Bol. So . Esp. Ma . Apl.
n
o
51(2010), 917
PULLBACK ATTRACTOR FOR A NON-AUTONOMOUS
REACTION-DIFFUSION EQUATION IN SOME UNBOUNDED
DOMAINS
MARÍA ANGUIANO
Dp o. Euaiones Di e eniales y Análisis Numé io
Uni e sidad de Se illa, Apdo. de Co eos 1160,
41080-Se illa (Spain)
anguianous.es
Abs a
The exis ene o a pullbak a a o in
L2(Ω)
o he ollowing non-
au onomous ea ion-diusion equa ion
8
>
<
>
:
∂u
∂ − △u= (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 neessa ily bounded bu
sa is ying he Poina é inequali y, and
h∈L2
loc(R;H−1(Ω))
. The main
onep used in he p o o is he asymp o i ompa ness o he p o ess
gene a ed by he p oblem.
Key wo ds:
pul lbak a a o , asymp o i ompa ness, e olu ion p oess,
non-au onomous ea ion-diusion equa ion.
AMS sub je lassia ions:
35B41, 35Q35, 35Q30, 35K90, 37L30.
1 In o du ion and se ing o he p oblem
Le
Ω⊂RN
b e an op en se , no neessa ily bounded and suppose ha
Ω
sa ises he Poina é inequali y, i.e., he e exis s a ons an
λ1>0
suh ha
ZΩ|u(x)|2dx ≤λ−1
1ZΩ|∇u(x)|2dx, ∀u∈H1
0(Ω)
. (2)
Le us onside he ollowing p oblem o a non-au onomous ea ion-
diusion equa ion wi h ze o Di ihle b ounda y ondi ion in
Ω
,





∂u
∂ −△u= (u) + h( ),
in
Ω×(τ, +∞)
,
u= 0,
on
∂Ω×(τ, +∞)
,
u(x, τ) = uτ(x), x ∈Ω
,
(3)
9
10
M. Anguiano
whe e
τ∈R
,
uτ∈L2(Ω)
,
h∈L2
loc(R;H−1(Ω))
and
∈C(R)
sa ises ha
he e exis ons an s
α1>0
,
α2>0
,
l≥0
, and
p > 2
suh ha
−α1|s|p≤ (s)s≤ −α2|s|p
, (4)
( (s)− ( ))(s− )≤l(s− )2∀ , s ∈R
. (5)
The aim o his pape is o show he exis ene o a pullbak a a o in he phase
spae
L2(Ω)
o he p oblem (3) in he ase o op en domains no neessa ily
b ounded bu sa is ying he Poina é inequali y. This, and he a ha he non-
au onomous
h
b elongs o he spae
L2
loc(R;H−1(Ω)),
a e he main no el ies o
ou p oblem.
The lak o ompa ness o he inje ion
H1
0(Ω) ⊂L2(Ω)
(in he ase o
unb ounded domains) implies ha he s anda d ehniques p e iously used,
pa iula ly he one in ol ing he so-alled a enning p op e y (see [6℄, [7℄, [12℄,
[14℄, amongs o he s), whih ha e b een suess ully used when
Ω
is b ounded
and
h∈L2
loc(R;L2(Ω))
, do no wo k in ou ase.
Ins ead, we will use he asymp o i ompa ness al eady used in he ase o
non-au onomous 2D-Na ie -S okes (see [1℄ and [2℄, see also [5℄ o a lose esul ),
and whih was p e iously used in [11℄ o he au onomous ase. We would like
o emphasize ha his ehnique seems o be he only one whih allows o p o e
he main esul o his pap e (namely Theo em 4) one ning he exis ene o
pullbak a a o o ou p oblem.
I is also wo h men ioning ha ou p oblem has eei ed muh a en ion
o e he las yea s in he ase o a b ounded domain o o a less gene al e m
h
(see [3℄, [7℄, [12℄, [14℄).
Finally, he eade an nd simila esul s o se e al a ian s o ou model in
he e e enes [9℄, [10℄, among o he s.
2 Exis ene and uniqueness o solu ion
We s a e in his se ion a esul on he exis ene and uniqueness o solu ion o
p oblem (3). By
|·|
we deno e he no m in
L2(Ω)
, by
|∇·|
he no m in
H1
0(Ω)
and by
k·k∗
he no m in
H−1(Ω)
. We will use
(·,·)
o deno e he sala p o du
in
L2(Ω)
and we will use
h·,·i
o deno e he duali y p o du b e ween
H−1(Ω)
and
H1
0(Ω)
.
Theo em 1
Suppose ha
Ω
sa ises (2). Assume ha
∈C(R)
sa ises (4)
and (5), and
h∈L2
loc(R;H−1(Ω))
. Then, o al l
τ∈R
,
uτ∈L2(Ω)
, he e
exis s a unique solu ion
u( ) = u( ;τ, uτ)
o (3) suh 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
D′(τ, ∞),∀ ∈H1
0(Ω) ∩Lp(Ω) ,
u(τ) = uτ.
A ea ion-diusion equa ion in some unb ounded domains
11
Mo eo e ,
u∈C([τ, ∞); L2(Ω)),
and
u
sa ises he ene gy equa ion,
1
2
d
d |u( )|2+|∇u( )|2=h (u( )), u( )i
+hh( ), u( )i
in
D′(τ, ∞).
(6)
P oo
. The p o o o his Theo em an b e done by he me ho d o mono ony
(see [8℄).

3 P elimina ies on he heo y o pullbak a a o s
Now, we will eall he main p oin s om he heo y o pullbak a a o s whih
will b e needed o p o e ou ob je i e (see [1℄ and [2℄ o mo e de ails).
Le us onside a p o ess (also alled a wo-pa ame e semig oup)
U
on a
me i spae
X
, i.e., a amily
{U( , τ); −∞ < τ ≤ < +∞}
o on inuous
mappings
U( , τ) : X→X
, suh ha
U(τ, τ)x=x,
and
U( , τ) = U( , )U( , τ) o all τ≤ ≤ .
(7)
Supp ose ha
D
is a nonemp y lass 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ni ion 1
The p oess
U(·,·)
is said o be pul lbak
D
-asymp o ial ly
ompa i o any
∈R
, any
b
D∈ D,
any sequene
τn→ −∞,
and any sequene
xn∈D(τn)
, he sequene
{U( , τn)xn}
is ela i ely ompa (i.e. p e-ompa )
in
X
.
Deni ion 2
I is said ha
b
B∈ D
is pul lbak
D
-abso bing o he p oess
U(·,·)
i o any
∈R
and any
b
D∈ D
, he e exis s a
τ0( , b
D)≤
suh ha
U( , τ)D(τ)⊂B( )
o al l
τ≤τ0( , b
D).
Deni ion 3
The amily
b
A={A( ); ∈R} ⊂ P(X)
is said o be a pul lbak
D
-a a o o
U(·,·)
i
1.
A( )
is ompa o al l
∈R
,
2.
b
A
is pul lbak
D
-a a ing, i.e.,
lim
τ→−∞dis (U( , τ)D(τ), A( )) = 0,
o al l
b
D∈ D
, and al l
∈R
,
3.
b
A
is in a ian , i.e.,
U( , τ)A(τ) = A( ),
o
−∞ < τ ≤ < +∞.
12
M. Anguiano
We ha e he ollowing esul (see [2℄ o mo e de ails).
Theo em 2
Suppose ha he p oess
U(·,·)
is pul lbak
D
-asymp o ial ly
ompa and ha
b
B∈ D
is a amily o pul lbak
D
-abso bing se s o
U(·,·)
.
Then, he amily
b
A={A( ); ∈R} ⊂ P(X)
dened by
A( ) = Λ( b
B, ), ∈
R,
whe e o eah
b
D∈ D
Λ( b
D, ) =
s≤ 
[
τ≤s
U( , τ)D(τ)
,
is a pul lbak
D
-a a o o
U(·,·)
whih sa ises in addi ion ha
A( ) =
Sb
D∈D Λ( b
D, ),
o
∈R.
Fu hemo e,
b
A
is minimal in he sense ha
i
b
C={C( ); ∈R} ⊂ P(X)
is a amily o losed se s suh ha
limτ→−∞ dis (U( , τ)B(τ), C( )) = 0,
hen
A( )⊂C( )
.
4 Exis ene o he pullbak a a o
Now, we an p o e ou main esul in his pap e . Fi s , we need a on inui y
esul whih is es ablished in he nex subse ion.
4.1 Weak Con inui y
Assume ha he un ion
∈C(R)
sa ises (4) and (5), and ha
h∈
L2
loc(R;H−1(Ω))
.
Thanks o Theo em 1, we an dene a p o ess
{U( , τ)
,
τ≤ }
in
L2(Ω)
,
as
U( , τ)uτ=u( ;τ, uτ)∀uτ∈L2(Ω) ,∀τ≤
. (8)
F om he uniqueness o solu ion o p oblem (3), i ollows ha (8) denes a
p o ess in
L2(Ω)
. In addi ion, i an b e p o ed ha he p o ess dened by (8)
is on inuous in
L2(Ω)
.
Mo eo e ,
U
is weakly on inuous, and mo e exa ly he ollowing esul
holds ue. We will deno e by 
⇀
 he weak on e gene in he o esp onding
india ed spae, while 
→
 will deno e he s ong on e gene, as usual.
P op osi ion 3
Le
{uτn} ⊂ L2(Ω)
be a sequene on e ging weakly in
L2(Ω)
o an elemen
uτ∈L2(Ω)
. Then, o al l
T > τ
, i ol lows
U( , τ)uτn⇀ U ( , τ)uτ
in
L2(Ω) ∀ ≥τ
,
(9)
U(·, τ)uτn⇀ U (·, τ)uτ
in
L2(τ, T ;H1
0(Ω)),
(10)
U(·, τ)uτn⇀ U (·, τ)uτ
in
Lp(τ, T ;Lp(Ω))
,
(11)
(U(·, τ)uτn)⇀ (U(·, τ)uτ)
in
Lp′(τ, T ;Lp′(Ω)).
(12)
I
Ω
is a bounded se , hen
U(·, τ)uτn−→ U(·, τ)uτ
in
L2(τ, T ;L2(Ω))
.
(13)
A ea ion-diusion equa ion in some unb ounded domains
13
P oo
. This esul may be p o ed in muh he same way as Theo em 1, and
using simila a gumen s o [11℄.

4.2 The exis ene o he global pullbak a a o
Le
Rλ1
b e he se o all un ions
:R→(0,+∞)
suh ha
lim
→−∞eλ1 2( ) = 0
,
and deno e by
Dλ1
he lass o all amilies
b
D={D( ) : ∈R} ⊂ P(L2(Ω) )
suh
ha
D( )⊂B(0, b
D( ))
, o some
b
D∈ Rλ1
, whe e
B(0, b
D( ))
deno es he
losed ball in
L2(Ω)
en e ed a ze o wi h adius
b
D( )
.
Now, we an p o e he ollowing esul .
Theo em 4
Suppose ha
Ω
sa ises (2), and suppose ha
∈C(R)
sa ises
(4) and (5) wi h
l= 0
. Le
h∈L2
loc(R;H−1(Ω))
be suh ha
Z
−∞
eλ1skh(s)k2
H−1(Ω) ds < +∞ ∀ ∈R
.
Then, he e exis s a unique global pul lbak
Dλ1
-a a o o he p oess
U
, whih
belongs o
Dλ1
, and is dened by (8).
P oo
. We only gi e he main ideas o he p o o . Le
τ∈R
, and
uτ∈L2(Ω)
b e xed, and deno e
u( ) = u( ;τ, uτ) = U( , τ)uτ∀ ≥τ
.
Le
b
D∈ Dλ1
b e gi en. Taking in o aoun (2), (4), he ene gy equali y and
in eg a ing b e ween
τ
and
,
|U( , τ)uτ|2≤e−λ1 Z
−∞
eλ1skh(s)k2
H−1(Ω) ds
+eλ1(τ− ) 2
D(τ)
, (14)
o all
uτ∈D(τ)
and o all
≥τ
.
Deno e by
Rλ1( )
he nonnega i e numbe gi en o eah
∈R
by
R2
λ1( ) = e−λ1 Z
−∞
eλ1skh(s)k2
H−1(Ω) ds + 1
. (15)
Obse e ha
Rλ1∈ Rλ1
. Now, onside he amily
b
Bλ1
o losed balls in
L2(Ω)
,
b
Bλ1={Bλ1( ) : ∈R}
, dened by
Bλ1( ) =  ∈L2(Ω) : | | ≤ Rλ1( )
. I is
s aigh o wa d o hek ha
b
Bλ1∈ Dλ1,
and mo eo e , by (14), he amily
b
Bλ1
is pullbak
Dλ1
-abso bing o he p o ess
U
.
Ao ding o Theo em 2, o nish he p o o o he heo em we only ha e o
p o e ha
U
is pullbak
Dλ1
-asymp o ially ompa .

14
M. Anguiano
Le us x
b
D∈ Dλ1
, a sequene
τn→ −∞
, a sequene
uτn∈D(τn)
, and
∈R
. We ha e o p o e ha om he sequene
{U( , τn)uτn}
we an ex a a
subsequene ha on e ges in
L2(Ω)
.
As he amily
b
Bλ1
is pullbak
Dλ1
-abso bing, by a diagonal p o edu e, i is no
diul o onlude ha he e exis a subsequene
τn′, uτn′⊂ {(τn, uτn)}
,
and a sequene
{wk;k≥0} ⊂ L2(Ω)
suh ha o all
k≥0
, and
wk∈
Bλ1( −k)
,
U( −k, τn′)uτn′⇀ wk
in
L2(Ω)
, (16)
and
|w0| ≤ lim in
n′→∞ U( , τn′)uτn′
. (17)
I we now p o e ha also
lim sup
n′→∞ U( , τn′)uτn′≤ |w0|
, (18)
hen we will ha e
lim
n′→∞U( , τn′)uτn′=|w0|
.
And his, oge he wi h he weak on e gene, will imply he s ong on e gene
in
L2(Ω)
o
U( , τn′)uτn′
o
w0
.
In o de o p o e (18), onside
[u] := |∇u|2−λ1
2|u|2−h (u), ui
. Taking in o
aoun (2), (4), he ene gy equali y and in eg a ing b e ween
τ
and
, i is
immedia e ha o all
k≥0
and all
τn′≤ −k
,
U( , τn′)uτn′2
(19)
=U( −k, τn′)uτn′2e−λ1k
+ 2 Z
−k
eλ1(s− )h(s), U(s, −k)U( −k, τn′)uτn′ds
−2Z
−k
eλ1(s− )U(s, −k)U( −k, τn′)uτn′ds
.
Now we will p o e ha
Z
−k
eλ1(s− )[U(s, −k)wk]ds
(20)
≤lim in
n′→∞ Z
−k
eλ1(s− )U(s, −k)U( −k, τn′)uτn′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
,
A ea ion-diusion equa ion in some unb ounded domains
15
and
J(2)
k( ) = −Z
−k
eλ1(s− )h ( ), ids
,
o all
∈L2( −k, ;H1
0(Ω)) ∩Lp( −k, ;Lp(Ω))
.
We also ob ain om (16) and using P op osi ion 3
lim in
n′→∞ (J(1)
k(U(·, −k)U( −k, τn′)uτn′)
≥J(1)
k(U(·, −k)wk)
. (21)
Using (5) wi h
l= 0
, om (16) and P op osi ion 3 we easily ob ain
lim
n′→∞in J(2)
k(U(·, −k)U( −k, τn′)uτn′)
≥J(2)
k(U(·, −k)wk)
.
The e o e (20) is easily ob ained om he las inequali y and (21).
Then, aking in o aoun ha he amily
b
Bλ1
is pullbak
Dλ1
-abso bing, om
(16), using P op osi ion 3 and hanks o (19) and (20), we ob ain
lim
n′→∞sup U( , τn′)uτn′2
≤R2
λ1( −k)e−λ1k+|w0|2
,
o all
k≥1.
Taking in o aoun (15), we easily ob ain
lim
n′→∞sup U( , τn′)uτn′2≤ |w0|2
.

Aknowledgemen s.
This wo k has b een pa ially supp o ed by Jun a de
Andaluía unde p o je P07-FQM-02468.
Re e enes
[1℄ T. Ca aballo, G. Lukaszewiz, and J. Real,
Pul lbak a a o s o
asymp o ial ly ompa non-au onomous dynamial sys ems
. Nonlinea
Analysis, 64 (2006), 484498.
[2℄ T. Ca aballo, G. Lukaszewiz, and J. Real,
Pul lbak a a o s o non-
au onomous 2D Na ie -S okes equa ions in unbounded domains
. C. R.
Ma h. Aad. Si. Pa is, 342 (2006), 263268.
[3℄ T. Ca aballo, J. A. Langa, and J. Vale o,
Dimension o a a o s o
nonau onomous ea ion-diusion equa ions
. ANZIAM J., 45 (2003), 207
222.
[4℄ H. C auel, A. Debusshe, and F. Flandoli,
Random a a o s
. J. Dyn.
Di. Eq., 9 (1995), no. 2, 307341.
16
M. Anguiano
[5℄ M.J. Ga ido-A ienza and P. Ma ín-Rubio,
Na ie -S okes equa ions wi h
delays on unbounded domains
. Nonlinea Anal., 64 (2006), no. 5, 1100
1118.
[6℄ P.E. Klo eden and J.A. Langa,
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