Manusc ip submi ed o Websi e h p://aimSciences.o g
AIMS jou nals
Volume X, Numbe X, XX 200X pp. X–XX
S abili y o delay e olu ion equa ions wi h s ochas ic pe u ba ions
Tom´
as Ca aballo
Dep 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
Leonid Shaikhe
Depa men o Highe Ma hema ics, Done sk S a e Uni e si y o Managemen
Chelyuskin se s ., 163-a, Done sk 83015, Uk aine
(Communica ed by Aim Sciences)
Dedica ed o he Memo y o Ma k I. Vishik
Abs ac . The in es iga ion o s abili y o he edi a y sys ems is o en ela ed o he con-
s uc ion o Lyapuno unc ionals. The gene al me hod o Lyapuno unc ionals cons uc-
ion, which was p oposed by V.Kolmano skii and L.Shaikhe , is used he e o in es iga e he
s abili y o s ochas ic delay e olu ion equa ions, in pa icula , o s ochas ic pa ial di e -
en ial equa ions. This me hod had al eady been success ully used o unc ional-di e en ial
equa ions, o di e ence equa ions wi h disc e e ime, and o di e ence equa ions wi h con-
inuous ime. I is shown ha he s abili y condi ions ob ained o s ochas ic 2D Na ie -
S okes model wi h delays a e essen ially be e han he known ones.
1. In oduc ion.
1.1. No a ions and de ini ions. Fi s o all, we in oduce he amewo k in which
ou analysis is going o be ca ied ou . Le U, H, K be eal, sepa able Hilbe spaces
such ha
U⊂H≡H∗⊂U∗,
whe e U∗is he dual o Uand he injec ions a e con inuous and dense. We deno e
by β he cons an sa is ying
|u| ≤ βkuk, u ∈U. (1.1)
In pa icula , we also assume bo h Uand U∗a e uni o mly con ex.
We deno e by k·k,|·| and k·k∗ he no ms in U,Hand U∗, espec i ely; by
h·,·i he duali y p oduc be ween U∗, U , and by (·,·) he scala p oduc in H.
Le W( ) be a Q- alued Wiene p ocess on a ce ain comple e p obabili y space
(Ω,F, P) which akes alues in he sepa able Hilbe space K, whe e Q∈L(K, K)
is a symme ic nonnega i e ope a o and EW( ) = 0, Co (W( )) = Q.
2000 Ma hema ics Subjec Classi ica ion. P ima y: 35K35, 60H15; Seconda y: 60G52.
Key wo ds and ph ases. Me hod o Lyapuno unc ionals cons uc ion, S ochas ic e olu ion
equa ions, Exponen ial s abili y, S ochas ic pa ial di e en ial equa ions, S ochas ic 2D Na ie -
S okes model wi h delays.
This esea ch was pa ially suppo ed by FEDER and Minis e io de Econom´ıa y Compe i i idad
(Sapin) unde g an MTM2011-22411.
1
2TOM ´
AS CARABALLO & LEONID SHAIKHET
Le (F ) ≥0be he σ-algeb as gene a ed by {W(s),0≤s≤ }, hen W( ) is a
ma ingale ela i e o (F ) ≥0and we ha e he ollowing ep esen a ion o W( ):
W( ) =
∞
X
i=1
βi( )ei,
whe e {ei}i≥1is an o hono mal se o eigen ec o s o Q,βi( ) a e mu ually inde-
penden eal Wiene p ocesses wi h inc emen al co a iance λi>0, Qei=λieiand
T Q =P∞
i=1 λi<∞(T deno es he ace o an ope a o , see, o ins ance, [9]).
Fo an ope a o G∈L(K, H), he space o all bounded linea ope a o s om K
in o H, we deno e by kGk2i s Hilbe -Schmid no m, i.e.,
kGk2
2=T (GW G∗).
Gi en h≥0, and T > 0, we deno e by Ip(−h, T;U), p > 0, he space o all
U– alued p ocesses (x( )) ∈[−h,T ](we will w i e x( ) o sho ) measu able ( om
[−h, T]×Ω in o U), and sa is ying:
1. x( ) is F -measu able almos su ely in , whe e we se F =F0 o ≤0;
2. RT
−hEkx( )kpd < +∞.
I is no di icul o check ha he space Ip(−h, T;U) is a closed subspace o Lp(Ω×
[−h, T],F⊗B([−h, T]), dP ⊗d ;U),whe e B([−h, T ]) deno es he Bo el σ–algeb a
on [−h, T]. We also w i e L2(Ω; C(−h, T ;H)) ins ead o L2(Ω,F, dP;C(−h, T;H)),
whe e C(−h, T;H) deno es he space o all con inuous unc ions om [−h, T] in o
H.
Le CH=C([−h, 0], H) be he space o all con inuous unc ions om [−h, 0]
in o Hwi h sup-no m kψkC= sup−h≤s≤0|ψ(s)|,ψ∈CH( he de ini ion is simila
o CU), L2
U=L2([−h, 0]; U) and L2
H=L2([−h, 0]; H).
Gi en a s ochas ic p ocess u( )∈I2(−h, T ;U)∩L2(Ω; C(−h, T;H)), we asso-
cia e wi h an L2
U∩CH- alued s ochas ic p ocess u : Ω →L2
U∩CH, ≥0, by
se ing u (s)(ω) = u( +s)(ω), s∈[−h, 0].
The aim o his pape is o analyze he s abili y p ope ies (by means o con-
s uc ing sui able Lyapuno unc ionals) o he ollowing class o nonlinea s ochas-
ic pa ial unc ional di e en ial equa ions
du( ) = (A( , u( )) + ( , u ))d +B( , u )dW ( ), ∈[0, T ]
u( ) = ψ( ), ∈[−h, 0],(1.2)
whe e, in gene al, he ope a o s a e assumed o be nonlinea . In ac , we a e
in e es ed in he case in which A( , ·) : U→U∗is a amily o nonlinea mono one
and coe ci i e ope a o s, ( , ·) : CU→U∗and B( , ·) : CU→L(K, H) sa is y
sublinea p ope ies.
The analysis o he exis ence and uniqueness o solu ions o his model has
al eady been ca ied ou , o ins ance, in [1,3], and we will no insis in his poin
he e. Howe e , we will explain now which is he concep o solu ion o be used in
ou s abili y analysis.
Fo a ixed T > 0, gi en an ini ial alue
ψ∈I2(−h, 0; U)∩L2(Ω; CH),
STABILITY OF DELAY STOCHASTIC EVOLUTION EQUATIONS 3
a ( a ia ional) solu ion o (1.2) is a p ocess u( )∈I2(−h, T;U)∩L2(Ω; C(−h, T;H))
such ha
u( ) =ψ(0) + Z
0
[A(s, u(s)) + (s, us)] ds
+Z
0
B(s, us)dW(s), P −a.s.,∀ ∈[0, T],
u( ) = ψ( ), P −a.s., ∀ ∈[−h, 0],
(1.3)
whe e he i s equali y is de ined in U∗.
F om now on, as we will be in e es ed in he long- ime beha io o he solu ions
o (1.2), we will assume ha (1.3) possesses solu ions o all T > 0.
Le us deno e by u(·;ψ) he solu ion o Eq. (1.2) co esponding o he ini ial
condi ion ψ.
De ini ion 1.1. The i ial solu ion o Eq. (1.2) is said o be mean squa e s able
i o any  > 0 he e exis s δ > 0 such ha E|u( ;ψ)|2< ε o all ≥0 i
kψk2
CH= sups∈[−h,0] E|ψ(s)|2< δ.
De ini ion 1.2. The i ial solu ion o Eq. (1.2) is said o be exponen ially mean
squa e s able i i is s able and he e exis s a posi i e cons an λsuch ha o any
ψ∈C(−h, 0, U) he e exis s C(which may depend on ψ) such ha E|u( ;ψ)|2≤
Ce−λ o > 0.
Now, as we will use he I ˆo o mula o he solu ions o (1.3), we need o de ine
an associa e ope a o Lwhich is usually called he “gene a o ”o equa ion (1.3).
To calcula e he s ochas ic di e en ial o he p ocess η( ) = ( , u( )), whe e u( )
is a solu ion o he equa ion (1.3) and he unc ion ( , u) : [0,∞)×U→R+has
con inuous pa ial de i a i es
0
( , u) = ∂ ( , u)
∂ , 0
u( , u) = ∂ ( , u)
∂u , 00
uu( , u) = ∂2 ( , u)
∂u2,
he I ˆo o mula (see, e.g. [10] o mo e de ails) is used
dη( ) = L ( , u( ))d +< 0
u( , u( )), B( , u )dW ( )>,
whe e he gene a o Lis de ined in he ollowing way
L ( , u( )) = 0
( , u( ))+ < 0
u( , u( )), A( , u )>
+1
2T [ 00
uu( , u( ))B( , u )QB∗( , u )].
The gene a o Lcan be applied also o some unc ionals V( , ϕ) : [0,∞)×H→
R+. Suppose ha a unc ional V( , ϕ) can be ep esen in he o m V( , ϕ) =
V( , ϕ(0), ϕ(θ)), θ < 0 and o ϕ=u (o ϕ(θ) = u( +θ)) pu
Vϕ( , u) = V( , ϕ) = V( , u, ϕ(θ)),
u=ϕ(0) = u( ), θ < 0.(1.4)
Deno e by D he se o he unc ionals, o which he unc ion Vϕ( , u) de ined by
(1.4) has a con inuous de i a i e wi h espec o and wo con inuous de i a i es
4TOM ´
AS CARABALLO & LEONID SHAIKHET
wi h espec o u. Fo unc ionals om D he gene a o Lo he equa ion (1.3) has
he o m LV ( , u ) =V0
ϕ ( , u( ))+ < V 0
ϕu( , u( )), A( , u )>
+1
2T [V00
ϕuu( , u( ))B( , u )QB∗( , u )].(1.5)
F om I ˆo’s o mula i ollows, ha o unc ionals om D,
E[V( , u )−V(s, us)] = Z
s
ELV (τ, uτ)dτ, ≥s. (1.6)
1.2. Lyapuno ype s abili y heo em. Le us now p o e a heo em which will
be c ucial in ou s abili y in es iga ion.
Theo em 1.1. Assume ha he e exis s a unc ional V( , u )such ha he ollow-
ing condi ions hold o some posi i e numbe s c1,c2and λ:
EV( , u )≥c1eλ E|u( )|2, ≥0,(1.7)
EV(0, u0)≤c2kψk2
CH,(1.8)
ELV ( , u )≤0, ≥0.(1.9)
Then he i ial solu ion o Eq. (1.2) is exponen ially mean squa e s able.
P oo . In eg a ing (1.9) ia (1.6) we ob ain EV( , u )≤EV(0, u0). F om his and
(1.7), (1.8) i ollows ha
c1E|u( )|2≤e−λ EV(0, u0)≤c2kψk2
CH.
The inequali y c1E|u( )|2≤c2kψk2
CHmeans ha he i ial solu ion o Eq. (1.2) is
s able. Besides, om he inequali y c1E|u( )|2≤e−λ EV(0, u0),i ollows ha he
i ial solu ion o Eq. (1.2) is exponen ially mean squa e s able. 
No e ha Theo em 1.1 implies ha he s abili y in es iga ion o Eq. (1.2) can
be educed o he cons uc ion o app op ia e Lyapuno unc ionals. The gene al
me hod o Lyapuno unc ionals cons uc ion is desc ibed in [6,7,11,12]. A o mal
p ocedu e o cons uc Lyapuno unc ionals is desc ibed below.
1.3. P ocedu e o Lyapuno unc ionals cons uc ion. The p ocedu e con-
sis s o ou s eps.
S ep 1. To ans o m Eq. (1.2) in o he o m
dz( , u )=(A1( , u( )) + A2( , u ))d + (B1( , u( )) + B2( , u ))dW( ),(1.10)
whe e z( , ·), A2( , ·) and B2( , ·) a e amilies o nonlinea ope a o s, z( , 0) = 0,
A2( , 0) = 0, B2( , 0) = 0, ope a o s A1( , ·) and B1( , ·), such ha A1( , 0) = 0,
B1( , 0) = 0, and depend only on and u( ), bu do no depend on he p e ious
alues u( +s), s < 0.
S ep 2. Assume ha he i ial solu ion o he auxilia y equa ion wi hou mem-
o y
dy( ) = A1( , y( ))d +B1( , y( ))dW( ),(1.11)
STABILITY OF DELAY STOCHASTIC EVOLUTION EQUATIONS 5
is exponen ially mean squa e s able and he e exis s a Lyapuno unc ion ( , y( )),
which sa is ies he condi ions o Theo em 1.1.
S ep 3. A Lyapuno unc ional V( , u ) o Eq.(1.10) is cons uc ed in he o m
V=V1+V2, whe e V1( , u ) = ( , z( , u )). He e he a gumen yo he unc ion
( , y) is eplaced on he unc ional z( , x ) om he le -hand pa o Eq. (1.10).
S ep 4. Usually, he unc ional V1( , u ) almos sa is ies he condi ions o Theo em
1.1. In o de o ully sa is y hese condi ions, i is necessa y o calcula e ELV1( , u )
and es ima e i . Then, he addi ional unc ional V2( , u ) can be chosen in a s anda d
way.
No e ha he ep esen a ion (1.10) is no unique. This ac allows, using di e en
ep esen a ions o he ype o (1.10) o di e en ways o es ima ing ELV1( , u ), o
cons uc di e en Lyapuno unc ionals and, as a esul , o ge di e en su icien
condi ions o exponen ial mean squa e s abili y.
2. Cons uc ion o Lyapuno unc ionals o equa ions wi h ime- a ying
delay. Conside he ollowing s ochas ic e olu ion equa ion
du( )=(A( , u( )) + F(u( −h( ))))d +B( , u( −τ( )))dW( ),
h( )∈[0, h0], τ( )∈[0, τ0], h = max[h0, τ0],
u(s) = ψ(s), s ∈[−h, 0].
(2.1)
which is a pa icula case o Eq. (1.2). He e A( , ·), F :U→U∗a e app op ia e
pa ial di e en ial ope a o s (see condi ions below), B( , ·) : U→H,W( ) is a
Q-Wiene p ocess.
We will apply he me hod desc ibed abo e o cons uc Lyapuno unc ionals
o Eq. (2.1), and, as a consequence, o ob ain su icien condi ions ensu ing he
s abili y o he i ial solu ion.
We will use wo di e en cons uc ions which will p o ide di e en s abili y e-
gions o he pa ame e s in ol ed in he p oblem.
2.1. The i s way o Lyapuno unc ionals cons uc ion. Fi s we conside
a qui e gene al si ua ion o he ope a o s in ol ed in Eq. (2.1).
Theo em 2.1. Assume ha ope a o s in Eq. (2.1) sa is y he condi ions
hA( , u), ui ≤ −γkuk2, γ > 0,
F:U→U∗,kF(u)k∗≤αkuk, u ∈U,
kB( , u)k2≤σkuk, u ∈U,
(2.2)
and
h( )∈[0, h0],˙
h( )≤h1<1,
τ( )∈[0, τ0],˙τ( )≤τ1<1.(2.3)
I
γ > α
√1−h1
+δ
1−τ1
, δ =1
2σ2,(2.4)
hen he i ial solu ion o Eq. (2.1) is exponen ially mean squa e s able.
P oo . Owing o he p ocedu e o Lyapuno unc ionals cons uc ion, le us conside
he auxilia y equa ion wi hou memo y o he ype o (1.11) as
˙y( ) = A( , y( )).(2.5)

6TOM ´
AS CARABALLO & LEONID SHAIKHET
The unc ion ( , y) = eλ |y|2,λ > 0, is a Lyapuno unc ion o Eq. (2.5), i.e., i
sa is ies he condi ions o Theo em 1.1. Ac ually, i is easy o see ha condi ions
(1.7), (1.8) hold o he unc ion ( , y( )). Besides, since γ > 0, he e exis s λ > 0
such ha 2γ > λβ2. Using (2.5), (1.1) and (2.2), we ob ain
d
d ( , y( )) = λeλ |y( )|2+ 2eλ hA( , y( )), y( )i≤−eλ (2γ−λβ2)ky( )k2≤0.
Acco ding o he p ocedu e, we now cons uc a Lyapuno unc ional V o Eq.
(2.1) in he o m V=V1+V2, whe e V1( , u ) = eλ |u( )|2. Fo Eq. (2.1) ia (1.5)
and some ε > 0 we ob ain
LV1( , u ) =λV1( , u )+2eλ hA( , u( )) + F(u( −h( ))), u( )i+eλ kB( , u( −τ( )))k2
2
≤eλ λ|u( )|2+ 2 −γku( )k2+αku( −h( ))kku( )k+σ2ku( −τ( ))k2
≤eλ λβ2ku( )k2−2γku( )k2+αεku( −h( ))k2+ε−1ku( )k2
+σ2ku( −τ( ))k2
=eλ hλβ2−2γ+α
εku( )k2+εαku( −h( ))k2+σ2ku( −τ( ))k2i.
Se now
V2( , u ) = εα
1−h1Z
−h( )
eλ(s+h0)ku(s)k2ds +σ2
1−τ1Z
−τ( )
eλ(s+τ0)ku(s)k2ds.
Then
LV2( , u ) = εα
1−h1eλ( +h0)ku( )k2−(1 −˙
h( ))eλ( −h( )+h0)ku( −h( ))k2
+σ2
1−τ1eλ( +τ0)ku( )k2−(1 −˙τ( ))eλ( −τ( )+τ0)ku( −τ( ))k2
≤εαeλ
1−h1eλh0ku( )k2−(1 −h1)eλ(h0−h( ))ku( −h( ))k2
+σ2eλ
1−τ1eλτ0ku( )k2−(1 −τ1)eλ(τ0−τ( ))ku( −τ( ))k2
≤eλ εα eλh0
1−h1ku( )k2−ku( −h( ))k2
+σ2eλτ0
1−τ1ku( )k2−ku( −τ( ))k2.
Thus, o V=V1+V2we ha e
LV ( , u )≤eλ λβ2−2γ+α1
ε+εeλh0
1−h1+σ2eλτ0
1−τ1ku( )k2.
Rew i e he exp ession in squa e b acke s as
−2γ+α1
ε+ε
1−h1+σ2
1−τ1
+λβ2+εαeλh0−1
1−h1
+σ2eλτ0−1
1−τ1
.
STABILITY OF DELAY STOCHASTIC EVOLUTION EQUATIONS 7
To minimize his exp ession in he b acke s, choose ε=√1−h1. As a consequence
we ob ain
LV ( , u )≤ −eλ 2γ−α
√1−h1−δ
1−τ1−ρ(λ)ku( )k2(2.6)
wi h
ρ(λ) = λβ2+αeλh0−1
√1−h1
+σ2eλτ0−1
1−τ1
.
Since ρ(0) = 0, hen by condi ion (2.4) he e exis s λ > 0 small enough such ha
2γ−α
√1−h1−δ
1−τ1≥ρ(λ).
F om he e and (2.6) i ollows ha ELV ( , u )≤0. So, he unc ional V( , u )
cons uc ed abo e sa is ies he condi ions in Theo em 1.1. This means ha he
i ial solu ion o Eq. (2.1) is exponen ially mean squa e s able. 
No e, in pa icula , i h( )≡h0,τ( )≡τ0 hen h1= 0, τ1= 0 and condi ion
(2.4) akes he o m γ > α +δ.
2.2. The second way o Lyapuno unc ionals cons uc ion. We now es-
ablish a second esul which implies ha he ope a o Fmus be less gene al han
in Theo em 2.1. Howe e , as we will show la e in he applica ions sec ion, he
s abili y egions p o ided by his heo em will be be e han he ones gi en by
Theo em 2.1.
Theo em 2.2. Suppose ha ope a o s in Eq. (2.1) sa is y he ollowing condi ions
hA( , u) + F(u), ui≤−γkuk2, γ > 0,
kA( , u) + F(u)k∗≤α1kuk,
F:U→U, kF(u)k∗≤α2kuk, u ∈U,
kB( , u)k2≤σkuk, u ∈U,
(2.7)
and
h( )∈[0, h0],˙
h( )≤h1<1,|˙
h( )| ≤ h2,
τ( )∈[0, τ0],˙τ( )≤τ1<1.(2.8)
I
γ > α1α2h0+ (1 + α2h0)α2h2
√1−h1
+δ
1−τ1
,(2.9)
hen he i ial solu ion o Eq. (2.1) is exponen ially mean squa e s able.
P oo . To use he p ocedu e o Lyapuno unc ionals cons uc ion, le us i s
ans o m Eq. (2.1) as
dz( , u ) =(A( , u( )) + F(u( )) + ˙
h( )F(u( −h( ))))d
+B( , u( −τ( )))dW( ),(2.10)
8TOM ´
AS CARABALLO & LEONID SHAIKHET
whe e
z( , u ) = u( ) + Z
−h( )
F(u(s))ds. (2.11)
Conside he ollowing auxilia y equa ion wi hou memo y, which is o he ype o
(1.11), and is gi en in he o m
˙y( ) = A( , y( )) + F(y( )).(2.12)
The unc ion ( , y) = eλ |y|2is a Lyapuno unc ion o Eq. (2.12). Ac ually, since
γ > 0 hen he e exis s λ > 0 such ha 2γ > λβ2. Using (2.12), (1.1), (2.7), we
ob ain
d
d ( , y( )) =λeλ |y( )|2+ 2eλ hA( , y( )) + F(y( )), y( )i
≤−eλ (2γ−λβ2)ky( )k2.
Nex , we cons uc a Lyapuno unc ional V o Eq. (2.10), (2.11) in he o m
V=V1+V2, whe e
V1( , u ) = eλ |z( , u )|2,(2.13)
and z( , u ) is de ined by (2.11). Using (2.7) o Eq. (2.10), (2.11) and some posi i e
εi,i= 1,2,3, we ha e
LV1( , u ) =λV1( , u )+2eλ DA( , u( )) + F(u( )) + ˙
h( )F(u( −h( ))), z( , u )E
+eλ kB( , u( −τ( )))k2
2
=λV1( , u )+2eλ DA( , u( )) + F(u( )) + ˙
h( )F(u( −h( ))), u( )
+Z
−h( )
F(u(s))ds++eλ kB( , u( −τ( )))k2
2
=λV1( , u )+2eλ *A( , u( )) + F(u( )), u( ) + Z
−h( )
F(u(s))ds+
+ 2eλ ˙
h( ) F(u( −h( ))), u( ) + Z
−h( )
F(u(s))ds!
+eλ kB(u( −τ( )))k2
2
STABILITY OF DELAY STOCHASTIC EVOLUTION EQUATIONS 9
≤λV1( , u )+2eλ "−γku( )k2+α1α2Z
−h( )ku( )kku(s)kds#
+ 2eλ |˙
h( )| α2ku( −h( ))kku( )k+α2
2Z
−h( )ku( −h( ))kku(s)kds!
+eλ σ2ku( −τ( ))k2
≤λV1( , u ) + eλ "−2γku( )k2+α1α2Z
−h( )1
ε1ku( )k2+ε1ku(s)k2ds#
+eλ |˙
h( )|α2ε2ku( −h( ))k2+1
ε2ku( )k2
+α2
2Z
−h( )ε3ku( −h( ))k2+1
ε3ku(s)k2ds#+eλ σ2ku( −τ( ))k2
=λV1( , u ) + eλ σ2ku( −τ( ))k2
+eλ −2γ+1
1
α1α2h( ) + 1
ε2
α2|˙
h( )|ku( )k2
+α2(ε2+ε3α2h( ))|˙
h( )|ku( −h( ))k2
+α2ε1α1+1
ε3
α2|˙
h( )|Z
−h( )ku(s)k2ds#.
F om (2.13) and (2.11) o some ε4>0 i ollows ha
e−λ V1( , u ) =|u( )|2+ 2 Z
−h( )
(u( ), F(u(s)))ds +Z
−h( )
F(u(s))ds
2
≤|u( )|2+ 2 Z
−h( )|u( )||F(u(s))|ds +h( )Z
−h( )|F(u(s))|2ds
≤|u( )|2+α2β2Z
−h( )ε4ku( )k2+1
ε4ku(s)k2ds
+α2
2h( )β2Z
−h( )ku(s)k2ds
≤(1 + ε4α2h( ))β2ku( )k2+α2β21
ε4
+α2h( )Z
−h( )ku(s)k2ds.
The e o e,
LV1( , u )≤eλ λβ2(1 + ε4α2h( )) −2γ+1
1
α1α2h( ) + 1
ε2
α2|˙
h( )|ku( )k2
+eλ α2(ε2+ε3α2h( ))|˙
h( )|ku( −h( ))k2+eλ σ2ku( −τ( ))k2
+eλ α2ε1α1+α2
ε3|˙
h( )|+λβ21
ε4
+α2h( )Z
−h( )ku(s)k2ds
16 TOM ´
AS CARABALLO & LEONID SHAIKHET
o equa ion (3.7)
ν > |µ|
λ1√1−h1
+σ2
2λ1(1 −τ1).(3.9)
No e ha in he pa icula case [a, b] = [0, π] i holds λ1= 1 and hese h ee
condi ions gi en by Theo em 2.1 a e he same.
Obse e ha Theo em 2.2 can be applied only o Eq. (3.7). Fo his equa ion
he pa ame e s o Theo em 2.2 a e γ=α1=ν−µλ−1
1,α2=|µ|λ−1/2
1. I gi es he
ollowing su icien s abili y condi ion:
ν > µ
λ1
+|µ|h2
pλ1(1 −h1)√λ1+|µ|h0
√λ1−|µ|h0
+σ2
2√λ1(√λ1−|µ|h0)(1 −τ1),|µ|<√λ1
h0
.
(3.10)
No e ha he s abili y condi ion (3.9) ha we ha e ob ained o equa ion (3.7)
imp o es he one in he pape [2]. Indeed, in he case [a, b] = [0, π] and cons an
delay, i.e., h( ) = τ( ) = h, he s abili y condi ion ob ained in [2] o ν= 1 is
1>3eh(µ2+σ2).(3.11)
Also, ou s abili y condi ion (3.9) imp o es he one in [8] since in his pape he
s abili y condi ion is ob ained in he o m
1>3(µ2+σ2),(3.12)
al hough he delay unc ions h(·) and τ(·) a e only assumed o be measu able.
Fig. 3.1 shows he s abili y egions o he equa ion (3.7) which ha e been ob-
ained o he alues o pa ame e s ν= 1, h1=τ1= 0, λ1= 1. The line (1)
ep esen s he condi ion (3.9), he line (2) ep esen s (3.10), (3) co esponds o
(3.11), and (4) o (3.12). One can see ha bo h condi ions (3.9) and (3.10) a e
essen ially be e han he condi ion (3.12) which is also be e han (3.11). On he
o he hand, he condi ion (3.10) is wo se han (3.9) o µ > 0, bu be e han (3.9)
o µ < 0.
In Fig. 3.2 and 3.3 one can see ha he condi ions (3.9) and (3.10) complemen
each o he o ν > 1 and ν < 1 (in Fig. 3.2 we ha e ν= 1.2, while ν= 0.7 in Fig.
3.3) wi h he same alues o o he pa ame e s.

STABILITY OF DELAY STOCHASTIC EVOLUTION EQUATIONS 17
Fig. 3.1.
Fig. 3.2.
18 TOM ´
AS CARABALLO & LEONID SHAIKHET
Fig. 3.3.
Le us hen conside he ollowing h ee p oblems:
du( , x) = ν∂2u( , x)
∂x2+µ∂2u( −h( ), x)
∂x2d +σ∂u( −τ( ), x)
∂x dW ( ),(3.13)
du( , x) = ν∂2u( , x)
∂x2+µ∂u( −h( ), x)
∂x d +σ∂u( −τ( ), x)
∂x dW ( ),(3.14)
du( , x) = ν∂2u( , x)
∂x2+µ u( −h( ), x)d +σ∂u( −τ( ), x)
∂x dW ( ) (3.15)
wi h he same condi ions (3.8), whe e ν > 0 and µis an a bi a y cons an . We
can again apply Theo em 2.1 o all hese examples yielding he ollowing su icien
s abili y condi ions.
Fo equa ion (3.13)
ν > |µ|
√1−h1
+σ2
2(1 −τ1),
o equa ion (3.14)
ν > |µ|
pλ1(1 −h1)+σ2
2(1 −τ1),
o equa ion (3.15)
ν > |µ|
λ1√1−h1
+σ2
2(1 −τ1).
STABILITY OF DELAY STOCHASTIC EVOLUTION EQUATIONS 19
No e ha in he pa icula case [a, b] = [0, π] as λ1= 1, and hese h ee condi ions
gi en by Theo em 2.1 a e he same.
Obse e ha Theo em 2.2 can be applied only o Eq. (3.15). Fo his equa ion
he pa ame e s o Theo em 2.2 a e γ=α1=ν−µλ−1
1,α2=|µ|λ−1/2
1. I gi es he
ollowing su icien s abili y condi ion:
ν > µ
λ1
+|µ|h2
pλ1(1 −h1)√λ1+|µ|h0
√λ1−|µ|h0
+σ2√λ1
2(√λ1−|µ|h0)(1 −τ1),|µ|<√λ1
h0
.
Rema k 3.2. Analogous examples ha e been analyzed in [13], and simila con-
di ions o ou s ha e been ob ained wi hou assuming ha he delay unc ion a e
con inuously di e en iable. Howe e , he concep o solu ion used in [13] is s onge
han he one we use in his pape , since hei p oo elies in an equali y s a ed in
Theo em 2.2 (see [13] page 492), which implies ha he solu ion mus belong o
C1(0, T;L2(Ω; H)), while in he usual si ua ion, he solu ion a e p o ed o belong
only o L2(Ω; C(0, T;H)) (see De ini ion 2.1 in [13] and ou de ini ion o solu ion in
his pape ) and, consequen ly, he echnique used in [13] canno be applied. Mo e-
o e , he ope a o in he di usion pa o he equa ion in [13] does no allow o
i s o de de i a i es while i does in ou case (see he p eceding examples).
Acknowledgemen s. We would like o hank he e e ees o hei help ul sugges-
ions which allowed us o imp o e he p esen a ion o his pape .
Re e ences
[1] T. Ca aballo, M.J. Ga ido-A ienza and J. Real, Asymp o ic s abili y o nonlinea s ochas ic
e olu ion equa ions, S och. Anal. Appl., 21 (2003), 301–327.
[2] T. Ca aballo and K. Liu, Exponen ial s abili y o mild solu ions o s ochas ic pa ial di e en ial
equa ions wi h delays, S och. Anal. Appl., 15 (1999), 743–763.
[3] T. Ca aballo, K. Liu and A. T uman, S ochas ic unc ional pa ial di e en ial equa ions: ex-
is ence, uniqueness and asymp o ic decay p ope ies, P oc. Roy. Soc. Lond. A, 456 (2000),
1775–1802.
[4] T. Ca aballo, J. Real and L. Shaikhe , Me hod o Lyapuno unc ionals cons uc ion in s abili y
o delay e olu ion equa ions, J. Ma h. Anal. Appl., 334(2) (2007), 1130–1145.
[5] H. Chen, Asymp o ic beha io o s ochas ic wo-dimensional Na ie -S okes equa ions wi h
delays, P oc. Indian Acad. Sci (Ma h.Sci), 122(2) (2012), 283–295.
[6] V. Kolmano skii and L. Shaikhe , A me hod o Lyapuno unc ionals cons uc ion o s ochas ic
di e en ial equa ions o neu al ype, Di e en ialniye u a neniya, 31(11) (2002), 691–716 (in
Russian). T ansla ion in: Di e en ial Equa ions, 31 (11) (1996) 1819-1825.
[7] V. Kolmano skii and L. Shaikhe , Cons uc ion o Lyapuno unc ionals o s ochas ic he edi-
a y sys ems: a su ey o some ecen esul s, Ma hema ical and Compu e Modelling, 36(6)
(1995), 1851–1857.
[8] J. Luo, Fixed poin s and exponen ial s abili y o mild solu ions o s ochas ic pa ial di e en ial
equa ions wi h delays, J. Ma h. anal. Appl., 342 (2008), 753–760.
[9] E. Pa doux, “Equa ions aux d´e i ´ees pa ielles s ochas iques nonlin´eai es mono ones,” Ph.D
hesis, Uni e si ´e Pa is Sud, 1975.
[10] G. Da P a o, J. Zabczyk, “ S ochas ic equa ions in in ini e dimensions. Encyclopedia o
ma hema ics and i s applica ions,” Camb idge Uni e si y P ess, 1992..
[11] L. Shaikhe , “ Lyapuno Func ionals and S abili y o S ochas ic Di e ence Equa ions,” Sp inge ,
London, Do d ech , Heidelbe g, New Yo k, 2011. 370p.
20 TOM ´
AS CARABALLO & LEONID SHAIKHET
[12] L. Shaikhe , Mode n s a e and de elopmen pe spec i es o Lyapuno unc ionals me hod in
he s abili y heo y o s ochas ic he edi a y sys ems, Theo y o S ochas ic P ocesses, 2(18)
(1996), 248–259.
[13] L. Wan and J. Duan, Exponen ial s abili y o non-au onomous s ochas ic pa ial di e en ial
equa ions wi h ini e memo y, S a is ics and P obabili y Le e s., 78 (2008), 490–498.
[14] M. Wei and T. Zhang, Exponen ial s abili y o s ochas ic 2D-Na ie -S okes equa ions wi h
ime delay, Appl. Ma h. J. Chinese Uni ., 24 (2009), 493–500 (in Chinese).
Recei ed XXXX ; e ised XXXX .
E-mail add ess: (T. Ca aballo) [email p o ec ed]
E-mail add ess: (L. Shaikhe ) [email p o ec ed]