Stability of delay evolution equations with stochastic perturbations

Manuscript submitted to Website http://aimSciences.org AIMS journals Volume X, Number X, XX 200X pp. X–XX Stability of delay evolution equations with stochastic perturbations Tom´ as Caraballo Depto. de Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla Apdo. de Correos 1160, 41080-Sevilla, Spain Leonid Shaikhet Department of Higher Mathematics, Donetsk State University of Management Chelyuskintsev str., 163-a, Donetsk 83015, Ukraine (Communicated by Aim Sciences) Dedicated to the Memory of Mark I. Vishik Abstract. The investigation of stability for hereditary systems is often related to the construction of Lyapunov functionals. The general method of Lyapunov functionals construction, which was proposed by V.Kolmanovskii and L.Shaikhet, is used here to investigate the stability of stochastic delay evolution equations, in particular, for stochastic partial differential equations. This method had already been successfully used for functional-differential equations, for difference equations with discrete time, and for difference equations with continuous time. It is shown that the stability conditions obtained for stochastic 2D Navier- Stokes model with delays are essentially better than the known ones. 1. Introduction. 1.1. Notations and definitions. First of all, we introduce the framework in which our analysis is going to be carried out. Let U, H, K be real, separable Hilbert spaces such that U⊂H≡H∗⊂U∗, where U∗is the dual of Uand the injections are continuous and dense. We denote by βthe constant satisfying |u| ≤ βkuk, u ∈U. (1.1) In particular, we also assume both Uand U∗are uniformly convex. We denote by k·k,|·| and k·k∗the norms in U,Hand U∗,respectively; by h·,·i the duality product between U∗, U , and by (·,·) the scalar product in H. Let W(t) be a Q-valued Wiener process on a certain complete probability space (Ω,F, P) which takes values in the separable Hilbert space K, where Q∈L(K, K) is a symmetric nonnegative operator and EW(t) = 0, Cov(W(t)) = tQ. 2000 Mathematics Subject Classification. Primary: 35K35, 60H15; Secondary: 60G52. Key words and phrases. Method of Lyapunov functionals construction, Stochastic evolution equations, Exponential stability, Stochastic partial differential equations, Stochastic 2D Navier- Stokes model with delays. This research was partially supported by FEDER and Ministerio de Econom´ıa y Competitividad (Sapin) under grant MTM2011-22411. 1 2TOM ´ AS CARABALLO & LEONID SHAIKHET Let (Ft)t≥0be the σ-algebras generated by {W(s),0≤s≤t}, then W(t) is a martingale relative to (Ft)t≥0and we have the following representation of W(t): W(t) = ∞ X i=1 βi(t)ei, where {ei}i≥1is an orthonormal set of eigenvectors of Q,βi(t) are mutually independent real Wiener processes with incremental covariance λi>0, Qei=λieiand Tr Q =P∞ i=1 λi<∞(Tr denotes the trace of an operator, see, for instance, [9]). For an operator G∈L(K, H), the space of all bounded linear operators from K into H, we denote by kGk2its Hilbert-Schmidt norm, i.e., kGk2 2=Tr(GW G∗). Given h≥0, and T > 0, we denote by Ip(−h, T;U), p > 0, the space of all U–valued processes (x(t))t∈[−h,T ](we will write x(t) for short) measurable (from [−h, T]×Ω into U), and satisfying: 1. x(t) is Ft-measurable almost surely in t, where we set Ft=F0for t≤0; 2. RT −hEkx(t)kpdt < +∞. It is not difficult to check that the space Ip(−h, T;U) is a closed subspace of Lp(Ω× [−h, T],F⊗B([−h, T]), dP ⊗dt;U),where B([−h, T ]) denotes the Borel σ–algebra on [−h, T]. We also write L2(Ω; C(−h, T ;H)) instead of L2(Ω,F, dP;C(−h, T;H)), where C(−h, T;H) denotes the space of all continuous functions from [−h, T] into H. Let CH=C([−h, 0], H) be the space of all continuous functions from [−h, 0] into Hwith sup-norm kψkC= sup−h≤s≤0|ψ(s)|,ψ∈CH(the definition is similar for CU), L2 U=L2([−h, 0]; U) and L2 H=L2([−h, 0]; H). Given a stochastic process u(t)∈I2(−h, T ;U)∩L2(Ω; C(−h, T;H)), we associate with an L2 U∩CH-valued stochastic process ut: Ω →L2 U∩CH,t≥0, by setting ut(s)(ω) = u(t+s)(ω), s∈[−h, 0]. The aim of this paper is to analyze the stability properties (by means of constructing suitable Lyapunov functionals) of the following class of nonlinear stochastic partial functional differential equations du(t) = (A(t, u(t)) + f(t, ut))dt +B(t, ut)dW (t), t ∈[0, T ] u(t) = ψ(t), t ∈[−h, 0],(1.2) where, in general, the operators are assumed to be nonlinear. In fact, we are interested in the case in which A(t, ·) : U→U∗is a family of nonlinear monotone and coercitive operators, f(t, ·) : CU→U∗and B(t, ·) : CU→L(K, H) satisfy sublinear properties. The analysis of the existence and uniqueness of solutions for this model has already been carried out, for instance, in [1,3], and we will not insist in this point here. However, we will explain now which is the concept of solution to be used in our stability analysis. For a fixed T > 0, given an initial value ψ∈I2(−h, 0; U)∩L2(Ω; CH), STABILITY OF DELAY STOCHASTIC EVOLUTION EQUATIONS 3 a (variational) solution of (1.2) is a process u(t)∈I2(−h, T;U)∩L2(Ω; C(−h, T;H)) such that u(t) =ψ(0) + Zt 0 [A(s, u(s)) + f(s, us)] ds +Zt 0 B(s, us)dW(s), P −a.s.,∀t∈[0, T], u(t) = ψ(t), P −a.s., ∀t∈[−h, 0], (1.3) where the first equality is defined in U∗. From now on, as we will be interested in the long-time behavior of the solutions of (1.2), we will assume that (1.3) possesses solutions for all T > 0. Let us denote by u(·;ψ) the solution of Eq. (1.2) corresponding to the initial condition ψ. Definition 1.1. The trivial solution of Eq. (1.2) is said to be mean square stable if for any  > 0 there exists δ > 0 such that E|u(t;ψ)|2< ε for all t≥0 if kψk2 CH= sups∈[−h,0] E|ψ(s)|2< δ. Definition 1.2. The trivial solution of Eq. (1.2) is said to be exponentially mean square stable if it is stable and there exists a positive constant λsuch that for any ψ∈C(−h, 0, U) there exists C(which may depend on ψ) such that E|u(t;ψ)|2≤ Ce−λt for t > 0. Now, as we will use the Itˆo formula for the solutions of (1.3), we need to define an associate operator Lwhich is usually called the “generator ”of equation (1.3). To calculate the stochastic differential of the process η(t) = v(t, u(t)), where u(t) is a solution of the equation (1.3) and the function v(t, u) : [0,∞)×U→R+has continuous partial derivatives v0 t(t, u) = ∂v(t, u) ∂t , v0 u(t, u) = ∂v(t, u) ∂u , v00 uu(t, u) = ∂2v(t, u) ∂u2, the Itˆo formula (see, e.g. [10] for more details) is used dη(t) = Lv(t, u(t))dt+< v0 u(t, u(t)), B(t, ut)dW (t)>, where the generator Lis defined in the following way Lv(t, u(t)) =v0 t(t, u(t))+ < v0 u(t, u(t)), A(t, ut)> +1 2Tr[v00 uu(t, u(t))B(t, ut)QB∗(t, ut)]. The generator Lcan be applied also for some functionals V(t, ϕ) : [0,∞)×H→ R+. Suppose that a functional V(t, ϕ) can be represent in the form V(t, ϕ) = V(t, ϕ(0), ϕ(θ)), θ < 0 and for ϕ=ut(or ϕ(θ) = u(t+θ)) put Vϕ(t, u) = V(t, ϕ) = V(t, u, ϕ(θ)), u=ϕ(0) = u(t), θ < 0.(1.4) Denote by Dthe set of the functionals, for which the function Vϕ(t, u) defined by (1.4) has a continuous derivative with respect to tand two continuous derivatives 4TOM ´ AS CARABALLO & LEONID SHAIKHET with respect to u. For functionals from Dthe generator Lof the equation (1.3) has the form LV (t, ut) =V0 ϕt(t, u(t))+ < V 0 ϕu(t, u(t)), A(t, ut)> +1 2Tr[V00 ϕuu(t, u(t))B(t, ut)QB∗(t, ut)].(1.5) From Itˆo’s formula it follows, that for functionals from D, E[V(t, ut)−V(s, us)] = Zt s ELV (τ, uτ)dτ, t ≥s. (1.6) 1.2. Lyapunov type stability theorem. Let us now prove a theorem which will be crucial in our stability investigation. Theorem 1.1. Assume that there exists a functional V(t, ut)such that the following conditions hold for some positive numbers c1,c2and λ: EV(t, ut)≥c1eλtE|u(t)|2, t ≥0,(1.7) EV(0, u0)≤c2kψk2 CH,(1.8) ELV (t, ut)≤0, t ≥0.(1.9) Then the trivial solution of Eq. (1.2) is exponentially mean square stable. Proof. Integrating (1.9) via (1.6) we obtain EV(t, ut)≤EV(0, u0). From this and (1.7), (1.8) it follows that c1E|u(t)|2≤e−λtEV(0, u0)≤c2kψk2 CH. The inequality c1E|u(t)|2≤c2kψk2 CHmeans that the trivial solution of Eq. (1.2) is stable. Besides, from the inequality c1E|u(t)|2≤e−λtEV(0, u0),it follows that the trivial solution of Eq. (1.2) is exponentially mean square stable.  Note that Theorem 1.1 implies that the stability investigation of Eq. (1.2) can be reduced to the construction of appropriate Lyapunov functionals. The general method of Lyapunov functionals construction is described in [6,7,11,12]. A formal procedure to construct Lyapunov functionals is described below. 1.3. Procedure of Lyapunov functionals construction. The procedure consists of four steps. Step 1. To transform Eq. (1.2) into the form dz(t, ut)=(A1(t, u(t)) + A2(t, ut))dt + (B1(t, u(t)) + B2(t, ut))dW(t),(1.10) where z(t, ·), A2(t, ·) and B2(t, ·) are families of nonlinear operators, z(t, 0) = 0, A2(t, 0) = 0, B2(t, 0) = 0, operators A1(t, ·) and B1(t, ·), such that A1(t, 0) = 0, B1(t, 0) = 0, and depend only on tand u(t), but do not depend on the previous values u(t+s), s < 0. Step 2. Assume that the trivial solution of the auxiliary equation without memory dy(t) = A1(t, y(t))dt +B1(t, y(t))dW(t),(1.11) STABILITY OF DELAY STOCHASTIC EVOLUTION EQUATIONS 5 is exponentially mean square stable and there exists a Lyapunov function v(t, y(t)), which satisfies the conditions of Theorem 1.1. Step 3. A Lyapunov functional V(t, ut) for Eq.(1.10) is constructed in the form V=V1+V2, where V1(t, ut) = v(t, z(t, ut)). Here the argument yof the function v(t, y) is replaced on the functional z(t, xt) from the left-hand part of Eq. (1.10). Step 4. Usually, the functional V1(t, ut) almost satisfies the conditions of Theorem 1.1. In order to fully satisfy these conditions, it is necessary to calculate ELV1(t, ut) and estimate it. Then, the additional functional V2(t, ut) can be chosen in a standard way. Note that the representation (1.10) is not unique. This fact allows, using different representations of the type of (1.10) or different ways of estimating ELV1(t, ut), to construct different Lyapunov functionals and, as a result, to get different sufficient conditions of exponential mean square stability. 2. Construction of Lyapunov functionals for equations with time-varying delay. Consider the following stochastic evolution equation du(t)=(A(t, u(t)) + F(u(t−h(t))))dt +B(t, u(t−τ(t)))dW(t), h(t)∈[0, h0], τ(t)∈[0, τ0], h = max[h0, τ0], u(s) = ψ(s), s ∈[−h, 0]. (2.1) which is a particular case of Eq. (1.2). Here A(t, ·), F :U→U∗are appropriate partial differential operators (see conditions below), B(t, ·) : U→H,W(t) is a Q-Wiener process. We will apply the method described above to construct Lyapunov functionals for Eq. (2.1), and, as a consequence, to obtain sufficient conditions ensuring the stability of the trivial solution. We will use two different constructions which will provide different stability regions for the parameters involved in the problem. 2.1. The first way of Lyapunov functionals construction. First we consider a quite general situation for the operators involved in Eq. (2.1). Theorem 2.1. Assume that operators in Eq. (2.1) satisfy the conditions hA(t, u), ui ≤ −γkuk2, γ > 0, F:U→U∗,kF(u)k∗≤αkuk, u ∈U, kB(t, u)k2≤σkuk, u ∈U, (2.2) and h(t)∈[0, h0],˙ h(t)≤h1<1, τ(t)∈[0, τ0],˙τ(t)≤τ1<1.(2.3) If γ > α √1−h1 +δ 1−τ1 , δ =1 2σ2,(2.4) then the trivial solution of Eq. (2.1) is exponentially mean square stable. Proof. Owing to the procedure of Lyapunov functionals construction, let us consider the auxiliary equation without memory of the type of (1.11) as ˙y(t) = A(t, y(t)).(2.5) 6TOM ´ AS CARABALLO & LEONID SHAIKHET The function v(t, y) = eλt|y|2,λ > 0, is a Lyapunov function for Eq. (2.5), i.e., it satisfies the conditions of Theorem 1.1. Actually, it is easy to see that conditions (1.7), (1.8) hold for the function v(t, y(t)). Besides, since γ > 0, there exists λ > 0 such that 2γ > λβ2. Using (2.5), (1.1) and (2.2), we obtain d dtv(t, y(t)) = λeλt|y(t)|2+ 2eλthA(t, y(t)), y(t)i≤−eλt(2γ−λβ2)ky(t)k2≤0. According to the procedure, we now construct a Lyapunov functional Vfor Eq. (2.1) in the form V=V1+V2, where V1(t, ut) = eλt|u(t)|2. For Eq. (2.1) via (1.5) and some ε > 0 we obtain LV1(t, ut) =λV1(t, ut)+2eλt hA(t, u(t)) + F(u(t−h(t))), u(t)i+eλtkB(t, u(t−τ(t)))k2 2 ≤eλt λ|u(t)|2+ 2 −γku(t)k2+αku(t−h(t))kku(t)k+σ2ku(t−τ(t))k2 ≤eλt λβ2ku(t)k2−2γku(t)k2+αεku(t−h(t))k2+ε−1ku(t)k2 +σ2ku(t−τ(t))k2 =eλt hλβ2−2γ+α εku(t)k2+εαku(t−h(t))k2+σ2ku(t−τ(t))k2i. Set now V2(t, ut) = εα 1−h1Zt t−h(t) eλ(s+h0)ku(s)k2ds +σ2 1−τ1Zt t−τ(t) eλ(s+τ0)ku(s)k2ds. Then LV2(t, ut) = εα 1−h1eλ(t+h0)ku(t)k2−(1 −˙ h(t))eλ(t−h(t)+h0)ku(t−h(t))k2 +σ2 1−τ1eλ(t+τ0)ku(t)k2−(1 −˙τ(t))eλ(t−τ(t)+τ0)ku(t−τ(t))k2 ≤εαeλt 1−h1eλh0ku(t)k2−(1 −h1)eλ(h0−h(t))ku(t−h(t))k2 +σ2eλt 1−τ1eλτ0ku(t)k2−(1 −τ1)eλ(τ0−τ(t))ku(t−τ(t))k2 ≤eλt εα eλh0 1−h1ku(t)k2−ku(t−h(t))k2 +σ2eλτ0 1−τ1ku(t)k2−ku(t−τ(t))k2. Thus, for V=V1+V2we have LV (t, ut)≤eλt λβ2−2γ+α1 ε+εeλh0 1−h1+σ2eλτ0 1−τ1ku(t)k2. Rewrite the expression in square brackets 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 this expression in the brackets, choose ε=√1−h1. As a consequence we obtain LV (t, ut)≤ −eλt 2γ−α √1−h1−δ 1−τ1−ρ(λ)ku(t)k2(2.6) with ρ(λ) = λβ2+αeλh0−1 √1−h1 +σ2eλτ0−1 1−τ1 . Since ρ(0) = 0, then by condition (2.4) there exists λ > 0 small enough such that 2γ−α √1−h1−δ 1−τ1≥ρ(λ). From here and (2.6) it follows that ELV (t, ut)≤0. So, the functional V(t, ut) constructed above satisfies the conditions in Theorem 1.1. This means that the trivial solution of Eq. (2.1) is exponentially mean square stable.  Note, in particular, if h(t)≡h0,τ(t)≡τ0then h1= 0, τ1= 0 and condition (2.4) takes the form γ > α +δ. 2.2. The second way of Lyapunov functionals construction. We now establish a second result which implies that the operator Fmust be less general than in Theorem 2.1. However, as we will show later in the applications section, the stability regions provided by this theorem will be better than the ones given by Theorem 2.1. Theorem 2.2. Suppose that operators in Eq. (2.1) satisfy the following conditions hA(t, u) + F(u), ui≤−γkuk2, γ > 0, kA(t, u) + F(u)k∗≤α1kuk, F:U→U, kF(u)k∗≤α2kuk, u ∈U, kB(t, u)k2≤σkuk, u ∈U, (2.7) and h(t)∈[0, h0],˙ h(t)≤h1<1,|˙ h(t)| ≤ h2, τ(t)∈[0, τ0],˙τ(t)≤τ1<1.(2.8) If γ > α1α2h0+ (1 + α2h0)α2h2 √1−h1 +δ 1−τ1 ,(2.9) then the trivial solution of Eq. (2.1) is exponentially mean square stable. Proof. To use the procedure of Lyapunov functionals construction, let us first transform Eq. (2.1) as dz(t, ut) =(A(t, u(t)) + F(u(t)) + ˙ h(t)F(u(t−h(t))))dt +B(t, u(t−τ(t)))dW(t),(2.10) 8TOM ´ AS CARABALLO & LEONID SHAIKHET where z(t, ut) = u(t) + Zt t−h(t) F(u(s))ds. (2.11) Consider the following auxiliary equation without memory, which is of the type of (1.11), and is given in the form ˙y(t) = A(t, y(t)) + F(y(t)).(2.12) The function v(t, y) = eλt|y|2is a Lyapunov function for Eq. (2.12). Actually, since γ > 0 then there exists λ > 0 such that 2γ > λβ2. Using (2.12), (1.1), (2.7), we obtain d dtv(t, y(t)) =λeλt|y(t)|2+ 2eλthA(t, y(t)) + F(y(t)), y(t)i ≤−eλt(2γ−λβ2)ky(t)k2. Next, we construct a Lyapunov functional Vfor Eq. (2.10), (2.11) in the form V=V1+V2, where V1(t, ut) = eλt|z(t, ut)|2,(2.13) and z(t, ut) is defined by (2.11). Using (2.7) for Eq. (2.10), (2.11) and some positive εi,i= 1,2,3, we have LV1(t, ut) =λV1(t, ut)+2eλt DA(t, u(t)) + F(u(t)) + ˙ h(t)F(u(t−h(t))), z(t, ut)E +eλtkB(t, u(t−τ(t)))k2 2 =λV1(t, ut)+2eλt DA(t, u(t)) + F(u(t)) + ˙ h(t)F(u(t−h(t))), u(t) +Zt t−h(t) F(u(s))ds++eλtkB(t, u(t−τ(t)))k2 2 =λV1(t, ut)+2eλt *A(t, u(t)) + F(u(t)), u(t) + Zt t−h(t) F(u(s))ds+ + 2eλt ˙ h(t) F(u(t−h(t))), u(t) + Zt t−h(t) F(u(s))ds! +eλtkB(u(t−τ(t)))k2 2 STABILITY OF DELAY STOCHASTIC EVOLUTION EQUATIONS 9 ≤λV1(t, ut)+2eλt "−γku(t)k2+α1α2Zt t−h(t)ku(t)kku(s)kds# + 2eλt|˙ h(t)| α2ku(t−h(t))kku(t)k+α2 2Zt t−h(t)ku(t−h(t))kku(s)kds! +eλtσ2ku(t−τ(t))k2 ≤λV1(t, ut) + eλt "−2γku(t)k2+α1α2Zt t−h(t)1 ε1ku(t)k2+ε1ku(s)k2ds# +eλt|˙ h(t)|α2ε2ku(t−h(t))k2+1 ε2ku(t)k2 +α2 2Zt t−h(t)ε3ku(t−h(t))k2+1 ε3ku(s)k2ds#+eλtσ2ku(t−τ(t))k2 =λV1(t, ut) + eλtσ2ku(t−τ(t))k2 +eλt −2γ+1 1 α1α2h(t) + 1 ε2 α2|˙ h(t)|ku(t)k2 +α2(ε2+ε3α2h(t))|˙ h(t)|ku(t−h(t))k2 +α2ε1α1+1 ε3 α2|˙ h(t)|Zt t−h(t)ku(s)k2ds#. From (2.13) and (2.11) for some ε4>0 it follows that e−λtV1(t, ut) =|u(t)|2+ 2 Zt t−h(t) (u(t), F(u(s)))ds +Zt t−h(t) F(u(s))ds 2 ≤|u(t)|2+ 2 Zt t−h(t)|u(t)||F(u(s))|ds +h(t)Zt t−h(t)|F(u(s))|2ds ≤|u(t)|2+α2β2Zt t−h(t)ε4ku(t)k2+1 ε4ku(s)k2ds +α2 2h(t)β2Zt t−h(t)ku(s)k2ds ≤(1 + ε4α2h(t))β2ku(t)k2+α2β21 ε4 +α2h(t)Zt t−h(t)ku(s)k2ds. Therefore, LV1(t, ut)≤eλt λβ2(1 + ε4α2h(t)) −2γ+1 1 α1α2h(t) + 1 ε2 α2|˙ h(t)|ku(t)k2 +eλtα2(ε2+ε3α2h(t))|˙ h(t)|ku(t−h(t))k2+eλtσ2ku(t−τ(t))k2 +eλtα2ε1α1+α2 ε3|˙ h(t)|+λβ21 ε4 +α2h(t)Zt t−h(t)ku(s)k2ds 16 TOM ´ AS CARABALLO & LEONID SHAIKHET for equation (3.7) ν > |µ| λ1√1−h1 +σ2 2λ1(1 −τ1).(3.9) Note that in the particular case [a, b] = [0, π] it holds λ1= 1 and these three conditions given by Theorem 2.1 are the same. Observe that Theorem 2.2 can be applied only to Eq. (3.7). For this equation the parameters of Theorem 2.2 are γ=α1=ν−µλ−1 1,α2=|µ|λ−1/2 1. It gives the following sufficient stability condition: 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) Note that the stability condition (3.9) that we have obtained for equation (3.7) improves the one in the paper [2]. Indeed, in the case [a, b] = [0, π] and constant delay, i.e., h(t) = τ(t) = h, the stability condition obtained in [2] for ν= 1 is 1>3eh(µ2+σ2).(3.11) Also, our stability condition (3.9) improves the one in [8] since in this paper the stability condition is obtained in the form 1>3(µ2+σ2),(3.12) although the delay functions h(·) and τ(·) are only assumed to be measurable. Fig. 3.1 shows the stability regions for the equation (3.7) which have been obtained for the values of parameters ν= 1, h1=τ1= 0, λ1= 1. The line (1) represents the condition (3.9), the line (2) represents (3.10), (3) corresponds to (3.11), and (4) to (3.12). One can see that both conditions (3.9) and (3.10) are essentially better than the condition (3.12) which is also better than (3.11). On the other hand, the condition (3.10) is worse than (3.9) for µ > 0, but better than (3.9) for µ < 0. In Fig. 3.2 and 3.3 one can see that the conditions (3.9) and (3.10) complement each other for ν > 1 and ν < 1 (in Fig. 3.2 we have ν= 1.2, while ν= 0.7 in Fig. 3.3) with the same values of other parameters. STABILITY OF DELAY STOCHASTIC EVOLUTION EQUATIONS 17 Fig. 3.1. Fig. 3.2. 18 TOM ´ AS CARABALLO & LEONID SHAIKHET Fig. 3.3. Let us then consider the following three problems: du(t, x) = ν∂2u(t, x) ∂x2+µ∂2u(t−h(t), x) ∂x2dt +σ∂u(t−τ(t), x) ∂x dW (t),(3.13) du(t, x) = ν∂2u(t, x) ∂x2+µ∂u(t−h(t), x) ∂x dt +σ∂u(t−τ(t), x) ∂x dW (t),(3.14) du(t, x) = ν∂2u(t, x) ∂x2+µ u(t−h(t), x)dt +σ∂u(t−τ(t), x) ∂x dW (t) (3.15) with the same conditions (3.8), where ν > 0 and µis an arbitrary constant. We can again apply Theorem 2.1 to all these examples yielding the following sufficient stability conditions. For equation (3.13) ν > |µ| √1−h1 +σ2 2(1 −τ1), for equation (3.14) ν > |µ| pλ1(1 −h1)+σ2 2(1 −τ1), for equation (3.15) ν > |µ| λ1√1−h1 +σ2 2(1 −τ1). STABILITY OF DELAY STOCHASTIC EVOLUTION EQUATIONS 19 Note that in the particular case [a, b] = [0, π] as λ1= 1, and these three conditions given by Theorem 2.1 are the same. Observe that Theorem 2.2 can be applied only to Eq. (3.15). For this equation the parameters of Theorem 2.2 are γ=α1=ν−µλ−1 1,α2=|µ|λ−1/2 1. It gives the following sufficient stability condition: 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 . Remark 3.2. Analogous examples have been analyzed in [13], and similar conditions to ours have been obtained without assuming that the delay function are continuously differentiable. However, the concept of solution used in [13] is stronger than the one we use in this paper, since their proof relies in an equality stated in Theorem 2.2 (see [13] page 492), which implies that the solution must belong to C1(0, T;L2(Ω; H)), while in the usual situation, the solution are proved to belong only to L2(Ω; C(0, T;H)) (see Definition 2.1 in [13] and our definition of solution in this paper) and, consequently, the technique used in [13] cannot be applied. Moreover, the operator in the diffusion part of the equation in [13] does not allow for first order derivatives while it does in our case (see the preceding examples). Acknowledgements. We would like to thank the referees for their helpful suggestions which allowed us to improve the presentation of this paper. References [1] T. Caraballo, M.J. Garrido-Atienza and J. Real, Asymptotic stability of nonlinear stochastic evolution equations, Stoch. Anal. Appl., 21 (2003), 301–327. [2] T. Caraballo and K. 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Shaikhet, Modern state and development perspectives of Lyapunov functionals method in the stability theory of stochastic hereditary systems, Theory of Stochastic Processes, 2(18) (1996), 248–259. [13] L. Wan and J. Duan, Exponential stability of non-autonomous stochastic partial differential equations with finite memory, Statistics and Probability Letters., 78 (2008), 490–498. [14] M. Wei and T. Zhang, Exponential stability for stochastic 2D-Navier-Stokes equations with time delay, Appl. Math. J. Chinese Univ., 24 (2009), 493–500 (in Chinese). Received XXXX ; revised XXXX . E-mail address: (T. Caraballo) [email protected] E-mail address: (L. Shaikhet) [email protected]