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PureMathematicsnØêÆ,2022,12(10),1615-1628
PublishedOnlineOctober2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.1210175
äk©ê‚5PÁš²;‡A*Ñ
•§)ìC5
êêê©©©¦¦¦§§§ÜÜÜJJJ
Ü“‰ŒÆêƆÚOÆ§[‹=²
ÂvFϵ2022c911F¶¹^Fϵ2022c1010F¶uÙFϵ2022c1018F
Á‡
ïÄ˜aäk©ê‚5PÁš²;‡A*Ñ•§)ìC51•.½ÂÜ·Lyapunov
•¼,y²š‚5‘f÷vO•5^‡,PÁ Øg¥•êP~ž,XÚ)´õ‘ªP~;‘
,A^Œ+nØ,y²)´š•ê-½.
'…c
š²;‡A*Ñ•§§©ê‚5PÁ§õ‘ªP~
TheAsymptoticBehaviorofSolutions
foraClassofNonclassical
Reaction-DiffusionEquationswith
FractionalLinearMemory
WenhuiMa,YingZhang
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Received:Sep.11
th
,2022;accepted:Oct.10
th
,2022;published:Oct.18
th
,2022
©ÙÚ^:ꩦ,ÜJ.äk©ê‚5PÁš²;‡A*Ñ•§)ìC5[J].nØêÆ,2022,12(10):
1615-1628.DOI:10.12677/pm.2022.1210175
ꩦ§ÜJ
Abstract
Theasymptoticbehaviorofsolutionsforaclassofnonclassicalreaction-diffusione-
quationswithfractionallinearmemoryisinvestigated.bydefininganappropriate
Lyapunovfunctional,itisprovedthatthesolutionofthesystemdecayspolynomially
whenthenonlineartermfsatisfiesthegrowthconditionandthememorykernelg
decaysexponentially.Afterthat,weachievethatthesolutionisnon-exponentially
stablebymeansofthesemigrouptheory.
Keywords
Non-ClassicalReactionDiffusionEquations,FractionalLinearMemory,Polynomially
Decay
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
•ÄXe‘©ê‚5PÁš²;‡A*Ñ•§)ìC5:







u
t
−∆u
t
−∆u−(g∗(−∆)
α
u)(t)+f(u) = h(x),x∈Ω,t>0,
u(x,t) = 0,x∈∂Ω,t≥0,
u(x,0) = u
0
(x),x∈Ω,
(1)
Ù¥Ω´R
n
¥äk1w>.∂Ωk.m«•,α∈[0,1),∗“LòÈ$Ž,š²;‡A*Ñ•§´
3²;‡A*Ñ•§¥•Ä0ŸÊ5!ØrσéXÚK•,dAifantis ïá,„
©z[1,2],§3šÚî6N,NåÆ99DnØ+•kX2•A^.¯K(1) ¥òÈ‘
Œ±‡NXÚLGéy39™5GK•,ùa¯K~~Ñy3§Ê5nØïÄ¥.
éPÁ‘œ¹,NõÆö?12•ïÄ,„©[3–6].Jaime<3©z[5]¥?Ø˜a‘
PÁÚDirichlet>.Ä–XÚ)õ‘ªP ~,/ÏT•{,©·‚̇•Ääk©ê
‚5PÁš²;‡A*Ñ•§)ìC51•.ÏL½ÂÜ·Lyapunov •¼,y²PÁ
Øg¥•êP~ž,¯K(1))´õ‘ªP~;‘,A^Œ+nØ,y²)´š•ê-½.
DOI:10.12677/pm.2022.12101751616nØêÆ
ꩦ§ÜJ
2.ý•£
Äk,•ÄHilbert˜mD(A
s/2
),∀s∈R,ÙSÈÚ‰ê©O½ÂXe
(·,·)
D(A
s/2
)
= (A
s/2
·,A
s/2
·), k·k
D(A
s/2
)
= kA
s/2
·k.

D(A
0
) = L
2
(Ω),D(A
1/2
) = H
1
0
(Ω),D(A
−1/2
) = H
−1
(Ω).
¿kXei\¤á
D(A
s/2
) →D(A
r/2
),∀s>r.
Ù¥A•H= L
2
(Ω)þî‚Žf,¿½Â
A= −∆,ٽ•D(A) = H
2
(Ω)∩H
1
0
(Ω),
©O^(·,·)Úk·k5L«L
2
(Ω)þSÈÚ‰ê;¿^k·k
p
5L«L
p
(Ω)‰ê.
Ó©[5],PÁØg∈C
2
(R
+
)∩W
2,1
(R
+
),é?¿s∈R
+
,÷vXeb
(h
1
)g(s) >0;
(h
2
)−c
0
g(s) ≤g
0
(s) ≤−c
1
g(s);
(h
3
)|g
00
(s)|≤c
2
g(s);
(h
4
)1−G(t) ≥c
3
>0.
Ù¥c
i
,i= 0,1,2,3,þ•~ê…G(t) :=
R
t
0
g(s)ds.
Ó©[7],š‚5‘f÷v
(f
1
)(ρ+1)F(s) ≤f(s)s,F(Z) :=
R
Z
0
f(s)ds, ∀s∈R.
(f
2
)|f(s
1
)−f(s
2
)|≤C(1+|s
2
|
ρ−1
+|s
2
|
ρ−1
)|s
1
−s
2
|, ∀s
1
,s
2
∈R,
Ù¥C>0, ρ≥1¿…k(n−2)ρ≤n,n≥3 .¿½ÂXeòÈ$Ž.
(g∗w)(t) :=
Z
t
0
g(t−s)w(s)ds.
••B¡$Ž,½Â
(gw)(t) :=
Z
t
0
g(t−s)|w(t)−w(s)|
2
ds,
(gw)(t) :=
Z
t
0
g(t−s)(w(t)−w(s))ds.
DOI:10.12677/pm.2022.12101751617nØêÆ
ꩦ§ÜJ
Ún1[5](Poincar´eØª)Ω´R
n
¥k.m«•,¿…k1w>.,bu∈
W
1,p
0
(Ω),1 ≤p<n.Kk
kuk
L
q
(Ω)
≤Ck∇uk
L
p
(Ω)
.
é?¿q∈[1,p
∗
],p
∗
= np/(n−p),Ù¥~êC =•6up,q,nÚΩ.
Ún2[5](YoungØª)a,b´šK¢ê,p>1,
1
p
+
1
q
= 1,@o
ab≤
a
p
p
+
b
q
q
.
3.)õ‘ªP~
Ún3[5]é?¿¼êk∈C(R)±9ϕ∈W
1,2
(0,T),Υ
(k∗ϕ)(t) = (kϕ)(t)+

Z
t
0
k(τ)dτ

ϕ(t).
Ún4[5]é?¿¼êk∈C
1
(R)±9ϕ∈W
1,2
(0,T),Œ
(k∗ϕ)(t)ϕ
t
(t) =−
1
2
k(t)|ϕ(t)|
2
+
1
2
(k
0
ϕ)(t)
−
1
2
d
dt

(kϕ)(t)−

Z
t
0
k(τ)dτ

|ϕ(t)|
2

.
Ún5[5]é?¿¼êk∈C(R)±9ϕ∈W
1,2
(0,T),K
|kϕ)(t)|
2
≤

Z
T
0
|k(τ)|dτ

(|K|ϕ)(t).
Ún6[5]3Hilbert˜mH¥,é?¿¼êk∈C(R),k>0±9ϕ∈W
1,2
(0,T),q∈H.
•3~êC
ε
>0,Ù¥ε>0,¦
|q(t)(kϕ)(t)|≤ε|q(t)|
2
+C
ε
(kϕ)(t).
Ún7[7]é?¿ü‡¼êk,w∈C
1
(R),Ù¥θ∈[0,1],ke¡Øª¤áµ
|(kw)(t)|
2
≤

Z
t
0
|k(s)|
2(1−θ)
ds

|k|
2θ
w.
½n1bg,f©O÷v^‡(h1) −(h4)Ú(f
1
) −(f
2
),K‰½ÐŠu
0
∈D(A)Úh∈H,¯
K(1)•3•˜)u,÷v
u∈C([0,T];D(A))∩C
1

[0,T];D(A
1/2
)

.
DOI:10.12677/pm.2022.12101751618nØêÆ
ꩦ§ÜJ
y²)·½5̇^Faedo−Galerkin•{±9GronwallÚn.
3ùp·‚½ÂXeUþ•¼
P(t) =
1
2
Z
Ω
h
|∇u(t)|
2
−2(u
t
,u)−G(t)|(−∆)
α
2
u(t)|
2
−|∇u
t
(t)|
2
+(g(−∆)
α
2
u)(t)
−|∇h|
2
i
dx+
Z
Ω
F(u)dx,
Q(t) =
1
2
Z
Ω

|(−∆)
1−
α
2
u(t)|
2
−G(t)|(−∆)
α
2
u(t)|
2
+(g∇u)(t)+|∇h|
2

dx
+
Z
Ω
(−∆)
1−α
F(u)dx.
Ún8u´¯K(1)),÷vЊu
0
∈D(A),Kke¡UþØª:
d
dt
P(t) ≤
Z
Ω

−2|u
t
|
2
+c
4
|∇h|
2
+(
c
4
η
1
−1)|∇u
t
|
2

dx
−
1
2
g(t)
Z
Ω
|(−∆)
α/2
u(t)|
2
dx+
1
2
Z
Ω
(g
0
(−∆)
α
2
u)(t)dx−(u
tt
,u),
9
d
dt
Q(t) ≤
Z
Ω

c
4
|∇h|
2
+
c
4
η
1
|∇u
t
|
2
−(c
5
+c
6
)|u
t
|
2

dx
−
1
2
g(t)
Z
Ω
|∇u(t)|
2
dx+
1
2
Z
Ω
(g
0
∇u)(t)dx.
y²^(1)¥•§†u
t
‰SÈ,Œ
1
2
d
dt
k∇uk
2
+ku
t
k
2
+k∇uk
2
−(g∗(−∆)
α
2
u,(−∆)
α
2
u
t
)+(f(u),u
t
) = (h,u
t
).
3Ún4¥-ϕ= (−∆)
α
2
u,·‚
1
2
d
dt

−(u
t
,u)−k∇u
t
k
2
+k∇uk
2
+g(−∆)
α
2
u
−G(t)k(−∆)
α
2
uk
2
+k∇hk
2
+
Z
Ω
F(u)dx

= −(u
tt
,u)+
Z
Ω
h(x)u
t
dx−2ku
t
k
2
−k∇u
t
k
2
+
1
2
g
0
(−∆)
α
2
u−
1
2
g(t)k(−∆)
α
2
uk
2
−(∇u
tt
,∇u
t
),
(2)
A^YoungØª9Poincar´eØª,k
Z
Ω
h(x)u
t
dx≤c
4
kh(x)k
2
+
c
4
η
1
ku
t
k
2
≤c
4
k∇h(x)k
2
+
c
4
η
1
k∇u
t
k
2
.
(3)
DOI:10.12677/pm.2022.12101751619nØêÆ
ꩦ§ÜJ
òª(3)“\ª(2) ¥,Œ
d
dt
P(t) ≤
Z
Ω

−|u
t
|
2
+c
4
|∇h|
2
+(
c
4
η
1
−1)|∇u
t
|
2

dx
−
1
2
g(t)
Z
Ω
|(−∆)
α/2
u(t)|
2
dx+
1
2
Z
Ω
(g
0
A
α
2
u)(t)dx−(u
tt
,u),
Ón,·‚^(1) ¥•§†(−∆)
1−α
u
t
‰SÈ.3Ún4 ¥-ϕ= (−∆)
1
2
u,k
1
2
d
dt
Z
Ω

|(−∆)
1−α/2
u|
2
+(−∆)
1−α
F(u)−G(t)|(−∆)
α
2
u(t)|
2
+(g∇u)(t)

dx
= −



(−∆)
1−α
2
u
t



2
−


(−∆)
1−α/2
u
t


2
+

(−∆)
1−α
2
h(x),(−∆)
1−α
2

−
1
2
g(t)
Z
Ω
|∇u(t)|
2
dx+
1
2
Z
Ω
(g
0
∇u)(t)dx.
UY¦^YoungØªÚPoincar´eØª,Œ
d
dt
Q(t) ≤
Z
Ω

c
4
|∇h|
2
+
c
4
η
1
|∇u
t
|
2
−(c
5
+c
6
)|u
t
|
2

dx
−
1
2
g(t)
Z
Ω
|∇u(t)|
2
dx+
1
2
Z
Ω
(g
0
∇u)(t)dx.
Úny²¤.
½ÂXeUþ•¼µ
F=
1
2
k(g∗∇u)(t)k
2
−(u(t),(g∗u)
t
(t)),(4)
K(t) = (u(t),u
t
(t))+k∇uk
2
.(5)
Ún93Ún8be,·‚
d
dt
F(t) ≤
g(0)
2
ku
t
k
2
+
c
2
0
g
2
g(0)
kuk
2
+
1
g(0)

Z
∞
0



g
00
(s)



ds





g
00



u

+η
2
k∇hk
2
+

c
7
η
2
+c
0

g(t)kuk
2
+

c
7
c
0
η
2
+
c
10
η
4
+
c
11
η
5

(g∇u)(t)
+(η
3
+η
5
)k∇u
t
k
2
+(c
8
η
2
+η
4
)k∇uk
2
.
y²(ÜÚn3¿?1¦$Ž,k
(g∗u)
t
(t) = g(0)u(t)+(g
0
∗u)(t)
= g(t)u(t)+(g
0
u)(t).
(6)
DOI:10.12677/pm.2022.12101751620nØêÆ
ꩦ§ÜJ
^(1)¥•§†(g∗u)
t
‰SÈ,2Šâ(6) ,·‚
d
dt
(u,(g∗u)
t
) = (h(x)−f(u)+(g∗(−∆)
α
u)(t)−∆u−∆u
t
,(g∗u)
t
)
+

g(0)u
t
+g
0
u+g
00
u,u

=

h(x),gu+g
0
u

−(f(u),(g∗u)
t
)
−

−∆u−∆u
t
,gu+g
0
u

+
1
2
d
dt
kg∗(−∆)
α
2
uk
2
+

g(0)u
t
+g
0
u+g
00
u,u

.
(7)
ò(5),(6)ª(Ü,²L{üOŽ,Œ
d
dt

1
2
kg∗(−∆)
α
2
uk
2
−(u(t),(g∗u)
t
)

= (f(u)−h(x),(g∗u)
t
)+

−∆u−∆u
t
,gu+g
0
u

−

g(0)u
t
+g
00
u,u

−g
0
kuk
2
.
(8)
e¡é(8)ªÒmà©O?1O,¦^YoungØª9Ún6,k
−

g(0)u
t
+g
00
u,u

≤c
0
g(u
t
,u)+




g
00
u,u




≤
g(0)
2
ku
t
k
2
+
c
2
0
g
2
g(0)
kuk
2
+
1
g(0)

Z
∞
0



g
00
(s)



ds





g
00



u

.
(9)
¦^YoungØªÚ(6),·‚k
(f(u)−h(x),(g∗u)
t
) ≤η
2
Z
Ω

|f(u)|
2
+|h|
2

dx+
c
7
η
2
Z
Ω

g(t)|u|
2
−g
0
u

dx.
(10)
d^‡(f
1
)−(f
2
),YoungØª,Poincar´eØªÚÚn7,Œ•
Z
Ω
|f(u)|
2
dx≤c
8

Z
Ω
|u|
2
dx+
Z
Ω
|u|
2ρ
dx

≤c
8

Z
Ω
|∇u|
2
dx+

Z
Ω
|∇u|
2
dx

ρ

≤c
8
Z
Ω
|∇u|
2
dx.
(11)
ò(11)“\(10),A^Poincar´eØªÚ^‡(h
2
),Œ
(f(u)−h(x),(g∗u)
t
) ≤c
8
η
2
k∇uk
2
+η
2
k∇hk
2
+
c
7
η
2
g(t)kuk
2
−
c
7
η
2
(g
0
u)
≤c
8
η
2
k∇uk
2
+η
2
k∇hk
2
+
c
7
η
2
g(t)kuk
2
+
c
7
c
0
η
2
(g∇u).
(12)
DOI:10.12677/pm.2022.12101751621nØêÆ
ꩦ§ÜJ
|^Poincar´eØª,YoungØªÚÚn6,Œ

−∆u−∆u
t
,gu+g
0
u

=(∇u,g
0
∇u)+(∇u
t
,g
0
∇u)
+g(t)k∇uk
2
+(∇u
t
,g(t)u(t))
≤

1+
c
9
η
3

g(t)k∇uk
2
+(η
3
+η
5
)k∇u
t
k
2
+

c
10
η
4
+
c
11
η
5

(g∇u)(t)+η
4
k∇uk
2
,
(13)
ò(9),(12)-(13)“\(8),nŒ
d
dt
F(t) ≤
g(0)
2
ku
t
k
2
+
c
2
0
g
2
g(0)
kuk
2
+
1
g(0)

Z
∞
0



g
00
(s)



ds





g
00



u

+(c
8
η
2
+η
4
)k∇uk
2
+η
2
k∇hk
2
+

c
7
η
2
+c
0

g(t)kuk
2
+

c
7
c
0
η
2
+
c
10
η
4
+
c
11
η
5

(g∇u)(t)
+(η
3
+η
5
)k∇u
t
k
2
.
(14)
Úny.
Ún103Ún8b^‡e, ·‚k
d
dt
K(t) ≤(u
tt
,u)−k∇u
t
k
2
+

1−c
3
+
c
12
η
6
+η
7

k∇uk
2
+
1
η
7
(
Z
∞
0
g(s)ds)(g(−∆)
α
2
u)+η
6
k∇hk
2
−(ρ+1)
Z
Ω
F(u)dx.
y²^(1)¥•§†u‰SÈ,3Ún4 ¥-ϕ= (−∆)
α
u,Œ
d
dt

(u,u
t
)+
1
2
kA
1
2
uk
2

= (u
tt
,u)−k∇u
t
k
2
−(f(u),u)+(h(x),u)
+(u,g(−∆)
α
u+G(t)(−∆)
α
u)
= (u
tt
,u)−k∇u
t
k
2
+G(t)k(−∆)
α
2
uk
2
+

(−∆)
α
2
u,gA
α
2
u

−(f(u),u)+(h(x),u)
≤(u
tt
,u)−k∇u
t
k
2
+(1−c
3
)k∇uk
2
+((−∆)
α
2
u,g(−∆)
α
2
u),
(15)
DOI:10.12677/pm.2022.12101751622nØêÆ
ꩦ§ÜJ
d^‡(f
1
),Υ
−(f(u),u) ≤−(ρ+1)
Z
Ω
F(u)dx,(16)
|^YoungØªÚPoincar´eØª, Œ
(h,u) ≤η
6
k∇hk
2
+
c
12
η
6
k∇uk
2
,(17)
ò(16)-(17)“\(15), 2ŠâÚn6 ÚPoincar´eØª, Œ•
d
dt
((u,u
t
)+
1
2
kA
1
2
uk
2
) ≤(u
tt
,u)−k∇u
t
k
2
+

1−c
3
+
c
12
η
6
+η
7

k∇uk
2
+
1
η
7
(
Z
∞
0
g(s)ds)(g(−∆)
α
2
u)
+η
6
k∇hk
2
−(ρ+1)
Z
Ω
F(u)dx.
(18)
½Â•¼
L(t) ≤NP(t)+NQ(t)+F(t)+NK(t).
½n2bg∈C
2
(R
+
)∩W
2,1
(R
+
),^‡(h
1
)−(h
4
)9(f
1
)−(f
2
)¤á,…Њ
u
0
∈D(A),
@o¯K(1))•õ‘ªP~,=•3˜‡~êC÷v
P(t)t≤C[P(0)+Q(0)].
y²d^‡(h
4
)Υ
R
∞
0
g(s)ds<1.òÚn8, 9, 10?1ƒAn,|^Poincar´eØª,
k
d
dt
L(t) ≤

N

2c
4
η
1
+
g(0)
2
−c
5
−c
6

+η
3
+η
5

k∇u
t
k
2
−

N

c
3
+
c
12
η
6
+η
7
−1

−c
8
η
2
−η
4

k∇uk
2
+(2Nc
4
+η
2
)k∇hk
2
−(ρ+1)
Z
Ω
F(u)dx
−

N
2
−
c
2
0
g(0)

g(t)


(−∆)
α/2
u(t)


2
−

N
2
−
c
7
η
2
−c
0

g(t)k∇u(t)k
2
−

Nc
1
2
−
1
η
7


g(−∆)
α
2
u

−

Nc
1
2
−
c
7
c
0
η
2
−
c
10
η
4
−
c
11
η
5

(g∇u),
DOI:10.12677/pm.2022.12101751623nØêÆ
ꩦ§ÜJ
3éÚn8, 9, 10?1nL§¥,·‚¦^^‡(h
2
),(h
3
)9Poincar´eØª.
ŠâP(t)½Â,XJN¿©Œ,K•3˜‡~êγ
0
,¦
d
dt
L(t) ≤−γ
0
P(t).
ØÒü>Óž'utÈ©,
L(t)−L(0)+γ
0
Z
t
0
P(τ)dτ≤0.
duN¿©Œ,…é∀t>0,ÑkL(t) >0.
Z
t
0
P(τ)dτ≤L(0),∀t>0.
ŠâÚn89^‡(h
1
)−(h
2
),(d/dt)P≤0.Œ•
d
dt
[tP(t)] = P(t)+t
d
dt
P(t) ≤P(t),∀t>0.
ØÒü>Óž'utÈ©,Œ
tP(t) ≤
Z
t
0
P(τ)dτ≤L(0),∀t>0,
•3~êC>0¦
P(t)t≤C[P(0)+Q(0)].
4.)š•ê-½
¯¢þ,3½n2Ä:þbPÁؼêg(s)´•êP~,@o¯K(1) 313 !¥•½
Щ^‡Ú>.^‡e,Ù)ØU±•êP~.
Ún11[8]X´˜‡Hilbert˜m, Š^u˜mX¥C
0
Œ+e
At
´•ê-½,…=,
iR⊂ρ(A)…•3M≥1,¦


(iλI−A)
−1


<M,∀λ∈R.
•ò¯K(1))Š^u·ƒ˜mŒ+,·‚Ú\¼ê
η
t
(s) = u(t)−u(t−s),
(19)
DOI:10.12677/pm.2022.12101751624nØêÆ
ꩦ§ÜJ
d(1),(19) Œ





















u
t
−∆u
t
−∆u−G
∞
(−∆)
α
u
+
R
∞
0
g(s)(−∆)
α
η(s)ds+f(u) = h(x),x∈Ω,t>0,
u(x,t) = 0,x∈∂Ω,t≥0,
η
t
(x,s) = 0,x∈∂Ω,s≥0,t≥0,
u(x,0) = u
0
(x),x∈Ω,
η
0
(x,s) = η
0
(x,s),x∈Ω,s≥0.
(20)
Ù¥G
∞
:=
R
∞
0
g(t)dt,Šâ^‡(h
4
)ΥG
∞
≤1−c
3
.
L
2
g

R
+
,D(A
α/2
)

´½Â3g-\˜mþgŒÈ¼ê˜m,ÙSȽÂXe
(η
1
,η
2
)
g
:=
Z
∞
0
g(s)((−∆)
α/2
η
1
(s),(−∆)
α/2
η
2
(s))ds.
…½ÂƒAHilbert˜m
Z:= D(A
1/2
)×H×L
2
g

R
+
,D((A)
α/2
)

.
-v:= u
t
,½Â
U(t) := [u(t),η]
>
,U
0
:= [u
0
,η
0
]
>
∈Z,
·‚ò¯K(20)=z•˜mZ¥Ä–‚5/ª
(
U
t
(t) = AU(t),
U(0) = U
0
.
(21)
‚5ŽfAkXe/ª
A
"
u
η
#
=
"
h−f−
R
∞
0
g(s)(−∆)
α
η(s)ds+G
∞
(−∆)
α
u+∆u+∆v
v−η
s
#
,
Ù¥½Â•L«•
D(A) :=
(
U∈Z:AU∈Z,
Z
∞
0
g(s)(−∆)
α
η(s)ds∈H,
η
s
∈L
2
g

R
+
,D(A
α/2
)

,η(0) = 0
)
.
ŠâÚn11,©•ÄäkDirichlet>.^‡ŽfAÌ,
(
Ae
i
= λ
i
e
i
,
e
i
|
∂Ω
= 0.
(22)
DOI:10.12677/pm.2022.12101751625nØêÆ
ꩦ§ÜJ
…
ke
i
k
L
2
(Ω)
= 1, i≥1.
Ù¥(λ
i
)
i≥1
´˜‡kk•‡)9kAŠª uá4OS;(e
i
)
i≥1
´ÙƒA8˜z
A¼êS.¿…b
λ
1
>G
∞
λ
α
1
.
½n33½n2 be,P ÁØg´•êP~,@o†•§(21) ƒ 'Š^uƒ˜mZ
þŒ+U(t)´š•ê-½.
y²F= [F
1
,F
2
]
>
∈Z,•ÄXe•§
(iλI−A)U= F,λ∈R.
=
(
iλv+f(u)+
R
∞
0
g(s)(−∆)
α
η(s)ds−G
∞
(−∆)
α
u−∆u−∆v= F
1
,
iλη−v+η
s
= F
2
.
(23)
F
1
= 0,F
2
= λ
−α+1−
2
ν
e
−λ
1−
ν
s
e
ν
.…½Â
u= pe
ν
,v= qe
ν
,η(s) = ϕ(s)e
ν
.
Ù¥p,q∈C,ϕ∈L
2
g
(R
+
).dd•§(23)ŒU•







iλpe
ν
−qe
ν
= 0,
iλqe
ν
+λ
ν
pe
ν
−G
∞
λ
α
ν
pe
ν
+
R
∞
0
g(s)λ
α
ν
ϕ(s)dse
ν
+λ
ν
qe
ν
= 0,
iλϕ(s)e
ν
−qe
ν
+ϕ
s
(s)e
ν
= λ
−α+1−
2
e
−λ
1−ε
ν
s
e
ν
.
(24)
d(24)
1
,(24)
2

−λ
2
pe
ν
+λ
ν
pe
ν
−G
∞
λ
α
ν
pe
ν
+
Z
∞
0
g(s)λ
α
ν
ϕ(s)dse
ν
= 0,
-λ=
√
λ
ν
,ÏL(24)
1,2
,Υ
G
∞
p=
Z
∞
0
g(s)ϕ(s)ds.(25)
|^(24)
1
,é~‡©•§(24)
3
?1¦),Œ
ψ(s) = Ce
−i
√
λ
ν
s
+p+
λ
−α+1−
2
ν
iλ
1
2
ν
−λ
1−
ν
e
−λ
1−
ν
s
.(26)
Њη(0) = 0ž,k
C= −p−
λ
−α+1−
2
ν
iλ
1
2
ν
−λ
1−
ν
.
DOI:10.12677/pm.2022.12101751626nØêÆ
ꩦ§ÜJ
d(26)Υ
ϕ(s) = (−p−
λ
−α+1−
2
ν
iλ
1
2
ν
−λ
1−
ν
)e
−i
√
λ
ν
s
+p+
λ
−α+1−
2
ν
iλ
1
2
ν
−λ
1−
ν
e
−λ
1−
ν
s
,(27)
ÏL(25)Ú(27),k
g(s) = e
−γs
,γ∈R
+
.

p=
λ
−α+1−
2
ν
γ+λ
1−
ν
.
Ïd,é?¿α∈[0,1),•3∈(0,1),>α¦
λ
ν
→∞ž,p≈Cλ
−α+1−
2
ν
,
-EÚ^u= pe
ν
,Υ
λ
ν
→∞ž,kuk
D(A
1
2
)
≈λ
−α
2
ν
→∞.
nþ¤ã,(ÜÚn11 ,½ny.
ë•©z
[1]Aifantis,E.C.(1980)OntheProblemofDiffusioninSolids.ActaMechanica,37,265-296.
https://doi.org/10.1007/BF01202949
[2]Aifantis,E.C.(2011)GradientNanomechanics:ApplicationstoDeformation,Fracture,and
DiffusioninNanopolycrystals.MetallurgicalandMaterialsTransactionsA,42,2985-2998.
https://doi.org/10.1007/s11661-011-0725-9
[3]Wang, X., Yang, L.and Zhong,C.K. (2010)Attractorsforthe NonclassicalDiffusionEquations
withFadingMemory.JournalofMathematicalAnalysisandApplications,362,327-337.
https://doi.org/10.1016/j.jmaa.2009.09.029
[4]Wang,X.andZhong,C.K.(2009)AttractorsfortheNon-AutonomousNonclassicalDiffusion
Equations withFading Memory. NonlinearAnalysis:Theory,MethodsApplications, 71,5733-
5746.https://doi.org/10.1016/j.na.2009.05.001
[5]Jaime,E.,Munoz,R.andMaria,G.N.(2003)AsymptoticBehaviorofEnergyforaClassof
WeaklyDissipativeSecond-OrderSystemswithMemory.JournalofMathematicalAnalysis
andApplications,286,692-704.https://doi.org/10.1016/S0022-247X(03)00511-0
[6]Al-Gharabli,M.M.(2019)ArbitraryDecayResultofaViscoelasticEquationwithInfinite
Memory and Nonlinear Frictional Damping. BoundaryValueProblems, 2019, Article No. 140.
https://doi.org/10.1186/s13661-019-1253-6
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[7]Cavalcanti, M.M. and Oquendo, H.P. (2003) Frictional versus Viscoelastic Damping in aSemi-
linearWaveEquation.SIAMJournalonControlandOptimization,42,1310-1324.
https://doi.org/10.1137/S0363012902408010
[8]Pruss, J. (1984) On the Spectrum of C
0
-Semigroups. TransactionsoftheAmericanMathemat-
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