Stability of delay evolution equations with stochastic perturbations

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]