Na ie -S okes Equa ions wi h Delays
By Tom´
as Ca aballo, & Jos´
e Real†
Depa amen 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
Some esul s on he exis ence and uniqueness o solu ions o Na ie -S okes equa ions
when he ex e nal o ce con ains some he edi a y cha ac e is ics a e p o ed.
Keywo ds: Na ie -S okes equa ions, a iable and dis ibu ed delays.
1. In oduc ion and s a emen o he p oblem
The Na ie -S okes equa ions go e n he mo ion o usual luids like wa e , ai , oil,
e c. These equa ions ha e been he objec o nume ous wo ks since he i s pape
o Le ay was published in 1933 (see Cons an in & Foias 1988; Lions 1969; Temam
1979, and he e e ences he ein). Howe e , up o da e, we ha e no ound in he
li e a u e any wo k which akes in o accoun he possibili y o appea ing some
kind o delay in hese equa ions. The main aim o his wo k is o conside se e al
si ua ions in which he ex e nal o ce con ains some he edi a y ea u es and p o e
exis ence o solu ions. These si ua ions may appea when we wan o con ol he
sys em (in ce ain sense) by applying a o ce which akes in o accoun no only he
p esen s a e o he sys em bu he his o y o he solu ion. I is wo h poin ing ou
ha a simila analysis was ca ied ou by A ola (1969) o gene al linea pa ial
di e en ial equa ions wi h delays.
Le Ω ⊂RN(N= 2 o 3) be an open and bounded se wi h egula bounda y
Γ, T > 0 gi en, and conside he ollowing unc ional Na ie -S okes p oblem ( o
u he de ails and no a ions see Lions 1969 and Temam 1979):















∂u
∂ −ν∆u+PN
i=1 ui
∂u
∂xi
= −∇p+g( , u ) in (0, T)×Ω,
di u= 0 in (0, T)×Ω,
u= 0 on (0, T)×Γ,
u(0, x) = u0(x), x ∈Ω,
u( , x) = φ( , x), ∈(−h, 0) x∈Ω,
whe e we assume ha ν > 0 is he kinema ic iscosi y, uis he eloci y ield o he
luid, p he p essu e, u0 he ini ial eloci y ield, a nondelayed ex e nal o ce ield,
gano he ex e nal o ce con aining some he edi a y cha ac e is ic and φ he ini ial
da um in he in e al o ime (−h, 0),whe e his a posi i e ixed numbe .
†E-mails: [email p o ec ed] ; eal@nume .us.es
A icle submi ed o Royal Socie y T
EX Pape
2T. Ca aballo, & J. Real
To s a , we conside he ollowing usual abs ac spaces:
V=nu∈(C∞
0(Ω))N: di u= 0o,
H= he closu e o Vin (L2(Ω))Nwi h he no m |·|,and inne p oduc (·,·)
whe e o u, ∈(L2(Ω))N,
(u, ) =
N
X
j=1 ZΩ
uj(x) j(x)dx,
V= he closu e o Vin (H1
0(Ω))Nwi h he no m k·k,and associa ed scala
p oduc ((·,·)),whe e o u, ∈(H1
0(Ω))N,
((u, )) =
N
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 .
Now we deno e a(u, ) = ((u, )), and de ine he ilinea o m bon V×V×Vby
b(u, , w) =
N
X
i,j=1 ZΩ
ui
∂ j
∂xi
wjdx ∀u, , w ∈V.
Le Xbe a Banach space. Gi en a unc ion u: (−h, T)→X, o each ∈(0, 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).
Finally, we will use k·k∗ o he no m in V0and h·,·i o he duali y hV0, V i.
In o de o s a e he p oblem in he co ec amewo k, le us i s ly es ablish
he sui able assump ions on he e m in which he delay is p esen .
In a gene al way, le Xand Ybe wo sepa able Banach spaces, and g: [0, T]×
C0([−h, 0]; X)→Ysuch ha
(I) o all ξ∈C0([−h, 0]; X), he mapping ∈[0, T ]→g( , ξ)∈Yis measu -
able,
(II) o each ∈[0, T], g( , 0) = 0,
(III) he e exis s Lg>0 such ha ∀ ∈[0, T ],∀ξ, η ∈C0([−h, 0]; X)
kg( , ξ)−g( , η)kY≤Lgkξ−ηkC0([−h,0];X),
(IV) he e exis s Cg>0 such ha ∀ ∈[0, T ],∀u, ∈C0([−h, T ]; X)
Z
0kg(s, us)−g(s, s)k2
Yds ≤CgZ
−hku(s)− (s)k2
Xds.
Obse e ha (I)-(III) imply ha gi en u∈C0([−h, T]; X), he unc ion gu: ∈
[0, T]→Yde ined by gu( ) = g( , u )∀ ∈[0, T ], is measu able (see Bensoussan e
al. 1992) and, in ac , belongs o L∞(0, T;Y). Then, hanks o (IV), he mapping
G:u∈C0([−h, T]; X)→gu∈L2(0, T ;Y)
A icle submi ed o Royal Socie y
Na ie -S okes equa ions wi h delays 3
has a unique ex ension o a mapping e
Gwhich is uni o mly con inuous om L2(−h, T;X)
in o L2(0, T;Y). F om now on, we will deno e g( , u ) = e
G(u)( ) o each u∈
L2(−h, T;X), and hus, ∀ ∈[0, T ],∀u, ∈L2(−h, T ;X),we will ha e
Z
0kg(s, us)−g(s, s)k2
Yds ≤CgZ
−hku(s)− (s)k2
Xds.
Wi h he con en ion abo e, assume ha u0∈H,φ∈L2(−h, 0; V), ∈L2(0, T;V0),
g1: [0, T ]×C0([−h, 0]; V)→(L2(Ω))Nsa is ies hypo heses (I)-(IV) wi h X=V,
Y= (L2(Ω))N,Lg1=L1and Cg1=C1, and g2: [0, T]×C0([−h, 0]; V)→V0
sa is ies hypo heses (I)-(IV) wi h X=V,Y=V0,Lg2=L2and Cg2=C2.
We a e in e es ed in he ollowing p oblem:









To ind u∈L2(−h, T ;V)∩L∞(0, T ;H) such ha , o all ∈V,
d
d (u( ), ) + νa(u( ), ) + b(u( ), u( ), ) = h ( ), i+ (g1( , u ), )
+hg2( , u ), i,
u(0) = u0, u( ) = φ( ), ∈(−h, 0),
(1.1)
whe e he equa ion in (1.1) mus be unde s ood in he sense o D0(0, T).
Rema k 1.1. Obse e ha he e ms in (1.1) a e well de ined. In pa icula , by
hypo heses (I)-(IV), i u∈L2(−h, T ;V) he e m g1( , u )de ines a unc ion in
L2(0, T; (L2(Ω)N), and he e m g2( , u )de ines a unc ion in L2(0, T;V0). Thus
(see Lions 1969), i u∈L2(−h, T ;V)∩L∞(0, T ;H)sa is ies he equa ion in (1.1), u
is weakly con inuous om [0, T ]in o H, and he e o e he ini ial condi ion u(0) = u0
makes sense. O cou se, o N= 2, i he e exis s a solu ion u o he p oblem (1.1),
i hen belongs o he space C0([0, T ]; H).
In he nex sec ion, we shall p o e exis ence o solu ions o (1.1) and he unique-
ness o solu ion o he p oblem in he case N= 2.In Sec ion 3, we show se e al
gene al si ua ions con aining delayed e ms including, in pa icula , hose wi h a i-
able and dis ibu ed delays, and we conclude he wo k by p o ing, in he Appendix,
a ini e-dimensional esul needed o he p oo o he exis ence o solu ions o (1.1).
2. Exis ence o solu ions
In his sec ion we will p o e a gene al heo em on he exis ence o solu ions when
N= 2 o 3, and uniqueness i N= 2.
Theo em 2.1. Le us conside u0∈H,φ∈L2(−h, 0; V), ∈L2(0, T ;V0),
and assume ha g1: [0, T ]×C0([−h, 0]; V)→(L2(Ω))Nsa is ies hypo heses (I)-
(IV) wi h X=V,Y= (L2(Ω))N,Lg1=L1and Cg1=C1, and g2: [0, T ]×
C0([−h, 0]; V)→V0sa is ies hypo heses (I)-(IV) wi h X=V,Y=V0,Lg2=L2
and Cg2=C2. Then:
a) I N= 2 and ν2> C2, he e exis s a mos one solu ion o p oblem (1.1).
b) I N∈ {2,3}and ν2> C2, he e exis s a solu ion o (1.1) i , in addi ion,
he ollowing assump ion (C) holds:
(C) I mcon e ges weakly o in L2(−h, T ;V)and s ongly in L2(−h, T;H),
hen gi(·, m
·)con e ges weakly o gi(·, ·)in L2(0, T ;V0) o i= 1,2.
A icle submi ed o Royal Socie y
4T. Ca aballo, & J. Real
P oo . a) I N= 2 and ν2> C2, le u, be wo solu ions o (1) and se w=u− .
Then, om he ene gy equali y, and he bounds o he ilinea o m (see Lions
1969), i ollows ha o all ∈(0, T)
|w( )|2+ 2νR
0||w(s)||2ds =−2R
0b(w(s), u(s), w(s)) ds
+2 R
0(g1(s, us)−g1(s, s), w(s)) ds
+2 R
0hg2(s, us)−g2(s, s), w(s)ids
≤2k1R
0|w(s)|(||w(s)||)||u(s)||ds
+2 R
0|g1(s, us)−g1(s, s)||w(s)|ds
+2 R
0kg2(s, us)−g2(s, s)k∗||w(s)|| ds.
Then, om assump ion (IV), aking in o accoun ha w(s) = 0 o s∈(−h, 0),
and deno ing 2ε=ν−√C2>0, we ha e o all ∈(0, T)
|w( )|2+ 2νR
0||w(s)||2ds ≤k2
1
εR
0|w(s)|2|| (s)||2ds +εR
0||w(s)||2ds
+C1
εR
0|w(s)|2ds +εR
0||w(s)||2ds
+2√C2R
0||w(s)||2ds,
and so,
|w( )|2+ 2εZ
0||w(s)||2ds ≤k2
1
εZ
0|w(s)|2|| (s)||2ds +C1
εZ
0|w(s)|2ds,
om which uniqueness ollows hanks o he G onwall lemma.
b) Now, we assume N∈ {2,3},ν2> C2and ha condi ion (C) holds. Fo he
p oo o exis ence, we will ollow a Gale kin scheme simila o he one in Cons an in
& Foias (1988), so we only emphasize he de ails in ol ing he new e ms gi.
Le us conside {wj} ⊂ V∩¡H2(Ω)¢N he o hono mal basis o Ho all he
eigen unc ions o he S okes p oblem in Ω wi h homogeneous Di ichle condi ions.
The subspace o Vspanned by w1, ..., wmwill be deno ed Vm. Conside he p ojec o
Pm:H→Vmgi en by Pmu=Pm
j=1(u, wj)wj,and de ine um( ) = Pm
j=1 γmj( )wj,
whe e









um∈L2(−h, T;Vm)∩C0([0, T ]; Vm)
d
d (um( ), wj) + νa(um( ), wj) + b(um( ), um( ), wj) = h ( ), wji+
+ (g1( , um
), wj) + hg2( , um
), wjiin D0(0, T),1≤j≤m,
um(0) = Pmu0, um( ) = Pmφ( ), ∈(−h, 0).
(2.1)
The p eceding is a sys em o o dina y unc ional di e en ial equa ions in he
unknown γm( ) = (γm1( ), ..., γmm( )). We can ge exis ence and uniqueness o
solu ion by applying Theo em 3.1 in he Appendix. Obse e ha , acco ding o
Theo em 3.1, we can ensu e ha p oblem (2.1) has one solu ion de ined in an
in e al [0, ∗] wi h 0 < ∗≤T. Howe e , as can be deduced by he a p io i es ima es
below, we can se ∗=T.
A icle submi ed o Royal Socie y
Na ie -S okes equa ions wi h delays 5
In ac , mul iplying in (2.1) by γmj( ) and summing in j, we ge o all ∈[0, ∗]
|um( )|2+ 2νR
0kum(s)k2ds ≤ |u0|2+ 2 R
0h (s), um(s)i
+2 R
0(g1(s, um
s), um(s)) ds
+2 R
0hg2(s, um
s), um(s)ids,
and a guing in a simila manne as we did in he p oo o uniqueness in he 2-
dimensional case, we easily ge wo cons an s (depending on φ, ν, , g1, g2, h, T, bu
no on mno ∗)K1and K2such ha
sup
∈[0, ∗]|um( )|2≤K1,Z ∗
0||um(s)||2ds ≤K2.
So we can ake ∗=T, and ob ain ha {um}is bounded in L2(0, T;V)∩L∞(0, T ;H).
Mo eo e , obse e ha um=Pmφin (−h, 0) and, by he choice o he basis {wj},
he sequence umcon e ges o φin L2(−h, 0; V), and, in pa icula , g1(·, um
·) +
g2(·, um
·) is bounded in L2(0, T;V0). Now, i is a s anda d ma e o bound he
nonlinea e m b(um, um,·), and using he same easoning ha in Cons an in &
Foias (1988) (see page 67), one can ob ain ha ©dum
d ªis bounded in L4/3(0, T;V0)
(in ac , i N= 2, ©dum
d ªis bounded in L2(0, T;V0)). Using he compac ness o he
injec ion o he space W={u∈L2(0, T;V) : du
d ∈L4/3(0, T;V0)}in o L2(0, T;H),
om he p eceding analysis and he assump ions on g1and g2, we can deduce ha
he e exis a subsequence (deno ed again um) and u∈L2(−h, T ;V) such ha :
um→uweakly in L2(−h, T;V),
um→uweakly s a in L∞(0, T ;H),
um→uin L2(−h, T;H),
gi(·, um
·)→gi(·, u·) weakly in L2(0, T ;V0), i = 1,2.
Now, as in he non-delay case, we can ake limi s in (2.1) a e in eg a ing o e
he in e al (0, ) ( o ∈(0, T)), ge ing ha uis a solu ion o ou p oblem (1.1)
(see once again, e.g., Cons an in & Foias 1988 o he comple e de ails).
Rema k 2.2. Obse e ha i g1sa is ies (I)-(IV) wi h X=Hand Y=¡L2(Ω)¢N,
hen, as a di ec consequence o (IV), g1sa is ies assump ion (C).
3. Some gene al si ua ions
In his sec ion, we a e going o show some si ua ions whe e ou heo y can be
applied. The cases conside ed include si ua ions such as dis ibu ed delay, a iable
delay, and delay in g adien o second o de de i a i es e ms.
(a)Case 1
Le G: [0, T]×RN→RNbe a measu able unc ion sa is ying G( , 0) = 0 o
all ∈[0, T],and assume ha he e exis s L1>0 such ha
|G( , u)−G( , )|RN≤L1|u− |RN,∀u, ∈RN.
A icle submi ed o Royal Socie y

6T. Ca aballo, & J. Real
Conside a unc ion ω( ), which is going o play he ole o he delay unc ion. We
suppose ha ω∈C1([0, T]), ω( )≥0 o all ∈[0, T], h= max ∈[0,T ]ω( )>0
and ω∗= max ∈[0,T ]ω0( )<1. Then, we de ine g1( , ξ)(x) = G( , ξ(−ω( ))(x)) o
each ξ∈C0([0, T]; H), x∈Ω and ∈[0, T]. No ice ha , in his case, he delayed
e m g1in ou p oblem u ns o g1( , u ) = G( , u( −ω( ))).Then, g1sa is ies he
hypo heses in Theo em 2.1 wi h X=Hand Y=L2(Ω)N.
Indeed, (I)-(III) ollow immedia ely. On he o he hand, i u, ∈L2(−h, T;H),
using he change o a iable τ=s−ω(s) i is easy o see ha
Z
0|g1(s, us)−g1(s, s)|2ds ≤Z
−h|u(τ)− (τ)|2dτ ∀ ∈[0, T],
and, consequen ly, (IV) and (C) a e ul illed.
(b)Case 2
Le now G: [0, T]×[−h, 0] ×RN→RNbe a measu able unc ion sa is ying
G( , s, 0) = 0 o all ( , s)∈[0, T ]×[−h, 0] and such ha he e exis s a unc ion
γ∈L2(−h, 0) such ha
|G( , s, u)−G( , s, )|RN≤γ(s)|u− |RN,∀u, ∈RN∀( , s)∈[0, T ]×[−h, 0].
Then, we de ine g1( , ξ)(x) = R0
−hG( , s, ξ(s)(x)) ds o each ξ∈C0([0, T]; H), x∈Ω
and ∈[0, T]. In his case, he delayed e m g1in ou p oblem becomes
g1( , u ) = Z0
−h
G( , s, u( +s)) ds.
As in Case 1, g1sa is ies he hypo heses in Theo em 2.1 wi h X=Hand Y=
¡L2(Ω)¢N.
Indeed, (I) and (II) can be deduced immedia ely. On he o he hand, i ξ, η ∈
C0([0, T]; H), o each ∈[0, T] we ob ain
|g1( , ξ)−g1( , η)|2≤RΩ³R0
−h|G( , s, ξ(s)(x)) −G( , s, η(s)(x))|RNds´2
dx
≤RΩ³R0
−hγ(s)|ξ(s)(x)−η(s)(x)|RNds´2
dx
≤RΩkγk2
L2(−h,0) ³R0
−h|ξ(s)(x)−η(s)(x)|2
RNds´dx
≤hkγk2
L2(−h,0)kξ−ηk2
C0([0,T ];H).
Finally, i u, ∈L2(−h, T ;H) hen, o each ∈[0, T ] i ollows
Z
0|g1(τ, uτ)−g1(τ, τ)|2dτ ≤hkγk2
L2(−h,0) Z
0µZ0
−h|u(s+τ)− (s+τ)|2ds¶dτ,
and, wi h he change =s+τ,
R
0|g1(τ, uτ)−g1(τ, τ)|2dτ ≤hkγk2
L2(−h,0) R
0³Rτ
τ−h|u( )− ( )|2d ´dτ
≤hTkγk2
L2(−h,0) R
−h|u( )− ( )|2d .
A icle submi ed o Royal Socie y
Na ie -S okes equa ions wi h delays 7
(c)Case 3
Now, we shall exhibi a si ua ion whe e ce ain delay can appea in e ms con-
aining pa ial de i a i es wi h espec o he spa ial a iables.
Le B(·)∈L∞(0, T;L(V;¡L2(Ω)¢N)) and ω∈C1([0, T]), such ha ω( )≥0
o all ∈[0, T ], h= max ∈[0,T ]ω( )>0 and ω∗= max ∈[0,T ]ω0( )<1. We now
de ine g1( , ξ) = B( )ξ(−ω( )) o each ξ∈C0([0, T]; V), and ∈[0, T ]. Thus, in
his case he delayed e m g1in p oblem (1.1) u ns o g1( , u ) = B( )u( −ω( )).
I is easy o see ha g1sa is ies he hypo heses in Theo em 2.1 wi h X=Vand
Y=¡L2(Ω)¢N.
Indeed, (I)-(IV) ob iously hold. On he o he hand, i mcon e ges o ze o
weakly in L2(−h, T;V) and ψ∈L2(0, T ;V) is gi en, we ha e
ZT
0hg1( , m
), ψ( )id =ZT
0hB∗( )ψ( ), m( −ω( ))id ,
wi h B∗(·)∈L∞(0, T;L(L2(Ω)N;V0)) ⊂L∞(0, T;L(V;V0)) he adjoin o B(·).
Using he change o a iables τ= −ω( ) = ρ( ), we ob ain
RT
0hg1( , m
), ψ( )id =Rρ(T)
ρ(0) B∗(ρ−1(τ))ψ(ρ−1(τ)), m(τ)®1
ρ0(ρ−1(τ)) dτ
=RT
−hhΨ(τ), m(τ)idτ,
wi h
Ψ(τ) = 




1
ρ0(ρ−1(τ))B∗(ρ−1(τ))ψ(ρ−1(τ)) i τ∈[ρ(0), ρ(T)],
0 i τ∈[−h, T] [ρ(0), ρ(T)].
Fo his unc ion Ψ i ollows
ZT
−hkΨ(τ)k2
∗dτ =Zρ(T)
ρ(0)
1
(ρ0(ρ−1(τ)))2kB∗(ρ−1(τ))ψ(ρ−1(τ))k2
∗dτ,
and hus, by means o he change τ=ρ( ) = −ω( ),
ZT
−hkΨ(τ)k2
∗dτ =ZT
0
1
1−ω0( )kB∗( )ψ( )k2
∗d ≤b2
0
1−ω∗ZT
0kψ( )k2d ,
whe e b0=kB∗(·)kL∞(0,T ;L(V;V0)) .Consequen ly, Ψ ∈L2(−h, T ;V0) and
lim
m→∞ ZT
0hg1( , m
), ψ( )id = lim
m→∞ ZT
−hhΨ(τ), m(τ)idτ = 0.
The e o e, hypo hesis (C) is sa is ied and once again we can apply ou heo y o
his si ua ion.
A icle submi ed o Royal Socie y
8T. Ca aballo, & J. Real
(d)Case 4
Le K∈L∞(−h, T;L(V;V0)) and conside in p oblem (1.1) a e m o he
o m g2( , u ) = R0
−hK( +s)u( +s)ds, de ined o all u∈L2(−h, T ;V). This e m
co esponds o he si ua ion g2( , ξ) = R0
−hK( +s)ξ(s)ds o each ∈[0, T] and ξ∈
C0([0, T]; V). In his case, i is easy o see ha g2is well de ined and sa is ies (I)-(IV)
wi h X=Vand Y=V0. In pa icula , i we deno e k=kK(·)kL∞(−h,T ;L(V;V0)),
we can see ha , o each ∈[0, T] and each u∈L2(−h, T;V), we ha e
Z
0kg2(s, us)k2
∗ds ≤k2hmin(h, T)Z
−hku(s)k2ds,
and hus, (IV) holds by se ing C2=k2hmin(h, T).
On he o he hand, le mbe weakly con e ging o ze o in L2(−h, T;V), and
ix ψ∈L2(0, T;V). Then
ZT
0hg2( , m
), ψ( )id =ZT
0¿Z
−h
K(τ) m(τ)dτ, ψ( )Àd ,
and, by Fubini’s heo em, i is easy o see ha
ZT
0hg2( , m
), ψ( )id =ZT
−hhΣ(τ), m(τ)idτ,
wi h Σ(τ) = K∗(τ)Ψ(τ) and
Ψ(τ) = 








Rτ+h
0ψ( )d i −h≤τ < 0,
Rτ+h
τψ( )d i 0 ≤τ < T −h,
RT
τψ( )d i T−h≤τ≤T,
in he case h≤T, and
Ψ(τ) = 








Rτ+h
0ψ( )d i −h≤τ < T −h,
RT
0ψ( )d i T−h≤τ < 0,
RT
τψ( )d i 0 ≤τ≤T,
in he case h > T . In bo h cases Ψ ∈C0([0, T ]; V), and in pa icula Σ ∈L2(0, T ;V0).
Consequen ly, i mcon e ges weakly o ze o in L2(−h, T;V), hen g2(·, m
·) con-
e ges weakly o ze o in L2(−h, T;V0) and hus, g2sa is ies hypo hesis (C).
Appendix
In his sec ion we will p o e a heo em on he exis ence o solu ions o a ini e-
dimensional p oblem. This esul has been used in he p oo o Theo em 2.1, and
is a a ian o Theo em 3.2 Chap e 4 (p. 213) in Bensoussan e al. (1992). We
p esen he p oo o he sake o comple eness. Howe e , i is wo h men ioning
ha a simila esul is p o ed in Hale & Lunel (1995) in he case o con inuous
ini ial da um, al hough his esul canno be applied o ou si ua ion.
A icle submi ed o Royal Socie y
Na ie -S okes equa ions wi h delays 9
Theo em 3.1. Le u0∈Rm,φ∈L2(−h, 0; Rm),k∈L2(0, T;Rm),g: [0, T ]×
C0([−h, 0]; Rm)→Rmsa is ying hypo heses (I)-(IV) wi h X=Y=Rm, and
: [0, T ]×Rm→Rma con inuous unc ion such ha ( , 0) = 0 and o all n > 0
he e exis s Ln>0such ha
| ( , u)− ( , )|Rm≤Ln|u− |Rm,∀|u|Rm≤n, | |Rm≤n, ∀ ∈[0, T ].
Then:
a) Fo each ∗∈(0, T ] he e exis s a mos one solu ion o he p oblem



To ind u∈L2(−h, ∗;Rm)∩C0([0, ∗]; Rm)such ha
u( ) = φ( ), ∈(−h, 0),
u( ) = u0+R
0 (s, u(s)) ds +R
0g(s, us)ds +R
0k(s)ds ∀ ∈[0, ∗].
(3.1)
b) The e exis s ∗∈(0, T ]such ha he e exis s one (and only one)solu ion o he
p oblem (3.1).
c) Suppose ha he e exis s a cons an C > 0such ha i ∗∈(0, T ]is such
ha he e is a solu ion uo (3.1), hen max ∈[0, ∗]|u( )|Rm≤C. Then, unde his
addi ional assump ion, he e exis s a solu ion o p oblem (3) wi h ∗=T.
P oo . a) I uand a e wo solu ions o (3.1) hen, deno ing w=u− , we ob ain
w= 0 in (−h, 0), and o all ∈[0, ∗]
|w( )|Rm≤LnZ
0|w(s)|Rmds +µCg ∗Z
0|w(s)|2
Rmds¶1/2
,
wi h n= max ¡max ∈[0, ∗]|u( )|Rm,max ∈[0, ∗]| ( )|Rm¢.
Consequen ly,
|w( )|2
Rm≤(Ln+C1/2
g)2 ∗Z
0|w(s)|2
Rmds, ∀ ∈[0, ∗],
and hus, w= 0 on [0, ∗].
b) Take any C > 0 such ha |u0|Rm≤C. Deno e
M= 1 + C+C1/2
gkφkL2(−h,0;Rm)+C(Cgh)1/2,
and ix ∗∈(0, T] such ha
2 ∗≤³1 + M(LM+C1/2
g) + kkkL2(0,T ;Rm)´−2
,
and T
∗being an in ege .
Le
X={u∈L2(−h, ∗;Rm)∩C0([0, ∗]; Rm); u=φin (−h, 0),|u( )|Rm≤M∀ ∈[0, ∗]},
wi h he me ic d(·,·) gi en by
d(u, ) = max
∈[0, ∗]|u( )− ( )|Rm.
A icle submi ed o Royal Socie y