S ong con e gence o solu ions and a ac o s o eac ion-di usion equa ions
go e ned by a ac ional Laplacian
Jiaohui Xu
Cen e o Nonlinea S udies, School o Ma hema ics,
No hwes Uni e si y, Xi’an 710127, P. R. China
Tom´
as Ca aballo1 2
Dp o. Ecuaciones Di e enciales y An´alisis Num´e ico,
Facul ad de Ma em´a icas, Uni e sidad de Se illa, c/ Ta ia s/n, 41012-Se illa, Spain
Depa men o Ma hema ics, Wenzhou Uni e si y,
Wenzhou, Zhejiang P o ince, 325035, P. R. China
Jos´e Vale o
Cen o de In es igaci´on Ope a i a, Uni e sidad Miguel He n´andez de Elche,
A enida de la Uni e sidad s/n, 03202-Elche, Spain
Abs ac
A nonlocal eac ion-di usion equa ion go e ned by a ac ional Laplace ope a o on a bounded
domain is s udied in his pape . Fi s , he s ong con e gence o solu ions o he equa ions go e ned
by ac ional Laplacian o he solu ions o he classical equa ions go e ned by a s anda d Laplace
ope a o is p o ed, when he ac ional pa ame e g ows o 1. Second, o he au onomous case,
he uppe semicon inui y o global a ac o s wi h espec o he a ac o s o he limi p oblem is
es ablished. Appa en ly, hese a e he i s esul s o his kind o p oblems on bounded domains.
Keywo ds: F ac ional Laplacian; S ong con e gence o solu ions; Global a ac o s.
AMS subjec classi ica ions. 35R11, 35A15, 35B41, 35K65
1. In oduc ion
In his pape , we s udy he p oblem







∂u
∂ + (−∆)γu= (u) + h( ),in O× (τ, ∞),
u= 0,on ∂O,
u(x, τ) = uτ(x),in O,
(1)
whe e (−∆)γ,γ∈(0,1), s ands o he ac ional Laplace ope a o ( ac ional Laplacian), Ois a
bounded subse o Rmwi h su icien ly smoo h bounda y, τ∈R,h∈L2
loc(R;L2(O)) and is a
con inuous unc ion sa is ying some app op ia e assump ions as speci ied la e .
Fi s o all, we need o decide which de ini ion o he ac ional Laplacian is mo e app op ia e o
ou analysis. I is well known ha , in he unbounded domain case, namely in Rm, he e a e se e al
equi alen de ini ions o ac ional Laplacian (see, o ins ance, [13] o en equi alen de ini ions),
bu in he bounded domain case, he e exis se e al de ini ions which a e no equi alen . We e e
he eade o he pape s [6, 16, 19] o mo e de ails. We will no discuss abou hem in his pape ,
bu will simply choose one o hose de ini ions o ou in es iga ion, mo e p ecisely he so called
“ egional”de ini ion (see [6]), in such a way ha we a e able o p o e he con e gence o solu ions
o (1) o he co esponding ones o he limi ing equa ion wi h s anda d Laplacian when γ→1−,
ensu ing his con e gence holds in he s ong sense, wha somehow jus i ies he sui abili y and
accu acy o he chosen de ini ion o ac ional Laplacian.
1Co esponding au ho
2E-mail add esses: [email protected] (J. Xu), [email p o ec ed] (T. Ca aballo), j [email p o ec ed] (J. Vale o).
1
2
The p ope ies o solu ions and a ac o s o p oblem (1) when γis ixed ha e been s udied
widely in he li e a u e. Fo example, he well-posedness and he exis ence o global a ac o s (in
bo h de e minis ic and andom se ing, wi h bounded o unbounded domains) o p oblem (1) ha e
been ex ensi ely in es iga ed in he li e a u e (see, e.g., [8, 17, 27, 28, 29, 30, 31] and he e e ences
he ein. Addi ionally, o a s ochas ic e sion o (1), he uppe semicon inui y o he a ac o s
was es ablished in [27, 28, 29] when a pa ame e in he noise e m a ies. I is ema kable ha
he de ini ion o ac ional Laplacian used in hese s ochas ic pape s is exac ly he “ egional”one
we will use in ou cu en analysis, wha ein o ces he idea ha he de ini ion we ha e chosen is
app op ia e.
Ou aim in his manusc ip is o analyze he beha io o solu ions o p oblem (1) when he
pa ame e γgoes o 1−, and he mo i a ion is based on he p ope ies o ac ional Laplace ope a o
(−∆)γas γ→1−, (see, o example, [8] and [21]). In [21, P oposi ion 4.4], he au ho s p o ed ha
(−∆)γu(x) con e ges o −∆u(x) as γ→1 o all x∈Rmwhen u∈C∞
0(Rm). In [8, P oposi ion
2.3], he au ho s p o ed ha , o e e y u∈C∞
0(Rm) and ∈W1,2
0(O), i holds
lim
γ→1−ZO
(−∆)γudx =−ZO
∆udx.
No ice ha in he pape s [3], [4] his ques ion was s udied o an abs ac pa abolic equa ion and
he Sch ¨odinge equa ion. Howe e , as a as we a e awa e, he e a e no esul s in he li e a u e
conce ning he con e gence o solu ions o (1) o he solu ions o he ollowing limi p oblem,







∂u
∂ −∆u= (u) + h( ),in O× (τ, ∞),
u= 0,on ∂O,
u(x, τ) = uτ(x),in O,
(2)
as he pa ame e γgoes o 1−. In [33], we comple ed a i s s ep by p o ing he con e gence o
solu ions wi h espec o he weak opology in L2(Rm) in he case ha he ex e nal unc ion is
sublinea . In his manusc ip , we will go much mo e u he o s udy he con e gence o solu ions
in he s ong opology sense o mo e gene al unc ions . Mo eo e , he uppe semicon inui y o
he global a ac o s will be es ablished as well.
The in e es in s udying p oblem (1) and i s s a iona y e sion comes om di e en ields as has
been desc ibed in ou pape [33] (see also [1, 2, 7, 9, 10, 11, 12, 18, 20, 23, 24, 25, 26, 32, 34, 35]
and he e e ences he ein).
This pape is o ganized as ollows. In Sec ion 2, we ecall he basic de ini ions and p ope ies
conce ning he ac ional Laplace ope a o s, es ablish he se ing o he p oblem we a e ackling
and impose he condi ions ha ensu e he exis ence and uniqueness o weak solu ions o p oblem
(1). In Sec ion 3, we imp o e i s he known esul s abou he con e gence o he ope a o (−∆)γ
by showing ha limγ→1−(−∆)γu=−∆us ongly in Lp((τ, τ +T)×Rm), o any p≥1 and
u∈C∞
0((τ, τ +T)×Rm), whe e τ∈R,T > 0. Also, limγ→1−(−∆)γu=−∆us ongly in
Lp(Rm), o any p≥1 and u∈C∞
0(Rm). A e ha , we p o e he main esul o he pape s a ing
ha he solu ions o p oblem (1) con e ge o he ones o he limi p oblem (2) in C([τ, τ +T]; L2(O)),
o any T > 0 as γ→1−. Finally, in Sec ion 4, we s udy he uppe semicon inui y o he global
a ac o s as he pa ame e γgoes o 1−in he au onomous si ua ion, ha is, when he unc ion h
belongs o L2(O).
3
2. Se ing o he p oblem
We conside he ollowing ac ional eac ion-di usion equa ion,







∂u
∂ + (−∆)γu= (u) + h( ),in O× (τ, ∞),
u= 0,on ∂O,
u(x, τ) = uτ(x),in O,
(3)
whe e (−∆)γ,γ∈(0,1), s ands o he ac ional Laplace ope a o , Ois a smoo h bounded subse
o Rm,τ∈Rand h∈L2
loc(R;L2(O)). Th oughou his pape , we assume ha ∈C(R) and he e
exis posi i e cons an s C ,κ,β1,β2and p≥2, such ha
( (s)− ( ))(s− )≤C (s− )2,∀s, ∈R,(4)
−κ−β1|s|p≤ (s)s≤κ−β2|s|p,∀s∈R.(5)
F om (5), we deduce ha he e exis s a cons an β3>0 such ha
| (s)| ≤ β3(|s|p−1+ 1),∀s∈R.(6)
We ix some µ∈(0, β2) and de ine he unc ion (u) = (u) + µu. Then, he equa ion in (3)
eads as ∂u
∂ + (−∆)γu+µu = (u) + h( ).(7)
I ollows om (5), (6) and he Young inequali y ha
(s)s≤κ−β2|s|p+µs2≤κ−β2|s|p,(8)
| (s)| ≤ β3(|s|p−1+ 1),(9)
o some posi i e cons an s κ, β2, β3.
Le Sdeno e he Schwa z space o apidly decaying C∞ unc ions on Rm. The ac ional Laplace
ope a o (−∆)γo u∈ S a poin x, o 0 < γ < 1, is de ined by,
(−∆)γu(x) = −1
2C(m, γ)ZRm
u(x+y) + u(x−y)−2u(x)
|y|m+2γdy, x ∈Rm,(10)
whe e C(m, γ) is he ollowing posi i e cons an
C(m, γ) = γ4γΓ(m+2γ
2)
πm
2Γ(1 −γ).(11)
I is well known [21] ha o any u∈ S, we ha e
(−∆)γu=F−1|ξ|2γFu,
whe e Fis he Fou ie ans o m gi en by
Fu=1
(2π)m/2ZRm
e−ix·ξu(x)dx,
and F−1is he in e se Fou ie ans o m. Fo 0 < γ < 1, we de ine he ac ional Sobole space
Wγ,2(Rm) := Hγ(Rm) by:
Hγ(Rm) = u∈L2(Rm) : ZRmZRm
|u(x)−u(y)|2
|x−y|m+2γdxdy < ∞,
endowed wi h he no m
kukHγ(Rm)=ZRm
|u(x)|2dx +ZRmZRm
|u(x)−u(y)|2
|x−y|m+2γdxdy1
2
.
4
In wha ollows, we deno e by k · kp he no m in Lp(Rm) o p > 2,whe eas by k · k and (·,·),
we deno e he no m and he inne p oduc o L2(Rm), espec i ely. By abusing o he no a ion,
(·,·) will be also used o he pai ing be ween Lq(Rm) and Lp(Rm). Mo eo e , we will conside he
Gaglia do semi-no m o Hγ(Rm),deno ed by k·k ˙
Hγ(Rm),which is gi en by,
kuk2˙
Hγ(Rm)=ZRmZRm
|u(x)−u(y)|2
|x−y|m+2γdxdy, u ∈Hγ(Rm).
The e o e, kuk2
Hγ(Rm)=kuk2+kuk2˙
Hγ(Rm) o all u∈Hγ(Rm). No ice ha Hγ(Rm) is a Hilbe
space wi h inne p oduc ,
(u, )Hγ(Rm)=ZRm
u(x) (x)dx +ZRmZRm
(u(x)−u(y))( (x)− (y))
|x−y|m+2γdxdy,
o u, ∈Hγ(Rm).Thanks o [21], he no m kukHγ(Rm)is equi alen o kuk2+k(−∆)γ
2uk21
2 o
u∈Hγ(Rm). Namely, i ollows ha
kuk2
Hγ(Rm)=kuk2+2
C(m, γ)k(−∆)γ
2uk2,∀u∈Hγ(Rm).
Since he ac ional Laplace ope a o (−∆)γgi en by (10) is nonlocal, we he e in e p e he homo-
geneous Di ichle bounda y p oblem (3) as ollows,







∂u
∂ + (−∆)γu= (u) + h( ),in O× (τ, ∞),
u= 0,on Rm O,
u(x, τ) = uτ(x),in O.
(12)
Na u ally, he limi p oblem (2) will be w i en as







∂u
∂ −∆u= (u) + h( ),in O× (τ, ∞),
u= 0,on Rm O,
u(x, τ) = uτ(x),in O,
(13)
so ha hey keep o mally consis en . Thus, we de ine he spaces
Vγ={u∈Hγ(Rm) : u= 0 a.e. on Rm O},
H={u∈L2(Rm) : u= 0 a.e. on Rm O},
V={u∈H1(Rm) : u= 0 a.e. on Rm O},
wi h hei dual spaces deno ed by V∗
γ,H∗and V∗, espec i ely. Mo eo e , we will use
kuk2
Vγ=kuk2+k(−∆)γ
2uk2,
o deno e he no m o space Vγ. The no m in Vwill be deno ed by k·kV.
Fu he mo e, le b:Vγ×Vγ→Rdeno e a bilinea o m gi en by,
bγ( 1, 2) = µ( 1, 2) + 1
2C(m, γ)ZRmZRm
( 1(x)− 1(y))( 2(x)− 2(y))
|x−y|m+2γdxdy, 1, 2∈Vγ,(14)
whe e C(m, γ) is he same as in (11). Fo con enience, we associa e wo ope a o s Aγ,Aγ:Vγ→V∗
γ
wi h b, such ha
< Aγ( 1), 2>(V∗
γ,Vγ)=µ( 1, 2)+ < Aγ( 1), 2>(V∗
γ,Vγ)=bγ( 1, 2),(15)
5
o all 1, 2∈Vγ,whe e <·,·>(V∗
γ,Vγ)is he duali y pai ing o V∗
γand Vγ. I is known [21] ha
he embeddings V⊂Vγ2⊂Vγ1a e con inuous o 0 < γ1≤γ2<1. Iden i ying Hwi h i s dual
space, we ob ain he chain o con inuous embeddings
V⊂Vγ⊂H⊂V∗
γ⊂V∗,∀γ∈(0,1) .
Pai ing be ween spaces Vand V∗will be deno ed by <·,·>(V∗,V ).
We conside wi hou loss o gene ali y ha h∈L2
loc(R;H) by se ing h( , x) = 0 o x∈Rm O.
De ini ion 1. Le τ∈R,uτ∈Hand γ∈(0,1). The unc ion u∈C([τ, ∞); H)is said o
be a weak solu ion o p oblem (12) i u(τ) = uτ,u∈L2
loc(τ, ∞;Vγ)∩Lp
loc(τ, ∞;Lp(Rm)),du
d ∈
L2
loc(τ, ∞;V∗
γ) + Lq
loc(τ, ∞;Lq(Rm)) and
d
d (u, ξ) + 1
2C(m, γ)ZRmZRm
(u( , x)−u( , y))(ξ(x)−ξ(y))
|x−y|m+2γdxdy
=ZO
( (u( , x)) + h( , x)) ξ(x)dx,
o any ξ∈Vγ∩Lp(Rm), in he sense o scala dis ibu ions in (τ, ∞).
In [27, Theo em 2.3], one can see ha p oblem (12) possesses a unique weak solu ion o any
uτ∈Hand γ∈(0,1). In addi ion, his solu ion is con inuous wi h espec o he ini ial da um uτ
in H, and sa is ies he ene gy equali y,
d
d kuk2+C(m, γ)kuk2˙
Hγ(Rm)= 2 ZO
( (u( , x)) + h( , x)) u( , x)dx, o a.a. ≥τ.
De ini ion 2. Le τ∈Rand uτ∈H. The unc ion u∈C([τ, ∞); H)is said o be a weak
solu ion o p oblem (13) i u(τ) = uτ,u∈L2
loc(τ, ∞;V)∩Lp
loc(τ, ∞;Lp(O)),du
d ∈L2
loc(τ, ∞;V∗) +
Lq
loc(τ, ∞;Lq(O)) and
d
d (u, ξ) + ZO
∇u·∇ξdx =ZO
( (u( , x)) + h( , x)) ξ(x)dx,
o any ξ∈V∩Lp(O), in he sense o scala dis ibu ions in (τ, ∞).
Recall ha (see, e.g., [5] o [22]) p oblem (2) possesses a unique weak solu ion o any uτ∈H,
which is con inuous wi h espec o he ini ial da um uτin H.
3. Con e gence o solu ions
Ou aim now is o p o e ha he solu ions o p oblem (12) con e ge, as γ→1−, o he unique
solu ion o he limi p oblem wi h γ= 1, ha is, o he unique solu ion o he s anda d eac ion-
di usion equa ion (13). I is well known [21, P oposi ion 4.4] ha i u∈C∞
0(Rm), hen
lim
γ→1−(−∆)γu(x) = −∆u(x), o all x∈Rm. (16)
Le C∞
0(X) be he space o in ini ely di e en iable unc ions u:X→Rwi h compac suppo ,
whe e Xis a Banach space. Deno e by D0 he space o dis ibu ions o D=C∞
0((τ, τ +T)×Rm)
and by <·,·>(D0,D) he duali y pai ing o D0and D. Deno e by BR he ball in Rmcen e ed a 0
wi h adius R.
Now, we ecall a esul p o ed in [33] which ex ends he con e gence esul (16) o a mo e gene al
one.

6
Lemma 3. (See [33, Lemma 3.1]) Fo any τ∈R,T > 0and u∈C∞
0((τ, τ +T)×Rm), he ollowing
s a emen holds,
lim
γ→1−(−∆)γu=−∆u, s ongly in Lp((τ, τ +T)×Rm),∀p≥1.
In pa icula , limγ→1−(−∆)γu=−∆uin he sense o dis ibu ions in D0.
Now, in a simila way we s a e he ollowing esul .
Lemma 4. (See [33, Lemma 3.2]) Fo any u∈C∞
0(Rm),limγ→1−(−∆)γu=−∆us ongly in
Lp(Rm) o any p≥1.
We also need he ollowing echnical lemma which is c ucial o ou analysis.
Lemma 5. (i) Fo any γ∈1
2,1and ∈Vγ, we ha e
k kV1
2
≤ 3
2k kV γ.(17)
(ii) The e exis s a posi i e cons an Ksuch ha , o any ∈Vγ,

Aγ 
V∗≤Kk kV γ, o any γ∈(0,1) .(18)
P oo . (i) Since
1
2<1
2γ<1,0<1−1
2γ<1
2.
By he Fou ie ans o m and Young inequali y, he es ima e (17) ollows om
k k2
V1
2
=k k2+k(−∆)1
4 k2=ZRm
(1 + |ξ|)|F (ξ)|2dξ
≤ZRm2−1
2γ+1
2γ|ξ|2γ|F (ξ)|2dξ
≤3
2ZRm1 + |ξ|2γ|F (ξ)|2dξ =3
2k k2
Vγ.
(i) Fo he second s a emen , le us i s p o e he exis ence o posi i e cons an Ksuch ha


(−∆)γ
2u

≤KkukV,∀u∈V. (19)
On he one hand, by Fubini’s Theo em, we ha e
ZRmZ|x−y|<1
|u(x)−u(y)|2
|x−y|m+2γdxdy ≤ZRmZB1
|u(y+z)−u(y)|2
|z|2|z|m+2(γ−1) dzdy
≤ZRmZB1
1
|z|m+2(γ−1) Z1
0
|∇u(y+ z)|d 2
dzdy
≤ZB1
1
|z|m+2(γ−1) Z1
0ZRm
|∇u(y+ z)|2dyd dz
≤ kuk2
VZB1
1
|z|m+2(γ−1) dz
≤ kuk2
Vωm−1Z1
0
1
ρ2γ−1dρ =ωm−1
2 (1 −γ)kuk2
V.
7
On he o he hand, he change o a iable implies
ZRmZ|x−y|≥1
|u(x)−u(y)|2
|x−y|m+2γdxdy ≤4ZRmZ|x−y|≥1
|u(y)|2
|x−y|m+2γdxdy
= 4 kuk2Z|z|≥1
1
|z|m+2γdz
≤4kuk2ωm−1Z∞
1
1
ρ2γ+1 dρ
=2ωm−1
γkuk2
V.
Thus, combining he abo e wo es ima es, we ob ain


(−∆)γ
2u


2=C(m, γ)
2kuk2˙
Hγ(Rm)
=C(m, γ)
2ZRmZRm
|u(x)−u(y)|2
|x−y|m+2γdxdy
≤ωm−1C(m, γ)
4 (1 −γ)+C(m, γ)
γkuk2
V
≤K2kuk2
V,
o some cons an s K > 0, whe e he las inequali y ollows om he ac
lim
γ→0+
C(m, γ)
γ= lim
γ→0+
C(m, γ)
γ(1 −γ)=2
ωm−1
,
see [21, Co olla y 4.2] o mo e de ails. Immedia ely, by he Young inequali y and (19), we ind

Aγ 
V∗= sup
u∈V, kukV≤1< Aγ( ), u >(V∗,V )
= sup
u∈V, kukV≤1< Aγ( ), u >(V∗
γ,Vγ)
= sup
u∈V, kukV≤1
1
2C(m, γ)ZRmZRm
( (x)− (y))(u(x)−u(y))
|x−y|m+2γdxdy
≤sup
u∈V, kukV≤1
1
2C(m, γ)k k˙
Hγ(Rm)kuk˙
Hγ(Rm)
= sup
u∈V, kukV≤1

(−∆)γ
2 



(−∆)γ
2u


≤Ksup
u∈V, kukV≤1

(−∆)γ
2 

kukV≤K

(−∆)γ
2 

≤Kk kV γ.
We inish he p oo o his lemma. 
We can now es ablish he main esul o his pape abou he con e gence o solu ions.
Theo em 6. Le un
τ→uτin H, le un(·)be he solu ion o p oblem (12) o γ=γnwi h ini ial
alue un
τ. Then un→uin C([τ, τ +T]; H)as n→ ∞ o any T > 0, whe e u(·)is he unique
solu ion o p oblem (13). I un
τ→uτweakly in H, hen un→uin C([τ+ε, τ +T]; H)as n→ ∞
o any 0< ε < T.
8
P oo . Conside a sequence γn→1−. Le un(·) be he unique solu ion o p oblem (12) wi h ini ial
alue un
τ. Mul iplying equa ion (7) by un, using he Young inequali y and on accoun o (8), we
de i e
1
2
d
d kunk2+µkunk2+1
2C(m, γn)kunk2˙
Hγn(Rm)
≤κ−β2kunkp
p+kh( )k kunk
≤κ−β2kunkp
p+1
2µkh( )k2+µ
2kunk2,(20)
which is equi alen o
d
d kunk2+µkunk2+C(m, γn)kunk2˙
Hγn(Rm)+ 2β2kunkp
p≤2κ+1
µkh( )k2.(21)
Mul iplying (21) by eµs and in eg a ing i o e he in e al (τ, ), we deduce
kun( )k2≤ kun
τk2e−µ( −τ)+2κ
µ1−e−µ( −τ)+1
µZ
τ
e−µ( −s)kh(s)k2ds. (22)
Then o T > 0, we in e ha
sup
∈[τ,τ+T]
kun( )k2≤MT,(23)
whe e MTdepends on T, and
Zτ+T
τC(m, γn)kun(s)k2˙
Hγn(Rm)+ 2β2kun(s)kp
pds ≤MT.(24)
The e o e, he sequence {un}is bounded in L∞(τ, τ +T;H)∩Lp(τ, τ +T;Lp(Rm)). By (23) and
(24), we ha e
Zτ+T
τ

(−∆)γn
2un(s)


2ds ≤Zτ+T
τ
C(m, γn)kun(s)k2˙
Hγn(Rm)ds ≤MT,(25)
Zτ+T
τ
kun(s)k2
Vγnds ≤MT+TMT.(26)
Hence, in iew o (26) and Lemma 5(i), he e exis s n0>0, such ha γn∈[1
2,1) o n≥n0. Then,
we ha e
Zτ+T
τ
kunk2
V1
2
ds ≤3
2MT+TMT,
which implies {un}is bounded in L2(τ, τ +T;V1
2). Also, (26) and (18) imply ha {Aγn(un)}is
bounded in L2(τ, τ +T;V∗). Mo eo e , by (24) and (9), we in e ha { (un)}is bounded in
Lq(τ, τ +T;Lq(Rm)). The e o e, {dun
d }is bounded in Lq(τ, τ +T;Lq(Rm))+L2(τ, τ +T;V∗). Then
he e exis unc ions u, χ, ξ such ha , up o a subsequence which we elabel he same, we deduce
he ollowing con e gences,
un→uweak-s a in L∞(τ, τ +T;H),(27)
un→uweakly in L2(τ, τ +T;V1
2),(28)
un→uweakly in Lp(τ, τ +T;Lp(Rm)),(29)
(un)→χweakly in Lq(τ, τ +T;Lq(Rm)),(30)
Aγn(un)→ξweakly in L2(τ, τ +T;V∗),(31)
dun
d →du
d weakly in Lq(τ, τ +T;Lq(Rm)) + L2(τ, τ +T;V∗).(32)
9
Since he embedding V1
2⊂His compac and he embedding H⊂(V∩Lp(Rm))∗is con inuous, a
s anda d Compac ness Theo em [22, Theo em 8.1] implies ha
un→us ongly in L2(τ, τ +T;H),(33)
un( , x)→u( , x) o a.a. ( , x)∈(τ, τ +T)× O.(34)
F om he con inui y o and [15, Lemma 1.3], we also ob ain ha χ= (u).
Fu he , we need o p o e ha ξ=−∆u. By Lemma 3, o a bi a y ϕ∈ D, we ha e
< Aγn(un), ϕ >(D0,D)=Zτ+T
τ
< Aγn(un), ϕ >(V∗,V )d
=Zτ+T
τ
< un, Aγn(un)>(V,V ∗)d =Zτ+T
τZRm
un(−∆)γnϕdxd
γn→1−
−→ Zτ+T
τZRm
u(−∆)ϕdxd =Zτ+T
τ
< u, −∆ϕ >(V,V ∗)d
=<−∆u, ϕ >(D0,D),
he abo e con e gence holds hanks o he ac s ha un→uin L2(τ, τ+T;H) and (−∆)γnϕ→ −∆ϕ
in L2(τ, τ +T;H). Consequen ly, Aγn(un)→ −∆uas γn→1−in he dis ibu ional sense, which
implies ha ξ=−∆u.
Thus, o any η∈L2(τ, τ +T;V)∩Lp(τ, τ +T;Lp(Rm)), he abo e con e gences yield ha
0 = Zτ+T
τ
<dun
d , η >(V∗+Lq(Rm),V ∩Lp(Rm)) d +Zτ+T
τ
< Aγn(un), η >(V∗,V )d
−Zτ+T
τZO
( (un) + h)ηdxd
−→ Zτ+T
τ
<du
d , η >(V∗+Lq(Rm),V ∩Lp(Rm)) d +Zτ+T
τ
<−∆u, η >(V∗,V )d
−Zτ+T
τZO
( (u) + h)ηdxd .
Mo eo e ,
u∈L∞(τ, τ +T;H)∩Lp(τ, τ +T;Lp(Rm)),
u∈L2(τ, τ +T;V1
2),
du
d ∈Lq(τ, τ +T;Lq(Rm)) + L2(τ, τ +T;V∗),
−∆u∈L2(τ, τ +T;V∗).
Since −∆u( )∈V∗, we de i e ha u( )∈V o a.a. ∈(τ, τ +T). Hence,
k∇u( )k2=<−∆u( ), u( )>(V∗,V )≤ k∆u( )kV∗k∇u( )k,
and, consequen ly,
u∈L2(τ, τ +T;V).
The e o e, u(·) is he weak solu ion o p oblem (13).
Since e e y con e ging sequence has he same limi , i ollows ha he con e gences (27)-(34)
hold o he whole sequence. Also, we deduce om (23) ha o n, 0∈[τ, τ +T] such ha n→ 0,
we ha e
un( n)→u( 0) weakly in H. (35)