An inverse problem for semilinear equations involving the fractional Laplacian
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An in e se p oblem o semilinea equa ions in ol ing he ac ional Laplacian
© 2023 IOP Publishing
Accep ed e sion (Final d a )
Kow, Pu-Zhao; Ma, Shiqi; Sahoo, Suman Kuma
Kow, P.-Z., Ma, S., & Sahoo, S. K. (2023). An in e se p oblem o semilinea equa ions in ol ing
he ac ional Laplacian. In e se P oblems, 39(9), A icle 095006. h ps://doi.o g/10.1088/1361-
6420/ace9 4
2023
AN INVERSE PROBLEM FOR SEMILINEAR EQUATIONS INVOLVING
THE FRACTIONAL LAPLACIAN
PU-ZHAO KOW, SHIQI MA, AND SUMAN KUMAR SAHOO
Abs ac . Ou wo k conce ns he s udy o in e se p oblems o hea and wa e equa ions
in ol ing he ac ional Laplacian ope a o wi h ze o h o de nonlinea pe u ba ions. We
eco e nonlinea e ms in he semilinea equa ions om he knowledge o he ac ional
Di ichle - o-Neumann ype map combined wi h he Runge app oxima ion and he unique
con inua ion p ope y o he ac ional Laplacian.
1. In oduc ion and main esul s
We in es iga e in e se p oblems o hea and wa e equa ions in ol ing he ac ional
Laplacian ope a o wi h ze o h o de nonlinea pe u ba ions. The s udy o in e se p ob-
lems in ol ing he ac ional Laplace began wi h he wo k [GSU20] by Ghosh, Salo and
Uhlmann. In [GSU20], hey p oposed and p o ed a Calde ´on ype in e se p oblem o a
linea ac ional Laplace ope a o . The Calde ´on p oblem was ini ia ed by Calde ´on in his
wo k [Cal06] o non- ac ional Laplace equa ions. The e is ample amoun o li e a u e a ail-
able on he non- ac ional Calde ´on p oblem and we e e he eade s o he su ey [Uhl09].
The key ool o s udying ac ional ype o in e se p oblems is he Runge app oxima ion
p ope y, which is a consequence o he ac ional unique con inua ion p ope y ( UCP),
i.e. i u= (−∆)su= 0 in ce ain open se , hen u= 0 e e ywhe e. U ilizing hese ools,
in e se p oblems in ol ing ac ional ope a o s ha e been g ea ly in es iga ed by nume ous
au ho s in ecen yea s. We e e eade s o [GRSU20,LL22a,LO22,Li21,Lin20,LL22b] o
some ecen wo ks in ol ing in e se p oblems o ac ional semilinea ellip ic equa ions.
Compa ed o he s udy o in e se p oblems in ol ing ac ional o de ope a o s, he
s udy o in e se p oblems in ol ing nonlinea e ms goes back o Isako [Isa01] and has
been unde ex ensi e s udy in he li e a u e. In [Isa01] he s udied he nonlinea in e se
p oblems o ellip ic and pa abolic equa ions using i s o de linea iza ion echniques. In
[LLLS21] he au ho s success ully implemen ed highe o de linea iza ion echniques o sol e
in e se p oblems o ellip ic equa ions in ol ing powe ype nonlinea i y. In he highe
o de linea iza ion, he idea is o use p oduc o he solu ions o “ ee equa ion” ∆u= 0
(i.e. only p incipal ope a o , no lowe o de e m is a ached). I was obse ed ha using
non-linea i y as a ool one can sol e ce ain in e se p oblems which a e no a ailable o
linea case. The me hod was also used o sol e se e al nonlinea in e se p oblems including
pa ial da a [KU20b,KU20a,HL22] and Riemannian mani olds [FO20,FLL21,LLST22].
In e se p oblems ela ed o mo e gene al nonlinea i ies we e e [CFK+21,MU20] and he
e e ences ci ed he e.
The s udy o in e se p oblems ela ed o semilinea wa e equa ions wi h quad a ic non-
linea i y s a ed wi h he undamen al wo k [KLU18] by Ku yle , Lassas and Uhlmann.
In [KLU18] he au ho s used p opaga ion o non-linea in e ac ion o non smoo h plane
wa es ha ing cono mal singula i ies. Then in [FO22] au ho s used wa e packe (some imes
2020 Ma hema ics Subjec Classi ica ion. 35R11, 35R30, 46T20.
Key wo ds and ph ases. ac ional Laplacian, ac ional Calde ´on p oblem, nonlocal semilinea equa ions,
ac ional di usion equa ion, ac ional wa e equa ion, Runge app oxima ion.
1
SEMILINEAR EQUATION FRACTIONAL LAPLACIAN 2
i is also called quasimode cons uc ion) cons uc ion o sol e ce ain non-linea hype bolic
in e se p oblems. This helps o a oid he need o use mic olocal analysis echniques. Fo a
compa ison be ween hese wo me hods men ioned abo e we e e [HUZ21]. In e se p ob-
lems o nonlinea pa abolic equa ions ha e been well s udied. We e e [LOST22,FKU22]
and he e e ences he ein o mo e esul s.
Mo i a ed by he wo ks men ioned abo e, in his a icle we conside an in e se p oblem
o nonlinea ac ional pa abolic equa ions. F ac ional pa abolic equa ions ha e applica-
ions in andom p ocesses [BBCK09]. We s udy he ac ional ype hea equa ions as well
as he ac ional ype wa e equa ions, and we s a wi h he hea equa ion i s .
Le n≥1 be a non-nega i e in ege and 0 < s < 1. Le Ω be a bounded Lipschi z domain
in Rnand Ωe:= Rn Ω. Le Wbe any bounded Lipschi z domain in Ωe. Le u=u( , x)
sa is y he ollowing ac ional di usion equa ion wi h nonlinea e m q=q( , x, z):
∂ u( , x) + (−∆)su( , x) + q( , x, u( , x)) = 0 in ΩT≡(0, T)×Ω,
u( , x) = ( , x) in Ωe
T≡(0, T)×Ωe,
u(0, x) = 0 ∀x∈Ω,
(1.1)
o ce ain app op ia e ex e io da a = ( , x)∈ C∞
c(WT), whe e WT:= (0, T)×W
and C∞
c(·) deno es he space o smoo h compac ly suppo ed unc ions on hei domain
o de ini ion. He e, he ac ional Laplacian (−∆)sis de ined ia he Fou ie ans o m:
F((−∆)s )(ξ) := |ξ|2sˆ (ξ) o all ξ∈Rn,whe e ˆ =F is he Fou ie ans o m o
dis ibu ion . Gi en any open se s Vand Win Ωe, we de ine he DN-map co esponding
o (1.1) as ollows:
Λhea
q( ) := (−∆)suVT o all “su icien ly small” ∈ C∞
c(WT),(1.2)
whe e uis he unique solu ion o (1.1), see P oposi ion 2.10. We now s a e he assump ions
on he coe icien unde which we s a e and p o e ou main esul s.
Assump ions 1.1. Le Ck(·)be he space o k- imes con inuously di e en iable unc ions
o all in ege s k≥0. Assume ha he unc ion q( , x, z)sa is ies ollowing condi ions.
(Q.1) Fo each ( , x)∈(0, T)×Ω, he mapping z7→ q( , x, z)is in Cm+1((−δ, δ)).
(Q.2) q( , x, 0) = 0 o all ( , x)∈ΩT.
(Q.3) The e exis s a non-dec easing unc ion Φ : (−δ, δ)→R+such ha
sup
( ,x)∈ΩT,|z|≤ǫ
|∂zq( , x, z)| ≤ Φ(ǫ)
o all 0< ǫ < δ and limǫ→0Φ(ǫ) = 0.
(Q.4) Gi en any k= 2,3,··· , m + 1, he e exis s Mk(depending on k) such ha
sup
( ,x)∈ΩT,|z|≤δ
|∂k
zq( , x, z)| ≤ Mk.(1.3)
Wi h hese assump ions on he coe icien , he ollowing is ou i s main esul :
Theo em 1.1 (Global uniqueness om DN-map).Choose any n∈Nand 0< s < 1. Le
Ω⊂Rnbe a bounded Lipschi z domain. Le W, V ⊂Ωebe any open se s, bo h wi h Lipschi z
bounda y, sa is ying V∩Ω = ∅and W∩Ω = ∅. Fix an in ege m≥2and a posi i e numbe
δ > 0. Assume ha each qj(j= 1,2) sa is ies (Q.1)-(Q.4). Then he e exis s a cons an
˜ǫ0= ˜ǫ0(n, s, Ω, T, δ)such ha , i
Λhea
q1( ) = Λhea
q2( ) o all ∈ C∞
c(WT)sa is ying k kex ≤˜ǫ0,
whe e he no m k · kex is de ined in (2.2)below, hen we ha e
∂k
zq1( , x, 0) = ∂k
zq2( , x, 0) ∀( , x)∈ΩT, k = 0,1,2,··· , m. (1.4)
SEMILINEAR EQUATION FRACTIONAL LAPLACIAN 3
Addi ionally, i we assume z7−→ q( , x, z)is analy ic o ( , x)∈ΩT, hen we ha e
q1( , x, z) = q2( , x, z)∀( , x)∈ΩT,∀z∈I.
Following he ideas om [GRSU20], one can s eng hen abo e esul and eco e he
coe icien s based on a ini e dimensional da a se . Ou nex co olla y is ela ed o a single
measu emen esul o linea ac ional Laplace equa ion, which can be p o ed by examining
ca e ully he p oo o Theo em 1.1.
Co olla y 1.2 (Reco e y o m-je om m-dimensional measu emen s).Suppose he as-
sump ions in Theo em 1.1 hold. We u he assume ha o j= 1,2
∂k
zqj(·,0) ∈ C0(Ω) is independen o ime a iable .
Fix any g1,··· , gm∈ C∞
c(WT)such ha g1( 0,·),··· , gm( 0,·)6≡ 0 o some 0∈(0, T).
Then Λhea
q1(ǫ1g1+···+ǫmgm) = Λhea
q2(ǫ1g1+···+ǫmgm), o all su icien ly small ǫj>0
(j= 1,··· , m), implies (1.4).
In his a icle, we also ake in o conside a ion a nonlinea in e se p oblem o ac ional
wa e equa ions in one spa ial dimension. Le u=u( , x) sa is y
∂2
u( , x) + (−∆)su( , x) + q( , x, u( , x)) = 0 in ΩT,
u( , x) = ( , x) in Ωe
T,
u(0, x) = ∂ u(0, x) = 0 o all x∈Ω,
(1.5)
o ce ain app op ia e ex e io da a. We can de ine he ollowing hype bolic DN-map
co esponding o (1.5) as ollows:
Λwa e
q( ) := (−∆)suVT o all “su icien ly small” ∈ C∞
c(WT),
whe e uis he unique solu ion o (1.5), see P oposi ion 5.4 o he well-posedness. The
ollowing esul can be p o ed adap ing he simila ideas:
Theo em 1.3 (Global uniqueness om DN-map).Le n= 1 and 1/2< s < 1. Le
Ω⊂Rbe a bounded open se , le W, V ⊂Ωebe any open se s sa is ying V∩Ω = ∅
and W∩Ω = ∅. Fix any in ege m≥2and a posi i e numbe δ > 0. Assume ha qj
(j= 1,2) sa is y (Q.1)–(Q.4). Then he e exis s a cons an ˜ǫ0= ˜ǫ0(s, Ω, T, δ)such ha , i
Λwa e
q1( ) = Λwa e
q2( ) o all ∈ C∞
c(WT)sa is ying (2.2), hen we ha e (1.4). Addi ionally,
i we assume z:→q( , x, z)is analy ic o ( , x)∈ΩT hen we ha e
q1( , x, z) = q2( , x, z)∀( , x)∈ΩT,∀z∈I.
The nex co olla y is analogous o Co olla y 1.2.
Co olla y 1.4 (Reco e y o m-je om m-dimensional measu emen s).Suppose he as-
sump ions in Theo em 1.3 hold. We u he assume ha
∂k
zqj(·,0) ∈ C0(Ω) is independen o ime a iable .
Fix any g1,··· , gm∈ C∞
c(WT)such ha g1( 0,·),··· , gm( 0,·)6≡ 0 o some 0∈(0, T).
I Λwa e
q1(ǫ1g1+···+ǫmgm) = Λwa e
q2(ǫ1g1+···+ǫmgm) o all su icien ly small ǫj>0
(j= 1,··· , m), hen we conclude (1.4).
The e a e only a ew wo k a ailable in he li e a u e abou he in e se p oblems o
ac ional hea equa ions as well as ac ional wa e equa ions. To mo i a e ou wo k, we
men ion se e al closely ela ed ones. In [Li21], he au ho sol ed ce ain in e se p oblems
o ac ional ype hea ope a o s, howe e he assump ions on he nonlinea e m in [Li21]
a e di e en om ou s. Then in [KLW21], he au ho s s udied an in e se p oblem in ol -
ing ac ional wa e equa ion, while in [LLL21], he au ho s sol ed an in e se p oblem o
hype bolic sys ems.
SEMILINEAR EQUATION FRACTIONAL LAPLACIAN 4
The es o he pape is o ganized as ollows. We discuss he o wa d p oblem o he
ac ional di usion equa ion in Sec ion 2. We p o e a Runge app oxima ion o he ac ional
di usion equa ion in Sec ion 3. Wi h hese ools a hand, Sec ion 4is dedica ed o he p oo
o Theo em 1.1. Finally, we in es iga e Theo em 1.3 in Sec ion 5. To make ou pape sel -
con ained, we also p esen he p oo o he well-posedness o he linea ac ional di usion
equa ion (P oposi ion 2.2) in Appendix A. Then in Appendix Bwe discuss he issue o
conside ing Theo em 1.3 in one spa ial dimension.
2. The o wa d p oblem o he ac ional di usion equa ion
In his sec ion, we p o e se e al p elimina ies ha will be use ul in his wo k.
2.1. F ac ional Sobole spaces. We use no a ions o ac ional Sobole spaces as in
[KLW21]. To make he pape sel -con ained, we gi e b ie in oduc ions o hem. Fo
α∈R, deno e as Hα(Rn) he s anda d L2-based ac ional Sobole spaces, which is de ined
ia Fou ie ans o m [DNPV12,Kwa17,S e16]. Fo s∈(0,1), in ac
Hs(Rn) = u∈L2(Rn)
|u(x)−u(y)|
|x−y|n
2+s∈L2(Rn×Rn)(as se s)
wi h equi alen no m: kuk2
Hs(Rn)=kuk2
L2(Rn)+ [u]2˙
Hs(Rn),whe e
[u]2˙
Hs(Rn)=ZZRn×Rn
|u(x)−u(y)|2
|x−y|n+2sdxdy. (2.1)
He e, (2.1) is called he A onszajn-Gaglia do-Slobodeckij semino m, see [DNPV12, equa-
ion (2.2)] o e e ence.
Le Obe any open se in Rn, and le α∈R. We de ine he ollowing Sobole spaces:
Hα(O) := {u|Ou∈Hα(Rn)},˜
Hα(O) := closu e o C∞
c(O) in Hα(Rn)
Hα
0(O) := closu e o C∞
c(O) in Hα(O), Hα
O:= {u∈Hα(Rn)supp (u)⊂O}.
The Sobole space Hα(O) is comple e unde he quo ien no m
kukHα(O):= in k kHα(Rn) ∈Hα(Rn) and |O=u.
I is easy o see ha ˜
Hα(O)⊂Hα
0(O), and ha Hα
Ois a closed subspace o Hα(Rn). I Ω
is a bounded Lipschi z domain, hen we also ha e ollowing iden i ica ions (wi h equi alen
no ms):
(˜
Hα(Ω) = Hα
Ω,(Hα
Ω)′=H−α(Ω) and (Hα(Ω))′=H−α
Ω∀α∈R,
Hs(Ω) = Hs
Ω=Hs
0(Ω) ∀ − 1/2< s < 1/2,
see e.g. [GSU20, Sec ion 2A], [McL00, Chap e 3], and [T i02]. Nex ollowing [E a10,
Chap e 5], we de ine ime dependen ac ional Sobole space o all in ege s p≥1 deno ed
by Lp((0, T); Hs). Then he ex e io no m o ∈ C∞
c(WT) is gi en by
k k2
ex := k k2
L∞(0,T;Hs(Rn))∩L∞(Rn
T)+k(−∆)s k2
L2(ΩT).(2.2)
Mo eo e , o any measu able se A⊂Rnwe use he ollowing no a ions:
( , g)L2(A):= ZA
g dx, (F, G)L2(AT):= ZT
0ZA
FG dxd .
SEMILINEAR EQUATION FRACTIONAL LAPLACIAN 5
2.2. Well-posedness o he linea equa ion. We s a e he well-posedness o he linea
ac ional di usion equa ion. Le T > 0, s∈(0,1), and a=a( , x)∈L∞(ΩT), and we
conside he ollowing ini ial-ex e io alue p oblem:
(∂ + (−∆)s+a)u=Fin ΩT,
u= in Ωe
T,
u=ϕin {0} × Rn,
(2.3)
whe e ∈ C∞
c(WT) o some open se wi h Lipschi z bounda y W⊂Ωesa is ying W∩Ω = ∅,
and ϕ∈˜
H0(Ω) = ϕ∈L2(Rn)supp ϕ⊂Ω. Se ing := u− , we hen conside he
ollowing linea equa ion wi h ze o ex e io da a:
(∂ + (−∆)s+a) =˜
Fin ΩT,
= 0 in Ωe
T,
=ϕin {0} × Rn,
(2.4)
whe e ˜
F=F−(−∆)s . Now i su ices o s udy he well-posedness o (2.4).
De ine unc ions : [0, T]→˜
Hs(Ω) and ˜
F: [0, T]→L2(Ω) by
[ ( )](x) := ( , x),[˜
F( )](x) := ˜
F( , x) o ( , x)∈[0, T]×Rn.(2.5)
Le h·,·i be he duali y pai ing on H−s(Ω) ⊕˜
Hs(Ω). Mul iplying (2.4) by any φ∈˜
Hs(Ω)
gi es
h ′( ), φi+B[ , φ; ] = ( ˜
F( ), φ)L2(Ω) o 0 ≤ ≤T,
whe e B[ , φ; ] is he bilinea o m gi en by
B[ , φ; ] := ZRn
(−∆)s/2 ( )(−∆)s/2φdx+ZΩ
a( , ·) ( )φdx.
De ini ion 2.1 (Weak solu ions).We say ha is a weak solu ion o (2.4), i
(a) ∈L2(0, T ;˜
Hs(Ω)) and ′∈L2(0, T;H−s(Ω));
(b) h ′( ), φi+B[ , φ; ] = ( ˜
F( ), φ)L2(Ω) o all φ∈˜
Hs(Ω) o (almos ) all 0 ≤ ≤T;
(c) (0) = ϕ,
whe e and ˜
Fa e de ined acco ding o (2.5).
P oposi ion 2.2 (Well-posedness).Gi en any n∈Nand 0< s < 1. Le Ω⊂Rnbe a
bounded Lipschi z domain in Rn. Le a∈L∞(ΩT). Fo any ˜
F∈L2(ΩT)and ϕ∈˜
H0(Ω),
he e exis s a unique weak solu ion o (2.4)and sa is ies he ollowing es ima e:
k k2
L∞(0,T;L2(Ω)) +k k2
L2(0,T;˜
Hs(Ω)) +k∂ k2
L2(0,T;H−s(Ω)) ≤C(kϕk2
L2(Ω) +k˜
Fk2
L2(ΩT)) (2.6)
o some cons an C=C(n, s, T, kakL∞(ΩT)). I we u he assume ϕ∈˜
Hs(Ω), hen
∈L∞(0, T;˜
Hs(Ω)) and ∂ ∈L2(ΩT). In his case, he unique weak solu ion also sa is ies
he ollowing es ima e:
k k2
L∞(0,T;˜
Hs(Ω)) +k∂ k2
L2(ΩT)≤C(kϕk2
˜
Hs(Ω) +k˜
Fk2
L2(ΩT)) (2.7)
o some cons an C=C(n, s, T, kakL∞(ΩT)).
The p oo o P oposi ion 2.2 is analogous o he s anda d well-posedness p oo o he
classical di usion equa ion. Howe e , o comple eness, we p esen a p oo in Appendix A.
Co olla y 2.3. Gi en any n∈Nand 0< s < 1. Le Ω⊂Rnbe a bounded Lipschi z
domain in Rn, and W⊂Ωebe any open se wi h Lipschi z bounda y sa is ying W∩Ω = ∅.
SEMILINEAR EQUATION FRACTIONAL LAPLACIAN 6
Le a∈L∞(ΩT). Then o any ˜
F∈L2(ΩT),ϕ∈˜
H0(Ω), and ∈ C∞
c(WT), he e exis s a
unique weak solu ion u= + o (2.3)sa is ying
ku− k2
L∞(0,T;L2(Ω)) +ku− k2
L2(0,T;˜
Hs(Ω)) +k∂ (u− )k2
L2(0,T;H−s(Ω))
≤C(kϕk2
L2(Ω) +kF−(−∆)s k2
L2(ΩT))
o some cons an C=C(n, s, T, kakL∞(ΩT)). I we u he assume ϕ∈˜
Hs(Ω), hen he
unique weak solu ion ualso sa is ies he ollowing es ima e:
ku− k2
L∞(0,T;Hs(Rn)) +k∂ uk2
L2(ΩT)≤C(kϕk2
˜
Hs(Ω) +kF−(−∆)s k2
L2(ΩT)) (2.8)
o some cons an C=C(n, s, T, kakL∞(ΩT)).
We skip he p oo o Co olla y 2.3 as i is a s aigh o wa d consequence o P oposi ion 2.2.
2.3. Maximum p inciple o he linea equa ion. Modi ying he ideas in [LL19, P opo-
si ion 3.1] o [RO16, P oposi ion 4.1], we can ob ain he ollowing p oposi ion:
P oposi ion 2.4 (Maximum p inciple).Gi en any n∈Nand 0< s < 1. Le Ω⊂Rnbe
a bounded Lipschi z domain in Rn. Le a∈L∞(ΩT). Suppose ha u∈L2(0, T;Hs(Rn)) ∩
H1(0, T;L2(Ω)) is a weak solu ion o (2.3). I F≥0in ΩT, ≥0in Ωe
T,ϕ≥0in Rn,
hen u≥0in ΩT.
P oo . Le Mbe a eal numbe which shall be de e mined la e . We de ine
uM( , x) := e−M u( , x), aM( , x) := a( , x) + M, FM( , x) := e−M F( , x) in ΩT,
M( , x) := e−M ( , x) in Ωe
T.(2.9)
We see ha uMsa is ies
(∂ + (−∆)s+aM)uM=FMin ΩT,
uM= Min Ωe
T,
uM=ϕon {0} × Rn.
(2.10)
We choose M=kakL∞(ΩT), hen aM≥0 in ΩT. Nex we w i e uM=u+
M−u−
M, whe e u+
M=
max{uM,0}and u−
M= max{−uM,0}. Since uM∈L2(0, T;Hs(Rn))∩H1(0, T;L2(Ω)), hen
u±
M∈L2(0, T;Hs(Rn)) ∩H1(0, T;L2(Ω)) and ha
∂ (u−
M) = (−∂ uMin {uM<0},
0 in {uM≥0}.
Since uM= M≥0 in Ωe
T, hence u−
M= 0 in Ωe
T, which implies u−
M∈L2(0, T;˜
Hs(Ω)) ∩
H1(0, T;L2(Ω)).Tes ing he i s equa ion o (2.10) by u−
M, we ha e
0≤(FM( ),u−
M( ))L2(Ω) (because FM≥0 and u−
M≥0 in ΩT)
=ZΩ
(∂ uM( ))u−
M( )dx+ZRn
(−∆)s
2uM( )(−∆)s
2u−
M( )dx+ZΩ
aM( , ·)uMu−
Mdx
=−d
d 1
2ZΩ
|u−
M( )|2dx+ZRn
(−∆)s
2uM( )(−∆)s
2u−
M( )dx−ZΩ
aM( , ·)|u−
M|2dx
o all 0 < < T . In [LL19, P oposi ion 3.1] o [RO16, P oposi ion 4.1], hey showed ha
ZRn
(−∆)s
2uM( )(−∆)s
2u−
M( )dx≤0 o all 0 < < T.
SEMILINEAR EQUATION FRACTIONAL LAPLACIAN 7
Combining he p eceding wo inequali ies, we hen conclude d
d RΩ|u−
M( )|2dx≤0 holds
ue o all 0 < < T. Since u−
M= 0 on Rn× {0}(because ϕ≥0 in Rn), hen we conclude
RΩ|u−
M( )|2dx= 0 o all 0 < < T, which comple es ou p oo .
Co olla y 2.5 (Compa ison p inciple).Gi en any n∈Nand 0< s < 1. Le Ω⊂Rnbe a
bounded Lipschi z domain in Rn, and le a∈L∞(ΩT). Le u1and u2be weak solu ions o
(∂ + (−∆)s+a)uj=Fjin ΩT,
uj= jin Ωe
T,
uj=ϕjon {0} × Rn,
o j= 1,2. I F1≥F2in ΩT, 1≥ 2in Ωe
T, ϕ1≥ϕ2in Rn, hen u1≥u2in ΩT.
P oo . By applying P oposi ion 2.4 wi h u=u1−u2, his can be p o ed immedia ely.
Rema k 2.1. P oposi ion 2.4 as well as Co olla y 2.5 also imply he uniqueness pa o
P oposi ion 2.2 and Co olla y 2.3.
2.4. L∞-bounds o solu ions o he linea equa ion. Fo ou pu poses, we equi e he
ollowing L∞-bound es ima e, which can be ound in [Li22, P oposi ion 3.3]:
P oposi ion 2.6. Gi en any n∈Nand 0< s < 1. Le Ω⊂Rnbe a bounded Lipschi z
domain in Rn, and le a∈L∞(ΩT). Suppose ha u∈L2(0, T;Hs(Rn)) ∩H1(0, T ;L2(Ω))
is a weak solu ion o
(∂ + (−∆)s+a)u=Fin ΩT,
u= in Ωe
T,
u= 0 on {0} × Rn,
wi h F∈L∞(ΩT)and ∈L∞(Ωe
T). Then kukL∞(ΩT)≤C(k kL∞(Ωe
T)+kFkL∞(ΩT)), o
some cons an C=C(n, s, T, Ω,kakL∞(ΩT)).
To make ou pape mo e sel -con ained, he e we ske ch he p oo o P oposi ion 2.6. The
ollowing lemma can be ound in [LL19, Lemma 3.4] (wi h a≡0) o [RO16, Lemma 5.1].
Lemma 2.7 (Ellip ic ba ie ).Gi en any n∈Nand 0< s < 1. Le Ωbe a bounded
Lipschi z domain in Rn. The e exis s a unc ion φ=φ(x)∈ C∞
c(Rn)such ha
(−∆)sφ≥1in Ω, φ ≥0in Rn, φ ≤Cin Ω, o some cons an C=C(n, s, Ω).
I we de ine Φ( , x) := e φ(x), we immedia ely ob ain he ollowing co olla y:
Co olla y 2.8 (Pa abolic ba ie ).Gi en any n∈Nand 0< s < 1. Le Ωbe a bounded
Lipschi z domain in Rn. The e exis s a unc ion Φ∈ C∞
c([0, T]×Rn)such ha
(∂ + (−∆)s)Φ ≥1in ΩTΦ≥0in [0, T)×Rn,Φ≤Cin ΩT,
o some cons an C=C(n, s, T, Ω).
Using he ba ie in Co olla y 2.8, we now can ob ain he ollowing L∞-bound o he
solu ion o (2.3).
P oo o P oposi ion 2.6.Using he unc ions gi en in (2.9) wi h M=kakL∞(ΩT), we know
ha
(∂ + (−∆)s+aM)uM=FMin ΩT,
uM= Min Ωe
T,
uM= 0 on {0} × Rn,
SEMILINEAR EQUATION FRACTIONAL LAPLACIAN 8
wi h aM≥0. Le ( , x) := k MkL∞(Ωe
T)+kFMkL∞(ΩT)Φ( , x)≥0 in [0, T)×Ω,whe e Φ
is he ba ie gi en in Co olla y 2.8. We see ha
(∂ + (−∆)s+aM) ≥(∂ + (−∆)s) =kFMkL∞(ΩT)(∂ + (−∆)s)Φ ≥ kFMkL∞(ΩT)
≥ ∓FM=∓(∂ + (−∆)s+aM)uMin ΩT.
Mo eo e , we also ha e
(∂ + (−∆)s+aM)( ±uM)≥0 in ΩT
±uM= ± M≥ k MkL∞(Ωe
T)± M≥0 in Ωe
T
±uM= ≥0 on {0} × Rn.
(2.11)
Combining ela ions in (2.11), and P oposi ion 2.4, we see ha ≥ ±uMin ΩT,which
u he implies ha kuMkL∞(ΩT)≤ k kL∞(ΩT)≤ k MkL∞(Ωe
T)+CkFMkL∞(ΩT),whe e C=
C(n, s, T, Ω) is he cons an gi en in he Co olla y 2.8. Finally, u ilizing
|u( , x)|=eM |uM( , x)| ≤ eTkakL∞(ΩT)kuMkL∞(ΩT)in ΩT,
|FM( , x)|=e−M |F( , x)| ≤ kFkL∞(ΩT)in ΩT,
| M( , x)|=e−M | ( , x)| ≤ k kL∞(Ωe
T)in Ωe
T,
we conclude he p oo .
We skip he p oo o he ollowing well-posedness esul as i ollows om combining
Co olla y 2.3 and P oposi ion 2.6.
P oposi ion 2.9. Gi en any n∈Nand 0< s < 1. Le Ω⊂Rnbe a bounded Lipschi z
domain in Rn, le W⊂Ωebe any open se wi h Lipschi z bounda y sa is ying W∩Ω = ∅.
Then o any ˜
F∈L∞(ΩT)and ∈ C∞
c(WT), he e exis s a unique weak solu ion uo
(∂ + (−∆)s+a)u=Fin ΩT,
u= in Ωe
T,
u= 0 in {0} × Rn,
sa is ying
kuk2
L∞(0,T;Hs(Rn))∩L∞(Rn
T)+k∂ uk2
L2(ΩT)
≤CkFk2
L∞(ΩT)+k k2
L∞(0,T;Hs(Rn))∩L∞(Rn
T)+k(−∆)s k2
L2(ΩT)
o some cons an C=C(n, s, T, kakL∞(ΩT),Ω).
2.5. Well-posedness o he nonlinea equa ion. We now s a e he well-posedness o
(1.1) o small ex e io da a:
P oposi ion 2.10. Gi en any n∈Nand 0< s < 1. Le Ω⊂Rnbe a bounded Lip-
schi z domain in Rn, and W⊂Ωebe any open se wi h Lipschi z bounda y sa is ying
W∩Ω = ∅. Fixing any pa ame e δ > 0. Assume ha qsa is ies (Q.1)–(Q.3). Then
he e exis s a su icien ly small pa ame e ˜ǫ0= ˜ǫ0(n, s, Ω, T, δ)>0such ha he ollowing
s a emen holds: Gi en any ∈ C∞
c(WT)wi h k kex ≤˜ǫ0, he e exis s a unique solu ion
u∈L∞(0, T;Hs(Rn)) ∩L∞(Rn
T)o (1.1)wi h
kukL∞(0,T;Hs(Rn))∩L∞(Rn
T)≤Ck kex (2.12)
o ce ain cons an C=C(n, s, T, Ω).
Rema k 2.2. In o de o p o e P oposi ion 2.10, we only need q o be C1-smoo h in z
a iable. Howe e o eco e m- h je o qwe need o assume (Q.1).
SEMILINEAR EQUATION FRACTIONAL LAPLACIAN 15
Since he e m ∂2
zqj(·, uǫ
j)(∂ǫ1uǫ
j)(∂ǫ2uǫ
j) is bounded in ΩT, using P oposi ion 2.9, he e exis s
a unique solu ion ǫ
j∈L∞(0, T;˜
Hs(Ω)) ∩L∞(Rn
T) o (4.12) wi h
k ǫ
jkL∞(0,T;˜
Hs(Ω))∩L∞(Rn
T)≤Ck∂2
zqj(·, uǫ
j)(∂ǫ1uǫ
j)(∂ǫ2uǫ
j)kL∞(ΩT)
≤CM2kg1kex kg2kex .(using (Q.4) and (4.11)) (4.13)
Again, ǫ
jis empo a y no a ion, which will be d opped a e showing ∂ǫ1ǫ2uǫ
jis well-de ined.
We emphasize ha we ha e al eady d opped ǫ
j, so his will no con lic wi h he one used
in Sec ion 4.2.
Lemma 4.4. The e exis s a cons an ǫ0=ǫ0(n, s, Ω, T, δ, g, m)>0wi h 0< ǫ0<˜ǫ0, whe e
˜ǫ0is gi en in P oposi ion 2.10, such ha o each ǫwi h |ǫ|< ǫ0, we ha e
lim
ǫ2→0k ǫ
j−δǫ2∂ǫ1uǫ
jkL∞(0,T;Hs(Rn))∩L∞(Rn
T)= 0,(4.14)
whe e δǫ2∂ǫ1uǫ
j=∂ǫ1uǫ+ǫ2e2
j−∂ǫ1uǫ
j
ǫ2in ΩT,p o ided |ǫ|+|ǫ2|< ǫ0.
P oo . Le ǫ2sa is ies |ǫ2| ≤ |ǫ|and |ǫ|+|ǫ2|< ǫ0. No e ha
((∂ + (−∆)s)( ǫ
j−δǫ2∂ǫ1uǫ
j) = G2in ΩT,
ǫ
j−δǫ2∂ǫ1uǫ
j= 0 in Ωe
Tand on {0} × Rn,
whe e
−G2=∂zqj(·, uǫ
j) ǫ
j+∂2
zqj(·, uǫ
j)(∂ǫ1uǫ
j)(∂ǫ2uǫ
j)
−∂zqj(·, uǫ+ǫ2e2
j)∂ǫ1uǫ+ǫ2e2
j−∂zqj(·, uǫ
j)∂ǫ1uǫ
j
ǫ2
.
A e some compu a ion we can w i e −G2=G21 +G22 +G23,whe e
G21 =∂zqj(·, uǫ
j) ǫ
j−δǫ2∂ǫ1uǫ
j,
G22 =∂2
zqj(·, uǫ
j)(∂ǫ2uǫ
j)−∂zqj(·,uǫ+ǫ2e2
j)−∂zqj(·,uǫ
j)
ǫ2(∂ǫ1uǫ+ǫ2e2
j),
G23 =∂2
zqj(·, uǫ
j)∂ǫ2uǫ
j∂ǫ1uǫ
j−∂ǫ1uǫ+ǫ2e2
j,
No e ha k ǫ
j−δǫ2∂ǫ1uǫ
jkL∞(0,T;Hs(Rn))∩L∞(Rn
T)≤CkG2kL∞(ΩT).Possibly choosing a smalle
ǫ0, we ha e kG21kL∞(ΩT)≤1
2k ǫ
j−δǫ2∂ǫ1uǫ
jkL∞(0,T;Hs(Rn))∩L∞(Rn
T).Using he mean alue
heo em and Lemma 4.2, we know ha limǫ2→0(kG22kL∞(ΩT)+kG23kL∞(ΩT)) = 0. We hen
conclude (4.14) by using a gumen s simila o Lemma 4.2.
Akin o Lemma 4.3, we nex demons a e he ollowing lemma.
Lemma 4.5. I Λq1( ) = Λq1( ) o all ∈ C∞
c(WT)wi h k kex ≤˜ǫ0, whe e ˜ǫ0is he
cons an gi en in P oposi ion 2.10, hen he e exis s a cons an ǫ0=ǫ0(n, s, Ω, T, δ, g, m)>0
wi h 0< ǫ0<˜ǫ0such ha
(−∆)s∂ǫ1ǫ2uǫ
1VT= (−∆)s∂ǫ1ǫ2uǫ
2VT∀ǫwi h |ǫ| ≤ ǫ0.(4.15)
P oo . Using simila a gumen s as in Lemma 4.3 (wi h Lemma 4.4), we can show ha (4.10)
implies (4.15). Then we conclude he lemma by Lemma 4.3.
P oo o Theo em 1.1 o m= 2.Using (Q.3) and (4.9), we know ha ∂ǫ1ǫ2uǫ
j|ǫ=0 sa is ies
(∂ + (−∆)s)(∂ǫ1ǫ2uǫ
j|ǫ=0)
+∂2
zqj(·,0)(∂ǫ1uǫ|ǫ=0)(∂ǫ2uǫ|ǫ=0) = 0 in ΩT,
∂ǫ1ǫ2uǫ
j|ǫ=0 = 0 in Ωe
Tand on {0} × Rn.
SEMILINEAR EQUATION FRACTIONAL LAPLACIAN 16
Hence, we know ha := ∂ǫ1ǫ2uǫ
1|ǫ=0 −∂ǫ1ǫ2uǫ
2|ǫ=0 sa is ies
((∂ + (−∆)s) + (∂2
zq1(·,0) −∂2
zq2(·,0))(∂ǫ1uǫ|ǫ=0)(∂ǫ2uǫ|ǫ=0) = 0 in ΩT,
= 0 in Ωe
Tand on {0} × Rn.
F om Lemma 4.5, we know ha (−∆)s VT= 0.Since = 0 in VT, using he unique
con inua ion p ope y o he ac ional Laplacian in Lemma 3.1, we conclude ha ≡0.
The e o e, we know ha
∂2
zq1(·,0) −∂2
zq2(·,0)(∂ǫ1uǫ|ǫ=0)(∂ǫ2uǫ|ǫ=0) = 0.(4.16)
Since g1, g2∈ C∞
c(WT) a e a bi a y, using (4.8) and he Runge app oxima ion o ac ional
di usion equa ion in P oposi ion 3.2, we conclude ∂2
zq1(·,0) −∂2
zq2(·,0) = 0 in ΩT, which
p o es Theo em 1.1 o m= 2.
P oo o Co olla y 1.2 o m= 2.Using he same a gumen as abo e, we each (4.16):
(∂2
zq1(·,0) −∂2
zq2(·,0))(∂ǫ1uǫ|ǫ=0)(∂ǫ2uǫ|ǫ=0) = 0.(4.17)
Using (4.8), he unique con inua ion p ope y o he ac ional Laplacian in Lemma 3.1 and
a simple con adic ion a gumen , o each x0∈Ω, we can ind a sequence {xk}k∈N⊂Ω wi h
xk→x0and a sequence { k}k∈N⊂(0, T ) such ha
∂ǫ1uǫ|ǫ=0( k, xk)6= 0 and ∂ǫ2uǫ|ǫ=0( k, xk)6= 0.
The e o e om (4.17) we know ha ∂2
zq1(xk,0) = ∂2
zq2(xk,0) (he e we ha e assumed ha
∂2
zq1(·,0) and ∂2
zq2(·,0) a e independen o ). Hence by con inui y o ∂2
zq1(·,0), ∂2
zq2(·,0)
and he a bi a iness o x0∈Ω, we conclude ∂2
zq1(·,0) −∂2
zq2(·,0) = 0 in Ω,which p o es
Co olla y 1.2 o m= 2.
4.4. Highe o de linea iza ion. Fo each 2 ≤p≤m, we deno e ∂(p)=∂ǫ1···∂ǫp. By
epea ing o mal di e en ia ions o he equa ion (4.12), we ob ain he ollowing p- h o de
linea iza ion
((∂ + (−∆)s)∂(p)uǫ
j+∂(p)qj(·, uǫ
j) = 0 in ΩT,
∂(p)uǫ
j= 0 in Ωe
Tand on {0} × Rn.
whe e we simply deno e ∂(p)=∂ǫ1···∂ǫp. By induc ion, we can e i y
∂(p)qj(·, uǫ
j) = ∂zqj(·, uǫ
j)∂(p)uǫ
j+
p−1
X
ℓ=2
∂ℓ
zqj(·, uǫ
j)Tℓ
p(uǫ
j) + ∂p
zqj(·, uǫ
j)
p
Y
ℓ=1
∂ǫℓuǫ
j,
whe e Tℓ
p(uǫ
j) is a gene ic no a ion (in o de plinea iza ion) signi ying a combina ion o
he e ms ∂α
ǫuǫ
jwi h mul i-index αsa is ying 1 ≤ |α| ≤ p−1. The ollowing ac s can be
p o ed using s ong induc ion on m:
(1) Func ions ∂(p)uǫ
j∈L∞(0, T;Hs(Rn))∩L∞(Rn
T) a e well-de ined o each 2 ≤p≤m.
(2) The e exis s ǫ0=ǫ0(n, s, Ω, T, δ, g, m)>0 wi h 0 < ǫ0<˜ǫ0, whe e ˜ǫ0is he cons an
gi en in P oposi ion 2.10, such ha
lim
ǫp→0k∂(p)uǫ
j−δǫp∂(p−1)uǫ
jkL∞(0,T;Hs(Rn))∩L∞(Rn
T)= 0 o all 2 ≤p≤m,
whe e δǫp∂(p−1)uǫ
j=∂(p−1)uǫ+ǫpep
j−∂(p−1)uǫ
j
ǫpin ΩT,p o ided |ǫ|+|ǫp|< ǫ0.
(3) Mo eo e , i Λq1( ) = Λq1( ) o all ∈ C∞
c(WT) wi h k kex ≤˜ǫ0, hen we ha e
(−∆)s∂(p)uǫ
1VT= (−∆)s∂(p)uǫ
2VT o all 2 ≤p≤m. (4.18)
Using he obse a ions abo e, we a e now eady o p o e ou main esul .
SEMILINEAR EQUATION FRACTIONAL LAPLACIAN 17
P oo o Theo em 1.1.We p o e by induc ion on m. We assume he ollowing hypo hesis:
∂p
zq1(·,0) = ∂p
zq1(·,0) o all 2 ≤p≤m−1.(4.19)
Using (4.9), we see ha
∂(m)qj(·, uǫ
j)|ǫ=0 =∂zqj(·,0)∂(m)uǫ
j|ǫ=0 +
m−1
X
ℓ=2
∂ℓ
zqj(·,0)Tℓ
m(uǫ
j)ǫ=0 +∂m
zqj(·,0)
m
Y
ℓ=1
∂ǫℓuǫ|ǫ=0.
Using (4.19), we see ha Pm−1
ℓ=2 ∂ℓ
zqj(·,0)Tℓ
m(uǫ
1)ǫ=0 =Pm−1
ℓ=2 ∂ℓ
zqj(·,0)Tℓ
m(uǫ
2)ǫ=0.Hence,
we know ha := ∂(m)uǫ
1|ǫ=0 −∂(m)uǫ
2|ǫ=0 sa is ies
((∂ + (−∆)s) + (∂m
zq1(·,0) −∂m
zq2(·,0)) Qm
ℓ=1 ∂ǫℓuǫ|ǫ=0 = 0 in ΩT
= 0 in Ωe
Tand on {0} × Rn.
Using (4.18), we know ha (−∆)s |VT= 0.Since = 0 in VT, by using he unique con-
inua ion p inciple o he ac ional Laplacian (see Lemma 3.1), we conclude ha ≡0.
Hence, we know ha (∂m
zq1(·,0)−∂m
zq2(·,0)) Qm
ℓ=1 ∂ǫℓuǫ|ǫ=0 = 0 in ΩT.Since g1,··· , gm∈
C∞
c(WT) a e a bi a y, using (4.8) and he Runge app oxima ion o he ac ional di u-
sion equa ion p o ed in P oposi ion 3.2, we conclude ∂m
zq1(·,0) = ∂m
zq2(·,0) in ΩT. This
comple es he p oo o Theo em 1.1.
5. Analogous esul o he ac ional wa e equa ion
The ollowing esul s can be p o ed by modi ying he ideas in [KLW21, Co olla y 2.2]
(o [E a10, Chap e 7]), which we use la e o p o e he well-posedness o (1.5) wi h small
ex e io da a and o sol e in e se p oblem as well.
Lemma 5.1. Gi en any n∈Nand 0< s < 1. Le Ω⊂Rnbe a bounded Lipschi z domain
in Rn, le W⊂Ωebe any open se wi h Lipschi z bounda y sa is ying W∩Ω = ∅. Le
a∈L∞(ΩT). Then o any F∈L2(ΩT), ∈ C∞
c(WT),ψ∈˜
H0(Ω),ϕ∈˜
Hs(Ω), he e
exis s a unique solu ion uo
(∂2
+ (−∆)s+a)u=Fin ΩT,
u= in Ωe
T,
u=ϕ, ∂ u=ψon {0} × Rn.
(5.1)
sa is ying
ku− kL∞(0,T;Hs(Rn)) +k∂ ukL∞(0,T ;L2(Ω))
≤C(kϕk˜
Hs(Ω) +kψkL2(Ω) +kF−(−∆)s kL2(ΩT)) (5.2)
o some cons an C=C(n, s, T, kakL∞(ΩT)).
Rema k 5.1. I is in e es ing o compa e (5.2) wi h (2.8): bo h solu ions (wa e and di -
usion) ha e egula i y L∞(0, T ;Hs(Rn)). In [KLW21, Co olla y 2.2], hey only conside
he case when ais independen o ime . The exis ence o solu ions can be p o ed using
exac ly he same a gumen , bu he p oo o uniqueness esul need ex a ca e. In con as
o [E a10, Chap e 7], he e we do no assume he W1,∞(0, T;L∞(Ω)) egula i y o he
coe icien a.
P oo o uniqueness esul o Lemma 5.1.Le u∈L2(0, T;Hs(Rn)) ∩H1(0, T;L2(Ω)) be
he solu ion o
(∂2
+ (−∆)s+a)u= 0 in ΩT,
u= 0 in Ωe
T,
u=∂ u= 0 on {0} × Rn.
(5.3)
SEMILINEAR EQUATION FRACTIONAL LAPLACIAN 18
We wan o show ha u≡0. Fix 0 ≤η≤Tand se
( , ·) := (Rη
u(τ, ·)dτi 0 ≤ ≤η,
0 i η≤ ≤T.
Then ( , ·)∈˜
Hs(Ω) o each 0 ≤ ≤T, and so
ZΩZη
0
(∂2
u) d dx+ZRnZη
0
(−∆)s
2u(−∆)s
2 d dx+ZΩZη
0
au d dx= 0.(5.4)
Since ∂ u(0) = (η) = 0 and ∂ =−u o all 0 ≤ ≤η, we see ha
ZΩZη
0
(∂2
u) d dx=−ZΩZη
0
(∂ u)(∂ )d dx=ZΩZη
0
(∂ u)ud dx
=1
2
d
d ZΩZη
0
|u|2d dx=1
2ku(η, ·)k2
L2(Ω).(5.5)
Using he ac ∂ =−u o all 0 ≤ ≤η, we also ha e
ZRnZη
0
(−∆)s
2u(−∆)s
2 d dx=−ZRnZη
0
∂ (−∆)s
2 (−∆)s
2 d dx
=−1
2ZRnZη
0
d
d |(−∆)s
2 |2d dx=1
2k(−∆)s
2 (0,·)k2
L2(Rn).(5.6)
Since a∈L∞(ΩT), combining (5.4), (5.5) and (5.6), oge he wi h he Ha dy-Li lewood-
Sobole inequali y (A.5), we ob ain
k (0,·)k2
˜
Hs(Ω) +ku(η, ·)k2
L2(Ω) ≤CZη
0k ( , ·)k2
L2(Ω) +ku( , ·)k2
L2(Ω)d . (5.7)
Le us w i e w( , ·) := R
0u(τ, ·)dτ. Since (0,·) = w(η, ·) and ( , ·) = w(η, ·)−w( , ·), om
(5.7) we know ha
kw(η, ·)k2
˜
Hs(Ω) +ku(η, ·)k2
L2(Ω) .Zη
0kw(η, ·)−w( , ·)k2
L2(Ω) +ku( , ·)k2
L2(Ω)d
.ηkw(η, ·)kL2(Ω) +Zη
0kw( , ·)k2
L2(Ω) +ku( , ·)k2
L2(Ω)d .
The e o e, we can choose T1, which is independen o η, such ha
kw(η, ·)k2
˜
Hs(Ω) +ku(η, ·)k2
L2(Ω) ≤CZη
0kw( , ·)k2
L2(Ω) +ku( , ·)k2
L2(Ω)d ,
o all 0 ≤η≤T1. Using he G ¨onwall’s inequali y in [E a10, Sec ion B.2], we know
ha u( , ·) = 0 o all ∈[0, T1]. Applying he same a gumen on he in e als [T1,2T1],
[2T1,3T1], e c., we conclude ha u≡0.
We need he ollowing Sobole embedding o ob ain L∞(ΩT)- egula i y o he solu ion,
which is a special case o [DNPV12, Theo em 8.2]:
Lemma 5.2 ([DNPV12]).Le n= 1 and 1/2< s < 1. The e exis s a cons an C=C(s, Ω)
such ha k kC0,α(Ω) ≤C(k k2
L2(Ω) + [ ]2˙
Hs(Ω))≤Ck k2
Hs(R), o any ∈L2(Ω) wi h α=
(2s−1)/2.
The e o e, Lemma 5.1 implies he ollowing esul .
SEMILINEAR EQUATION FRACTIONAL LAPLACIAN 19
P oposi ion 5.3. Le n= 1 and 1/2< s < 1. Le Ω⊂Rbe a bounded open se in Rn, le
W⊂Ωebe any open se sa is ying W∩Ω = ∅. Le a∈L∞(ΩT). Then o any ˜
F∈L∞(ΩT)
and ∈ C∞
c(WT), he e exis s a unique weak solu ion uo (5.1)sa is ying
kukL∞(0,T;Hs(R1))∩L∞(R1
T)+k∂ ukL∞(0,T;L2(Ω))
≤Ckϕk˜
Hs(Ω) +kψkL2(Ω) +kFkL2(ΩT)
+k kL∞(0,T;Hs(R1))∩L∞(R1
T)+k(−∆)s kL2(ΩT)
o ce ain cons an C=C(s, T, kakL∞(ΩT),Ω).
Using he same a gumen as in P oposi ion 2.10, we also can p o e he well-posedness o
(1.5) o small ex e io da a:
P oposi ion 5.4. Le n= 1 and 1
2< s < 1. Le Ω⊂Rbe a bounded open se , le
W⊂Ωebe any open se sa is ying W∩Ω = ∅. Fix any pa ame e δ > 0. Assume qsa is ies
(Q.1)–(Q.3). The e exis s a su icien ly small pa ame e ˜ǫ0= ˜ǫ0(s, Ω, T, δ)>0such ha he
ollowing s a emen holds: Gi en any ∈ C∞
c(WT)wi h k kex ≤˜ǫ0, he e exis s a unique
solu ion u∈L∞(0, T ;Hs(R1)) ∩L∞(R1
T)o (1.5)wi h
kukL∞(0,T;Hs(R1))∩L∞(R1
T)≤Ck kex (5.8)
o ce ain cons an C=C(s, T, Ω).
Finally, he in e se p oblem o he nonlinea ac ional wa e equa ion (1.5), i.e. Theo-
em 1.3 and Co olla y 1.4, can be p o ed using exac ly he same idea as in Theo em 1.1
and Co olla y 1.2, espec i ely, see Sec ion 4.
Appendix A. Well-posedness o he linea ac ional di usion equa ion
A.1. Uniqueness o weak solu ion. We i s p o e he uniqueness o weak solu ion o
(2.3) as well as (2.4). I su ices o p o e he ollowing s a emen : I ua weak solu ion o
((∂ + (−∆)s+a) = 0 in ΩT,
= 0 in Ωe
Tand on {0} × Rn.(A.1)
hen u≡0. Mul iplying he i s equa ion o (A.1) by , we ob ain
0 = h ′, i+B[ , ; ] = d
d 1
2k ( )k2
L2(Ω)+B[ , ; ]
≥d
d 1
2k ( )k2
L2(Ω)− kakL∞(ΩT)k ( )k2
L2(Ω),
ha is, d
d k ( )k2
L2(Ω)≤2kakL∞(ΩT)k ( )k2
L2(Ω).Using he G ¨onwall’s inequali y in [E a10,
Sec ion B.2], we conclude k ( )k2
L2(Ω) = 0 o all 0 ≤ ≤T, hence u≡0. The uniqueness
is p o ed.
A.2. Exis ence o weak solu ion. Now i su ices p o e ha he e exis s a weak solu ion
o (2.4).
S ep 1: Gale kin app oxima ion. We now se up he Gale kin app oxima ion o
(2.4). Simila o [KLW21, Appendix A], we conside an eigenbasis {wk}k∈Nassocia ed wi h
he Di ichle ac ional Laplacian in a bounded domain Ω. We no malize hese eigen unc-
ions so ha
{wk}k∈Nbe an o hogonal basis in ˜
Hs(Ω),
{wk}k∈Nbe an o hono mal basis in L2(Ω).
SEMILINEAR EQUATION FRACTIONAL LAPLACIAN 20
Gi en any ixed in ege m∈N, we conside he ollowing ansa z:
m( ) :=
m
X
k=1
dk
m( )wk.(A.2)
Plugging he ansa z (A.2) in o De ini ion 2.1(b), we ob ain
(( ′
m( ), wk)L2(Ω) +B[ m, wk; ] = ( ˜
F( ), wk)L2(Ω) o all 0 ≤ ≤T,
dk
m(0) = ( ˜ϕ, wk)L2(Ω).(A.3)
No e ha ( ′
m, wk)L2(Ω) = (dk
m)′( ), B[ m, wk; ] = Pm
ℓ=1 ekℓ( )dk
m( ) wi h he coe icien s
ekℓ( ) := B[wℓ, wk; ].This shows ha dk
m( ) sa is ies he ollowing linea sys em o o dina y
di e en ial equa ion (ODE):
((dk
m)′( ) + Pm
ℓ=1 ekℓ( )dk
m( ) = ( ˜
F( ), wk)L2(Ω) o all 0 ≤ ≤T,
dk
m(0) = ( ˜ϕ, wk)L2(Ω).
The e o e, he s anda d ODE heo y gua an ees he exis ence and uniqueness o such dk
m( ),
and hus (A.2) is a alid disc e iza ion o (2.4).
S ep 2: Ene gy es ima e. Mul iplying (A.3) by dk
m( ), and summing o e index
k= 1,··· , m, we ha e
( ′
m, m)L2(Ω) +B[ m, m; ] = ( ˜
F, m)L2(Ω).(A.4)
The ollowing Ha dy-Li lewood-Sobole inequali y can be ound in [Pon16, P op. 15.5] o
in [KLW21, equa ion (A.11)]:
k mkL2(R1)=k mkL2(Ω) ≤C(n, s)kφkL2n
n−s(Rn)≤C(n, s)k(−∆)s
2φkL2(R1)(A.5)
o n= 1 and o all φ∈˜
Hs(Ω). On he o he hand, we obse e ha ( ′
m, m)L2(Ω) =
d
d 1
2k mk2
L2(Ω).Hence, om (A.4) we ha e
d
d 1
2k mk2
L2(Ω)+k mk2
˜
Hs(Ω) ≤C(n, s, kakL∞(ΩT))k mk2
L2(Ω) +k˜
Fk2
L2(Ω)(A.6)
o all 0 ≤ ≤T. Using he G ¨onwall’s inequali y in [E a10, Sec ion B.2], we ha e
k m( )k2
L2(Ω) ≤eC k m(0)k2
L2(Ω) +CZ
0
k˜
F(s)k2
L2(Ω) ds o all 0 ≤ ≤T.
Since k m(0)k2
L2(Ω) =Pm
k=1 |( ˜ϕ, wk)L2(Ω)|2≤P∞
k=1 |( ˜ϕ, wk)L2(Ω)|2=kϕk2
L2(Ω), hen we
ha e
sup
0≤ ≤T
k m( )k2
L2(Ω) ≤Cs,T,kak∞kϕk2
L2(Ω) +k˜
Fk2
L2(ΩT).(A.7)
In eg a ing (A.6) on ∈[0, T], we ob ain
k mk2
L2(0,T;˜
Hs(Ω)) ≤Cs,kak∞k mk2
L2(ΩT)+k˜
Fk2
L2(ΩT).(A.8)
Combining (A.7) and (A.8), we ob ain he ollowing ene gy es ima e:
sup
0≤ ≤T
k m( )k2
L2(Ω) +k mk2
L2(0,T;˜
Hs(Ω)) ≤C(n, s, T, kakL∞(ΩT))(kϕk2
L2(Ω) +k˜
Fk2
L2(ΩT)).
(A.9)
Fixing any φ∈˜
Hs(Ω) wi h kφk˜
Hs(Ω) ≤1, we w i e φ=φ1+φ2, whe e φ1∈span {wk}m
k=1
and (φ2, wk)L2(Ω) = 0 o k= 1,··· , m. Using (A.3), we see ha
( ′
m( ), φ)L2(Ω) = ( ′
m( ), φ1)L2(Ω) = ( ˜
F, φ1)− B[ m, φ1; ].
SEMILINEAR EQUATION FRACTIONAL LAPLACIAN 21
Since kφ1k˜
Hs(Ω) ≤1, his implies |( ′
m( ), φ)L2(Ω)| ≤ C(k˜
F( )k2
L2(Ω) +k mk2
˜
Hs(Ω)). Hence
we know ha
k ′
m( )k2
H−s(Ω) := sup
kφk˜
Hs(Ω)≤1
|( ′
m( ), φ)L2(Ω)| ≤ Ckak∞(k˜
F( )k2
L2(Ω) +k mk2
˜
Hs(Ω)).
In eg a ing he inequali y abo e on ∈[0, T], and combining he esul wi h (A.9), we
ob ain
sup
0≤ ≤T
k m( )k2
L2(Ω) +k mk2
L2(0,T;˜
Hs(Ω)) +k ′
mk2
L2(0,T;H−s(Ω))
≤Cs,T,kak∞(kϕk2
L2(Ω) +k˜
Fk2
L2(ΩT)).(A.10)
S ep 3: Passing o he limi . By (A.10), we can ex ac a subsequence o { m}m∈N,
s ill deno ed by { m}m∈N( o simplici y), such ha
( m⇀ weakly in L2(0, T;˜
Hs(Ω)),
′
m⇀ ′weakly in L2(0, T;H−s(Ω)).(A.11)
Gi en any ixed in ege N, we w i e ˜
( ) := PN
k=1 dk( )wk,whe e dk( ) (k= 1,··· , N) a e
a bi a y smoo h unc ions (no he one in (A.2)). Choosing m≥N, mul iplying (A.3) by
dk( ), and summing o e k= 1,··· , N, we ob ain
ZT
0( ′
m( ),˜
( ))L2(Ω) +B[ m,˜
; ]d =ZT
0
(˜
F( ),˜
( ))L2(Ω) d . (A.12)
Taking m→+∞in (A.12), and om (A.11), we know ha
ZT
0h ′( ),˜
( )i+B[ ,˜
; ]d =ZT
0
(˜
F( ),˜
( ))L2(Ω) d . (A.13)
Due o he a bi a iness o Nand {dk}N
k=1, we ha e
h ′, φi+B[ , φ; ] = ( ˜
F( ), φ)L2(Ω) o all φ∈˜
Hs(Ω).
This oge he wi h (A.10) e i ies De ini ion 2.1(a)(b).
I emains o show e i ies De ini ion 2.1(c). To ha end, le us choose any ˜
∈
C1(0, T;˜
Hs(Ω)) wi h ˜
(T) = 0. F om (A.13), we ha e
ZT
0(˜
′( ), ( ))L2(Ω) +B[ ,˜
; ]d =ZT
0
(˜
F( ),˜
( ))L2(Ω) d + (˜
′(0), (0))L2(Ω).(A.14)
Simila ly, om (A.12), we ha e
ZT
0(˜
′( ), m( ))L2(Ω) +B[ m,˜
; ]d =ZT
0
(˜
F( ),˜
( ))L2(Ω) d + (˜
′(0), m(0))L2(Ω).
(A.15)
Combining (A.11) and (A.15), we ob ain
ZT
0(˜
′( ), ( ))L2(Ω) +B[ ,˜
; ]d =ZT
0
(˜
F( ),˜
( ))L2(Ω) d + (˜
′(0), ϕ)L2(Ω).
Compa ing his wi h (A.14), we see ha (˜
′(0), (0))L2(Ω) = (˜
′(0), ϕ)L2(Ω).Due o he
a bi a iness o ˜
, we conclude ha e i ies De ini ion 2.1(c).
S ep 4. Highe egula i y. We now u he assume ϕ∈˜
Hs(Ω). Mul iplying (A.3) by
(dk
m)′( ), and summing o e k= 1,··· , m, we ha e
( ′
m, ′
m)L2(Ω) +B[ m, ′
m] = ( ˜
F, ′
m)L2(Ω).(A.16)
SEMILINEAR EQUATION FRACTIONAL LAPLACIAN 22
No e ha we ha e
B[ m, ′
m] = ZRn
(−∆)s
2 m( )(−∆)s
2 ′
m( )dx+ZΩ
a( , x) m( , x) ′
m( , x)dx
=d
d 1
2ZR1
|(−∆)s
2 m( )|2dx+ZΩ
a( , x) m( , x) ′
m( , x)dx,
and RΩa( , x) m( ) ′
m( )dx≤ǫk ′
m( )k2
L2(Ω) +Cǫ−1k m( )k2
L2(Ω) and |(˜
F, ′
m)L2(Ω)| ≤
ǫk ′
m( )k2
L2(Ω) +Cǫ−1k˜
F( )k2
L2(Ω).These along wi h (A.16) imply
k ′
m( )k2
L2(Ω) +d
d 1
2ZRn
|(−∆)s
2 m( )|2dx
≤2ǫk ′
m( )k2
L2(Ω) +Cǫ−1k m( )k2
L2(Ω) +Cǫ−1k˜
F( )k2
L2(Ω).
Choosing ǫ= 1/4, we ob ain
k ′
m( )k2
L2(Ω) +d
d ZRn
|(−∆)s
2 m( )|2dx≤C(k m( )k2
L2(Ω) +k˜
F( )k2
L2(Ω)).(A.17)
Gi en any 0 ≤˜
≤T, we in eg a e (A.17) on ∈[0,˜
],
Z˜
0
k ′
m( )k2
L2(Ω) d +ZRn
|(−∆)s
2 m(˜
)|2dx−ZRn
|(−∆)s
2 m(0)|2dx
≤CZ˜
0
k m( )k2
L2(Ω) +Z˜
0
k˜
F( )k2
L2(Ω)≤Ck mk2
L2(ΩT)+k˜
Fk2
L2(ΩT).
Combining his inequali y wi h (A.5), we ob ain
k ′
mk2
L2(ΩT)+k mk2
L∞(0,T;˜
Hs(Ω)) ≤Ck ′
mk2
L2(ΩT)+ sup
0≤˜
≤TZRn
|(−∆)s
2 m(˜
)|2dx
≤CZRn
|(−∆)s
2 m(0)|2dx+k mk2
L2(ΩT)+k˜
Fk2
L2(ΩT)
≤C(k m(0)k2
˜
Hs(Ω) +k mk2
L2(ΩT)+k˜
Fk2
L2(ΩT).(A.18)
Since k m(0)k2
˜
Hs(Ω) ≤P∞
k=1 |(g, wk)L2(Ω)|2kwkk2
˜
Hs(Ω) =kϕk2
˜
Hs(Ω), (A.18) implies
k ′
mk2
L2(ΩT)+k mk2
L∞(0,T;˜
Hs(Ω)) ≤C(k˜ϕk2
˜
Hs(Ω) +k mk2
L2(ΩT)+k˜
Fk2
L2(ΩT)).
The e o e, combining his inequali y wi h (A.10), we ob ain
k ′
mk2
L2(ΩT)+k mk2
L∞(0,T;˜
Hs(Ω)) ≤C(kϕk2
˜
Hs(Ω) +k˜
Fk2
L2(ΩT)).
Finally, aking he limi m→ ∞, we comple e ou p oo .
Appendix B. Some discussions
The main di icul y in p o ing Theo em 1.3 is he egula i y o he solu ions. Due o
his di icul y, we a e only able o p o e Theo em 1.3 in one dimension. The me hod we
used equi es he L∞(ΩT)- egula i y o he linea ac ional wa e equa ion. The L∞(ΩT)-
egula i y is equi ed o gua an ee he well-posedness o (1.5), and i is essen ial o p o e
ha he linea iza ion is well-de ined as we see in Sec ion 2. Howe e , we a e only able
o ob ain his egula i y in he case when n= 1 and 1
2< s < 1. I one can p o e he
well-posedness o (1.5) o gene al n∈Nand 0 < s < 1, hen Theo em 1.3 immedia ely
ex ends o gene al n∈Nand 0 < s < 1.
In iew o s anda d ellip ic egula i y esul s (see [GT01] o [JLS17, P oposi ion A.1] in
e ms o o he no ms) as well as Sobole embedding, an a emp o imp o e he esul in
SEMILINEAR EQUATION FRACTIONAL LAPLACIAN 23
Theo em 1.3 is o y o ob ain he L∞(0, T;H2s(Rn)) egula i y o he solu ion. Howe e ,
his idea is less likely o be easible. Using [GSU20, Lemma 2.3], we know he e exis s a
unique solu ion w∈˜
Hs(Ω) o
((−∆)sw=Fin Ω,
w= 0 in Ωe,(B.1)
o F∈H−s(Ω). Choose Ω o be he uni disk and F o be a posi i e cons an in Ω.
Then he bes egula i y esul o (B.1) we know is Cs(Rn) [RO16, P oposi ion 7.2]. In ac ,
[RO16, Lemma 5.4] gi es an explici solu ion w(x) = (1−|x|2)s
+∈ Cs(Rn), and such wdoes
no belong o Cs′(Rn) o any s′> s. When n= 1, we ha e he con inuous embedding
H2s(R)֒→C2s−1
2(R). The e o e, a leas when n= 1 and s > 1
2, such a solu ion wo (B.1)
canno be in H2s(R).
Acknowledgemen s
All he au ho s we e pa ly suppo ed by he Academy o Finland (Cen e o Excellence
in In e se Modelling and Imaging, g an 284715) and by he Eu opean Resea ch Council
unde Ho izon 2020 (ERC CoG 770924).
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