A DELAY NONLOCAL QUASILINEAR CHAFEE-INFANTE PROBLEM: AN
APPROACH VIA SEMIGROUP THEORY
TOM´
AS CARABALLO, A. N. CARVALHO AND YESSICA JULIO
Abs ac . In his wo k we s udy a dissipa i e one dimensional scala pa abolic p oblem wi h
non-local nonlinea di usion wi h delay. We conside he gene al si ua ion in which he unc-
ions in ol ed a e only con inuous and solu ions may no be unique. We es ablish condi ions o
global exis ence and p o e he exis ence o global a ac o s. All esul s a e p esen ed only in
he au onomous since he non-au onomous case ollows in he same way, including he exis ence
o pullback a ac o s. A pa icula ly in e es ing ea u e is ha he e is a semilinea p oblem
(nonlocal in space and in ime) om which one can ob ain all solu ions o he associa ed quasi-
linea p oblem and ha o his semilinea p oblem he delay depends on he ini ial unc ion
making i s s udy mo e in ol ed.
1. In oduc ion
Reac ion-di usion equa ions wi h non-local e ms ha e a ac ed g ea a en ion du ing he
las wen y yea s. A ew ep esen a i e e e ences a e [11, 19, 10, 12, 17, 3, 7, 1]).
In o de o explain he p oblems we wish o conside , le us s a wi h an example o he ype
o models we ha e in mind, ha is, conside he ollowing quasilinea ini ial alue p oblem wi h
delay:
(1)
∂w
∂τ −a(l(w))∂w2
∂x2=λ (w) + γw(τ−ρ) + h(τ), τ > 0, x ∈Ω,
w(τ, 0) = w(τ, 1) = 0,
w(τ) = φ(τ), τ ∈[−ρ, 0],
whe e Ω = (0,1), γ, λ, ρ > 0, la con inuous ope a o om H1
0(Ω) in o R+,a∈C(R) wi h
a(R)⊂[m, M]⊂(0,∞), ∈C(R,R), h∈C(R, L2(0,1)) is bounded and φ∈C([−ρ, 0], H1
0(Ω)).
The local p oblem, i.e. a≡1, wi hou delay, has i s shown o ha e e y in e es ing p ope ies
in 1974 in he seminal wo ks o N. Cha ee and E. In an e (see [9, 8]). Th ough he wo k o many
au ho s, his has become he bes unde s ood in ini e dimensional dynamical sys em (see, o
example, [14, 13] o he au onomous case and [2] o he non-au onomous case).
1991 Ma hema ics Subjec Classi ica ion. 35Q30, 35B41, 35K58, 76D05.
Key wo ds and ph ases. non-local quasilinea pa abolic p oblems wi h delay wi hou uniqueness, exis ence and
egula i y o solu ions, compa ison esul s, mul i alued p ocesses, global a ac o s, uni o m bounds.
[TC] Pa ially suppo ed by he Spanish Minis e io de Ciencia e Inno aci´on (MCI), Agencia Es a al de In es-
igaci´on (AEI) and Fondo Eu opeo de Desa ollo Regional (FEDER) unde he p ojec PID2021-122991NB-C21.
[ANC] Pa ially suppo ed by FAPESP G an # 20/14075-6 and by CNPq G an # 308902/2023-8, B azil.
[YJ] Pa ially suppo ed by CAPES G an # 88887.695331/2022-00 and by he Colombian Minis e io de
Ciencia, Tecnolog´ıa e Inno aci´on (Minciencias).
2 TOM´
AS CARABALLO, A. N. CARVALHO AND YESSICA JULIO
Fo he non-local p oblem (anon-cons an ), wi hou delay, many in e es ing new ea u es
ha e been disco e ed ela i e o wha was known o he local case making his p oblem a e y
in e es ing one om he poin o iew o dynamics (see o example [1, 7, 17, 3]). O cou se, non-
local p oblems a e qui e challenging om he analy ical poin o iew making any new disco e y
e en mo e in e es ing.
In [5] we ha e deal wi h a p o o ype o non-au onomous scala one dimensional pa abolic
p oblem wi h he non-local nonlinea di usion being only con inuous and h ough he semi-
g oup heo y. The in oduc ion o he ime a iable makes he p oblem qui e challenging and
in e es ing al eady.
In his pape we go one s ep u he conside ing models simila o hose ea ed in [5] wi h
delay. As we will see nex , his b ings up a new and in e es ing ea u e o he p oblem, ha is,
he ini ial alue p oblem has o be conside ed wi h a delay depending on he ini ial unc ion.
To ob ain he local exis ence and egula i y o mild solu ions (con inuous in ime unc ions
aking alue in a sui able phase space and sa is ying he a ia ion o cons an s o mula) o (1)
we will use he esul s o [5] bu , o ha end, we will need o deal wi h he e y in e es ing new
ea u e o p oblems wi h delay depending on he ini ial unc ion φ. To ob ain egula i y, some
addi ional assump ion is needed on ,a◦l(as in [5]) bu also on he ini ial unc ion φ.
To ensu e ha solu ions a e globally de ined and o be able o apply he me hod o s eps we
impose he s uc u al condi ion
(S) Assume ha he e exis C0, C1∈Rsuch ha
u (u)⩽−νC0u2+|u|C1
o all u∈Rand o bo h ν=m
λand ν=M
λ.
Finally, o ob ain he exis ence o a global a ac o we assume, he dissipa i i y condi ion
(D) Assume ha (S) holds o some C0such ha he i s eigen alue ωo A+νC0Iis posi i e
and sa is ies e−ωρ/m +γ
ωm <1.
The nonlinea nonlocal di usion a(l(·)) : H1
0(0,1) →[m, M] makes he abo e p oblem a
nonlocal (in space) quasilinea p oblem.
Ou aim will be o es ablish a gene al local exis ence and egula i y esul o solu ions o
(1), p o e ha i condi ion (S) is sa is ied, solu ions a e globally de ined and, i condi ion (D)
is sa is ied, he mul i alued semi low associa ed o (1) has a global a ac o . We also use
compa ison esul s o ob ain uni o m bounds o he solu ions in he global a ac o . We could
wo k wi h he nonlinea i y as in (1) (being ime dependen ) bu all esul s would ha e iden ical
p oo s and he e o e we ha e decided o conside only he case h≡0. I will be clea om
he p oo s ha adding a bounded con inuous unc ion h:R→L2(0,1),o e en mo e gene al
non-au onomous nonlinea i ies, will no change he p oo s so we choose o omi i o he sake
o simplici y in he no a ion.
A DELAY NONLOCAL QUASILINEAR CHAFEE-INFANTE PROBLEM 3
Be o e we p oceed, le us wo k a li le mo e wi h he model (1) in o de o unde s and he
in e es ing new ea u e i b ings. Gi en a solu ion w: [−ρ, ∞)→H1
0(0,1) o he p oblem (1),
making := α−1
(τ) = Zτ
0
a(l(w( )))−1d ,τ∈[−ρ, ∞), we ha e ha he unc ion ude ined by
u( ) = w(τ) will be a solu ion o he p oblem
(2)
u =uxx +λ (u) + γu(α( )−ρ)
a(l(u)) , > 0, x ∈Ω,
u( , 0) = u( , 1) = 0,
u( ) = φ(α0( )), ∈[α−1(−ρ),0],
whe e α−1(−ρ)=Z−ρ
0
a(l(φ( )))−1
d <0 and, since a(R)⊂[m, M]⊂(0,∞), ρ
M⩽−α−1(−ρ)⩽ρ
m.
We no e ha , i τ⩾0
α( ) = Z
0
a(l(u( ))d
and α0: [α−1(−ρ),0] →[−ρ, 0] is he only solu ion o he in eg al equa ion
α(s) = Zs
0
a(l(φ◦α( )))d , s ∈(α−1(−ρ),0].
On he o he hand, gi en φ∈C(−ρ, 0], H1
0(Ω)) we de ine α0and α−1(ρ) as abo e. I u:
[α−1(−ρ),∞)→H1
0(0,1) is a solu ion o he semilinea delay di e en ial p oblem (2), making
τ=α( ) = Z
0
a(l(u( )))d ,
he unc ion w(τ) := u( ) will be a solu ion o p oblem (1).
P oblem (2) is a non-local (in ime and in space) non-au onomous semilinea p oblem.
E en hough, o a ixed ini ial condi ion φ∈C([−ρ, 0], H1
0(Ω)), he solu ion o he p oblem
(2) may no be unique, he delay is always he same, de e mined by he ini ial unc ion φ. In ac ,
i is a s iking ea u e o his model ha α0and α−1(ρ) a e uniquely de e mined by φonly and
ha , o sol e he (1) wi h ini ial unc ion φ, co esponds o sol e he nonlocal non-au onomous
semilinea delay di e en ial p oblem (2) wi h delay de e mined by φ.
Wi h his, we will p o e he exis ence o solu ions o p oblem (2) and hose will gi e us solu-
ions o (1). We will use he me hod o s eps which consis s o sol ing he p oblem i e a i ely,
in in e als o ime o leng h α−1(ρ) = Z0
−ρ
a(l(φ( )))−1
d . In each s ep we will apply he esul s
ob ained in [5] o he semilinea non-local (in ime and in space) non-au onomous p oblem (2).
As in [5], we will p oceed as abs ac as possible in o de ha he heo y can be applied o o he
simila models wi h li le e o .
Le us now ecall he esul s o [5]. Fo a Banach space X, le C(X) (L(X)) deno e he space
o con inuous (linea con inuous) ans o ma ions om Xin o i sel . Fo u∈C([0, T ], Xα), le
4 TOM´
AS CARABALLO, A. N. CARVALHO AND YESSICA JULIO
u (·) = u(·)[0, ], ∈[0, T]. Conside he Cauchy p oblem
(3)
du( )
d +Au( ) = g( , u (·)) > 0,
u(0) = w0∈Xα,
whe e −Ahas compac esol en and is he in ini esimal gene a o o an exponen ially decaying
analy ic semig oup {e−A : ⩾0},Xα, α ⩾0, is he ac ional powe spaces associa ed o A
[14, 16], and g: [0, T]×C([0, T], Xα)→Xis a con inuous unc ion.
Wi h his p elimina ies, we in oduce he de ini ions o s ong and mild solu ion o (3).
De ini ion 1.1. [20] A unc ion u: [0, T )→Xis a s ong solu ion o (3) on [0, T),i uis
con inuous on [0, T )and con inuously di e en iable on (0, T ), u( )∈D(A) o 0< < T, u(0) =
w0and (3) is sa is ied on [0, T ).
De ini ion 1.2. [20] Le Abe he in ini esimal gene a o o an analy ic semig oup S( ),w0∈Xα
and g: [0, T)×C([0, T], Xα)→X. A unc ion u∈C([0, T ]; Xα)such ha
(4) u( ) = S( )w0+Z
0
S( −s)g(s, us(·))ds, 0⩽ ⩽T,
is called a mild solu ion o he ini ial alue p oblem (3) on [0, T].
We summa ize he esul s o [5] nex .
Theo em 1.3 ([5]).Le Xbe a Banach space, Aa sec o ial ope a o such ha (λ+A)−1is
compac o all λ∈ρ(−A), and le {S( ) : ⩾0}be he analy ic semig oup gene a ed by −A.
I g: [0, T]×C([0, T], Xα)→X, 0⩽α⩽1is a con inuous map, hen o each w0∈Xα
he e exis s a T1=T1(w0)∈(0, T ]such ha he ini ial alue p oblem (3) has a mild solu ion
u∈C([0, T1]; Xα). Fu he mo e, T1may be chosen uni o mly o w0in bounded subse s o Xα.
In addi ion, i g: [0, T ]×C([0, T ], Xα)→X, 0⩽α⩽1, is such ha , gi en u∈C([0, T ], Xα)
(5) kg( , u (·)) −g(s, us(·))kX⩽w(ku( )−u(s)kα) + w(| −s|β),0< β < 1−α,
whe e w: [0,∞)→[0,∞)is an inc easing con inuous unc ion such ha w(0) = 0 and
(6) Z
0
u−1w(uβ)du < ∞.
A con inuous unc ion u: [0, T1]→Xαsa is ying
u( ) = S( )w0+Z
0
S( −s)g(s, us(·))ds, 0⩽ ⩽T1,
is a s ong solu ion o (3).
Nex we will show how o apply he esul s in [5] o es ablish exis ence and egula i y o
solu ions o (2). O cou se, since :R→R,a:R+→[m, M]⊂(0,∞), l:H1
0(0,1) →Rand
φ∈C([−ρ, 0], H1
0(0,1)) we ha e ha , o T∈[0, α−1(ρ)),
g( , u (·))(x) = λ (u( )(x)) + γφ(α( )−ρ)
a(l(u ( )))
A DELAY NONLOCAL QUASILINEAR CHAFEE-INFANTE PROBLEM 5
is a con inuous map om [0, T ]×C([0, T ], H1
0(0,1) in o X=L2(0,1). The only poin ha has
o be analyzed mo e ca e ully is he con inui y o [0, α−1(ρ)] 3 7→ φ(α( )−ρ)∈L2(0,1), bu
ha ollows om he con inui y o φand om he ac ha
|α( )−α( 0)|=|Z
0
a(l(u( )))d | → 0
−→ 0.
I ollows ha he ollowing heo em holds.
Theo em 1.4. Le X=L2(0,1),D(A) = H2(0,1) ∩H1
0(0,1) and A:D(A)⊂X→Xbe
he ope a o de ined by Au =uxx. Then −Ais posi i e and sel -adjoin (hence sec o ial), wi h
ac ional powe spaces Xα:= D(−Aα)wi h he g aph no m, α⩾0,X1
2=H1
0(0,1),(λ+A)−1
has compac esol en , λ∈ρ(−A)and he semig oup {S( ) : ⩾0}gene a ed by Ais a
compac and exponen ially decaying analy ic semig oup. I :R→Ris a con inuous unc ion
sa is ying (S), gi en φ∈C([−ρ, 0], H1
0(0,1)) he ini ial alue p oblem (2) has a mild solu ion
u∈C([0,∞); H1
0(0,1)), u he mo e u∈C([0,∞); Xα), o all α∈(0,1).
Fo egula i y, we assume also ha l:H1
0(0,1) →Ris con inuous and ha :R→R,
a:R+→[m, M], l:H1
0(0,1) →Rand φ: (−ρ, 0] →H1
0(0,1) sa is y
| (s)− (s0)|⩽w(|s−s0|)
|a(l(u)) −a(l( ))|⩽w(ku− kH1
0(0,1)),
|h(s)−h(s0)|⩽w(|s−s0|β),
kφ( )−φ( 0)kX⩽w(| − 0|β),
(7)
whe e w: [0,∞)→[0,∞) is a con inuous inc easing unc ion, w(0) = 0 and
Z1
0
u−1w(uβ)du < ∞, o some β∈(0,1−α).
We only need o pay a en ion o he unc ion [0, α−1(ρ)] 3 7→ φ(α( )−ρ)∈L2(0,1) and check
ha , unde he abo e condi ions, his unc ion sa is ies he condi ions o Theo em 1.3 o he
exis ence o a mild solu ion.
Tha ollows om
kφ(α( )−ρ)−φ(α( 0)−ρ)kX⩽w(|α( )−α( 0)|)
=w(|Z
0
a(l(u( )))d |)⩽w(M| − 0|).
Theo em 1.5. Unde he assump ions o Theo em 1.4 and ha (7) is sa is ied, he ini ial alue
p oblem (1) has a s ong solu ion w∈C([0,∞); H2(0,1) ∩H1
0(0,1)).
2. Exis ence o Solu ions
P oo o Theo em 1.5. We a e going o use he me hod o s eps. Le us e iew he easoning
and he needed de ails o apply Theo em 1.3. We know ha w(τ) = φ(τ) o τ∈(−ρ, 0]. Then,
o τ∈(0, ρ] , τ−ρ∈(−ρ, 0] and hen we ha e w(τ−ρ) = φ(τ−ρ), so ha equa ion (1)
6 TOM´
AS CARABALLO, A. N. CARVALHO AND YESSICA JULIO
becomes he ollowing non-au onomous quasilinea non-local scala one-dimensional pa abolic
pa ial di e en ial equa ion
(8)
∂w
∂τ =a(l(w))∂w2
∂x2+λ (w) + γφ(τ−ρ), τ > 0, x ∈Ω,
w(τ, 0) = w(τ, 1) = 0,
w(0) = φ(0).
One can pe o m a change in he ime scale in o de o ob ain he semilinea p oblem
(9)
u =uxx +λ (u) + γφ(α( )−ρ)
a(l(u)) , > 0, x ∈Ω,
u( , 0) = u( , 1) = 0,
u(0) = φ(0),
whe e τ=Z
0
a(l((u( )))d =: α( ).
I is clea , om he discussion in he In oduc ion, ha his ini ial alue p oblem has a leas
a mild solu ion ha we will call g1(see [5]). Unde Assump ion (S), since [0, α−1(ρ)] 3 7→
φ(α( )−ρ)∈H1
0(0,1) is con inuous, his solu ion exis s in he in e al [0, α−1(ρ)]. Thus, we
jus need o check egula i y.
To ha end, se 1: [0, α−1(ρ)] ×C([0, α−1(ρ)], Xα)→Xde ined by
1( , u (·)) = λ (u( )) + γφ(α( )−ρ)
a(l(u( ))) , α( ) = Z
0
a(l(u(θ)))dθ.
Then, o g1( ), g1(s)∈V, o Va neighbo hood o φ(0),we ha e ha
k 1( ,g
1(·))− 1(s, g
1(·))k
⩽
λ (g1( ))+γφ(α( )−ρ)
a(l(g1( ))) −λ (g1(s))+γφ(α(s)−ρ)
a(l(g1(s))) ±λ (g1(s))+γφ(α(s)−ρ)
a(l(g1( )))
⩽m−1kλ( (g1( )) − (g1(s)))k+m−1kγ(φ(α( )−ρ)−φ(α(s)−ρ))k
+k(λ (g1(s)) + γφ(α(s)−ρ))k
a(l(g1(s))) −a(l(g1( )))
a(l(g1( )))a(l(g1(s)))
.
(10)
Since o >swe ha e
|α( )−α(s)|=Z
s
a(l(g1(θ)))dθ
⩽M| −s|.
(11)
I ollows ha
(12) k 1( , g
1(·)) − 1(s, gs
1(·))k⩽K1w(kg1( )−g1(s)kα) + K2w(c| −s|β),
o some posi i e numbe s K1, K2and c, and since
Z
0
u−1w(uβ)du < ∞, o 0< β < 1−α,
we can apply Theo em 1.3 and conclude ha g1is a s ong solu ion o equa ion (9) in he
in e al [0, α−1(ρ)]. Also w(τ) = g1( ) is a solu ion o (8) in he in e al [0, ρ].
A DELAY NONLOCAL QUASILINEAR CHAFEE-INFANTE PROBLEM 7
Now, o τ∈[ρ, 2ρ], τ −ρ∈[0, ρ] and we ha e ha w(τ) = u(α−1(τ)) = g1(τ), so equa ion
(1) becomes
(13)
∂w
∂τ =a(l(w))∂w2
∂x2+λ (w) + γg1(τ−ρ), τ > ρ,
w(τ, 0) = w(τ, 1) = 0,
w(ρ) = g1(ρ).
Again, by making a change in he ime scale we ha e he ini ial alue p oblem
(14)
∂u
∂ =∂u2
∂x2+λ (u) + γg1(α( )−ρ)
a(l(u)) , ∈(α−1(ρ), α−1(2ρ)],
u( , 0) = u( , 1) = 0,
u(α−1(ρ)) = g1(α−1(ρ)).
Since g1∈C((0, α−1(ρ)], Xα), we ha e ha he applica ion
1( , u( ), u(·)) = λ (u( )) + γg1((α( )−ρ))
a(l(u( ))
is con inuous as long as α( )∈[ρ, 2ρ], which means ha ∈[α−1(ρ), α−1(2ρ)]. Thus, again we
jus need o check he egula i y o he solu ion. Hence, we need o see ha
k 1( , u (·)) − 1(s, us(·))k⩽w(ku( )−u(s)kα) + w(| −s|β),
bu ha ollows exac ly as be o e. To p oceed, we make g2(τ) = u(α−1(τ)) o τ∈[ρ, 2ρ].
Con inuing in his way, i gn−1(τ) = u(α−1(τ)) o τ∈[(n−2)ρ, (n−1)ρ] we ha e, o n⩾3,
(15)
∂w
∂τ −a(l(w))∂w2
∂x2=λ (w) + γgn−1(τ−ρ), τ ∈((n−1)ρ, nρ],
w(τ, 0) = w(τ, 1) = 0,
w((n−1)ρ) = gn−1((n−1)ρ).
has a mild solu ion gn: [(n−1)ρ, nρ]→H1
0(0,1). Now i w:[−ρ, ∞)→H1
0(0,1) is gi en by
(16) w(τ) =
φ(τ), τ ∈[−ρ, 0];
gn(τ), τ ∈((n−1)ρ, nρ], n ⩾1,
i is a s ong solu ion o (1).
Using he a ia ion o cons an s o mula we ha e
g1(τ) = u( ) = S( )φ(0) + Z
0
S( −s)λ (g1(α(s))) + γφ(α(s)−ρ)
a(l(g1(α(s)))) ds, 0⩽ ⩽α−1(ρ)
g2(τ) = u( ) = S( −α−1(ρ))g1(ρ) + Z
α−1(ρ)
S( −s)(λ (g2(α(s))) + γg1(α(s)−ρ))
a(l(g2(α(s)))) ds,
o ∈[α−1(ρ), α−1(2ρ)]. We know ha
g1(ρ) = S(α−1(ρ))φ(0) + Zα−1(ρ)
0
S(α−1(ρ)−s)λ (g1(α(s))) + γφ(α(s)−ρ)
a(l(g1(α(s)))) ds,
8 TOM´
AS CARABALLO, A. N. CARVALHO AND YESSICA JULIO
hen, we ob ain ha
S( −α−1(ρ))g1(ρ) = S( )φ(0) + Zα−1(ρ)
0
S( −s)λ (g1(α(s))) + γφ(α(s)−ρ)
a(l(g1(α(s)))) ds
and
g2(τ) = u( ) = S( )φ(0) + Zα−1(ρ)
0
S( −s)λ (g1(α(s))) + γφ(α(s)−ρ)
a(l(g1(α(s)))) ds
+Z
α−1(ρ)
S( −s)(λ (g2(α(s))) + γg1(α(s)−ρ))
a(l(g2(α(s)))) ds.
Simila ly, o n > 1 we ob ain ha
gn(τ) = u( ) = S( )φ(0) +
n−1
X
m=1 Zα−1(mρ)
α−1((m−1)ρ)
S( −s)(λ (gm(α(s))) + γgm−1(α(s)−ρ))
a(l(gm(α(s)))) ds
+Z
α−1((n−1)ρ)
S( −s)(λ (gn(α(s))) + γgn−1(α(s)−ρ))
a(l(gn(α(s)))) ds.
3. Compa ison esul s and global exis ence
Th oughou his sec ion, φis he ini ial unc ion o p oblem (1), K > 0 is such ha −K⩽
φ( )⩽Kand “ ⩽” is a pa ial o de ing in H1
0(Ω), ha is:
u⩽ in H1
0(Ω) ⇔u(x)⩽ (x) a.e. o xin Ω.
We conside he ini ial alue p oblems :
∂u
∂ =Au+λ (u)−K
M, >0, x∈Ω,
u(0) = −K,
u|∂Ω= 0,
(17)
∂u
∂ =Au+λ (u)+K
m, >0, x∈Ω,
u(0) = K,
u|∂Ω= 0,
(18)
and we w i e
−(u) = λ (u)−γK
M, u −(u)⩽νC0u2+γK
M+C1|u|,
+(u) = λ (u) + γK
m, u +(u)⩽νC0u2+γK
m+C1|u|,
g( , u (·)) = λ (u( )) + γu(R
0a(l(u(θ))) −ρ)dθ
a(l(u( ))) ,
whe e A:D(A)→L2(Ω) is he linea ope a o de ined in he ollowing way, D(A) = H2(Ω) ∩
H1
0(Ω) and Au =uxx, u ∈D(A); :R→Rand g: [0, T ]×C([0, T ], Xα)→Xa e con inuous
unc ions. Fo u(·)∈C([0, T ], Xα), u (·) = u(·)|[0, ], u( ) = u ( ). We shall p o e ha , unde
some s uc u al condi ion on and assuming ha o each > 0 he e is a κ=κ( ) such ha
u7→ κu + (u) is an inc easing unc ion in [− , ], hen o each n∈N, he e is a cons an Kn
such ha he solu ions u( , φ) o (2) a e globally de ined, and he e a e u( , Kn), u( , −Kn),
A DELAY NONLOCAL QUASILINEAR CHAFEE-INFANTE PROBLEM 9
solu ions o (18) and (17) wi h K eplaced by Kn, such ha u( , −Kn)⩽u( , φ)⩽u( , Kn),
∈[α−1((n−1)ρ), α−1(nρ)].
Tha is, hanks o he ac ha he solu ions o (17) and (18) a e globally de ined, we can
gua an ee ha he solu ions o (2) a e de ined in he in e al [0, α−1(ρ)] and he e is a posi i e
cons an K1such ha −K1⩽u( , φ)⩽K1, 0 ⩽ ⩽α−1(ρ). We can he e o e i e a e his
p ocedu e o ob ain ha he solu ions o (2) a e globally de ined.
To p o e esul s desc ibed abo e, we need o impose he s uc u al condi ion (S) s a ed in he
in oduc ion o he non-linea o cing e m. This condi ion ensu es ha he solu ions o (18)
and (17) a e global and consequen ly, using he p ocedu e desc ibed abo e we ob ain ha he
solu ions o (1) a e global.
Obse e ha condi ion (D) imposes a es ic ion on he cons an C0 ha appea s in condi ion
(S). I was used in [5] o ensu e he exis ence o pullback a ac o . He e some special ca e
needs o be aken due o he p ocedu e desc ibed abo e wi h changing cons an s Ka in e als
o leng h ρ. In ac , he ollowing esul holds
Theo em 3.1. Assume ha (S) holds o a con inuous unc ion :R→Rsuch ha , o e e y
> 0, he e exis s a cons an κ=κ( )>0such ha s7→ κs + (s)is inc easing in [− , ]. Le
φ: [−ρ, 0] →H1
0(0,1) and K > 0such ha −K⩽φ(τ)(x) = φ(α( ))(x)⩽K o all τ∈[−ρ, 0]
and x∈[0,1]. I u( , φ)is a solu ion o
(19)
u =uxx +g( , u (·)), > 0, x ∈Ω,
u( , 0) = u( , 1) = 0,
u( ) = φ(α( )), ∈[α−1(−ρ),0],
he e a e u( , −K),u( , K)solu ions o (17) and (18) such ha u( , −K)⩽u( , φ)⩽u( , K),
o ∈[0, α−1(ρ)]. As an immedia e consequence o his, u(·, φ)is de ined o all ⩾0.
P oo . We will use he i e a i e s ep me hod. Fo ha , i s we conside ∈(0, α−1(ρ)], hen,
α( )−ρ∈(−ρ, 0],and w(τ−ρ) = φ(α( )−ρ), so ha he equa ion (1) becomes he ini ial alue
p oblem
(20)
u =uxx +λ (u) + γφ(α( )−ρ)
a(l(u)) , > 0, x ∈Ω,
u( , 0) = u( , 1) = 0,
u(0) = φ(0).
Using [5, Co olla y 4.4], since o −K⩽φ(0) ⩽K, we ob ain he exis ence o u+( , K),and
u−( , −K), solu ions (18) and (17), de ined in [0, α−1(ρ)], such ha u−( , −K)⩽u( , φ)⩽
u+( , K) o ∈[0, α−1(ρ)]. Now, aking K1= sup ∈[0,α−1(ρ)],x∈[0,1] |u( , φ)(x)|, we may epea
his p ocedu e o ensu e ha u( , φ) is de ined in [α−1(ρ), α−1(2ρ)] and, by induc ion, o all
⩾0. To ha , in each s ep, we use compa ison in he in e al [α−1(iρ), α−1((i+ 1)ρ)] o ensu e
16 TOM´
AS CARABALLO, A. N. CARVALHO AND YESSICA JULIO
[17] Li, Y., Ca alho, A. N., Luna, T. L. M., and Mo ei a, E. M. A non-au onomous bi u ca ion p oblem o
a non-local scala one-dimensional pa abolic equa ion. Commun. Pu e Appl. Anal. 19, 11 (2020), 5181–5196.
[18] Melnik, V. S., and Vale o, J. On a ac o s o mul i alued semi- lows and di e en ial inclusions. Se -
Valued Analysis 6, 1 (1998), 83–111.
[19] Michel, C., and Lue, M. Asymp o ic beha iou o some nonlocal di usion p oblems. Applicable Analysis
80, 3-4 (2001), 279–315.
[20] Pazy, A. Semig oups o Linea Ope a o s and Applica ions o Pa ial Di e en ial Equa ions. Applied ma h-
ema ical sciences. Sp inge , 1983.
(TC) Dep o. Ecuaciones Di e enciales y Anal. Num., Facul ad de Ma em´
a icas, Uni e sidad de
Se illa, C/ Ta ia s/n, 41012-Se illa (Spain)
Email add ess, TC: [email p o ec ed]
(ANC and YJ) Ins i u o de Ciˆ
encias Ma em´
a icas e de Compu ac¸˜
ao Uni e sidade de S˜
ao Paulo,
Campus de S˜
ao Ca los, Caixa Pos al 668, S˜
ao Ca los SP, B azil.
Email add ess, ANC: [email p o ec ed]
Email add ess, YJ: [email p o ec ed]
