具有时间依赖记忆核的粘弹性方程的吸引子 Attractors for the Viscoelastic Equationwith Time-Dependent Memory Kernel

doi:10.12677/PM.2022.127122

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PureMathematicsnØêÆ,2022,12(7),1103-1124

PublishedOnlineJuly2022inHans.http://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2022.127122

äkžm•6PÁØÊ5•§áÚf

°°°ÿÿÿ§§§ààà

∗

Ü“‰ŒÆêÆ†ÚOÆ§[‹=²

ÂvFÏµ2022c525F¶¹^FÏµ2022c628F¶uÙFÏµ2022c75F

Á‡

éuäkžm•6PÁØÊ5•§§©ïÄT•§)•žmÄåÆ1•"š‚5‘f

O••êp÷v16p65ž§3#nØµee§|^È©O•{)·½5"Ó

ž§p÷v1 6p<5 ž§y²žm•6ÛáÚf•35±9ØC5"

'…c

Ê5•§§žm•6PÁØ§žm•6ÛáÚf

AttractorsfortheViscoelasticEquation

withTime-DependentMemoryKernel

HaiyanYuan,XuanWang

∗

CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu

Received:May25

,2022;accepted:Jun.28

,2022;published:Jul.5

,2022

Abstract

Inthispaper,weinvestigatethelong-timedynamicalbehaviorofsolutionsforthe

∗ÏÕŠö

©ÙÚ^:°ÿ,à.äkžm•6PÁØÊ5•§áÚf[J].nØêÆ,2022,12(7):1103-1124.

DOI:10.12677/pm.2022.127122

°ÿ§à

viscoelasticequationwithtime-dependentmemorykernel.Whenthegrowthexponent

pofnonlinearityf(u)isupto1 6p65,thewell-posednessofthesolutionsisproved

byusingtheintegralestimationmethod,andweobtainedaninvarianttime-dependent

globalattractorwhen1 6p<5 .

Keywords

ViscoelasticEquation,Time-DependentMemoryKernel,Time-DependentGlobal

Attractors

2022byauthor(s)andHansPublishersInc.

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

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

1.Úó

©ïÄäkžm•6PÁØÊ5•§ÄåXÚ)•žm1•











−∆u

−h

(0)∆u−

∞

∂

(s)∆u(t−s)ds+f(u) = g,(x,t) ∈Ω×(τ,+∞),

u(x,t) = 0,x∈∂Ω, t>τ,

u(x,t) = u

(x),x∈Ω, t6τ,

(x,t) = v

(x),x∈Ω, t6τ,

(1.1)

Ù¥Ω ⊂R

•‘k1w>.k.•.

žm•6¼êh

(s) =k

(s)+k

∞

= 1,k

(s) >0,∂

(s) 60,∀s∈R

,t∈R.?˜Ú, b

µ

(s) = −∂

(s),¿…N(t,s) 7→µ

(s) : R×R

→R

= [0,∞)÷v±e^‡:

)éu?¿½t∈R, Ns7→µ

(s)´šO,ýéëY…ŒÚ. ½Â

κ(t) =

∞

(s)ds,inf

t∈R

κ(t) >0.

)éu?¿τ∈R,•3˜‡ëY¼êK

: [τ,∞) →R

¦

(s) 6K

(t)µ

(s),∀t>τ, a.e.s∈R

DOI:10.12677/pm.2022.1271221104nØêÆ

°ÿ§à

)éuz˜‡½s>0, Nt7→µ

(s)éu¤kt∈R´Œ‡,¿…éu?¿;

8K⊂R×R

(t,s) 7→µ

(s) ∈L

∞

(K),(t,s) 7→∂

(s) ∈L

∞

(K).

)•3δ>0 ¦

∂

(s)+∂

(s)+δκ(t)µ

(s) 60,∀t∈R

, a.e.s∈R

)¼êt7→∂

(s)÷v

sup

t∈R

[κ(t)]

∞

|∂

(s)|ds<∞.

)éu?¿t∈R,µ

(0) <C, ¿…

sup

t∈R

(0)

[κ(t)]

<+∞.

)éu?¿a<b∈R,•3ν>0,¦

(s)ds>

κ(t)

,∀t∈[a,b].

bå‘g∈L

(Ω),…š‚5‘f∈C

(R)÷vf(0) = 0,¿…÷v

|f(u

)−f(u

)|6C(1+|u

p−1

+|u

p−1

)|u

−u

|,∀u

∈R,(1.2)

Ù¥1 6p65, C•~ê, …•3θ: 0 <θ61,¦

hf(u),ui>hF(u),1i−

(1−θ)kuk

−c

,(1.3)

hF(u),1i>−

(1−θ)kuk

−c

,∀u∈H

(Ω),(1.4)

Ù¥F(u) =

f(s)ds,c

>0.

PÁØk

(s) ≡k(s)ž, •§(1.1) =z•˜„PòPÁ.Ê5•§.Cc5,kNõÆö

Ñ3l¯'uÊ5.ïÄ,¿¼´aïÄ¤J[1–8].~X, ©z[1] ¥,Cavalcanti<

32001cïáäkPÁ‘Úš‚5‘|u

Ê5.,¿y²f)N•35.

•§•¹PÁØk

(s) ž,L«Ê5áÊ5‘Xžm6²¬Åìž”, =ÑyPzy

–.32018 c,Conti<3©z[8] ¥ïÄäkžm•6PÁØÊ5.

∂

u−h

(0)∆u−

∞

∂

(s)∆u(t−s)ds+f(u) = g,(1.5)

DOI:10.12677/pm.2022.1271221105nØêÆ

°ÿ§à

Ù¥PÁØ¼êh

(·) •‘XžmCzŒÿ¼ê, Conti<•§(1.5) f)·½5. 3

dÄ:þ, ¦‚3©z[9]¥y²d•§(1.5) ¤)¤uÐL§žm •6ÛáÚf•35

±9K5.

É±þ©zéu, ©3#nØµee, |^È©O•{)·½5, ¿y²ž

m•6ÛáÚf•35†ØC5. ©(Xe:31!, 0˜ò‡^VgÚ(Ø;

31n!,?Ø·½5,¿y²•§(1.1) žm•6ÛáÚf•35†ØC5.

3‘Øã¥,•{Bå„,½ÂC•?¿~ê.

2.ý•£

-µ

(s) = −∂

(s)…k

(∞) = 1,K•§(1.1) Œ=z•

−∆u

−∆u−

∞

(s)∆η

(s)ds+f(u) = g.(2.1)

éu?¿t>τ,k

(s) =

(

u(t)−u(t−s),s6t−τ,

(s−t+τ)+u(t)−u

,s>t−τ.

(2.2)

ƒAÐ->Š^‡•:











u(x,t) = 0,x∈∂Ω,t>τ,

(x,s) = 0,(x,s) ∈∂Ω×R

,t>τ,

u(x,τ) = u

(x,t),x∈Ω,

(x,τ) = v

(x,t),x∈Ω,

(x,s) = η

(x,s),(x,s) ∈Ω×R

(2.3)

XÓ©z[10],A= −∆ …½Â•D(A) = H

(Ω)∩H

(Ω).•ÄHilbert ˜mxD(A

),k∈

R,¿DƒƒASÈ†‰ê

h·,·i

D(A

)

= hA

·,A

i,k·k

D(A

)

= kA

·k,

ùph·,·iÚk·k•L

(Ω)SÈ†‰ê.

Ïd, éu?¿k>r, k;i\D(A

)→D(A

), ±9éu¤kk∈[0,

), këYi\

D(A

) →L

n−2k

(Ω).

éu0 6k<3,PH

= D(A

),k·k

= k·k

D(A

)

,KH= L

(Ω),H

= H

(Ω),H

(Ω)∩H

(Ω).ŠâPÁØ¼ê÷v^‡,0 6r<3 ž,½ÂXePÁ˜m

= L

1+σ

) = {ξ

: R

→H

1+σ

∞

(s)kξ

(s)k

1+σ

ds<+∞},

DOI:10.12677/pm.2022.1271221106nØêÆ

°ÿ§à

¿DƒƒASÈ†‰ê

hη

,ξ

∞

(s)hη

(s),ξ

(s)i

1+σ

ds,

kξ

∞

(s)kξ

(s)k

1+σ

ds.

Šâ(H

),éu?¿η

∈M

kη

(t)kη

,∀t>τ,(2.4)

…këYi\

⊂M

,∀t>τ.

AO/,½Âžm•6ƒ˜m

= H

1+σ

×H

1+σ

×M

¿…Dƒ‰ê

kzk

= k(u,v,η

= kuk

1+σ

+kvk

1+σ

+kη

éu?¿r>0 ±9?¿t∈R, -

(R) = {z∈H

:kzk

6R}.

½Â2.1[11]{H

}

t∈R

´˜xD‰‚5˜m, éuVëêŽfxU(t,τ) :H

→H

,t>τ∈

R,XJ÷v±e5Ÿ:

(i)é?¿τ∈R,U(τ,τ) = Id´H

þðN;

(ii)é?¿t>s>τ,τ∈R,kU(t,s)U(s,τ) = U(t,τ),

K¡U(t,τ)´˜‡L§.

Ún2.2[9](È©.GronwallØª)Γ : [τ,∞) →R´ëY¼ê,éu,ε>0 ±9?¿

b>a>τ, eÈ©Øª¤á:

Γ(b)+2ε

Γ(y)dy6Γ(a)+

(y)Γ(y)dy+

(y)dy,

Ù¥q

>0 …q

∈L

loc

[τ,∞)(i= 1,2) ÷v,•3c

>0, ¦

(y)dy6ε(b−a)+c

,sup

t>τ

t+1

(y)dy6c

DOI:10.12677/pm.2022.1271221107nØêÆ

°ÿ§à

Γ(t) 6e



|Γ(τ)|e

−ε(t−τ)

1−e

−ε



,∀t>τ.

½Â2.3[12,13]XJ˜‡8xB= {B

}

t∈R

´˜—k.,=

sup

t∈R

= sup

t∈R

sup

ξ∈B

kξk

<+∞, ¿…éu z˜‡R>0, •3~êτ

= τ

(R)>0, ¦



U(t,τ)B

(R) ⊂B

,∀t−τ>τ

K¡B= {B

}

t∈R

•žm•6áÂ8.

½Â2.4[12,13]e˜‡8xA= {A

}

t∈R

÷vXe5Ÿ:

(i)é?¿t∈R,z˜‡A

¥Ñ´;;

(ii)A´.£áÚ, =A•˜—k.,¿…é?¿˜—k.8xC= {C

}

t∈R

lim

τ→−∞

dist

(U(t,τ)C

) = 0;

(iii)(•5)e•3˜‡8xD= {D

}

t∈R

÷v(i)Ú(ii), @oA

⊂D

,∀t∈R,

K¡A= {A

}

t∈R

•L§U(t,τ)žm•6ÛáÚf.

Ún2.5[12]L§U(t,τ)žm•6áÚfA={A

}

t∈R

•3…•˜…=8ÜO=

}

t∈R

•š˜,Ù¥O

∈X

´;,¿…O´.£áÚ.

Ún2.6[14]U(t,τ)´Š^užm•6ƒ˜mX

˜‡L§, é?¿t>τ, U(t,τ):

→X

´ëY,…Pkžm•6ÛáÚfA= {A(t)}

t∈R

.@oA´ØC,=,U(t,τ)A

,∀t>τ.

Ún2.7[9]PI= [τ,T],bu∈W

1,∞

(I;H

1+σ

)…η

∈M

.@o, éu?¿τ6a6b6

T,±eØª¤á:

kη

−

∞

[∂

(s)+∂

(s)]kη

(s)k

1+σ

dsdt6kη

(t),η

dt.

Ún2.8[9]éu?¿b>a>τ±9z‡ω∈(0,1],½Â•¼

(t) = 2hu(t),u

(t)i

(t) = −

κ(t)

∞

(s)hψ

(s),u

(t)i

ds.

¼êΦ

÷v

(b)+(2−ω)

kp(t)k

dt6Φ

(a)+2

k∂

p(t)k

κ(t)kϕ

dt−2

hγ(t),p(t)idt,

DOI:10.12677/pm.2022.1271221108nØêÆ

°ÿ§à

¿…Ψ

÷v

(b)+

k∂

p(t)k

dt6Ψ

(a)−M

∞

[∂

(s)+∂

(s)]kϕ

(s)k

dsdt

+ω

kp(t)k

dt+

κ(t)kϕ

κ(t)

∞

(s)hγ(t),ϕ

(s)idsdt,

Ù¥MÚC•=•6uPÁØ~ê.

3.Ì‡(J

3.1.·½5

½Â3.1éu?¿ T>τ∈R,g∈H,…z

=(u

,η

)∈H

,XJ(u(τ),u

(τ),η

,η

),¿…

(i)z(t) ∈H

a.e.t∈[τ,T];

(ii)u∈W

2,∞

(τ,T;H

),u

∈C([τ,T];H

),η

÷vª(2.2);

(iii)éu?¿ω∈H

,ωi+hu

,ωi

+hu,ωi

∞

(s)hη

(s),ωi

ds+hf(u),ωi= hg,ωi, a.e.t∈(τ,T],

K¡z(t) = (u(t),u

(t),η

)•¯K(2.1) 3žm«mIþ÷vÐŠz(τ) = z

f).

½n3.2(·½5)PI=[τ,T],∀T>τ.b(1.2)-(1.4)ª±9^‡(H

)-(H

)¤á,g∈

H,Kéu?¿ÐŠz

∈H

÷vkz

6R,¯K(2.1)-(2.3)•3•˜f)z(t)=

(u(t),u

(t),η

) = U(t,τ)z

,Kéu?¿½t, k

sup

t>τ

kz(t)k

κ(y)kη

dy+sup

t>τ

t+1

(s)k

ds6Q(R),

Ù¥Q(R) >0 ´Rþ˜‡ëYO¼ê, ¿…

k¯z(t)k

6Q(R)e

C(R,T)(t−τ)

k¯z(τ)k

, t∈[τ,T],

Ù¥¯z(t)=z

(t)−z

(t),…z

(t),z

(t)´¯K(2.1)-(2.3)÷vÐŠz

=(u

,η

),z

,η

2τ

)f).

y²•§(2.1)¦±u

,Œ

(kuk

+ku

+2hF(u),1i−2hg,ui)+2hu

,η

= 0.(3.1)

DOI:10.12677/pm.2022.1271221109nØêÆ

°ÿ§à

½ÂN(t) = kuk

+ku

+2hF(u),1i−2hg,ui.

|^(1.4)ª, Œ•

N(t) >

(kuk

+ku

)−Q

,(3.2)

Ù¥θd(1.4)ª½Â, Q

θλ

kgk

é(3.1)ª3[τ,t]þÈ©,

N(t)+2

,η

dy= N(τ) 6Q(R),∀t>τ.(3.3)

ŠâÚn2.7,Œ•

kη

−

∞

[∂

(s)+∂

(s)]kη

(s)k

dsdy6kη

,η

dy.(3.4)

d(1.2)ª, Œ•

2hF(u),1i6C(1+kuk

p+1

(Ü(3.2)ª, 

(kuk

+ku

+kη

)−Q

6N(t) = N(t)+kη

6C(1+kuk

+kuk

p+1

+2ku

+kgk

), ∀t>τ.(3.5)

Ïd,

k(u(t),η

κ(y)kη

dy6Q(R),∀t>τ,(3.6)

•§(2.1)¦±u

,

+ku

= −hu,u

−

∞

(s)hη

(s),u

ds−hf(u),u

i+hg,u

i.(3.7)

d(1.2)ª, Œ

|hf(u),u

i|6kf(u)k

p+1

6C(1+ku(t)k

)ku

DOI:10.12677/pm.2022.1271221110nØêÆ

°ÿ§à

¿…d(H

)



−

∞

(s)hη

(s),∂



6ku

∞

(s)kη

(s)k

6ku



∞

(s)ds





∞

(s)kη

(s)k



6ku

(t)

κ(τ)kη

d(3.7)ªŒ

6C(1+ku(t)k

+ku(t)k

(t)

κ(τ)kη

+kgk)ku

6Q(R)(1+

(t)

κ(τ)kη

)ku

+C(R,T),∀t∈[τ,T].(3.8)

qÏ•

+Q(R)(1+κ(t)kη

sup

t>τ

t+1

(s)k

ds6Q(R)



1+sup

t>τ

t+1

κ(s)kη



6Q(R).(3.9)

z

(t) = (u

,η

),•3H

Ä{w

}

∞

j=1

,…Aw

= λ

,j= 1,2,···,¦

h∂

i+h∂

+hu

∞

(s)hη

(s),w

ds+hf(u

),w

=hg,w

i, j= 1,···,n, t>τ,(3.10)

Ù¥u

= Σ

j=i

(t)(w

),¿…

(s) =







(t)−u

(t−s),s6t−τ,

(s−t+τ)+u

(t)−u

,s>t−τ,

±9ƒAÐ©^‡

(τ),∂

(τ),η

) = (u

,η

) →(u

,η

'u(3.10)ª3[τ,t]þÈ©,

(h∂

i+h∂

+hu

∞

(s)hη

(s),w

+hf(u

),w

i−hg,w

i)dt= 0,j= 1,···,n.(3.11)

DOI:10.12677/pm.2022.1271221111nØêÆ

°ÿ§à

qdu

kf(u

6C(1+ku

) 6Q(R),

KkXe(J

∞

(τ,T;H

×H

)þ(u

,∂

) →(u,∂

u,∂

u)f∗Âñ,

×H

þ(u

(t),∂

(t)) →(u(t),∂

u(t),∂

u(t))fÂñ,

3C([τ,T];H×H) þ(u

(t),∂

(t)) →(u(t),∂

u(t)),

3Ω×(τ,T)þ(u

,∂

) →(u,∂

u),

∞

(τ,T;L

)þf(u

) →f(u)f∗Âñ,

¿…d(3.16)ª•, •3q

∈M

¦

þη

→q

f∗Âñ, ∀t∈[τ,T].

q

¯η

= η

−η

,¯u

= u

−u,¯η

= η

−η

,¯u

= u

−u

d(2.4)ª, 

k¯η

−1

(t)k¯η

−1

=C(T)



t−τ

(s)k¯u

(t)−¯u

(t−s)drk

∞

t−τ

(s)k¯η

(s−t+τ)+¯u

(t)−¯u



6C(T)(k¯u

C([τ,T];H)

κ(τ)+k¯η

) →0,∀t∈[τ,T].

Šâ4••˜5,•q

= η

¯η

(s) =

(

¯u

(t)−¯u

(t−s),s6t−τ,

¯η

(s−t+τ)+¯u

(t)−¯u

,s>t−τ,

∞

(s)h¯η

(s),w

t−τ

(s)h¯u

(t),w

ds−

t−τ

(s)h¯u

(t−s),w

∞

t−τ

(s)h¯η

(s−t+τ)−¯u

ds+

∞

t−τ

(s)h¯u

(t),w

=κ(t)h¯u

(t),w

−

t−τ

(s)h¯u

(t−s),w

∞

(s+t−τ)h¯η

(s),w

ds−h¯u

(s),w

∞

t−τ

(s)ds.(3.12)

DOI:10.12677/pm.2022.1271221112nØêÆ

°ÿ§à

|µ

(s)h¯u

(t−s),w

|6µ

(s)k¯u

(t−s)k

6Q(R)µ

(s) ∈L



∞

(s+t−τ)h¯η

(s),w



6CK

(t)

κ(τ)k¯η

→0,

Ïd,n→∞ž, A^Lebesgue ››Âñ½n,Œ

lim

n→∞

∞

(s)h¯η

(s),w

ds= 0,j= 1,2,···,n, a.e.t∈(τ,T].

qdu



∞

(s)h¯η

(s),w



∞

(s)k¯η

(s)k

κ(t)k¯η

6Q(R)(1+κ(t)) ∈L

(τ,t).

ŠâLebesgue ››Âñ½n,k

lim

n→∞

∞

(s)h¯η

(s),w

dsdt= 0.

-(3.12)ª¥n→∞Œ•z= (u,u

,η

)•¯K(2.1)-(2.3) f).

e¡y²f)Lipschitz -½5.

= (u

(t),∂

(t),η

),z

= (u

(t),∂

(t),η

)

•¯K(2.1)-(2.3)ü‡f). @o¯z(t) = (¯u(t),∂

¯u(t),¯η

) = z

(t)−z

(t)÷v

¯u

+A¯u

+A¯u+

∞

(s)A¯η

(s)ds= −f(u

)+f(u

),(3.13)

Ù¥

¯η

(s) =

(

¯u(t)−¯u(t−s),s6t−τ,

¯η

(s−t+τ)+¯u(t)−¯u

,s>t−τ,

•§(3.13)¦±¯u

,

F(t)+

∞

(s)h¯η

(s),¯u

(t)i

=−hf(u

)−f(u

),¯u(t)i

6C(1+ku

p−1

p+1

+ku

p−1

p+1

)k¯uk

p+1

k¯u

p+1

6C(R,T)F(t), t∈[τ,T],

Ù¥F(t) = (k¯uk

+k¯u

DOI:10.12677/pm.2022.1271221113nØêÆ

°ÿ§à

Ïd,éþª3[τ,t] þÈ©,

F(t)+2

h¯u

(y),¯η

dy6F(τ)+C(R,T)

F(y)dy,t∈[τ,T].(3.14)

dÚn2.7Œ•

k¯η

−

∞

[∂

(s)+∂

(s)]k¯η

(s)k

dsdy

6k¯η

h¯u

,¯η

dy.(3.15)

F(t) = F(t)+k¯η

,Kk

k¯z(t)k

6F(t) 6Q(R)k¯z(t)k

l

F(t) 6F(τ)+C(R,T)

F(y)dy,

F(t) 6F(τ)e

C(R,T)(t−τ)

k¯z(t)k

6Q(R)e

C(R,T)(t−τ)

k¯z(τ)k

, t∈[τ,T].

Ù¥,k¯z

6R.

Šâ½n3.2,Œ±½Â¯K(2.1)-(2.3) 3H

þ)L§,=

U(t,τ) : H

→H

,U(t,τ)z

= z(t),∀z

∈H

,t>τ,(3.16)

…{U(t,τ)}•Š^uH

þL§x.

3.2.žm•6ÛáÚf

½n3.3(ÑÑ5){U(t,τ)}

t>τ

•(3.16)ª½Â)L§,¿…(1.2)-(1.4) ª±9^‡(H

)¤á,g∈H, …z

∈H

÷vkz

6R,K•3ε>0,R

>0 ¦

kU(t,τ)z

6Q(R)e

−ε(t−τ)

,∀t>τ.

y²ò•§(2.1)¦±2u(t),Œ

Φ(t)−2ku

−2ku

+2hf(u)−g,ui= −2

∞

(s)hη

(s),ui

6−2

∞

(s)kη

(s)k

kuk

ku(t)k

κ(t)kη

,(3.17)

DOI:10.12677/pm.2022.1271221114nØêÆ

°ÿ§à

Ù¥

Φ(t) = hu

,ui+hu

,ui

6Q(R)(kuk

+ku

).(3.18)

é(3.17)ª3[a,b] þÈ©,¿|^(1.3)ª, Œ•

Φ(b)+(1+

)

kuk

dt+2

hF(u),1idt−2

hg,uidt

6Φ(a)+2

dt+2

dt+

κ(t)kη

dt+c

(b−a).(3.19)

•§(2.1)ˆ‘¦±−

κ(t)

(s)η

(s),'us∈(0,∞) È©,

−

κ(t)

∞

(s)



,η

(s)i+hu

,η

(s)i



κ(t)

∞

(s)hu,η

(s)i

ds+

κ(t)



∞

(s)A

(s)ds



κ(t)

∞

(s)hf(u)−g,η

(s)ids.(3.20)

•O(3.20)ª†à, ½Â•¼

Ψ(t) = −

κ(t)

∞

(s)



(t),η

(s)i+hu

(t),η

(s)i



ds,(3.21)

…

|Ψ(t)|6

κ(t)

∞

(s)



(t)k

kη

(s)k

+ku

(t)kkη

(s)k



κ(t)

κ(t)ku

(t)k

kη

κ(t)ku

(t)kkη

6Q(R)



(t)k

+kη



.(3.22)

½Â©ã¼ê

(s) =











0,0 6s<ε,

−1,ε6s<2ε,

1,2ε6s6

2−εs,

6s<

6s,

DOI:10.12677/pm.2022.1271221115nØêÆ

°ÿ§à

Ù¥0 <ε<

,¿…

(s) = φ

(s)µ

(s) 6µ

(s),∀s∈R

w,,suppµ

(s) = [ε,

].

(t) =

∞

(s)ds=

(s)ds6κ(t).

|^(3.18)Ú(3.21) ª,Œ•

|Φ(t)|+|Ψ(t)|6Q(R)E(t),(3.23)

Ù¥E(t) =

kU(t,τ)z

½Â•¼

(t) = −

(t)

∞

(s)



(t),η

(s)i+hu

(t),η

(s)i



ds,

(t) = J

,(3.24)

Ù¥

= −2



(t)



∞

(s)hu

(t),η

(s)i

ds−

(t)

∞

(s)hu

(t),η

(s)i

ds,

= −2



(t)



∞

(s)hu

(t),η

(s)ids−

(t)

∞

(s)hu

(t),η

(s)ids.

(t) =



κ(t)

(t)



>1.

d^‡(H

),ε¿©, k

sup

t∈[a,b]

(t) 64,∀t∈[a,b].(3.25)

aqu©z[9]¥y², 

(t) 6−(2−R

(t))ku

−M

∞

[∂

(s)+∂

(s)]kη

(s)k

+Q(R)

∞

(t,s)ds−

(t)

κ(t)

∞

(s)hu

,η

(s)ids

+Cκ(t)kη

(s)k

.(3.26)

DOI:10.12677/pm.2022.1271221116nØêÆ

°ÿ§à

(t) 6−(2−R

(t))ku

−M

∞

[∂

(s)+∂

(s)]kη

(s)k

+Q(R)

∞

(t,s)ds−

(t)

κ(t)

∞

(s)hu

,η

(s)ids

+Cκ(t)kη

(s)k

.(3.27)

òª(3.25)-(3.27)“\ª(3.24), ¿3[a,b] þÈ©,

(b)+

[2−R

(t)]ku

dt+

[2−R

(t)]ku

6Ψ

(a)−2M

∞

[∂

(s)+∂

(s)]kη

(s)k

dsdt+C

κ(t)kη

(s)k

+Q(R)

∞

(t,s)dsdt−

(t)

κ(t)

∞

(s)hu

+Au

,η

(s)idsdt.(3.28)

ε→0 ž,w,k

(s) →µ

(s),κ

(t) →κ(t),R

(t) →1,∀(t,s) ∈[a,b]×R

é?¿δ>0Ú?¿½s>0,•3δ

>0,¦ε<δ

ž,

[ε,2ε]

(s) = 0 <δ,lim

ε→0

(t,s) = 0.

-(3.28)ª¥ε→0,¿|^Lebesgue››Âñ½n, Œ•

Ψ(b)+

dt+

6Ψ(a)−2M

∞

[∂

(s)+∂

(s)]kη

(s)k

dsdt+C

κ(t)kη

(s)k

−

κ(t)

∞

(s)hu

+Au

,η

(s)idsdt.(3.29)

d•§(2.1)Œ•

−

κ(t)

∞

(s)hu

+Au

,η

(s)ids

κ(t)

∞

(s)hu(t),η

(s)i

ds+



∞

(s)A

(s)ds



∞

(s)hf(u)−g,η

(s)ids



κ(t)



(ku(t)k

+kf(u)k

+kgk

−1

)

∞

(s)kη

(s)k

Ω



∞

(s)ds

∞

(s)(A

(s))





DOI:10.12677/pm.2022.1271221117nØêÆ

°ÿ§à

κ(t)

(ku(t)k

+(1+kuk

)kuk

+kgk)kη

∞

(s)

Ω



(s)



dxds

ku(t)k

kgk

+Q(R)kη

ku(t)k

kgk

+Q(R)κ(t)kη

,(3.30)

Ù¥A^Øª1 6

inf

t∈R

κ(t)

κ(t) 6Cκ(t).

ò(3.30)ª“\(3.29) ª,

Ψ(b)+

dt+

6Ψ(a)−2M

∞

[∂

(s)+∂

(s)]kη

(s)k

dsdt+

kuk

+Q(R)

κ(t)kη

(s)k

dt+

kgk

(b−a).(3.31)



Υ(t) = N(t)+2ε[Φ(t)+4Ψ(t)]+Q

Ù¥N(t) d(3.5) ª¤½Â,Φ(t) ÚΨ(t) d(3.23) ª¤½Â.

E(t)−Q

6N(t) 6Q(R)E(t)+Ckgk

E(t) 6Υ(t) 6Q(R)E(t)+Q

,(3.32)

Ù¥E(t) d(3.23) ª½Â,…Q

= C(Q

+kgk

) = C(c

+kgk

Šâ(3.3),(3.19) Ú(3.29) ª,¿|^^‡(H

)±9(3.32) ªŒ

Υ(b)+2ε

Υ(t)dt+P

6Υ(a)+εQ

(b−a),

…

= ε

(θE(t)−4ε[Φ(t)+4Ψ(t)])dt>0,

= −(1−16εM)

∞

[∂

(s)+∂

(s)]kη

(s)k

dsdt−εQ(R)

κ(t)kη

>[δ(1−16εM)−εQ(R)]

κ(t)kη

dt>0.

DOI:10.12677/pm.2022.1271221118nØêÆ

°ÿ§à

•,A^Ún1.2 

Υ(t) 6Υ(τ)e

−ε(t−τ)

εQ

1−e

−ε

E(t) 6

Υ(t) 6Q(R)e

−ε(t−τ)

Ù¥R

sup

ε∈(0,1]

εe

1−e

−ε

½n3.4e½n3.3b¤á, …c

= 0,g= 0,KR

= Q

= 0,¿…

kU(t,τ)k

6Q(R)e

−ε(t−τ)

,∀kz

6R,t>τ.

?¿½τ∈R…(p

,ϕ

) ∈H

.•Ä•§

+Ap+

∞

(s)Aϕ

(s)ds+γ(t) = 0,(3.33)

Ù¥γ•å‘,…

(s) =







p(t)−p(t−s)dr,

(s−t+τ)+p(t)−p

(3.34)

ƒAÐŠ^‡•

(p(τ),p

(τ),ϕ

) = (p

,ϕ

).(3.35)

f(s) = f

(s)+f

(s),

Ù¥f

•Lipschitz ëY¼ê…f

(0) = 0,¿…

(s) = 0,∀s∈[−1,1],f

(s)s>F

(s) :=

(y)dy,∀s∈R,

)−f

)|6C|s

−s

|(1+|s

|+|s

p−1

,∀s

∈R.

½n3.3 L²L§U(t,τ) kžm•6áÂ8B={B

}

t∈R

. éu?¿z

=(u

,η

)∈B

U(t,τ)z

= U

(t,τ)z

DOI:10.12677/pm.2022.1271221119nØêÆ

°ÿ§à

Ù¥U

(t,τ)z

= z

(t)…U

(t,τ)z

= z

(t),=z= (u,u

,η

) = z

(t)+z

(t),z

(t)÷v











+Av

+Av+

∞

(s)Aξ

(s)ds+f

(v) = 0,

(τ,τ)z

= z

…

(s) =







v(t)−v(t−s)dr,s6t−τ,

(s−t+τ)+v(t)−v

,s>t−τ,

(t)÷v











+Aw

+Aw+

∞

(s)Aζ

(s)ds+f(u)−f

(v) = g,

(τ,τ)z

= 0,

(3.36)

…

(s) =







w(t)−w(t−s),s6t−τ,

(s−t+τ)+w(t)−w

,s>t−τ,

Ù¥η

= ξ

+ζ

Šâ½n3.4,Œ•

(t,τ)z

6Ce

−ε(t−τ)

,∀t>τ,(3.37)

¿…d(3.9)ª, Œ

sup

t>τ

t+1

(t)k

dt6Q(R).(3.38)

Ún3.6b(1.2)-(1.4)ª±9^‡(H

)-(H

)¤á,g∈H,…z

=(u

,η

)∈H

,÷v

6R.K

(t,τ)z

6Q(R),∀t>τ.(3.39)

Ù¥0 <σ6

y²ò•§(3.36)¦±A

,Œ

(t)+hu

−v

wi+2hζ

(t)i

=2hg−f

(u),A

wi−2hf

(u)−f

(v),A

wi,(3.40)

Ù¥N

(t) = kw(t)k

1+σ

+kw

(t)k

1+σ

DOI:10.12677/pm.2022.1271221120nØêÆ

°ÿ§à

dÚn3.5,½n3.2 ±9Sobolev i\,

2|hg−f

(u),A

i|6C(kgk+kuk)kA

1+σ

Q(R)

,(3.41)

2|hf

(u)−f

(v),A

i|6C(1+kuk

p−1

3(p−1)

+kvk

p−1

3(p−1)

)kwk

6Q(R)(kwk

1+σ

)

1+σ

Q(R)

kwk

1+σ

(kwk

1+σ

+kw

1+σ

Q(R)

,(3.42)

−v

i6ε

(t)k

1+σ

Q(R)

(ku

+kv

).(3.43)

Ù¥A^Øª

3(p−1)

2(p+1)

<1.

ò(3.41)-(3.43)ª“\(3.40) ª,k

(t)+2hζ

(t)i

6ε

(kw(t)k

1+σ

+kw

(t)k

1+σ

Q(R)

(t),

(b)+2

hζ

(a)+ε

(kwk

1+σ

+kw

1+σ

)dt+

Q(R)

(t)dt,(3.44)

Ù¥y

(t) = 1+ku

+kv

,…sup

t>τ

t+1

(r)dr6Q(R).



(t) :=

(t,τ)z

(kw(t)k

1+σ

+kw

(t)k

1+σ

+kζ

¿|^Ún2.7,Œ

(b)−

∞

[∂

(s)+∂

(s)]kζ

(s)k

1+σ

dsdt

62E

(a)+ε

(kwk

1+σ

+kw

1+σ

)dt+

Q(R)

(t)dt.(3.45)



(p,p

,ϕ

) = (A

w,A

),γ(t) = g−f(u)+f

(v)−u

DOI:10.12677/pm.2022.1271221121nØêÆ

°ÿ§à

(t) = Φ

(t) = 2hw

(t),w(t)i

1+σ

,Ψ

(t) = Ψ

(t) = −

κ(t)

∞

(s)hζ

(s),w

(t)i

1+σ

ds.

aqu½n3.3y², 

|Φ

(t)|+|Ψ

(t)|6CE

(t).(3.46)

dÚn2.8(-w=

),Œ•

(b)+4Ψ

(b)+

kw(t)k

1+σ

dt+2

(t)k

1+σ

dt+

kζ

6Φ

(a)+4Ψ

(a)−4M

∞

[∂

(s)+∂

(s)]kζ

(s)k

1+σ

dsdt

(κ(t)+1)kζ

dt−2

hγ(t),A

w(t)idt

κ(t)

∞

(s)hγ(t),A

(s)idsdt.(3.47)

d(3.41)-(3.43),ε



hγ(t),A

w(t)idt



kw(t)k

1+σ

dt+Q(R)

(t)dt,(3.48)

¿…

hγ(t),A

(s)i6

kw(t)k

1+σ

+Ckζ

(s)k

1+σ

+Q(R)y

(t),

κ(t)

∞

(s)hγ(t),A

(s)idsdt

kw(t)k

1+σ

dt+C

κ(t)kζ

dt+Q(R)

(t)dt.(3.49)

ò(3.48)-(3.49)ª“\(3.47) ª,Œ

(b)+4Ψ

(b)+

(t)dt+

(kw(t)k

1+σ

+kw

(t)k

1+σ

)dt

6Φ

(a)+Ψ

(a)+C

κ(t)kζ

dt+Q(R)

(t)dt

−4M

∞

[∂

(s)+∂

(s)]kζ

(s)k

1+σ

dsdt.(3.50)

DOI:10.12677/pm.2022.1271221122nØêÆ

°ÿ§à



(t) = N

(t)+2ε[Φ

(t)+4Ψ

(t)].

(t) 6Υ

(t) 6

(t).(3.51)

ε

,Œ

(b)+2ε

(t)dt+P(t) 6Υ

(a)+

(t)dt,(3.52)

Ù¥

P(t) = −(1−8εM)

∞

[∂

(s)+∂

(s)]kζ

(s)k

1+σ

dsdt−εC

κ(t)kζ

>[δ(1−8εM)−εC]

κ(t)kζ

dt>0,

¿…éuvε>0,k

sup

t>τ

t+1

(r)dr= sup

t>τ

t+1



2εQ(R)y

(r)+

Q(R)

(r)



dr6

Q(R)

|^Ún1.2±9(3.51) ª,Œ•

(t) 6Q(R),∀t>τ.

Ïd,(3.39)ª¤á.

½n3.7e(1.2)-(1.4)ª±9^‡(H

)-(H

)¤á,g∈H,…z

=(u

,η

)∈H

,÷v

6R.L§U(t,τ)Pkžm•6ÛáÚfA={A

}

t∈R

.d,áÚf´ØC,=,

U(t,τ)A

= A

,∀t>τ.

y²½n3.3L²U(t,τ)kžm•6áÚ8B={B

}

t∈R

.½n3.6L²,é?¿r>

2Q(R),8xB= {B

(r)}

t∈R

´.£áÚ,Ù¥B

(r) = {ξ∈H

|kξk

6r}.

τ→−∞ž,k

dist

(U(t,τ)B

(r)) 6sup

∈B

dist

(t,τ)z

(r))

6sup

∈B

(t,τ)z

6Ce

−

(t−τ)

→0.

éuz‡t∈R,3˜mH

þ•3;8C

⊂B

(r),¦8ÜC= {C

}

t∈R

´.£áÚ.Ïd,

dÚn2.5 •,U(t,τ)kÛáÚfA= {A

}

t∈R

.dL§ëY5±9Ún2.6,Œ•A´ØC.

DOI:10.12677/pm.2022.1271221123nØêÆ

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