τ,T]) ¿…3[τ,T] þ'u(3.25)ª?1È©, ·‚uy

ϕh∂

(r),ψidr+

ϕhu

(r),ψi

∞

(s)hη

(s),ψi

dsdr+

ϕhf(u

),ψidr=

ϕhg,ψidr.

(3.27)

é²w,S{z

},OŠ(3.13), (3.20),(3.22)´¤á. @o,

∂

([τ,T];H) k.;

∞

([τ,T];V

)k.;

([τ,T];V

)k.;

∞

([τ,T];M

)k..

Ï•kf(u

6C(1+ku

) 6C, ·‚íäÑ

f(u

)3L

(Ω)k..

|^Galerkin %C)z

= (u

,η

),·‚••3|z= (u,η

),¦(7‡žf)

∂

→∂

u†L

([τ,T];H

)fÂñ;(3.28)

→u†L

∞

([τ,T];V

∗

Âñ;(3.29)

→u†L

([τ,T];V

)fÂñ;(3.30)

→q

†L

∞

([τ,T];M

∗

Âñ;(3.31)

f(u

) →f(u)†L

(Ω)fÂñ.(3.32)

A^Ún2.1, ·‚Œ±l(3.28) Ú(3.29)¥¼

→u†C([τ,T];V

),(3.33)

¿…Å:Âñ

DOI:10.12677/pm.2023.1351501469nØêÆ

oŒA§à

(x,t) →u(x,t) a.e.†Ω×[τ,T].

ŠâfëY5, Œ

f(u

(x,t)) →f(u(x,t))a.e. †Ω×[τ,T]

•´¤á.

|^(3.28) Ú(3.30) ª, ·‚Œ±N´/(3.27) †à1˜‘1‘Âñ5.·‚ò

?nÙ{ü‘.

Ï•ψ∈H

⊂V

,éN´ψ∈P

p+1

(Ω).Ïd, Šâ(3.32) ª,Œ

hf(u

)−f(u),ψidr→0

¤á.duf(u

)Úf(u) 3L

(Ω)¥k.5, A^››Âñ½n,·‚íäÑ

ϕhf(u

)−f(u),ψidr→0.

·‚Œ±y²ù‡Âñ

∞

(s)hη

(s),ψi

dsdr.•,·‚

¯η

= η

−η

,¯u

= u

−u

éz‡t∈[τ,T],

¯η

= η

−η

,¯u

(t) = u

(t)−u(t).

•Ä(H

)¿¦^

¯η

(s) =







¯u

(t−ζ)dζ,0 <s6t−τ,

¯η

(s−t+τ)+

t−τ

¯u

(t−ζ)dζ,s>t−τ,

·‚k

k¯η

(t)k¯η

=C(T)(

t−τ

(s)k

¯u

(t−ζ)dζk

∞

t−τ

(s)k¯η

(s−t+τ)+

t−τ

¯u

(t−ζ)dζk

ds)

6C(T)(3(T−τ)

k¯u

C([τ,T];V

)

∞

(s)ds+2

∞

(s+t−τ)k¯η

(s)k

ds)

6C(T)(3(T−τ)

k¯u

C([τ,T];V

)

κ(τ)+2k¯η

) →0,∀t∈[τ,T].

du4••˜5,·‚q

= η

DOI:10.12677/pm.2023.1351501470nØêÆ

oŒA§à

w,,

∞

(s)h¯η

(s),ψi

t−τ

(s)h

¯u

(t−ζ)dζ,ψi

ds+

∞

t−τ

(s)h¯η

(s−t+τ),ψi

∞

t−τ

(s)h

t−τ

¯u

(t−ζ)dζ,ψi

t−τ

(s)

h¯u

(t−ζ),ψi

dζds+

∞

(s+t−τ)h¯η

(s),ψi

∞

t−τ

(s)s

h¯u

(ζ),ψi

dζds.

2g¦^(H

),·‚

t−τ

(s)

h¯u

(t−ζ),ψi

dζds

t−τ

(s)s

k¯u

(t−ζ)k

kψk

dζds

6k¯u

C([τ,T];V

)

kψk

(T−τ)

(t)κ(τ) →0,a.e.t∈[τ,T],

∞

t−τ

(s)

h¯u

(ζ),ψi

dζds

∞

t−τ

(s)s

k¯u

(ζ)k

kψk

dζds

6k¯u

C([τ,T];V

)

kψk

(T−τ)

(t)κ(τ) →0,a.e.t∈[τ,T],

∞

(s+t−τ)h¯η

(s),ψi

ds6kψk

(t)

κ(τ)k¯η

→0,a.e.t∈[τ,T].

(J,

lim

n→∞

∞

(s)h¯η

(s),ψi

ds= 0,a.e.t∈[τ,T].

¿…



∞

(s)h¯η

(s),ψi



∞

(s)k¯η

(s)k

kψk

6kψk

(t)κ(τ)k¯η

∈L

([τ,T]).

A^Lebesgue››Âñ½n, Œ

lim

n→∞

∞

(s)h¯η

(s),ψi

dsdr= 0.

DOI:10.12677/pm.2023.1351501471nØêÆ

oŒA§à

•,·‚¯K(2.2) Ú(2.3)f)z= (u,η

y3,·‚y²f)'uÐŠëY•65, •=•˜5.b

(t) = (u

(t),η

),z

(t) = (u

(t),η

)

´¯K(2.2) Ú(2.3) 3[τ,T] þü‡f).@o¯z(t) = z

(t)−z

(t) = (¯u(t),¯η

)÷v

∂

¯u+A¯u+

∞

(s)A¯η

(s)ds= −f(u

)+f(u

(3.34)

Ù¥

¯η

(s) =







¯u(t−r)dr,s6t−τ,

¯η

(s−t+τ)+

t−τ

¯u(t−r)dr,s>t−τ.

(3.35)

‰(3.35) ¦±¯u, ·‚k

F(t)+2

∞

(s)h¯η

(s),¯u(t)i

= −2k¯uk

−2hf(u

)−f(u

),¯u(t)i

6−

k¯uk

+C(1+ku

p−1

p+1

+ku

p−1

p+1

)k¯uk

p+1

6−

k¯uk

+C(1+ku

p−1

+ku

p−1

)k¯uk

6C(R,λ

)F(t),t∈[τ,T],

Ù¥F(t) = (k¯uk

).3[τ,t] þÈ©,·‚uy

F(t)+2

h¯u(y),¯η

dy6F(τ)+C(R,λ

)

F(y)dy, t∈[τ,T].

(3.36)

Šâ½n3.6, ·‚•

k¯η

+δ

κ(y)k¯η

(s)k

dy6k¯η

h¯u,¯η

dy.

(3.37)

F(t) = F(t)+k¯η

,·‚k

k¯z(t)k

6F(t) 6Ck¯z(t)k

(Ü(3.36) Ú(3.37), ·‚k

F(t) 6F(τ)+C(R,λ

)

F(y)dy.

DOI:10.12677/pm.2023.1351501472nØêÆ

oŒA§à

A^Gronwall Øª,·‚¼

k¯z(t)k

6Ce

C(R,λ

)(t−τ)

k¯z(τ)k

, t∈[τ,T].

†dÓž,·‚y²¯K(2.2) Ú(2.3)f)•˜5.

du½n3.8, L§U(t,τ)ÎÜ¯K(2.2) Ú(2.3)½ÂXe:

z(t) = U(t,τ)z(τ) : H

→H

ù‡lH

H

L§´ëY.

4.žm•6ÛáÚf•35

4.1.žm•6áÂ83H

¥•35

½n4.1(ÑÑ5)b(1.3)Ú(1.4)ª±9^‡(H

)-(H

)¤á,g∈L

(Ω),•3S{z

(τ)}∈

, ¦z

(τ)→z(τ).éu?¿Ð©^‡z(τ)∈B

(R)⊂H

, @o•3R

>0, ¦ƒA¯K

(2.2),(2.3) L§U(t,τ)Pk˜‡žm•6áÂ8, ¿›xB

= {B

)}

t∈R

y²¦^Poincar´eØªÚ(H

),·‚Ul(3.18) ª¥¼

N(t)+

θλ

ku(r)k

dr+

ku(r)k

dr+δ

κ(r)kη

(s)k

dsdr

6N(τ)+Q

(t−τ).

(4.1)

¿›X,

N(t)+2ε

N(r)dr6N(τ)+ε

N(r)dr+Q

(t−τ),

ùp,ε= min{

θλ

θ,δinf

r∈[τ,t]

κ(r)}.A^Ún2.4 ,·‚íäÑ

N(t) 6N(τ)e

−ε(t−τ)

1−e

−ε

d,

kz(t)k

6N(t) 6(1+

)kz(τ)k

−ε(t−τ)

(4.2)

Ù¥R

= 2

1−e

−ε

.@oéz‡R>0,•3˜‡t

= t

(R) =

2(1+

6t¿…R

>0 ¦

τ6t−t

⇒U(t,τ)B

(R) ⊂B

4.2.žm•6ÛáÂ83H

¥•35

e5,·‚òy²)L§U(t,τ) ƒA(2.2), (2.3) éAìC;5. •d, ·‚I‡éš‚

DOI:10.12677/pm.2023.1351501473nØêÆ

oŒA§à

f(s) = f

(s)+f

(s),

Ù¥f

∈C

(R)¿…÷v:

(u)|6C(1+|u|

p−1

),∀u∈R,1 6p63,(4.3)

(u)u>0,∀u∈R,(4.4)

(u)|6C(1+|u|

),∀u∈R,1 6γ<2,(4.5)

liminf

|u|→∞

(u) >−λ

.(4.6)

É[20]gŽK•,ò¯K(2.2),(2.3) )z(t) = (u(t),η

)©)•:

z(t) = z

(t)+z

(t),u(t) = v(t)+w(t),η

= ζ

+ξ

ùp,z

(t) = (v(t),ζ

)…z

(t) = (w(t),ξ

))ûeã¯K:











∂

v+Av+

∞

(s)Aζ

(s)ds+f

(v) = 0,

∂

+∂

= v(t),

v(x,t)|

∂Ω

= 0,v(x,τ) = u

(x,t),

(x,s)|

∂Ω

= 0,ζ

(x,s) = η

(x,s),

(4.7)

Ù¥,

(s) =







v(t−r)dr,0 <s6t−τ,

(s−t+τ)+

t−τ

v(t−r)dr,s>t−τ,

¿…











∂

w+Aw+

∞

(s)Aξ

(s)ds+f(u)−f

(v) = g,

∂

+∂

= w(t),

w(x,t)|

∂Ω

= 0,w(x,τ) = 0,

(x,s)|

∂Ω

= 0,ξ

(x,s) = 0,

(4.8)

Ù¥,

(s) =







w(t−r)dr,0 <s6t−τ,

t−τ

w(t−r)dr,s>t−τ.

aqu½n3.8 y²,•§(4.7)Ú(4.8))•35Ú•˜5Œ±¼.

DOI:10.12677/pm.2023.1351501474nØêÆ

oŒA§à

?˜Ú,N´•, L§U

(t,τ) ÚU

(t,τ) éA•§(4.7) Ú(4.8).{üå„, ·‚

U(t,τ)z

= U

(t,τ)z

(τ)+U

(t,τ)z

(τ) = z

(t)+z

(t).

aqu½n4.1, Œ±Xe(J.

Ún4.2bf

÷v(4.3) Ú(4.4) .XJ(H

)-(H

)¤á, @o•§(4.7) ÷vO:

(i)•3S{z

(τ)}∈H

,¦z

(τ) →z(τ).ÐŠz

(τ) ∈H

…kz

(τ)k

(t)k

6C(R

−ε

(t−τ)

(4.9)

(ii)ÐŠz

(τ) ∈H

…kz

(τ)k

(t)k

6C(R

−ε

(t−τ)

(4.10)

ùpR

ÚR

þ•~ê.

y²ò•§(4.7) ¦±v,

(kvk

)+2kvk

+2hv,ζ

+2hf

(v),vi= 0.

(4.11)

·‚½Â

F(t) = kvk

•Ä(4.4), ·‚íäÑ

F(t)+2kvk

+2hv,ζ

60.

(4.12)

F(t)+2

kv(r)k

dr+2

hv,ζ

dr6F(τ),∀t>τ.

(4.13)

du½n3.6, ·‚k

F(t)+kζ

kv(r)k

dr+δ

κ(r)kζ

(s)k

6F(τ)+kζ

, ∀t>τ.

½Â

F(t) = F(t)+kζ

@o,

F(t)+2

kv(r)k

dr+δ

κ(r)kζ

(s)k

dr6F(τ).

(4.14)

DOI:10.12677/pm.2023.1351501475nØêÆ

oŒA§à

•Ò´`,

F(t)+2ε

F(r)dr6F(τ)+ε

F(r)dr,

ùp,ε

= min{λ

,1,δinf

r∈[τ,t]

κ(r)}.dÚn2.4, ·‚¼

F(t) 6F(τ)e

−ε

(t−τ)

?˜Ú,

(t)k

= F(t) 6C(R

,λ

−ε

(t−τ)

(4.15)

Ù¥kz(τ)k

ÐŠz

(τ) ∈H

ž,ò•§(4.7) ¦±−∆v,

(kvk

)+2kvk

+2hv,ζ

+2hf

(v),vi= 0.

(4.16)

·‚½Â

(t) = kvk

•Ä(4.4), ·‚íäÑ

(t)+2kvk

+2hv,ζ

60.

(4.17)

(t)+2

kv(r)k

dr+2

hv,ζ

dr6F

(τ),∀t>τ.

(4.18)

du½n3.6, ·‚k

(t)+kζ

kv(r)k

dr+δ

κ(r)kζ

(s)k

(τ)+kζ

, ∀t>τ.

½Â

(t) = F

(t)+kζ

@o,

(t)+2

kv(r)k

dr+δ

κ(r)kζ

(s)k

dr6F

(τ).

(4.19)

•Ò´`,

(t)+2ε

(r)dr6F

(τ)+ε

(r)dr,

ùp,ε

= min{λ

,1,δinf

r∈[τ,t]

κ(r)}.dÚn2.4, ·‚¼

DOI:10.12677/pm.2023.1351501476nØêÆ

oŒA§à

(t) 6F

(τ)e

−ε

(t−τ)

?˜Ú,

(t)k

= F

(t) 6C(R

,λ

−ε

(t−τ)

(4.20)

Ù¥kz(τ)k

Ún4.3bš‚5‘f÷v(1.3), (1.4) Ú(4.3)-(4.6).XJg∈L

(Ω) ¿…(H

)-(H

) ¤á,

@oéuzãžmT>0, •3S{z

(τ)}∈H

,¦z

∈H

,•3

˜‡~êI= I(kgk,kz

,T,λ

),¦(4.8) )÷v:

(T+τ,τ)z

(τ)k

T+τ

= kz

(T+τ)k

T+τ

6I.

(4.21)

y²ò•§(4.8) 1˜‡•§¦±A

w,Œ

G(t)+2kw(t)k

+2hξ

,w(t)i

= 2hg,A

wi−2hf

(v),A

wi−2hf(u)−f(v),A

wi,

(4.22)

Ù¥G(t) = kw(t)k

.N´•

2|hg,A

wi|6

kwk

4kgk

(4.23)

·‚Œ±l(4.5) Ú(1.3) ¥¼

−2hf

(v),A

wi6C

Ω

(1+|v|

)|A

w|dx

6C(

Ω

(1+|v|

18γ

)dx)

(

Ω

dx)

6C(1+kvk

)kA

6C(R,λ

)kwk

kwk

(4.24)

…

−2hf(u)−f(v),A

wi6C

Ω

(1+|u|

p−1

+|v|

p−1

)|w||A

w|dx

6C(kuk

p−1

3(p−1)

+kvk

p−1

3(p−1)

)kwk

6C(kuk

p−1

+kvk

p−1

)kwk

kwk

(4.25)

Ù¥c

= c

),¿…·‚¦^i\V

→L

Ïd,r(4.23)-(4.25) \(4.22),·‚k

G(t)+2hξ

,w(t)i

6(c

−

)kw(t)k

+C.

(4.26)

DOI:10.12677/pm.2023.1351501477nØêÆ

oŒA§à

G(T+τ)+2

T+τ

hξ

,w(r)i

dr6G(τ)+(c

−

)

T+τ

kw(r)k

dr+CT.

(4.27)

½Â

G(t) = kw(t)k

+kξ

du½n3.6, ·‚k

G(T+τ)+δ

T+τ

κ(r)kξ

(s)k

dr6G(τ)+(c

−

)

T+τ

kw(r)k

dr+CT.

(4.28)

•Ò´`

G(T+τ) 6G(τ)+c

T+τ

G(r)dr+CT.

(4.29)

dGronwall Øª,·‚íäÑ

G(T+τ) 6e

(G(τ)+CT) = CTe

aq,

(T+τ)k

T+τ

6G(T+τ) 6CTe

= I.

d,éu?¿ξ

∈L

),Cauchy ¯K(„[3,12,18])







∂

= −∂

+w,t>τ,

= ξ

(4.30)

k•˜)ξ

∈C([τ,+∞);L

))…kwªLˆ:

(s) =







w(t−r)dr,0 <s6t−τ,

t−τ

w(t−r)dr,s>t−τ.

(4.31)

·‚^B

L«k½n4.1 ¼žm•6áÂ8.@o, ·‚

= ΠU

(T,τ)B

ùp,Π : V

×L

) →L

)´˜‡ÝKŽf.

Ún4.4z

(t) = (w(t),ξ

)´¯K(4.8)).bš‚5‘÷v(1.3),(1.4)…(4.3)−(4.6).X

Jg∈L

(Ω) ¿…(H

)-(H

) ¤á, @o, éz‡‰½T>τ, •3˜‡~êI

(kB

¦

(i)K

);V

)∩H

)k.;

DOI:10.12677/pm.2023.1351501478nØêÆ

oŒA§à

(ii)sup

∈K

kξ

(s)k

y²3(4.31) *:e,·‚íäÑ

∂

(s) =







w(t−s),0 <s6t−τ,

0,s>t−τ.

(4.32)

duÚn4.3, §Œ±y²(i) ¤á.

e5,´

kξ

(s)k







kw(T−r)k

dr6

T−τ

kw(T−r)k

dr,0 <s6T−τ,

T−τ

kw(T−r)k

dr,s>T−τ,

(4.33)

¤á.d(4.22) ª,(ii)•Œ±y.

Ún4.53Ún4.4 b¤áe.@oéuz‡½T>τ, U

(T,τ)B

ƒé;.

y²¯¢þ,A^Ún2.2 ·‚•K

)ƒé;. ¿…2g¦^(H

)b, ·

‚K

)ƒé;. d,k;i\: V

→→V

,·‚íäÑ

(T,τ)B

ƒé;.

Ún4.6U(t,τ)´¯K(2.2), (2.3))L§.bš‚5‘f÷v(1.3), (1.4)Ú

(4.3)-(4.6).XJg∈L

(Ω)¿…(H

)-(H

)¤á,@oL§U(t,τ)Pkžm•6áÂ8A= {A

}

t∈R

inH

.d, áÚfA´ØC, ¿›X,

U(t,τ)A

= A

,∀t>τ.

y²B

= {B

)}

t∈R

´k½n4.1¼žm•6áÂ8. dÚn4.2ÚÚn4.3, éuv

Œ~êR

,´

= {B

)}

t∈R

´.£áÚ,

ùpB

) = {ξ|kξk

¯¢þ,(Ü(4.9) Ú(4.21),·‚íäÑ

dist

(U(t,τ)B

) 6dist

(t,τ)B

)

= dist

(t,τ)B

)

6C(kB

−ε

(t−τ)

ùp,ε

= min{λ

,1,δinf

r∈[τ,t]

κ(r)}.

DOI:10.12677/pm.2023.1351501479nØêÆ

oŒA§à

é?Ûk.8(3H

= {B

(R)}

τ∈R

,k½n4.1, •3˜‡t

= t

(R) ¦

τ6t−t

⇒U(t,τ)B

(R) ⊂B

Ïd,

dist

(U(t,τ)B

) 6e

−ε

(t−τ)

Ù¥=sup

06t−τ6t

kU(t,τ)B

A^Ún2.3 Ú½n3.8, ·‚Œ±¼

dist

(U(t,τ)B

) 6C(kB

−ε

(t−τ)

(ÜÚn4.5, ·‚k¯K(2.2), (2.3) ƒAL§U(t,τ) 3H

¥ìC;5. Ïd,A^½n2.7,

½n2.10 Ú½n3.8,·‚Œ±y²žm•6áÂ8A={A

}

t∈R

ØC5Ú•35, •Ò´

U(t,τ)A

= A

¿…

A= {Z|t→Z(t) ∈H

…Z(t) ´L§U(t,τ)CBT}.

Ä7‘8

I[g,‰ÆÄ7‘8(1OÒ:11961059; 12061062)"

ë•©z

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