A ac o s o 2D-Na ie -S okes models wi h
delays
T. Ca aballo ∗and J. Real
Dp o. de 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
Abs ac
The exis ence o an a ac o o a 2D-Na ie -S okes sys em wi h delay is p o ed.
The heo y o pullback a ac o s is success ully applied o ob ain he esul s since
he abs ac unc ional amewo k conside ed u ns ou o be nonau onomous. How-
e e , on some occasions, he a ac o s may a ac no only in he pullback sense
bu in he o wa d one as well. Also, his o mula ion allows o ea , in a uni ied way,
e ms con aining a ious classes o delay ea u es (cons an , a iable, dis ibu ed de-
lays, e c.). As a consequence, some esul s o he au onomous model a e deduced
as pa icula cases o ou gene al o mula ion.
Key wo ds: 2D-Na ie -S okes equa ions, pullback a ac o , o wa d a ac o ,
a iable delay, dis ibu ed delay
1991 MSC: 35R10, 35B40, 47H20, 58F39, 73K70
1 In oduc ion
Na ie -S okes equa ions ha e ecei ed e y much a en ion o e he las decades
due o hei impo ance in he unde s anding o luids mo ion and u bulence
(see [1], [10], [12], [15], [18], [26], amongs o he s). Ve y ecen ly, in [7],[8] we
s a ed an in es iga ion in ol ing Na ie -S okes models in which he o cing
e m con ains some he edi a y ea u es. These si ua ions may appea , o in-
s ance, when we wan o con ol he sys em by applying a o ce which akes
?Pa ly suppo ed by Minis e io de Ciencia y Tecnolog´ıa (Spain) and FEDER
(Eu opean Communi y), p ojec s HA2001-0075 and BFM2002-03068.
∗Co esponding au ho
Email add esses: [email p o ec ed] (T. Ca aballo), [email p o ec ed] (J. Real).
A icle o be published in J. Di e en ial Equa ions 28 Oc obe 2003
in o accoun no only he p esen s a e o he sys em bu he his o y o he
solu ions.
No doub a all, he asymp o ic beha iou o dynamical sys ems is an in e -
es ing and challenging p oblem, since i can p o ide use ul in o ma ion on
he u u e e olu ion o he sys em. This will be he main aim o his pape .
To his espec , some su icien condi ions ensu ing he exponen ial beha iou
o solu ions o a 2D-Na ie -S okes delay model we e p o ed in [8]. Roughly
speaking, when he iscosi y is la ge, he e exis s a unique s a iona y solu ion
o some models and his solu ion is exponen ially s able (which means ha
he global a ac o o hese si ua ions becomes he unique s a iona y solu-
ion). Howe e , when he iscosi y is small i is expec ed some hing simila
o wha happens in he non-delay amewo k, i.e., he exis ence o a com-
pac in a ian a ac ing se (a global a ac o o he associa ed semig oup).
Bu on his occasion, we need o be ca e ul wi h ou analysis since we ha e
o conside he semig oup in a di e en phase space. In ac , he dynamical
sys em needs o be de ined in a phase space o ajec o ies ( o a simila ap-
p oach o nondelay models see [20]). To be mo e p ecise, ou in en ion is o
conside an abs ac unc ional model o he delay so ha a wide ange o
he edi a y cha ac e is ics (cons an o a iable delay, dis ibu ed delay, e c)
can be ea ed in a uni ied way. Al hough o some pa icula cases, he e-
sul ing abs ac equa ion becomes au onomous (e.g. o cons an delays) and
he s anda d echnique o au onomous dynamical sys ems can be adap ed o
sol e he p oblem, mos cases need o a nonau onomous model o desc ibe he
sys em and, consequen ly, a nonau onomous echnique is necessa y o han-
dle he p oblem. Being possible a ious op ions o deal wi h he p oblem o
a ac o s o nonau onomous sys ems (ke nel sec ions [10], skew-p oduc o -
malism [24], e c.), o ou pa icula si ua ion we ha e p e e ed o choose ha
o pullback a ac o (see [9], [16], [17], [23]) which has also p o ed ex emely
ui ul, pa icula ly in he case o andom dynamical sys ems (see [13], [14],
[23]). The main eason is ha , al hough when one knows he explici depen-
dence o he delay (e.g. as in he cases o a iable o dis ibu ed delays) i
could be possible o cons uc he pa ame e s se which is needed o ha e a
skew-p oduc low (o he symbols se in he heo y o ke nel sec ions), i is no
known how o cons uc hem when one is ying o de elop a gene al heo y
conce ning abs ac delay e ms, i.e. unde a gene al unc ional o mula ion
(see [6] o mo e de ails). I is also wo h poin ing ou ha , a e p o ing ou
heo y o he nonau onomous delay model, we will ob ain simila esul s o
an au onomous e sion in a s aigh o wa d way.
As a as we know, no many pape s ha e been published dealing wi h he
exis ence o a ac o s o pa ial di e en ial equa ions wi h delay. We would
like o men ion ha , o ins ance, a linea pa ial di e en ial equa ion con ain-
ing a nonlinea au onomous e m wi h ini e delay is conside ed in [11], and a
class o e a ded pa ial di e en ial equa ions o second o de wi h espec o
2
he ime a iable is analyzed in [3]. Howe e , we do no know any wo k con-
ce ning nonau onomous delay e ms. Some esul s in he ini e dimensional
con ex can be ound in [6], [5] (see also Malle -Pa e and Sell [21], [22] o
some p elimina y and in e es ing esul s on he s uc u e o he a ac o s o
o dina y di e en ial delay sys ems).
In Sec ion 2, we will ecall some p elimina y esul s on he exis ence, unique-
ness and egula i y o solu ions o ou model as well as some esul s on he
heo y o pullback a ac o s. Sec ion 3 is de o ed o p o e he exis ence o he
a ac o o ou nonau onomous delay models. In ac , unde sui able uni o m
assump ions we p o e he exis ence o a pullback a ac o . In addi ion, some
applica ions a e exhibi ed ( a iable and dis ibu ed delays), and we also poin
ou how can be ob ained co esponding esul s o he au onomous amewo k
as a pa icula case o ou gene al model.
2 P elimina ies
In his sec ion we will include some p elimina ies on he exis ence and unique-
ness o solu ions o ou p oblem and ecall some ac s om he heo y o
pullback a ac o s.
2.1 Exis ence and uniqueness o solu ions
The gene al o mula ion o ou model is he ollowing. Le Ω ⊂R2be an open
bounded se wi h egula bounda y Γ, and conside he ollowing unc ional
2D−Na ie -S okes p oblem ( o u he de ails and no a ions see Lions [19]
and Temam [25]):































∂u
∂ −ν∆u+
2
X
i=1
ui
∂u
∂xi
= − ∇p+g( , u ) in (τ, +∞)×Ω,
di u= 0 in (τ, +∞)×Ω,
u= 0 on (τ, +∞)×Γ,
u(τ, x) = u0(x), x ∈Ω,
u( , x) = φ( −τ, x), ∈(τ−h, τ)x∈Ω,
whe e ν > 0 is he kinema ic iscosi y, uis he eloci y ield o he luid, p he
p essu e, τ∈R he ini ial ime, u0 he ini ial eloci y ield, a nondelayed
ex e nal o ce ield, gano he ex e nal o ce wi h some he edi a y cha ac e is-
3
ics and φ he ini ial da um in he in e al o ime (−h, 0),whe e his a ixed
posi i e numbe .
To se ou p oblem in he abs ac amewo k, we conside he ollowing usual
abs ac spaces:
V=nu∈(C∞
0(Ω))2: di u= 0o,
H= he closu e o Vin (L2(Ω))2wi h no m |·| ,and inne p oduc (·,·) whe e
o u, ∈(L2(Ω))2,
(u, ) =
2
X
j=1 ZΩuj(x) j(x)dx,
V= he closu e o Vin (H1
0(Ω))2wi h no m k·k ,and associa ed scala p oduc
((·,·)),whe e o u, ∈(H1
0(Ω))2,
((u, )) =
2
X
i,j=1 ZΩ
∂uj
∂xi
∂ j
∂xi
dx.
I ollows ha V⊂H≡H0⊂V0,whe e he injec ions a e dense and compac .
Finally, we will use k·k∗ o he no m in V0and h·,·i o he duali y pai ing
be ween Vand V0.
Now we de ine he ilinea o m bon V×V×Vby
b(u, , w) =
2
X
i,j=1 ZΩui
∂ j
∂xi
wjdx∀u, , w ∈V.
Gi en T > τ and u: (τ−h, T)→(L2(Ω))2, o each ∈(τ, T) we deno e by u
he unc ion de ined on (−h, 0) by he ela ion u (s) = u( +s), s ∈(−h, 0).We
also deno e CH=C0([−h, 0]; H), CV=C0([−h, 0]; V), L2
H=L2(−h, 0; H)
and L2
V=L2(−h, 0; V).
Now, we es ablish sui able hypo heses on he e m con aining he delay. Le
g:R×CH→(L2(Ω))2sa is y he ollowing assump ions:
(I) ∀ξ∈CH, ∈R→g( , ξ)∈(L2(Ω))2is measu able,
(II) ∀ ∈R,g( , 0) = 0,
(III) ∃Lg>0 s. .∀ ∈R,∀ξ, η ∈CH
|g( , ξ)−g( , η)| ≤ Lgkξ−ηkCH,
4
(IV) ∃m0≥0, Cg>0: ∀m∈[0, m0], τ ≤ , u, ∈C0([τ−h, ]; H)
Z
τems |g(s, us)−g(s, s)|2ds≤C2
gZ
τ−hems |u(s)− (s)|2ds.
Obse e ha (I)-(III) imply ha gi en u∈C0([τ−h, T]; H), he unc ion
gu: ∈[τ, T]→(L2(Ω))2de ined by gu( ) = g( , u )∀ ∈[τ, T], is measu able
(see Bensoussan e al. [2]) and, in ac , belongs o L∞(τ, T; (L2(Ω))2). Then,
hanks o (IV), he mapping
G:u∈C0([τ−h, T]; H)→gu∈L2(τ, T; (L2(Ω))2)
has a unique ex ension o a mapping e
Gwhich is uni o mly con inuous om
L2(τ−h, T;H) in o L2(τ, T; (L2(Ω))2). F om now on, we will deno e g( , u ) =
e
G(u)( ) o each u∈L2(τ−h, T;H), and hus, ∀ ∈[τ, T],∀u, ∈L2(τ−
h, T;H),we will ha e
Z
τ|g(s, us)−g(s, s)|2
(L2(Ω))2ds≤C2
gZ
τ−h|u(s)− (s)|2ds.
Assume now ha u0∈H,φ∈L2
H, ∈L2
loc(R;V0), and g:R×CH→
(L2(Ω))2sa is ies hypo heses (I)-(IV). Fo example, when he unc ion gis
de ined by g( , φ) = G(φ(−ρ( )) o a sui able di e en iable delay unc ion
ρand a Lipschi z con inuous mapping G:R2→R2, he assump ions abo e
hold (see Ca aballo & Real [7] o mo e de ails and examples). Se A:V→
V0as hAu, i= ((u, )), B :V×V→V0by hB(u, ), wi=b(u, , w),
∀u, , w ∈V, and B(u) = B(u, u).Deno ing D(A)=(H2(Ω))2∩V, hen
Au =−P∆u, ∀u∈D(A),(P he o ho-p ojec o om (L2(Ω))2on o H). Fo
each τ∈Rwe conside he p oblem:













To ind u∈L2(τ−h, T;H)∩L2(τ, T;V)∩L∞(τ, T;H)∀T > τ,
d
d u( ) + νAu( ) + B(u( )) = ( ) + g( , u ) in D0(τ, +∞;V0),
u(τ) = u0, u( ) = φ( −τ), ∈(τ−h, τ),
(1)
The ollowing esul can be p o ed as Theo em 2.3 in Ca aballo & Real [8].
Theo em 1 Le us conside u0∈H,φ∈L2
H, ∈L2
loc(R;V0), and assume
ha g:R×CH→(L2(Ω))2sa is ies hypo heses (I)-(IV). Then, o each
τ∈R,
a) The e exis s a unique solu ion o (1) which, in addi ion, belongs o he
space C0([τ, +∞); H).
b) I ∈L2
loc(R; (L2(Ω))2)and u0∈V, hen he solu ion u o (1) is a s ong
5

solu ion, ha is,
u∈L2(τ, T;D(A)) ∩C0([τ, T]; V)and u0∈L2(τ, T;H)∀T > τ. (2)
In pa icula , i φ∈CVand u0=φ(0), hen u∈C0([τ−h, +∞); V).
2.2 P elimina ies on pullback a ac o s
We now discuss he heo y o pullback a ac o s, as de eloped in Kloeden and
S onie [16], Kloeden and Schmal uss [17], and C auel e al. [14]. As i is well
known, in he case o nonau onomous di e en ial equa ions he ini ial ime is
jus as impo an as he inal ime, and he classical semig oup p ope y o
au onomous dynamical sys ems is no longe a ailable.
Ins ead o a amily o one ime-dependen maps S( ) we need o use a wo-
pa ame e p ocess U( , τ) on he comple e me ic space X(which in ou case
will be CHo H×L2
H) (c . Sell [24]); U( , τ)ψuses o deno e he alue o he
solu ion a ime which was equal o he ini ial alue ψa ime τ.
The semig oup p ope y is eplaced by he p ocess composi ion p ope y
U( , τ)U(τ, ) = U( , ) o all ≥τ≥ ,
and, ob iously, he ini ial condi ion implies U(τ, τ) =Id. As wi h he semi-
g oup composi ion S( )S(τ) = S( +τ), his jus exp esses he uniqueness o
solu ions.
I is also possible o p esen he heo y wi hin he mo e gene al amewo k o
cocycle dynamical sys ems. In his case he second componen o Uis iewed
as an elemen o some pa ame e space J, so ha he solu ion can be w i en as
U( , p)φ, and a shi map θ :J→Jis de ined so ha he p ocess composi ion
becomes he cocycle p ope y,
U( +τ, p) = U( , θτp)U(τ, p).
Howe e , when one ies o de elop a heo y which can include se e al kinds
o he edi a y cha ac e is ics unde a uni ied abs ac o mula ion, wha means
ha we do no know a p io i he explici exp ession o he delay appea ing in
he p oblem, he con ex o cocycle (o skew-p oduc lows) may no be he
mos app op ia e o deal wi h he p oblem, since i is no known how o con-
s uc he se J( he same happens wi h he cons uc ion o he symbols se i
one wishes o apply he heo y o ke nel sec ions as de eloped by Chepyzho
and Vishik [10]). Fo his eason, we do no pu sue his app oach he e, bu
no e ha i has p o ed ex emely ui ul, pa icula ly in he case o andom
6
dynamical sys ems. Fo a ious examples using his gene al se ing, see Kloe-
den and Schmal uss [17], o Sell [24]. Fo his eason, pullback a ac o s a e
o en e e ed o as ‘cocycle a ac o s’.
As in he s anda d heo y o a ac o s, we seek an in a ian a ac ing se .
Howe e , since he equa ion is nonau onomous his se also depends on ime.
De ini ion 2 Le Ube a p ocess on a comple e me ic space X. A amily o
compac se s {A( )} ∈Ris said o be a (global) pullback a ac o o Ui , o
all τ∈R, i sa is ies
i) U( , τ)A(τ) = A( ) o all ≥τ, and
ii) lims→∞ dis (U( , −s)D, A( )) = 0, o all bounded subse s Do X.
The pullback a ac o is said o be uni o m i he a ac ion p ope y is
uni o m in ime, i.e.
lim
s→∞ sup
∈R
dis (U( , −s)D, A( )) = 0, o all bounded subse s D⊂X.
De ini ion 3 A amily o compac se s {A( )} ∈Ris said o be a (global) o -
wa d a ac o o Ui , o all τ∈R, i sa is ies
i) U( , τ)A(τ) = A( ) o all ≥τ, and
ii) lim →∞ dis (U( , τ)D, A( )) = 0, o all bounded subse s Do X.
The o wa d a ac o is said o be uni o m i he a ac ion p ope y is
uni o m in ime, i.e.
lim
→∞ sup
τ∈R
dis (U( +τ, τ)D, A( +τ)) = 0, o all bounded subse s D⊂X.
The eade is e e ed o Cheban e al. [9] o a de ailed analysis on he ela-
ionship be ween hese concep s. We emphasize ha he p ope y o uni o m
pullback a ac ion is equi alen o ha o uni o m o wa d a ac ion.
In he de ini ion, dis (A, B) is he Hausdo semidis ance be ween Aand B,
de ined as
dis (A, B) = sup
a∈A
in
b∈Bd(a, b), o A, B ⊆X.
P ope y i) is a gene aliza ion o he in a iance p ope y o au onomous dy-
namical sys ems. The pullback a ac ing p ope y ii) conside s he s a e o
he sys em a ime when he ini ial ime −sgoes o −∞ (see also Chepyzho
and Vishik [10])
The no ion o an a ac o is closely ela ed o ha o an abso bing se .
De ini ion 4 The amily {B( )} ∈Ris said o be (pullback) abso bing wi h
espec o he p ocess Ui , o all ∈Rand all D⊂Xbounded, he e exis s
7
TD( )>0such ha o all s≥TD( )
U( , −s)D⊂B( ).
The abso p ion is said o be uni o m i TD( )does no depend on he ime
a iable .
Indeed, jus as in he au onomous case, he exis ence o compac abso bing
se s is he c ucial p ope y in o de o ob ain pullback a ac o s. Fo he
ollowing esul see C auel and Flandoli [13] o Schmal uss [23].
Theo em 5 Le U( , τ)be a wo-pa ame e p ocess, and suppose U( , τ) :
X→Xis con inuous o all ≥τ. I he e exis s a amily o compac (pull-
back) abso bing se s {B( )} ∈R, hen he e exis s a pullback a ac o {A( )} ∈R,
and A( )⊂B( ) o all ∈R. Fu he mo e,
A( ) = [
D⊂X
bounded
ΛD( ),
whe e
ΛD( ) =
n∈N[
s≥n
U( , −s)D.
Rema k 6 I is wo h men ioning ha he uniqueness o he pullback a ac-
o , as de ined abo e, does no hold in gene al (see Ca aballo and Langa [4]).
Howe e , he one gi en in he p eceding heo em is minimal wi h espec o se
inclusion (see C auel and Flandoli [13]). Bu , i we impose in he de ini ion
o pullback a ac o ha he amily {A( )} ∈Ris uni o mly bounded (i.e. he e
exis s a bounded se B⊂Xsuch ha A( )⊂B o all ∈R) o we a e
in e es ed in inding uni o mly bounded a ac o s, hen he uniqueness o his
a ac o ollows immedia ely. A su icien condi ion ensu ing his is ha he
amily o compac abso bing se s in Theo em 5 is also uni o mly bounded. Fi-
nally, he e exis s ano he possibili y o ensu e he uniqueness o he pullback
a ac o which is ela ed o he ac ha he a ac o is asked o belong o a
ce ain class o se alued unc ions which a e a ac ed by he a ac o (see
[9]).
3 Exis ence o he a ac o
We deno e by λ1 he i s eigen alue o he ope a o A.
8
3.1 Cons uc ion o he associa ed p ocess
Now we will apply he heo y in he p e ious sec ion o p o e he exis ence
o an a ac o o ou nonau onomous Na ie -S okes model wi h delay. To
his end, we conside g:R×CH→(L2(Ω))2sa is ying (I)-(IV) and assume
ha u0∈H,φ∈L2
Hand ∈L2
loc(R;V0). Then, o each ini ial ime τ∈R,
Theo em 1 ensu es ha p oblem (1) possesses a unique solu ion u(·;τ, (u0, φ))
which belongs o he space L2(τ, T;V)∩L2(τ−h, τ;H)∩C0([τ, T]; H) o all
T > τ. We can now p oceed in wo di e en o ms o cons uc he e olu ion
p ocess which can help us in he analysis o he long- ime beha iou o ou
model. On he one hand, we can de ine a p ocess in he phase space CHas
he amily o mappings U( , τ) : CH→CHgi en by
U( , τ)φ=u (·;τ, (φ(0), φ)), o any φ∈CH,and any τ≤ . (3)
Howe e , i may seem ha he p oduc space M2
H=H×L2
Hcan be mo e
con enien since his is he usual space whe e he ini ial da a a e aken. This
space is a Hilbe space wi h associa ed no m
k(u0, φ)k2
M2
H=|u0|2+Z0
−h|φ(s)|2ds, o (u0, φ)∈M2
H.
In his way, we can de ine he co esponding p ocess as
S( , τ)(u0, φ) = (u( ;τ, (u0, φ)), u (·;τ, (u0, φ))), o (u0, φ)∈M2
H,τ≤ . (4)
Al hough, due o he con inui y o ajec o ies, i seems sensible o conside
only he i s case, wi h a li le mo e o addi ional wo k we will be able o
handle bo h si ua ions a he same ime. O cou se, i is sensible o expec
ha he a ac o s o bo h si ua ions should be ela ed. We will p o e ha
his is indeed he case.
Rema k 7 Associa ed o he p ocesses U(·,·)and S(·,·)we will conside he
amily o mappings ˜
U(·,·) : M2
H→L2
Hde ined as
˜
U( , τ)(u0, φ) = u (·;τ, (u0, φ)), o (u0, φ)∈M2
H,and τ≤ . (5)
Obse e ha
U( , τ)φ=˜
U( , τ)(φ(0), φ) o any ≥τ, and any φ∈CH.(6)
In his way, we hen ha e ha he p ocess S( , τ)can be ew i en as
S( , τ)(u0, φ) = (u( ;τ, (u0, φ)),˜
U( , τ)(u0, φ)).(7)
These ac s will allow us o p o e he es ima es o he p ocesses Uand Sin
a s aigh o wa d way by using he p e iously ob ained ones o he p ocess ˜
U.
9
I we now conside he ime −sins ead o τ(i.e. u(·) deno es now u(·; −s, (u0, φ)) ,
so ha we can use mo e easily he de ini ion o abso bing se s) we ha e
°
°
°˜
U( , −s)(u0, φ)°
°
°CH
=ku k2
CH≤| |2
mσ +˜
d2emh (1 + Cg)e−ms o all , and s≥h.
and deno ing by ˜ρ2=| |2
mσ and ˜ρ2
H= 2˜ρ2,i easily ollows ha he e exis s
e
T˜
D( )(= e
T˜
D)≥hsuch ha o all s≥e
T˜
D( ) and all (u0, φ)∈M2
H, i holds
°
°
°˜
U( , −s)(u0, φ)°
°
°CH
≤˜ρH,which means ha he balls B( ) = BCH(0,˜ρH)
o m an abso bing amily o bounded se s o he mappings ˜
U( , τ).
Co olla y 13 Unde he assump ions in Theo em 12, he e exis s a amily
{B( )} ∈Ro bounded abso bing se s in CH o he p ocess U, which is gi en by
B( ) = B1=BCH(0,˜ρH) o all ∈R. Mo eo e , he amily {B( )} ∈Rgi en
by B( )=BH(0,˜ρH)×BL2
H(0, h1/2˜ρH)⊂M2
H o all ∈Ris abso bing o he
p ocess S.
PROOF. The i s pa ollows om he p e ious Theo em 12 and Lemma 11.
As o he second, obse e ha {j(B( ))} ∈Ris a amily o bounded abso bing
se s o S(·,·).On he o he hand, as kφk2
L2
H≤hkφk2
CHand
j(B( )) = {(φ(0), φ) : φ∈BCH(0,˜ρH)},
i ollows ha
j(B( )) ⊂BH(0,˜ρH)×BL2
H(0, h1/2˜ρH) = B( ),
wha implies ha he amily {B( )} ∈Ris abso bing o he p ocess S(·,·).
Rema k 14 I we assume ha ∈V0, he p e ious esul s also hold ue by
modi ying sligh ly he p oo s and subs i u ing | |by k k∗.
3.3 Exis ence o an abso bing amily o se s in CV
We now p o e he exis ence o an abso bing amily o se s in CVand a neces-
sa y bound on he e m R +θ2
+θ1|Au(s)|2ds. We p oceed in a simila way as we
ha e al eady done in he p e ious subsec ion.
Theo em 15 Unde he assump ions in Theo em 12, he e exis posi i e con-
s an s ˜ρV,e
β1,e
β2such ha o any bounded se ˜
D⊂M2
Hand o e
T˜
D he ab-
16

so bing ime co esponding o he se B1in Theo em 12, i ollows
°
°
°˜
U( , −s)(u0, φ)°
°
°2
CV
= max
θ∈[−h,0] ku( +θ; −s, (u0, φ))k2≤˜ρ2
V,
Z +θ2
+θ1
|Au(σ; −s, (u0, φ))|2dσ ≤e
β1|θ2−θ1|+e
β2,
o all s≥e
T˜
D+1+h, ∈R,(u0, φ)∈˜
D, and θ1, θ2∈[−h, 0].
PROOF. As in he p oo o Theo em 12, le ˜
D⊂M2
Hbe a bounded se ,
i.e. he e exis s ˜
d > 0 such ha k(u0, φ)kM2
H≤˜
d o all (u0, φ)∈˜
D. Deno e
u(·) = u(·; 0−s, (u0, φ)) o (u0, φ)∈˜
D, whe e 0∈Ris a ixed numbe ,
and le us ake s≥e
T˜
D,whe e we ha e chosen he same σand m han in ha
p oo . We can hen in eg a e in (11) be ween and +1 o ≥ 0and s≥e
T˜
D.
We ob ain
|u( + 1)|2− |u( )|2+³2ν−(σ+Cg)λ−1
1´Z +1
ku( )k2d
≤| |2
σ+1
CgZ +1
|g( , u )|2d
≤| |2
σ+1
Cg·C2
gZ +1
−h|u( )|2d ¸
≤| |2
σ+CgZ
−h|u( )|2d +CgZ +1
|u( )|2d
≤| |2
σ+CgZ
−h|u( )|2d +Cgλ−1
1Z +1
ku( )k2d ,
and
³2ν−(σ+ 2Cg)λ−1
1´Z +1
ku( )k2d ≤| |2
σ+CgZ
−h|u( )|2d +|u( )|2
≤| |2
σ+CgZ
−hku k2
CHd + ˜ρ2
H
≤| |2
σ+ (1 + hCg) ˜ρ2
H.
The e o e, Z +1
ku( )k2d ≤e
IV,∀ ≥ 0,(15)
whe e
e
IV=1
2ν−(σ+ 2Cg)λ−1
1Ã| |2
σ+ (1 + hCg) ˜ρ2
H!.
On he o he hand, we ake he inne p oduc wi h Au and ob ain o ≥ 0
1
2
d
d kuk2+ν|Au|2+b(u, u, Au)≤( , Au)+(g( , u ), Au).(16)
17
Now we e alua e he e ms. Fi s , no ice ha
|( , Au)|+|(g( , u ), Au)| ≤ |Au|(| |+|g( , u )|)
≤ν
4|Au|2+2
ν³| |2+|g( , u )|2´.(17)
Nex ,
|b(u, u, Au)| ≤ c1|u|1/2kuk |Au|3/2(18)
≤ν
4|Au|2+c0
1
ν3|u|2kuk4.
Thanks o (17)-(18), and he ac ha kϕk ≤ λ−1
1|Aϕ| o ϕ∈D(A), we can
deduce om Eq. (16)
d
d kuk2+ν|Au|2≤ν
4³| |2+L2
gku k2
CH´+2c0
1
ν3|u|2kuk4,(19)
and
d
d kuk2+νλ1kuk2≤ν
4³| |2+L2
gku k2
CH´+2c0
1
ν3|u|2kuk4
≤ν
4³| |2+L2
g˜ρ2
H´+2c0
1
ν3|u|2kuk4.
Now, we can apply he uni o m G onwall lemma o s≥e
T˜
D(see Temam [26]).
Then,
ku( )k2≤(a3+a2)ea1, o all ≥ 0+ 1, p o ided s≥e
T˜
D,
whe e
a3=e
IV
a2=ν
4³| |2+L2
g˜ρ2
H´
a1=2c0
1
ν3˜ρ2
He
IV,
and, consequen ly, i we ake s≥e
T˜
D+1+h,
sup
θ∈[−h,0]
ku( 0+θ)k2≤(a3+a2)ea1= ˜ρ2
V, (20)
whe e he cons an s appea ing in (20) a e independen o he ixed ime 0∈R.
So, (20) holds ue o all 0∈R. Deno ing om now on
u(·) = u(·; −s, (u0, φ)),
and, aking in o accoun ha pa b) in Theo em 1 ensu es ha u (·)∈CV
o s > h, we indeed ha e
ku kCV≤˜ρV, o all ∈R, p o ided s≥e
T˜
D+1+h.
18
Finally we will ob ain he bound on he e m R +θ2
+θ1|Au( )|2d . Indeed, om
(19) i ollows
|Au|2≤α1+α2|u|2kuk4−1
ν
d
d kuk2.
I we choose s≥e
T˜
D+1+hand θ1, θ2∈[−h, 0] wi h e.g. θ2> θ1,we ha e
Z +θ2
+θ1
|Au( )|2d ≤α1|θ2−θ1|+α2Z +θ2
+θ1
|u( )|2ku( )k4d
−1
νku( +θ2)k2+1
νku( +θ1)k2
≤³α1+α2˜ρ2
H˜ρ4
V´|θ2−θ1|+1
ν˜ρ2
V,
as desi ed.
Co olla y 16 Unde he assump ions in Theo em 12, he e exis posi i e con-
s an s ρV, β1, β2such ha o any bounded se D⊂CHand o TD=˜
Tj(D)
wi h ˜
Tj(D) he abso bing ime co esponding o he se B1in Theo em 12, i
ollows
kU( , −s)φk2
CV=ku (·; −s, j(φ))k2
CV= max
θ∈[−h,0] ku( +θ; −s, j(φ))k2≤ρ2
V,
Z +θ2
+θ1
|Au(σ; −s, j(φ))|2dσ≤β1|θ2−θ1|+β2,
o all s≥TD+1+h, ∈R, φ ∈D, and θ1, θ2∈[−h, 0].In pa icula ,
he amily {B2( )} ∈R,whe e B2( ) = B2=BCV(0, ρV),is abso bing o he
p ocess U(·,·).
Mo eo e , he amily {BS( )} ∈R, whe e BS( ) = BCV(0, ρV)×BL2
V(0, h1/2ρV),
is abso bing o S(·,·).
PROOF. The p oo ollows he same lines as hose o Co olla y 13.
3.4 Exis ence o he pullback a ac o s
Now we can p o e he ollowing esul .
Theo em 17 Unde he assump ions in Theo em 12, he e exis a unique
uni o mly bounded pullback a ac o {ACH( )} ∈R o he p ocess U(·,·)in CH,
and a unique uni o mly bounded pullback a ac o {AM2
H( )} ∈R o S(·,·)in
M2
H. Fu he mo e, AM2
H( )⊂H×CH o all ∈Rand bo h a ac o s a e
ela ed by means o
AM2
H( ) = j(ACH( )) , o all ∈R.
19
PROOF. Le us conside he amily {B2( )} ∈R,whe e B2( ) = B2=BCV(0; ρV)
o all ∈R. This is a amily o bounded se s in CV, which is also (uni o mly)
abso bing o ˜
U(·,·).Take now ˜
B2=j(B2).Then, using he p e ious no a-
ion, he e exis s ˜
T0
˜
B2=TB2+1+h > 0 such ha
˜
U( , −s)˜
B2⊂B2, o all ∈R, and all s≥˜
T0
˜
B2.
Now, o each ∈R, conside he se
B3( ) = [
s≥˜
T0
˜
B2
˜
U( , −s)˜
B2⊂B2⊂CV.
Thus, {B3( )} ∈Ris a amily o uni o mly bounded se s in CVwhich is (uni-
o mly) abso bing o ˜
U(·,·).
I we p o e ha each B3( ) is ela i ely compac in CH, hen {B3( )} ∈R(whe e
he closu e is aken in CH) is a amily o compac abso bing se in CH o
˜
U(·,·).Consequen ly, i is also a amily o compac (uni o m) abso bing se s
o he p ocess U(·,·) in CH,and {j³B3( )´} ∈Ris ano he amily o compac
(uni o m) abso bing se s o S(·,·) in M2
H,wha ensu es he exis ence o he
pullback a ac o s o he p ocesses. The uniqueness o hese a ac o s holds
since hey a e uni o mly bounded (see Rema k 6).
Le us now p o e his compac ness p ope y. To his end, we will use he
Ascoli-A zel`a heo em, in o he wo ds, we ha e o check
(A) The se [
s≥˜
T0
˜
B2
˜
U( , −s)˜
B2is equicon inuous (i.e. ∀ε > 0,∃δ > 0 such ha i
|θ1−θ2| ≤ δ, hen ¯¯¯˜
U( , −s) (j(φ)) (θ1)−˜
U( , −s) (j(φ)) (θ2)¯¯¯≤ε, ∀ ∈
R,s≥˜
T0
˜
B2,∀φ∈B2.)
(B) Fo each θ∈[−h, 0],
[
s≥˜
T0
˜
B2[
φ∈B2
˜
U( , −s) (j(φ)) (θ) is a compac se in H.
To p o e (B) we need o check ha , o any ixed θ∈[−h, 0] and ∈R, he
se nu( +θ; −s, j (φ)) : s≥˜
T0
˜
B2, φ ∈B2o
is ela i ely compac . Bu his holds since his se is bounded in V(see The-
o em 15) and he injec ion V⊂His compac .
20
Finally, in o de o p o e (A) we p oceed by es ima ing
¯¯¯˜
U( , −s) (j(φ)) (θ1)−˜
U( , −s) (j(φ)) (θ2)¯¯¯
=|u( +θ1; −s, j(φ)) −u( +θ2; −s, j(φ))|
o ∈R,θ1, θ2∈[−h, 0], s ≥˜
T0
˜
B2and φ∈B2.Then we ob ain (deno ing o
simplici y u(·; −s, j(φ)) by u(·) and assuming θ2> θ1)
|u( +θ1)−u( +θ2)|=¯¯¯¯¯Z +θ2
+θ1
u0( )d ¯¯¯¯¯
≤Z +θ2
+θ1
|u0( )|d
≤Z +θ2
+θ1
(ν|Au( )|+|B(u( ))|+| |+|g( , u )|) d
≤ | | |θ1−θ2|
+Z +θ2
+θ1³ν|Au( )|+c1|Au( )| ku( )k+Lgku kCH´d
≤ | | |θ1−θ2|
+Z +θ2
+θ1³(ν+c1ku( )k)|Au( )|+Lgku kCH´d ,
(21)
and, consequen ly, o ∈R,s≥˜
T0
˜
B2
|u( +θ1)−u( +θ2)| ≤ | | |θ1−θ2|
+Z +θ2
+θ1³(ν+c1ku( )k)|Au( )|+Lgku kCH´d
≤(| |+ρHLg)|θ1−θ2|+Z +θ2
+θ1
(ν+c1ρV)|Au( )|d
≤(| |+ρHLg)|θ1−θ2|
+ (ν+c1ρV)|θ1−θ2|1/2Z +θ2
+θ1
|Au( )|2d
≤(| |+ρHLg)|θ1−θ2|
+ (ν+c1ρV) (β1|θ1−θ2|+β2)|θ1−θ2|1/2,
which implies he needed equicon inui y.
Finally, we will p o e he in e es ing ela ionship ha he e exis s be ween
he a ac o s {ACH( )} ∈Rand {AM2
H( )} ∈R. Obse e ha om he p op-
e ies o he mapping j(·), he esul s in Lemma 11 and Co olla y 16, i is
s aigh o wa d o check ha {j(ACH( ))} ∈Ris a uni o mly bounded amily
o compac se s in M2
Hwhich is pullback a ac ing o he p ocess S(·,·),
and i is also in a ian . Taking in o accoun he uniqueness o he uni o mly
21

bounded a ac o s, i ollows immedia ely ha
AM2
H( ) = j(ACH( )) o all ∈R.
The p oo is now comple e.
Rema k 18 As we ha e al eady men ioned, ou analysis can be ex ended o
deal wi h mo e gene al nonau onomous and g. The echnique we ha e used in
he p e ious subsec ions can be pe o med o ea his case in a s aigh o wa d
way, al hough wi h addi ional di icul ies in he compu a ions. Fo ins ance, i
we assume ha ∈L2
loc(R;L2(Ω)2),and sa is ies
Z
−∞ ems| (s)|2ds < +∞, o all ∈R, and m > 0,
hen, unde assump ions (I)-(IV) wi h m0>0,and νλ1> Cg, i is no di -
icul o check ha he e exis s a amily {B( )} ∈Ro bounded abso bing se s
in CH o ˜
U(·,·). To be mo e p ecise, B( ) = BCH(0, ρH( )) whe e ρ2
H( ) =
2emhe−m R
−∞ ems| (s)|2ds, o a posi i e bu small enough m. Unde hese as-
sump ions, we can hen p o e simila ly he exis ence o he nonau onomous
abso bing amily in CV,and conclude wi h he exis ence o he pullback a ac-
o . We lea e he de ails o he eade .
3.5 An applica ion: a o cing e m wi h a iable delay
Conside ha ope a o gis gi en by
g( , u ) = G(u( −ρ( ))),
wi h G:R2→R2a unc ion sa is ying G(0) = 0 and such ha he e exis s
L1>0 o which
|G(u)−G( )|R2≤L1|u− |R2,∀u, ∈R2,
and ρ∈C1(R), ρ( )≥0 o all ∈R,h= sup ∈Rρ( )∈(0,+∞) and
ρ∗= sup ∈Rρ0( )<1. This si ua ion is wi hin ou amewo k and sa is ies
ou assump ions (Condi ions (I)-(IV)) ensu ing he exis ence and uniqueness
o solu ions (see Ca aballo & Real [7]). Mo eo e , (IV) is ul illed by se ing
22
C2
g=L2
1em0h/(1 −ρ∗) o any m0>0.Indeed, i ollows o ≥τ
Z
τems|g(s, us)−g(s, s)|2ds=Z
τems|G(u(s−ρ(s))) −G( (s−ρ(s)))|2ds
≤L2
1Z
τems|u(s−ρ(s)) − (s−ρ(s))|2ds
≤L2
1emh
1−ρ∗Z −ρ( )
τ−ρ(τ)emσ|u(σ)− (σ)|2dσ
≤L2
1em0h
1−ρ∗Z
τ−hems|u(s)− (s)|2ds, m ∈[0, m0).
Obse e ha i νλ1> L1/(1 −ρ∗)1/2,ou esul ensu es he exis ence o
a pullback a ac o ACH( )⊂CH o he p ocess U(·,·) (and also ano he
pullback a ac o AM2
H( ) o S(·,·)).Indeed, we only need o check ha
νλ1> Cg=L1em0h/2/(1 −ρ∗)1/2.Bu , i νλ1> L1/(1 −ρ∗)1/2, hen o a
su icien ly small bu posi i e m0, we ha e ha νλ1> L1em0h/2/(1 −ρ∗)1/2.
No ice ha he analysis done in Ca aballo and Real [8] ensu es ha i he
iscosi y νis la ge , i.e., i o ins ance, o ce ain posi i e cons an s k1and
k2(depending only on Ω), i holds ha
2νλ1>(2 −ρ∗)L1
(1 −ρ∗)+k1| |
ν−λ−1
1L1
+k2| |3
ν2(ν−λ−1
1L1)3,
hen, he e exis s a unique s a iona y solu ion u∞∈V o ou p oblem and
e e y solu ion app oaches his s a iona y solu ion exponen ially as . In o he
wo ds, ACH( ) consis s o his unique s a iona y solu ion. No ice ha in he
pa icula case ρ∗= 0 (which means ha he delay unc ion ρis no inc easing)
we ob ain an a ac o o ou model i νλ1> L1,and his a ac o becomes
a unique poin i
2νλ1>2L1+k1| |
ν−λ−1
1L1
+k2| |3
ν2(ν−λ−1
1L1)3.
3.6 Rema ks on he au onomous case
We a e now in e es ed in he ollowing au onomous e sion o ou p oblem













To ind u∈L2(0, T;H)∩L2(−h, T;V)∩L∞(0, T;H)∀T > 0, s. .
d
d u( ) + νAu( ) + B(u( )) = +g(u ) in D0(0,+∞;V0),
u(0) = u0, u( ) = φ( ), ∈(−h, 0),
(22)
whe e ∈(L2(Ω))2,and g:CH→(L2(Ω))2sa is ies (II), (III) and (IV) in
Sec ion 2. Owing o he ac ha gdoes no explici ly depend on he ime
23
a iable , hese condi ions can be ew i en as ollows:
(g1) g(0) = 0
(g2) he e exis s Lg>0 such ha ∀ξ, η ∈CH
|g(ξ)−g(η)| ≤ Lgkξ−ηkCH,
(g3) ∃m0≥0, Cg>0: ∀m∈[0, m0],0≤ , u, ∈C0([−h, ]; H)
Z
0ems|g(us)−g( s)|2ds≤C2
gZ
−hems|u(s)− (s)|2ds.
Fo each ini ial unc ion φ∈CHand aking as ini ial alue u0=φ(0), he e ex-
is s a unique solu ion u(·;φ) o p oblem (22) such ha u∈C0([−h, +∞); H).
Then, o any ≥0 we can de ine an ope a o U0( ) : CH→CHas
U0( )φ=u (·;φ).
Bea ing in mind he analysis done in he p e ious sec ion, we can p oceed
only on he phase space CHsince he exis ence o an a ac o in CHenables
us o ob ain ano he one in M2
H.In his sense, i is no di icul o p o e,
in a simila ashion as we ha e done in he p eceding subsec ions, ha his
dynamical sys em U0(·) possesses a global a ac o in CH.Bu , we no e ha ,
conside ing his p oblem as a nonau onomous one and se ing U( , τ) o i s
associa ed p ocess, i holds ha
U( , τ) = U( −τ, 0), o all ≥τ,
and, consequen ly,
U0( ) = U( , 0), o all ≥0,
is a semig oup o nonlinea con inuous ope a o s.
Now, ou p e iously de eloped heo y allows he eade o p o e as an easy
exe cise he ollowing esul .
Theo em 19 (Exis ence o global a ac o ) Assume ha (g1),(g2) and (g3)
hold wi h m0>0. I , in addi ion, νλ1> Cg, hen he e exis s he global
a ac o ACH⊂CH o he semig oup U0( ).
Rema k 20 Needless o say ha a simila esul can also be p o ed i we
conside he semig oup S0( ) = S( , 0).
As an applica ion, we will now conside an example in which he o cing e m
con ains a dis ibu ed delay.
Le G: [−h, 0] ×R2→R2be a measu able unc ion sa is ying G(s, 0) = 0
o all s∈[−h, 0] and assume ha he e exis s a unc ion γ∈L2(−h, 0) such
24
ha
|G(s, u)−G(s, )|RN≤γ(s)|u− |RN,∀u, ∈RN∀s∈[−h, 0].
Then, we de ine g(ξ)(x) = R0
−hG(s, ξ(s)(x)) ds o each ξ∈C0([0, T]; H) and
x∈Ω. In his case, he delayed e m gin ou p oblem becomes
g(u ) = Z0
−hG(s, u( +s)) ds.
I holds ha gsa is ies he hypo heses in Theo em 19.
Indeed, (g1) is e iden . As o (g2), no ice ha , i ξ, η ∈CH, we ob ain
|g(ξ)−g(η)|2≤RΩ³R0
−h|G(s, ξ(s)(x)) −G(s, η(s)(x))|RNds´2dx
≤RΩ³R0
−hγ(s)|ξ(s)(x)−η(s)(x)|RNds´2dx
≤RΩkγk2
L2(−h,0) ³R0
−h|ξ(s)(x)−η(s)(x)|2
RNds´dx
≤hkγk2
L2(−h,0)kξ−ηk2
CH.
Finally, i u, ∈C0([−h, T]; H) hen, o each > 0, m0>0 and all m∈
[0, m0],i ollows
Z
0emτ |g(uτ)−g( τ)|2dτ ≤ kγk2
L2(−h,0) Z
0emτ µZ0
−h|u(s+τ)− (s+τ)|2ds¶dτ
≤ kγk2
L2(−h,0) Z0
−hµZ
0emτ |u(s+τ)− (s+τ)|2dτ¶ds
≤ kγk2
L2(−h,0) Z0
−hµZ +s
sem( −s)|u( )− ( )|2d ¶ds
≤ kγk2
L2(−h,0) Z0
−he−ms µZ
−hem |u( )− ( )|2d ¶ds
≤ kγk2
L2(−h,0)hem0hZ
−hem |u( )− ( )|2d .
Consequen ly, Theo em 19 ensu es he exis ence o he global a ac o in CH
p o ided νλ1>kγkL2(−h,0)h1/2em0h/2.Bu , we no e ha i νλ1>kγkL2(−h,0)h1/2,
we can choose m0small enough such ha νλ1>kγkL2(−h,0)h1/2em0h/2.I is
also ema kable ha when h→0 he su icien condi ion ensu ing he exis-
ence o he global a ac o becomes νλ1>0, which is he usual one in he
case wi hou delays (and i ially ul illed).
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