EXISTENCE OF PULLBACK ATTRACTOR
FOR A REACTION-DIFFUSION EQUATION
IN SOME UNBOUNDED DOMAINS WITH
NON-AUTONOMOUS FORCING TERM IN H−1
MAR´
IA ANGUIANO, TOM´
AS CARABALLO, & JOS´
E REAL
Dp o. Ecuaciones Di e enciales y An´alisis Num´e ico
Uni e sidad de Se illa
Apdo. de Co eos 1160
41080-Se illa (Spain)
E-mails: [email p o ec ed], ca abal[email p o ec ed], j e[email p o ec ed]
To he Memo y o P o esso Vale y S. Melnik
The exis ence o a pullback a ac o in L2(Ω) o he ollowing non-au onomous eac ion-
di usion equa ion 




∂u
∂ − 4u= (u) + h( ),in Ω ×(τ, +∞),
u= 0,on ∂Ω×(τ, +∞),
u(x, τ) = uτ(x), x ∈Ω,
(1)
is p o ed in his pape , when he domain Ω is no necessa ily bounded bu sa is ying he
Poinca ´e inequali y, and h∈L2
loc(R;H−1(Ω)). The main concep used in he p oo is he
asymp o ic compac ness o he p ocess gene a ed by he p oblem.
Keywo ds: pullback a ac o , asymp o ic compac ness, e olu ion p ocess, non-au onomous
eac ion-di usion equa ion.
Ma hema ics Subjec Classi ica ions (2000): 35B41, 35Q35, 35Q30, 35K90, 37L30
1. In oduc ion and se ing o he p oblem
Le Ω ⊂RNbe an open se , no necessa ily
bounded and suppose ha Ω sa is ies he Poinca ´e
inequali y, i.e., he e exis s a cons an λ1>0 such
ha
ZΩ
|u(x)|2dx ≤λ−1
1ZΩ
|∇u(x)|2dx, ∀u∈H1
0(Ω) .
(2)
Le us conside he ollowing p oblem o a non-
au onomous eac ion-di usion equa ion wi h ze o
Di ichle bounda y condi ion in Ω,





∂u
∂ − 4u= (u) + h( ),in Ω ×(τ, +∞),
u= 0,on ∂Ω×(τ, +∞),
u(x, τ) = uτ(x), x ∈Ω,
(3)
whe e τ∈R,uτ∈L2(Ω), h∈L2
loc(R;H−1(Ω))
and ∈C(R) sa is ies ha he e exis cons an s
α1>0, α2>0, l≥0, and p > 2 such ha
−α1|s|p≤ (s)s≤ −α2|s|p, (4)
( (s)− ( ))(s− )≤l(s− )2∀ , s ∈R. (5)
1
2M. Anguiano, T. Ca aballo & J. Real
Using (4), i ollows ha
| (s)| ≤ α1|s|p−1∀s∈R. (6)
The aim o his pape is o show he exis ence
o a pullback a ac o in he phase space L2(Ω)
o he p oblem (3) in he case o open do-
mains no necessa ily bounded bu sa is ying
he Poinca ´e inequali y. This, and he ac ha
he non-au onomous hbelongs o he space
L2
loc(R;H−1(Ω)),a e he main no el ies o ou
p oblem.
The lack o compac ness o he injec ion
H1
0(Ω) ⊂L2(Ω) (in he case o unbounded
domains) implies ha he s anda d ech-
niques p e iously used, pa icula ly he one
in ol ing he so-called la enning p ope y (see
[Kloeden & Langa, 2007], [Li & Zhong, 2007],
[Song & Wu, 2007], [Wang & Zhong, 2008],
amongs o he s), which ha e been success ully
used when Ω is bounded and h∈L2
loc(R;L2(Ω)),
do no wo k in ou case.
Ins ead, we will use he asymp o ic compac -
ness al eady used in he case o non-au onomous
2D-Na ie -S okes (see [Ca aballo e al., 2006] and
[Ca aballo e al., 2006b]), and which was p e i-
ously used in [Rosa, 1998] o he au onomous case.
We would like o emphasize ha his echnique
seems o be he only one which allows o p o e he
main esul o his pape (namely Theo em 4.4) con-
ce ning he exis ence o pullback a ac o o ou
p oblem.
I is also wo h men ioning ha ou p oblem
has ecei ed much a en ion o e he las yea s in
he case o a bounded domain o o a less gene al
e m h, as we will ecall now.
In [Ca aballo e al., 2003] i is p o ed he exis ence
o pullback a ac o in he space L2(Ω) (and ha
i possesses ini e Hausdo dimension) when he
domain is bounded and his unbounded bu wi h
polynomial g ow h, i.e
kh( )kL2(Ω) ≤k1| |α+k2
whe e k1, k2and αa e nonnega i e cons an s.
When Ω is bounded and h∈L2
loc(R;L2(Ω)) and is
ansla ion bounded, i.e.
sup
m∈RZm+1
m
kh(s)kL2(Ω) ds < ∞,(7)
he exis ence o a pullback a ac o in he space
H1
0(Ω) is p o ed in [Song & Wu, 2007], while in
[Li & Zhong, 2007] he ansla ion bounded condi-
ion (7) is weakened o
kh(s)k2
L2(Ω) ≤Meα|s|,
whe e 0 ≤α≤λ1, and λ1deno es he i s eigen-
alue o he Laplacian.
In [Wang & Zhong, 2008], he exis ence o pullback
a ac o in H1
0(Ω) is shown o a bounded domain
and o a h∈L2
loc(R;L2(Ω)) such ha
Z
−∞
eσs kh(s)k2
L2(Ω) +kh0(s)k2
L2(Ω)ds < +∞
o all ∈Rand ce ain σ≥0.
Fo a bounded domain Ω, and a ansla ion
bounded unc ion h∈L2
loc(R;L2(Ω)), he exis ence
o a uni o m a ac o in Lp(Ω) is demons a ed in
[Song & Zhong, 2008].
Finally, he eade can ind simila esul s
o se e al a ian s o ou model in he
e e ences [Wang e al., 2007], [P izzi, 2003],
[Mo illas & Vale o, 2005], [Sun & Zhong, 2005],
among o he s.
We will p o ide in his pape a su icien condi-
ion ensu ing he exis ence o pullback a ac o in
L2(Ω) when he domain is no necessa ily bounded
and h∈L2
loc(R;H−1(Ω)). A case ha has no been
conside ed in he li e a u e ye , as a as we know.
2. Exis ence and uniqueness o solu ion
We s a e in his sec ion a esul on he exis ence
and uniqueness o solu ion o p oblem (3). Ins ead
o wo king di ec ly wi h ou equa ion, we will es-
ablish a gene al esul which, in pa icula , can be
applied o handle ou p oblem.
2.1. An abs ac esul
Le Hbe a sepa able Hilbe space wi h scala
p oduc (·,·) and no m |·|. Le Vi,i= 1, ..., m,
be m≥1 e lexi e and sepa able Banach spaces
such ha
m
[
i=1
Vi⊂H,
m
i=1
Viis dense in H, and Vi,
i= 1, ..., m is included in Hwi h con inuous injec-
ion.
Pullback A ac o s o eac ion-di usion equa ions 3
By k·kiwe deno e he no m in Vi, by k·k∗i he
no m in V0
i,i= 1, ..., m and, by V he space
V=
m
i=1
Vi,
wi h he no m
k k=
m
X
i=1
k ki,∀ ∈V.
I is easy o see ha Vis a sepa able Banach space.
We will use h·,·i o deno e he duali y p oduc
be ween V0
iand Vi, o each i= 1, ..., m.
We iden i y Hwi h i s dual H0using he Riesz
Theo em, bu i Viis a Hilbe space, we do no
iden i y Viwi h V0
i. Le u∈H, we iden i y uwi h
Tu∈ ∩m
i=1V0
isuch ha
hTu, i= (u, ),∀ ∈Vi,∀i= 1, ..., m.
Le τ∈Rbe an ini ial ime, and le Ai: (τ, ∞)×
Vi→V0
i,i= 1, ..., m, be mope a o s, in gene al
nonlinea , such ha
A1) Fo each ∈Vi, he unc ion ∈
(τ, ∞)7−→ Ai( , )∈V0
iis Lebesgue measu able.
A2) Each ope a o Aiis hemicon inuous, i.e,
o all ∈(τ, ∞) and u, , w ∈Vi, he unc ion
θ∈R7−→ hAi( , u +θ ), wi ∈ Ris con inuous.
Suppose ha he e exis 2 ≤pi<+∞,i=
1, ..., m, and he e exis cons an s c > 0, α > 0 and
λ≥0, and a nonnega i e unc ion C( )∈L1(τ, T ),
o all T > τ, such ha o each i= 1, ..., m,
A3) (Boundedness)
kAi( , )k∗i≤c(1 + k kpi−1
i),
o all ∈(τ, ∞), ∈Vi.
A4) (Mono onici y)
hAi( , )−Ai( , w), −wi+λ| −w|2≥0,
o all ∈(τ, ∞) and o all , w ∈Vi.
A5) (Coe ci i y)
Fo each i he e exis s a semino m [·]iin Vi,
such ha he e exis s λi≥0 o which [ ]i+λi| |
is ano he no m in Vi. Mo eo e , [ ]i+λi| |and
k·kia e equi alen , and
hAi( , ), i+λ| |2+C( )≥α[ ]pi
i,
o all ∈(τ, ∞) and o all ∈Vi.
Conside m unc ions
hi( )∈Lp0
i(τ, T ;V0
i),∀T > τ, i = 1, ..., m, (8)
and he ini ial condi ion
uτ∈H. (9)
I we se
A( , ) =
m
X
i=1
Ai( , ), h( ) =
m
X
i=1
hi( ),
we can conside he ollowing p oblem









u∈
m
i=1
Lpi(τ, T ;Vi)∀T > τ,
u0( ) + A( , u( )) = h( ),in D0(τ, ∞;V0),
u(τ) = uτ.
(10)
The p oo o he ollowing esul is simila o ha
o Theo em 1.4, Chap e 2 in [Lions, 1969].
Theo em 2.1. Assume A1)-A5), (8) and (9).
Then, he e exis s a unique solu ion uo (10), such
ha
u∈C([τ, ∞); H),u0∈
m
X
i=1
Lp0
i(τ, T ;V0
i),(11)
o all T > τ.
2.2. Exis ence and uniqueness o solu ion
o p oblem (3)
We use Theo em 2.1 o show he exis ence and
uniqueness o solu ion o (3).
Conside m= 2, H =L2(Ω), V1=H1
0(Ω) and
V2=Lp(Ω) ∩L2(Ω) wi h p > 2, and deno e V0
1=
H−1(Ω), V0
2=Lp0(Ω) + L2(Ω).
Recall ha |·| deno es he no m in H, by k·k1=
|∇·| we will deno e he no m in V1, and by k·k2=
k·kLp(Ω) +|·| he no m in V2.
I we se
A1( , u) = −∆u,
A2( , u) = − (u),
and
h1( ) = h( ), h2( ) = 0,
hen, i is no di icul o apply Theo em 2.1 wi h
p1= 2 and p2=p, and we ob ain
4M. Anguiano, T. Ca aballo & J. Real
Theo em 2.2. Assume ha ∈C(R)sa is ies (4)
and (5), and h∈L2
loc(R;H−1(Ω)). Then, o all
τ∈R,uτ∈L2(Ω), he e exis s a unique solu ion
u( ) = u( ;τ, uτ)o (3) such ha
u∈L2(τ, T ;H1
0(Ω)) ∩Lp(τ, T ;Lp(Ω)) ∀T > τ,
d
d (u( ), )− h∆u( ), i=h (u( )), i
+hh( ), i,in D0(τ, ∞),∀ ∈H1
0(Ω) ∩Lp(Ω) ,
u(τ) = uτ.
Mo eo e ,
u∈C([τ, ∞); L2(Ω)),
and usa is ies he ene gy equa ion,
1
2
d
d |u( )|2+|∇u( )|2=h (u( )), u( )i
+hh( ), u( )iin D0(τ, ∞).(12)
3. P elimina ies on he heo y o pullback
a ac o s
Now, we will ecall he main poin s om he heo y
o pullback a ac o s which will be needed in o de
o p o e ou objec i e (see [Ca aballo e al., 2006]
and [Ca aballo e al., 2006b] o mo e de ails).
Le us conside a p ocess (also called a wo-
pa ame e semig oup) Uon a me ic space X, i.e., a
amily {U( , τ); −∞ < τ ≤ < +∞} o con inuous
mappings U( , τ) : X→X, such ha U(τ, τ)x=x,
and
U( , τ) = U( , )U( , τ) o all τ≤ ≤ . (13)
Suppose Dis a nonemp y class o pa ame e ized
se s b
D={D( ); ∈R} ⊂ P(X),whe e P(X)
deno es he amily o all nonemp y subse s o X.
De ini ion 3.1. The p ocess U(·,·) is said o be
pullback D-asymp o ically compac i o any ∈
R, any b
D∈ D,any sequence τn→ −∞,and any
sequence xn∈D(τn), he sequence {U( , τn)xn}is
ela i ely compac (i.e. p e-compac ) in X.
De ini ion 3.2. I is said ha b
B∈ D is pullback
D-abso bing o he p ocess U(·,·) i o any ∈R
and any b
D∈ D, he e exis s a τ0( , b
D)≤ such
ha
U( , τ)D(τ)⊂B( ) o all τ≤τ0( , b
D).
De ini ion 3.3. The amily b
A={A( ); ∈R} ⊂
P(X) is said o be a pullback D-a ac o o U(·,·)
i
1. A( ) is compac o all ∈R,
2. b
Ais pullback D-a ac ing, i.e.,
lim
τ→−∞ dis (U( , τ)D(τ), A( )) = 0,
o all b
D∈ D, and all ∈R,
3. b
Ais in a ian , i.e.,
U( , τ)A(τ) = A( ), o −∞ < τ ≤ < +∞.
We ha e he ollowing esul .
Theo em 3.4. Suppose ha he p ocess U(·,·)is
pullback D-asymp o ically compac and ha b
B∈ D
is a amily o pullback D-abso bing se s o U(·,·).
Then, he amily b
A={A( ); ∈R}⊂P(X)
de ined by A( ) = Λ( b
B, ), ∈R,whe e o each
b
D∈ D
Λ( b
D, ) =
s≤ 
[
τ≤s
U( , τ)D(τ)
,
is a pullback D-a ac o o U(·,·)which sa is ies
in addi ion ha A( ) = Sb
D∈D Λ( b
D, ), o ∈R.
Fu hemo e, b
Ais minimal in he sense ha i
b
C={C( ); ∈R} ⊂ P(X)is a amily o closed
se s such ha limτ→−∞ dis (U( , τ)B(τ), C( )) = 0,
hen A( )⊂C( ).
4. Exis ence o he pullback a ac o
Now, we can p o e ou main aim in his pape .
Fi s , we need a con inui y esul which is es ab-
lished in he nex subsec ion.
4.1. Weak Con inui y
Le ∈C(R) be a unc ion, and suppose ha
sa is ies (4) and (5), and h∈L2
loc(R;H−1(Ω)).
Thanks o Theo em 2.2, we can de ine a p ocess
{U( , τ), τ≤ }in L2(Ω), as
U( , τ)uτ=u( ;τ, uτ)∀uτ∈L2(Ω) ,∀τ≤ . (14)
F om he uniqueness o solu ion o p oblem (3), i
ollows ha (14) de ines a p ocess in L2(Ω). In
Pullback A ac o s o eac ion-di usion equa ions 5
addi ion, i can be p o ed ha he p ocess de ined
by (14) is con inuous in L2(Ω).
Mo eo e , Uis weakly con inuous, and mo e
exac ly he ollowing esul holds ue. We will
deno e by “*” he weak con e gence in he co -
esponding indica ed space, while “→” will deno e
he s ong con e gence, as usual.
P oposi ion 4.1. Le {uτn} ⊂ L2(Ω) be a se-
quence con e ging weakly in L2(Ω) o an elemen
uτ∈L2(Ω). Then, o all T > τ, i ollows
U( , τ)uτn* U ( , τ)uτin L2(Ω) ∀ ≥τ,(15)
U(·, τ)uτn* U (·, τ)uτin L2(τ, T;H1
0(Ω)),(16)
U(·, τ)uτn* U (·, τ)uτin Lp(τ, T;Lp(Ω)),(17)
(U(·, τ)uτn)* (U(·, τ)uτ)in Lp0(τ, T;Lp0(Ω)).
(18)
I Ωis a bounded se , hen
U(·, τ)uτn−→ U(·, τ)uτin L2(τ, T ;L2(Ω)).(19)
P oo . F om he assump ions, we deduce ha he e
exis s a posi i e cons an Csuch ha
|uτn| ≤ C∀n≥1.(20)
Fix τ∈R, and se
un( ) = U( , τ)uτn, u( ) = U( , τ)uτ. (21)
Using (4), i ollows
d
d |un( )|2+ 2 |∇un( )|2≤ −2α2kun( )kp
Lp(Ω)
+2 hh( ), uni.
In eg a ing be ween τand , we ob ain
|un( )|2+ 2 Z
τ
|∇un(s)|2ds
+ 2α2Z
τ
kun(s)kp
Lp(Ω) ds
≤ |uτn|2+ 2 Z
τ
hh(s), unids ∀ ≥τ. (22)
On he o he hand,
Z
τ
hh(s), unids ≤Z
τ
khkH−1(Ω) |∇un|ds
≤1
2ZT
τ
khk2
H−1(Ω) ds
+1
2Z
τ
|∇un|2ds,
which, join ly wi h (20) and (22) imply
|un( )|2+Z
τ
|∇un(s)|2ds + 2α2Z
τ
kun(s)kp
Lp(Ω) ds
≤C2+Z
τ
khk2
H−1(Ω) ds ∀ > τ.
We deduce ha {un}is bounded in
L2(τ, T ;H1
0(Ω))∩Lp(τ, T ;Lp(Ω))∩C([τ, T ]; L2(Ω)),
(23)
o all T > τ.
Fix T > τ. In pa icula , {un(T)}is bounded
in L2(Ω). On he o he hand, om (6) we ha e
k (un( ))kp0
Lp0(Ω) ≤αp0
1kun( )kp
Lp(Ω) ,
whence (un) is bounded in Lp0(τ, T;Lp0(Ω)).
Then, he e exis s a subsequence {uµ}⊂{un}
such ha
uµ∗
* weak-s a in L∞(τ, T ;L2(Ω)),
uµ* in Lp(τ, T ;Lp(Ω)), (24)
uµ(T)* ξ in L2(Ω) , (25)
uµ* in L2(τ, T ;H1
0(Ω)), (26)
and
(uµ)* χ in Lp0(τ, T ;Lp0(Ω)). (27)
Now (26) imply ha
∆uµ*∆ in L2(τ, T ;H−1(Ω)).
F om (26), (27), and hanks o he equa ion
u0
µ( ) = ∆uµ( ) + (uµ( ))+h( ), (28)
i is a s anda d ma e o p o e ha we can pick
an elemen in he equi alence class o sa is ying
( ) = uτ+Z
τ
(∆ (s) + χ(s) + h(s))ds, (29)
o all ∈[τ, T ].
We a e now in posi ion o show ha ξ= (T)
and χ( ) = ( ( )).
Le w∈H1
0(Ω) ∩Lp(Ω). In eg a ing (28) be-
ween τand T, we ob ain
(uµ(T), w) = uτµ, w
+ZT
τ
h∆uµ(s) + (uµ(s))+h(s), wids,

6M. Anguiano, T. Ca aballo & J. Real
and hus
(ξ, w) = (uτ, w) + ZT
τ
h∆ (s) + χ(s) + h(s), wids,
when µ−→ +∞. By densi y and using (29), i
ollows
(T) = ξ. (30)
To p o e ha χ( ) = ( ( )),we a gue simila ly o
[Rosa, 1998]. In eg a ing he equali y
d
ds (uµ(s), w) = −(∇uµ(s),∇w)
+h (uµ(s)), wi+hh(s), wi,
be ween and +a, wi h a∈(0, T −τ), ∈
(τ, T −a),and using he H¨olde inequali y, we ob-
ain
(uµ( +a)−uµ( ), w)
≤Z +a
|∇uµ(s)| |∇w|ds
+Z +a
k (uµ(s)kLp0(Ω) kwkLp(Ω) ds
+Z +a
kh(s)kH−1(Ω) |∇w|ds
≤ |∇w|a1/2kuµkL2(τ,T ;H1
0(Ω))
+kwkLp(Ω) a1/p k (uµ)kLp0(τ,T ;Lp0(Ω))
+|∇w|a1/2khkL2(τ,T ;H−1(Ω)) ,
and hanks o (23), we deduce ha he e exis s a
cons an C(1) such ha
(uµ( +a)−uµ( ), w)
≤C(1)(a1/2+a1/p)(|∇w|+kwkLp(Ω)).
I we ake in he las inequali y w=uµ( +a)−
uµ( )∈H1
0(Ω) ∩Lp(Ω) a.e. ∈(τ, T −a),we
ob ain
|uµ( +a)−uµ( )|2
≤C(1)(a1/2+a1/p)|∇uµ( +a)− ∇uµ( )|
+C(1)(a1/2+a1/p)kuµ( +a)−uµ( )kLp(Ω) ,
a.e. ∈(τ, T −a).
In eg a ing be ween τand T−a,
ZT−a
τ
|uµ( +a)−uµ( )|2d
≤C(1)(a1/2+a1/p)ZT−a
τ
|∇uµ( +a)|d
+C(1)(a1/2+a1/p)ZT−a
τ
|∇uµ( )|d
+C(1)(a1/2+a1/p)ZT−a
τ
kuµ( +a)kLp(Ω) d
+C(1)(a1/2+a1/p)ZT−a
τ
kuµ( )kLp(Ω) d ,
i ollows
ZT−a
τ
|uµ( +a)−uµ( )|2d
≤2C(1)(a1/2+a1/p)ZT
τ
|∇uµ(s)|ds
+ 2C(1)(a1/2+a1/p)ZT
τ
kuµ(s)kLp(Ω) ds,
and using he H¨olde inequali y, we ob ain
ZT−a
τ
|uµ( +a)−uµ( )|2d
≤2C(1)(a1/2+a1/p)(T−τ)1/2kuµkL2(τ,T ;H1
0(Ω))
+ 2C(1)(a1/2+a1/p)(T−τ)1/p0kuµkLp(τ,T ;Lp(Ω)) .
Thanks o (23) we deduce ha he e exis s a con-
s an e
CTsuch ha
ZT−a
τ
|uµ( +a)−uµ( )|2d ≤e
CT(a1/2+a1/p),
o all µ, and all a∈(0, T −τ),and hus
lim
a→0sup
µZT−a
τ
|uµ( +a)−uµ( )|2d = 0. (31)
Now, o all m∈Z,m≥1, we deno e
Ωm= Ω ∩x∈RN:|x|RN< m,
whe e |·|RNdeno es he Euclidean no m in RN.
Le φ∈C1([0,+∞)) be a unc ion such ha
0≤φ(s)≤1, φ(s)=1 ∀s∈[0,1], and φ(s) =
0∀s≥2.
Fo each µand m≥1, we de ine
µ,m(x, ) = φ |x|2
RN
m2!uµ(x, ) .
Pullback A ac o s o eac ion-di usion equa ions 7
F om (23), o all m≥1, we ob ain ha
{ µ,m}µ≥1is bounded in L2(τ, T ;H1
0(Ω2m)) ∩
Lp(τ, T ;Lp(Ω2m)) ∩L∞(τ, T ;L2(Ω2m)).
In pa icula ,
lim
a→0sup
µZτ+a
τ
| µ,m( )|2
L2(Ω2m)d +
+ZT
T−a
| µ,m( )|2
L2(Ω2m)d = 0. (32)
On he o he hand, om (31) we deduce ha o
all m≥1,
lim
a→0sup
µZT−a
τ
| µ,m( +a)− µ,m( )|2
L2(Ω2m)d = 0.
(33)
Mo eo e , as Ω2mis a bounded se , hen H1
0(Ω2m)
is included in L2(Ω2m) wi h compac injec ion.
Then, by he compac ness Theo em 13.3 o
[Temam, 1983] wi h X=L2(Ω2m), Y=H1
0(Ω2m),
= 2 and G={ µ,m}µ≥1, we ob ain ha { µ,m}µ≥1
is ela i ely compac in L2τ, T;L2(Ω2m), and
hus, aking in o accoun ha µ,m(x, ) = uµ(x, )
o all x∈Ωm,we deduce ha , in pa icula , o
all m≥1
nuµ|Ωmoµ≥1is p e-compac in L2τ, T ;L2(Ωm).
(34)
I is no di icul o conclude om (34), (26) and
(2), ia a diagonal p ocedu e, he exis ence o a
subsequence {uµ
µ}µ≥1⊂ {uµ}µ≥1such ha
uµ
µ→ a.e. in Ωm×(τ, T) as µ−→ ∞ ∀m≥1.
Then, as is con inuous,
(uµ
µ)→ ( ) a.e. in Ωm×(τ, T ) ,
and as { (uµ
µ)}is bounded in Lp0(Ωm×(τ, T)),by
Lemma 1.3, Chap e 1 in [Lions, 1969], we ob ain
(uµ
µ)* ( ) weakly in Lp0τ, T;Lp0(Ωm).
F om (27)
(uµ)* χ|Ωm×(τ,T )weakly in Lp0τ, T ;Lp0(Ωm).
By he uniqueness o he weak limi , we ha e
χ= ( ) a.e. in Ωm×(τ, T )∀m≥1,
and hus, aking in o accoun ha
∞
[
m=1
Ωm= Ω, we
ob ain
χ= ( ) a.e. in Ω ×(τ, T ) . (35)
F om (29) and (35), and by he uniqueness o so-
lu ions we ha e ( ) = u( ) o all ∈[τ, T ]. And
hen, i we conside (30) in (26) and (35) in (27),
we ha e
uµ* u in L2(τ, T ;H1
0(Ω)),
uµ* u in Lp(τ, T ;Lp(Ω)),
uµ(T)* u(T) in L2(Ω) ,
(uµ)* (u) in Lp0(τ, T ;Lp0(Ω)).
Then, by a con adic ion a gumen we deduce
un* u in L2(τ, T ;H1
0(Ω)),
un* u in Lp(τ, T ;Lp(Ω)),
un(T)* u(T) in L2(Ω) ,
(un)* (u) in Lp0(τ, T ;Lp0(Ω)),
and, as T > τ has been aken a bi a ily, he i s
pa o he p oo is inished.
Now, i Ω is bounded, we deduce om (34) ha
uµ|Ωµ≥1is p e-compac in L2τ, T ;L2(Ω). (36)
Finally, (19) ollows om (36).
Rema k 4.2. F om he p oo , i is clea ha o any
m≥1,
U(·, τ)uτn→U(·, τ)uτin L2(τ, T ;L2(Ωm)).
Mo eo e , i is possible o p o e ha
U( , τ)uτn→U( , τ)uτin L2(Ωm),
o all > τ.
Rema k 4.3. No ice ha all he esul s ob ained in
he p e ious analysis hold ue o a gene al non-
emp y open subse Ω ⊂RN.
8M. Anguiano, T. Ca aballo & J. Real
4.2. The exis ence o he global pullback a -
ac o
Le Rλ1be he se o all unc ions :R→(0,+∞)
such ha
lim
→−∞ eλ1 2( ) = 0,
and deno e by Dλ1 he class o all amilies
b
D={D( ) : ∈R}⊂P(L2(Ω) ) such ha D( )⊂
B(0, b
D( )), o some b
D∈ Rλ1, whe e B(0, b
D( ))
deno es he closed ball in L2(Ω) cen e ed a ze o
wi h adius b
D( ).
Now, we can p o e he ollowing esul .
Theo em 4.4. Suppose ha Ωsa is ies (2), and
suppose ha ∈C(R)sa is ies (4) and (5) wi h
l= 0. Le h∈L2
loc(R;H−1(Ω)) such ha
Z
−∞
eλ1skh(s)k2
H−1(Ω) ds < +∞ ∀ ∈R.
Then, he e exis s a unique global pullback Dλ1-
a ac o o he p ocess U, which belongs o Dλ1,
and is de ined by (14).
P oo . Le τ∈R, and uτ∈L2(Ω) be ixed, and
deno e
u( ) = u( ;τ, uτ) = U( , τ)uτ∀ ≥τ.
Taking in o accoun (4) and he ene gy equali y,
d
d eλ1 |u( )|2+ 2eλ1 |∇u( )|2
=λ1eλ1 |u( )|2+ 2eλ1 h (u( )), u( )i
+ 2eλ1 hh( ), u( )i(37)
≤λ1eλ1 |u( )|2
+eλ1 |∇u( )|2+eλ1 kh( )k2
H−1(Ω) ,
and hus, om (2), we ob ain
d
d eλ1 |u( )|2≤eλ1 kh( )k2
H−1(Ω) .
In eg a ing be ween τand , i ollows
eλ1 |u( )|2≤Z
τ
eλ1skh(s)k2
H−1(Ω) ds +eλ1τ|uτ|2
≤Z
−∞
eλ1skh(s)k2
H−1(Ω) ds +eλ1τ|uτ|2.
Le b
D∈ Dλ1be gi en. Then
|U( , τ)uτ|2≤e−λ1 Z
−∞
eλ1skh(s)k2
H−1(Ω) ds
+eλ1(τ− ) 2
D(τ), (38)
o all uτ∈D(τ) and o all ≥τ.
Deno e by Rλ1( ) he nonnega i e numbe gi en o
each ∈Rby
R2
λ1( ) = e−λ1 Z
−∞
eλ1skh(s)k2
H−1(Ω) ds + 1. (39)
Obse e ha
lim
→−∞ eλ1 R2
λ1( ) = 0,
and, consequen ly,
Rλ1∈ Rλ1.
Now, conside he amily b
Bλ1o closed balls in
L2(Ω) b
Bλ1={Bλ1( ) : ∈R},
de ined by
Bλ1( ) =  ∈L2(Ω) : | | ≤ Rλ1( ).
I is s aigh o wa d o check ha
b
Bλ1∈ Dλ1,
and mo eo e , by (38), he amily b
Bλ1is pullback
Dλ1-abso bing o he p ocess U.
Acco ding o Theo em 3.4, o inish he p oo o he
heo em we only ha e o p o e ha Uis pullback
Dλ1-asymp o ically compac .
Le us ix b
D∈ Dλ1, a sequence τn→ −∞, a se-
quence uτn∈D(τn), and ∈R. We ha e o p o e
ha om he sequence {U( , τn)uτn}we can ex ac
a subsequence ha con e ges in L2(Ω).
As he amily b
Bλ1is pullback Dλ1-abso bing, o
each in ege k≥0, he e exis s a τD(k)≤ −k
such ha
U( −k, τ)D(τ)⊂Bλ1( −k)∀τ≤τD(k). (40)
Again, by a diagonal p ocedu e, i is no di icul
o conclude om (40), ha he e exis a subse-
quence τn0, uτn0⊂ {(τn, uτn)}, and a sequence
{wk;k≥0} ⊂ L2(Ω) such ha o all k≥0, and
wk∈Bλ1( −k),
U( −k, τn0)uτn0* wkin L2(Ω) . (41)
Pullback A ac o s o eac ion-di usion equa ions 9
Obse e ha , by P oposi ion 4.1.
w0=weak −lim
n0→∞ U( , τn0)uτn0
=weak −lim
n0→∞ U( , −k)U( −k, τn0)uτn0
=U( , −k)weak −lim
n0→∞ U( −k, τn0)uτn0.
i.e.,
U( , −k)wk=w0∀k≥0. (42)
Then, by he lowe semi-con inui y o he no m,
using (41) we ob ain
|w0| ≤ lim in
n0→∞ U( , τn0)uτn0. (43)
I we now p o e ha also
lim sup
n0→∞ U( , τn0)uτn0≤ |w0|, (44)
hen we will ha e
lim
n0→∞ U( , τn0)uτn0=|w0|.
And his, oge he wi h he weak con e gence,
will imply he s ong con e gence in L2(Ω) o
U( , τn0)uτn0 o w0.
In o de o p o e (44), conside
[u] := |∇u|2−λ1
2|u|2− h (u), ui. (45)
F om (37), and in eg a ing be ween τand ,
eλ1 |u( )|2−eλ1τ|uτ|2=−2Z
τ
eλ1s[u(s)] ds
+2 Z
τ
eλ1shh(s), u(s)ids,
i.e.,
|U( , τ)uτ|2=eλ1(τ− )|uτ|2(46)
+ 2 Z
τ
eλ1(s− )(hh(s), U(s, τ)uτi − [U(s, τ)uτ]) ds.
F om (46) i is immedia e ha o all k≥0 and all
τn0≤ −k,
U( , τn0)uτn02
=U( , −k)U( −k, τn0)uτn02(47)
=U( −k, τn0)uτn02e−λ1k
+ 2 Z
−k
eλ1(s− )h(s), U(s, −k)U( −k, τn0)uτn0ds
−2Z
−k
eλ1(s− )U(s, −k)U( −k, τn0)uτn0ds.
As, hanks o (40),
U( −k,τn0)uτn0∈Bλ1( −k)∀τn0≤τD(k), k≥0,
we ha e
lim sup
n0→∞ U( −k, τn0)uτn02e−λ1k
≤R2
λ1( −k)e−λ1k∀k≥0. (48)
On he o he hand, om (41) and P oposi ion 4.1
we deduce ha
U(·, −k)U( −k, τn0)uτn0* U(·, −k)wk(49)
in L2( −k, ;H1
0(Ω)).
Taking in o accoun ha , in pa icula ,
eλ1(s− )h(s)∈L2( −k, ;H−1(Ω)),
we ob ain om (49),
lim
n0→∞ Z
−k
eλ1(s− )h(s), U(s, −k)U( −k, τn0)uτn0ds
=Z
−k
eλ1(s− )hh(s), U(s, −k)wkids. (50)
Now we will p o e ha
Z
−k
eλ1(s− )[U(s, −k)wk]ds (51)
≤lim in
n0→∞ Z
−k
eλ1(s− )U(s, −k)U( −k, τn0)uτn0ds.
Deno e
Jk( ) = J(1)
k( ) + J(2)
k( ),
whe e
J(1)
k( ) = Z
−k
eλ1(s− )|∇ (s)|2−λ1
2| (s)|2ds,
and
J(2)
k( ) = −Z
−k
eλ1(s− )h ( ), ids,
o all ∈L2( −k, ;H1
0(Ω)) ∩Lp( −k, ;Lp(Ω)).
Then, we wan o p o e
Jk(U(·, −k)wk)
≤lim in
n0→∞ Jk(U(·, −k)U( −k, τn0)uτn0),