The Exponential Stability of Neutral Stochastic Delay Partial Differential Equations

Manuscript submitted to Website: http://AIMsciences.org AIMS’ Journals Volume X, Number 0X, XX 200X pp. X–XX THE EXPONENTIAL STABILITY OF NEUTRAL STOCHASTIC DELAY PARTIAL DIFFERENTIAL EQUATIONS T. Caraballo1, J. Real1, & T. Taniguchi2 1Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080–Sevilla, Spain 2Division of Mathematical Sciences, Graduate School of Comparative Culture, Kurume University, Miimachi Kurume, Fukuoka 839-8502, Japan (Communicated by Aim Sciences) Abstract. In this paper we analyse the almost sure exponential stability and ultimate boundedness of the solutions to a class of neutral stochastic semilinear partial delay differential equations. This kind of equations arises in problems related to coupled oscillators in a noisy environment, or in viscoeslastic materials under random or stochastic influences. 1. Introduction. In this paper we are mainly interested in the analysis of the exponential stability of some types of stochastic neutral partial functional differential equations. To motivate and justify our work, let us first describe the general framework in which our study will be carried out. Let Zbe a Banach space, Cr=C([−r, 0], Z) the Banach space of all continuous mappings φfrom [−r, 0] into Zwhere r > 0 is a real number.Let A:Z→Zbe a closed operator and let A0:Z→Zbe a bounded linear operator. Define the functional difference operator D(·) : [0,∞)×Cr→Z by D(t)ϕ=ϕ(0) −g(t, ϕ), t ∈[0,∞), ϕ ∈Cr, where g: [0,∞)×Cr→Zis a continuous function. Let G: [0,∞)×Z→Zbe a continuous function and let Xt∈Crbe defined by Xt(θ) = X(t+θ) for θ∈[−r, 0]. If g6= 0,then the following stochastic problem d[D(t)Xt] = [AX(t) + A0Xt]dt +G(t, Xt)dW(t), t ≥0, X(t) = φ(t), t ∈[−r, 0], φ ∈Cr,(1) 2000 Mathematics Subject Classification. Primary 60H20; Secondary 60G48 . Key words and phrases. Neutral stochastic partial differential equations, existence of solutions, ultimate boundedness, exponential stability. Partially supported by Ministerio de Educaci´on y Ciencia (Spain) and FEDER (European Community) grant MTM2005-01412. This paper is in final form and no version of it will be submitted for publication elsewhere. 1 2 T. CARABALLO, J. REAL, & T. TANIGUCHI where, for simplicity, W(t) is a standard Brownian motion, is termed a stochastic neutral functional differential equation. When Z=Rnand G≡0,the deterministic neutral equation has been extensively studied in the literature (see, e.g. [10], [11], [13], [12] and the references therein). As for the stochastic version (i.e., for G6= 0), one can find some results in [14] and [18] amongst others. However, when Zis an infinite-dimensional Hilbert space, i.e. when operators A, A0,and Ginvolve partial derivatives, only a few results have been obtained in this field despite the importance and interest of the model (1). In this respect, it is worth mentioning that this kind of neutral equations arises from problems related to coupled oscillators in a noisy environment, or in problems of viscoeslastic materials under random or stochastic influences (see [24] for a description of these problems in the deterministic case). To the best of our knowledge, there exist only a few papers already published in this field. To be more precise, in [16] (see also [17]) it is considered a linear version of (1) in the particular case in which the delays are constant, and some stability properties of the mild solutions are analysed in a similar way as Dakto proved in [8] in the deterministic case, while in [9] it is studied the existence and uniqueness of mild solutions to a semilinear model, as well as some results on the stability of the null solution. Here we are concerned with a non-autonomous semilinear model containing different types of finite delays (constant, variable, distributed, etc), treated within a variational formulation (as in [19], [1], [2]), and we analyse the ultimate boundedness and asymptotic behaviour of solutions even when zero is not a solution to our model. In addition, the partial differential operators appearing in the right-hand side of our equation are more general than the ones in [16] and [9]. Our main objective is to provide sufficient conditions ensuring the exponential pathwise stability of solutions to (1). Nevertheless, we will also prove a result on the existence and uniqueness of solution to our model by using the method of steps. In particular, our stability results can also be applied to stochastic delay partial differential equations, extending in some sense, some of the results previously proved in that field (see, e.g. [3] and [1]). The contents of this paper are as follows. In Section 2 we recall some preliminaries on the Hilbert-valued stochastic integral. In Section 3 we discuss the existence and uniqueness of solution to our neutral semilinear stochastic partial differential equation. We present some auxiliary lemmata in Section 4. The moment exponential stability and ultimate exponential boundedness of the solutions, as well as the almost sure exponential stability of solutions are analysed in Section 5. Finally, in Section 6 we present an application example which illustrates the theory previously developed in this paper. 2. Preliminaries. In this section we first introduce the variational framework in which our analysis will be developed. Let Vand Hbe a reflexive Banach space and a separable Hilbert space with their respective norms k · k and | · |, and such that V⊂H∼ =H∗⊂V∗, where Vis a dense subspace of Hand the injections are continuous. We denote by h·,·i the duality between Vand V∗,and by λ1a constant satisfying λ1|u|2≤ ||u||2,for all u∈V. THE EXPONENTIAL STABILITY FOR NSDDE 3 Assume that {Ω,F, P }is a complete probability space, and let {Ft}t≥0be an increasing and right continuous family of sub σ-algebras of F, such that F0contains all the P-null sets of F. We will denote Ft=F0for t < 0.Let {βj(t), t ≥ 0, j = 1,2, ...}be a given sequence of mutually independent standard real Ft- Wiener processes defined on this space, and suppose given K, another separable Hilbert space, and {ej;j= 1,2, ...}, an orthonormal basis of K. We denote by {W(t); t≥0}, the cylindrical Wiener process with values in Kdefined formally as W(t) = P∞ j=1 βj(t)ej. It is well known that this series does not converge in K, but rather in any Hilbert space e Ksuch that K⊂e K, being the injection of Kinto e KHilbert-Schmidt (see e.g. Da Prato & Zabczyk [7] for more details). Given real numbers a < b, and a separable Hilbert space Hwe will denote by M2 Ft(a, b;H) the space of all processes X∈L2(Ω ×(a, b),F ⊗ B((a, b)), dP ⊗ dt;H) (where B((a, b)) denotes the Borel σ−algebra on (a, b)) such that X(t) is Ft−measurable a.e. t∈(a, b). The space M2 Ft(a, b;H) is a closed subspace of L2(Ω× (a, b),F ⊗B((a, b)), dP ⊗dt;H). We will write L2 Ft(Ω; C([a, b]; H)) to denote the space of all continuous and Ftprogressively measurable H-valued processes {ϕt;a≤t≤b}satisfying Esupa≤t≤bkϕtk2 H<∞. For our separable Hilbert space H, with scalar product (·,·), let us denote by L2 0(K;H) the separable Hilbert space of Hilbert-Schmidt operators from Kinto H, and by ((·,·))L2 0and k·kL2 0the scalar product and its associated norm in L2 0(K;H), where for all Rand Sin L2 0(K;H), ((R, S))L2 0= ∞ X j=1 (Rej, Sej). For any process Ψ ∈M2 Ft(0, T;L2 0(K;H)),we define the stochastic integral of Ψ with respect to the cylindrical Wiener process W(t), denoted by Rt 0Ψ(s)dW(s), 0≤t≤T, as the unique continuous H-valued Ft-martingale such that for all h∈H, (Zt 0 Ψ(s)dW(s), h) = ∞ X j=1 Zt 0 (Ψ(s)ej, h)dβj(s),0≤t≤T, where the integral with respect to βj(s) is understood in the sense of Itˆo, and the series converges in L2(Ω; C([0, T])). See e.g. Da Prato & Zabczyk [7] for the properties of the stochastic integral so defined. In particular, we note that if Ψ ∈M2 Ft(0, T;L2 0(K;H)) and φ∈L2(Ω; L∞(0, T;H)) is Ft-progressively measurable, then the series P∞ j=1 Rt 0(Ψ(s)ej, φ(s)) dβj(s),0≤t≤T, converges in L1(Ω; C([0, T])),and defines a real valued continuous Ft-martingale. We will use the notation Zt 0 (Ψ(s), φ(s)dW(s)) = ∞ X j=1 Zt 0 (Ψ(s)ej, φ(s)) dβj(s),0≤t≤T. Let r > 0 denote a positive real number and let Cr=C([−r, 0], H) be the Banach space of all continuous mappings from [−r, 0] into Hequipped with the sup-norm. Given u∈C([−r, +∞), H),for any t≥0, we denote by utthe element in Cr defined by ut(θ) = u(t+θ), θ ∈[−r, 0]. 4 T. CARABALLO, J. REAL, & T. TANIGUCHI In this paper we consider the exponential stability and exponential ultimate boundedness of solutions to the following neutral stochastic delay partial differential equation: d[X(t)−k(t, X(t−r))] = [A(t)X(t) + f(t, Xt)] dt +g(t, Xt)dW (t) X(t) = ϕ(s),−r≤t≤0,(2) where ϕ(s, ·) is F0−measurable for all s∈[−r, 0], A(t) : V−→ V∗is a linear operator for almost all t∈[0,∞), k : [−r, ∞)×V−→ Vis a globally Lipschitz function,and f: (0,∞)×Cr→V∗and g: (0,∞)×Cr−→ L2 0(K, H) are continuous and globally Lipschitz mappings. 3. Existence and uniqueness of solutions. Although the existence and uniqueness of solutions of a problem more general than (2) (for instance, in which the term ktakes the form k(t, Xt) and A(t, ·) is a family of nonlinear operators) will be the aim of our forthcoming paper [5], we will content ourselves now with this case since, as far as we know, there are only a few works dealing with stochatic neutral partial differential equations (e.g. Liu [16] and [9] for the existence of mild solutions to semilinear neutral equations), but none of them analyse the variational framework in the present paper. On the other hand, it is remarkable that, allowing the initial datum to be defined in the interval [−2r, 0] instead of [−r, 0],a suitable method of steps will give us the existence and uniqueness of solutions in a straighforward way. Bearing in mind that our main objective in this paper is to study the asymptotic behaviour of (2), we will consider this situation. Let A(t) : V→V∗be a family of linear operators defined a.e. t∈(0,+∞) such that A(·)∈L∞(0, T;L(V, V ∗)) for all T > 0, and satisfying the following coercivity assumption: (C) there exist α > 0 and λ∈Rsuch that −2hA(t)u, ui+λ|u|2≥αkuk2,∀u∈V, a.e. t ∈(0,+∞). Let f: Ω ×(0,+∞)×Cr→V∗and g: Ω ×(0,+∞)×Cr→ L2 0(K, H) be two families of nonlinear operators defined a.e. t∈(0,+∞) such that: (f.1) ∀φ∈Cr,the stochastic process f(·, φ) is Ft−progressively measurable, (f.2) f(·,0) ∈M2 Ft(0, T;V∗),for all T > 0, (f.3) there exists Cf>0 such that kf(t, φ1)−f(t, φ2)k2 ∗≤Cf|φ1−φ2|2 Cr,∀φ1, φ2∈Cr,a.e. t ∈(0,+∞), (g.1) ∀φ∈Cr,the stochastic process g(·, φ) is Ft−progressively measurable, (g.2) g(t, 0) ∈M2 Ft(0, T;L2 0(K;H)) for all T > 0, (g.3) there exists Cg>0 such that kg(t, φ1)−g(t, φ2)k2 L2 0≤Cg|φ1−φ2|2 Cr,∀φ1, φ2∈Cr,a.e. t ∈(0,+∞), Let us now consider the problem for any T > 0,              u∈M2 Ft(−r, T ;V)∩L2(Ω; C(−r, T;H)), u(t) = ψ(0) + Rt 0A(s)u(s)ds +Rt 0(f(s, us) + e f(s)) ds +Rt 0(g(s, us) + eg(s)) dW(s), t ∈[0, T], u(t) = ψ(t), t ∈[−r, 0]. (3) THE EXPONENTIAL STABILITY FOR NSDDE 5 Caraballo et al. proved in [1] the following result∗: Theorem 1. Assume that hypotheses (C),(f.1)-(f.3) and (g.1)-(g.3) hold. Then, for every ψ∈M2 F0(−r, 0; V)∩L2(Ω; Cr),e f∈M2 Ft(0, T;V∗)and eg∈M2 Ft(0, T;L2 0(K;H)), there exists a unique solution uto the problem (3). Based on this result we can prove the existence and uniqueness of solutions to our problem (2) but for an initial value in [−2r, 0]. Theorem 2. Assume that A, f and gsatisfy assumptions in Theorem 1 for any T > 0. Suppose that k: [−r, +∞)×H→His a globally Lipschitz function (with respect to its second variable) such that k([−r, +∞)×V)⊂V, and is also globally Lipschitz in V, i.e., there exists Lk>0such that |k(t, u)−k(t, v)| ≤ Lk|u−v|, for all u, v ∈H, t ≥ −r, ||k(t, u)−k(t, v)|| ≤ Lk||u−v||, for all u, v ∈V, t ≥ −r, and it satisfies that, k(·,0) ∈L2(−r, T ;V)for all T > 0,and for each v∈H, the mapping k(·, v)∈C([−r, +∞); H).Then, for any initial datum ϕ∈M2 F0(−2r, 0; V)∩ L2(Ω; C2r)there exists a unique solution to the problem      X∈M2 Ft(−2r, T ;V)∩L2(Ω; C([−2r, T]; H)for all T > 0, d[X(t)−k(t, X(t−r))] = [A(t)X(t) + f(t, Xt)] dt +g(t, Xt)dW (t), X(t) = ϕ(s),−2r≤t≤0, (4) or, in its integral form,      X(t)−k(t, X(t−r)) = ϕ(0) −k(0, ϕ(−r)) +Rt 0[A(s)X(s) + f(s, Xs)] ds +Rt 0g(s, Xs)dW(s), t ≥0, X(t) = ϕ(t),−2r≤t≤0. (5) Proof. Let us first consider the problem for tin the interval [0, r].Let us denote by h(t) = k(t, ϕ(t−r)) for t∈[−r, r]. Then, problem (5) can be rewritten as        X(t) = X(0) + h(t)−h(0) + Rt 0[A(s)X(s) + f(s, Xs)] ds +Rt 0g(s, Xs)dW(s), t ∈[0, r], X(t) = ϕ(t),−2r≤t≤0, (6) and making the change of variables Y(t) = X(t)−h(t),we obtain        Y(t) = Y(0) + Rt 0[A(s)Y(s) + A(s)h(s) + f(s, Ys+hs)] ds +Rt 0g(s, Ys+hs)dW(s), t ∈[0, r], Y(t) = ϕ(t)−k(t, ϕ(t−r)),−r≤t≤0. (7) It is straightforward to check that the operators and the initial value in this problem satisfy the assumptions in Theorem 1, so that we can ensure the existence of a unique solution Y(·) of (7) in M2 Ft(−r, r;V)∩L2(Ω; C(−r, r;H)),and thus a unique solution X(·) = Y(·) + h(·) of (5) in [−r, r]. It is clear that we can iterate again in the interval [r, 2r] and, recursively, in any [nr, (n+ 1)r] for n∈N. Thus there exists a unique solution in [−r, +∞). ∗In [1] the result was proved for a standard Wiener process but it can be easily extended to the actual situation. 6 T. CARABALLO, J. REAL, & T. TANIGUCHI In what follows we assume that our problem (2) possesses a unique solution defined globally in time (what may happen under different assumptions than those in Theorem 2). 4. Some auxiliary lemmata. In this section we prove several lemmata which will be helpful for our later stability analysis. We start with a result which is based on an inequality for real numbers. Lemma 1. Let 0< c < 1.Then, for any u, v ∈H, it follows that |u|2≤1 1−c|u−v|2+1 c|v|2.(8) Proof. Observe that, for any u, v ∈H, if follows from the Young inequality that (u, u) = (u−v, u −v) + 2 (u, v)−(v, v) ≤ |u−v|2−|v|2+c|u|2+1 c|v|2, whence (8) holds. Given a H−valued stochastic process X(t), t ≥ −rwith X(s) = ϕ(s),−r≤s≤ 0,let us first assume that there exists a positive real number B0>0 such that sup −r≤t≤0 E|ϕ(t)|2≤B0.(9) Then, we have the following result. Lemma 2. Let (9) hold and let Mδand Mγbe nonnegative constants. Assume that the continuous function k: [−r, ∞)×H→Hsatisfies the following inequality |k(t, x)|2≤c2|x|2+δ(t),0< c < 1, t ≥ −r, (10) where δ: [−r, +∞)→[0,+∞)is a continuous function with 0≤δ(t)≤Mδ. If a stochastic process X(t), t ≥ −rwith X(s) = ϕ(s),−r≤s≤0,satisfies the following inequality: E|X(t)−k(t, X(t−r))|2≤M0e−θt +γ(t), M0>0, θ > 0, t ≥0,(11) where γ: [−r, +∞)→[0,+∞)is another continuous function with 0≤γ(t)≤Mγ, then there exist positive real numbers ρ > 0and M1>0such that E|X(t)|2≤M1e−ρt +c0, t ≥0, where c0=Mδ c(1−c)+Mγ (1−c)2. Proof. By Lemma 1 we have that E|X(t)|2≤1 1−cE|X(t)−k(t, X(t−r))|2+1 cE|k(t, X(t−r))|2, and, by (10)-(11), E|X(t)|2≤M0 1−ce−θt +cE|X(t−r)|2+Mδ c+Mγ 1−c. THE EXPONENTIAL STABILITY FOR NSDDE 7 Since 0 < c < 1,we can take a β∈(c, 1) such that 0 <1 rlog β c< θ. Let α=1 rlog β c. We note that c < ceαr =β < 1 and cje−α(t−jr)=βje−αt.For any fixed t > 0, there exists an integer n≥0 such that −r≤t−nr < 0.Thus, since 0 < α < θ, E|X(t)|2≤M0 1−ce−αt +cE|X(t−r)|2+Mδ c+Mγ 1−c ≤M0 1−c  n−1 X j=0 cje−α(t−jr) +cnE|ϕ(t−nr)|2+  n−1 X j=0 cj Mδ c+Mγ 1−c ≤M0 1−c  n−1 X j=0 βj e−αt +cnB0+  n−1 X j=0 cj Mδ c+Mγ 1−c. On the other hand, since c=βe−αr and −r≤t−nr < 0,we have that cn=βne−nαr < βne−αt < e−αt. Hence we obtain that E|X(t)|2≤M0β0 1−c+B0e−αt +c0, where β0=1 1−β.This completes the proof of the lemma. Next we assume that Esup −r≤t≤0|ϕ(t)|2≤B1.(12) Then we can establish the following lemma. Lemma 3. Let (12) hold and let Nbe a natural number. Assume that all the conditions of Lemma 2 are fulfilled except condition (11) which is replaced by E sup Nr≤t≤(N+1)r|X(t)−k(t, X(t−r))|2!≤M0e−θN +γ(t), t ≥0.(13) Then, there exist positive real numbers ρ > 0and M2>0such that E sup Nr≤t≤(N+1)r|X(t)|2!≤M2e−ρN +c0, t ≥0. Proof. By Lemma 1 it follows that Esup N≤t≤N+1 |X(t)|2 ≤1 1−cE sup Nr≤t≤(N+1)r|X(t)−k(t, X(t−r))|2! +1 cE sup Nr≤t≤(N+1)r|k(t, X(t−r))|2!. 8 T. CARABALLO, J. REAL, & T. TANIGUCHI Thus, we have E sup Nr≤t≤(N+1)r|X(t)|2! ≤M0 1−ce−θNr +cE sup Nr≤t≤(N+1)r|X(t−r)|2!+Mδ c+Mγ 1−c. Since the real number csatisfies 0 < c < 1,we can choose µ > 0 such that 0< µ < θ and 0 < µ < 1 rlog 1 c. Then we obtain that E sup Nr≤t≤(N+1)r|X(t)|2!≤M0 1−c  N−1 X j=0 (ceµr)j e−µNr +cNE sup Nr≤t≤(N+1)r|X(t−Nr)|2!+c0  N−1 X j=0 cj . Thus, since cN< e−µNr,we obtain that E sup Nr≤t≤(N+1)r|X(t)|2!≤M0e−µNr (1 −c)(1 −ceθr)+cNB1+c0 1−c ≤M0 (1 −c)(1 −ceµr)+B1e−µNr +c0 1−c, which completes the proof of the lemma. Now, thanks to these previous lemmata, we can discuss the ultimate exponential boundedness and the exponential stability of the solutions to the neutral stochastic delay differential equation (2). 5. Exponential stability under a coercivity condition. It is known that some kind of coercivity conditions may play the rˆole of stability criteria (see, e.g., [3] or [2]) in dealing with the analysis of stability of solutions to partial differential equations (either deterministic or stochastic). In this section we will show that the same is true in our neutral case. However, we will formulate this condition under an integral form which is, in fact, weaker. To this end, we assume that the operators in the neutral stochastic delay partial differential equation (2) satisfy the following condition: There exists m0>0 such that for each m∈[0, m0] there exist constants α1(m)> 0, α2(m)≥0 and a function ξ(·, m) : R+→R+such that for all T > 0 and all u∈L2(−r, T ;V)∩C(−r, T;H) it holds Zt t0 ems h2hu(s)−k(s, u(s−r)), A(s)u(s) + f(s, us)i+||g(s, us)||2 L2 0ids ≤ −α1(m)Zt t0 ems||u(s)||2ds +α2(m)Zt0 t0−r ems||u(s)||2ds +ξ(t, m),(14) THE EXPONENTIAL STABILITY FOR NSDDE 9 for a.e. t∈(t0, T),and all t0∈[0, T ]. We assume in this section that k(t, ·) : H→H is continuous, maps Vinto itself, satisfies (10) and ||k(t, u)||2≤c2||u||2+δ(t),0< c < 1,for u∈V, (15) for the same constant cand function δas in (10). Remark 1. Observe that a stronger condition implying (14) is the following: There exist a constant α1>0 and a nonnegative function ξ1(·) such that, for any φ∈Crit follows 2hφ(0) −k(t, φ(−r)), A(t)φ(0) + f(t, φ)i+||g(t, φ)||2 L2 0≤ −α1||φ(0)||2+ξ1(t).(16) Notice that (16) seems to be a strong condition since it indicates that the final value φ(0) plays a dominant rˆole in the analysis, what in addition may restrict the possibilities for the operators fand g. Nevertheless, some types of integral inequalities imposed on fand gallow that more general cases fall within this framework. Indeed, assume that fand gsatisfy the following assumptions: There exists m0>0 such that for all m∈[0, m0], T > 0, and any u, v ∈ L2(−r, T ;V)∩C(−r, T;H) Zt t0 ems||f(s, us)−f(s, vs)||2 ∗ds ≤L2 fZt t0−r ems|u(s)−v(s)|2ds, (17) Zt t0 ems||g(s, us)−g(s, vs)||2 L2 0ds ≤L2 gZt t0−r ems|u(s)−v(s)|2ds. (18) We remark that these conditions are fulfilled for functions f, g which contain constant, variable or distributed bounded delays (see [4] for some illustrative examples). In addition, let us suppose that ksatisfies (10), (15) and that Asatisfies (C) with λ= 0 (or with α−λλ−1 1>0). Then I=Zt t0 ems h2hu(s)−k(s, u(s−r)), A(s)u(s) + f(s, us)i+||g(s, us)||2 L2 0ids ≤Zt t0 2ems hu(s), A(s)u(s)ids +Zt 0 2ems hu(s), f(s, us)ids −Zt t0 2ems hk(s, u(s−r)), A(s)u(s)ids −Zt t0 2ems hk(s, u(s−r)), f(s, us)ids +Zt t0 ems||g(s, us)||2 L2 0ds ≤ −αZt t0 ems||u(s)||2ds +Zt t0 2ems||u(s)||||f(s, us)||∗ds | {z } =I1 +Zt t0 2ems||k(s, u(s−r))||||A(s)u(s)||∗ds | {z } =I2 +Zt t0 2ems||k(s, u(s−r))||||f(s, us)||∗ds | {z } =I3 +Zt t0 ems||g(s, us)||2 L2 0ds |{z } =I4 . 16 T. CARABALLO, J. REAL, & T. TANIGUCHI where n2>0 is a suitable constant. Then, we have E sup Nr≤t≤(N+1)r|X(t)−k(t, X(t−r))|2! ≤2E|X(Nr)−k(Nr, X(Nr −r)) |2 + 2n2"Z(N+1)r Nr g0E|X(s))|2ds +g1ZNr Nr−r E||X(s)||2ds +g2e−ρ3(N+1)r# + sup t∈[Nr,(N+1)r] ξ(t, 0) Therefore by condition (21), Lemma 2, Lemma 4 and Lemma 5, there exist positive real numbers M5>0, ρ1>0 such that E sup Nr≤t≤(N+1)r|X(t)−k(t, X(t−r))|2!≤M5e−ρ1N Thus by Lemma 3 we obtain that E sup Nr≤t≤(N+1)r|X(t)|2!≤M4e−ρ2N, M4>0, ρ2>0. Now, thanks to the Borel Cantelli lemma, it is a standard matter to prove the pathwise exponential stability. As a corollary of our previous results (i.e. by simply setting k= 0) we can establish a result for the exponential stability of solutions to the following stochastic delay partial differential equation (see [2] for similar results) dX(t) = [A(t)X(t)) + f(t, Xt)] dt +g(t, Xt)dW (t) (22) X(t) = ϕ(s),−r≤t≤0,(23) where ER0 −r||ϕ(s)||2ds < ∞and the function gsatisfies the same condition as in Theorem 4. Then we have Corollary 1. Let α1, η > 0and ξ1(t) = δ0(t)e−ηt with δ0(t)a bounded integrable function. Assume that there exists m0>0such that for all m∈[0, m0],T > 0, and any u∈L2(−r, T ;V)∩C(−r, T ;H)the following inequality holds: Zt t0 ems h2hu(s), A(s)u(s) + f(s, us)i+||g(s, us)||2 L2 0ids ≤ −Zt t0 emsα1(m)||u(s)||2ds +Zt0 t0−r emsα2(m)||u(s)||2ds +ξ1(t), for a.e. t∈(t0, T ),and all t0∈[0, T).Then, any solution to the stochastic delay partial differential equation (22)-(23) converges to zero exponentially almost surely. Remark 2. The reader is referred to the papers [2], [3] in order to compare Corollary 1 with the results in these papers. THE EXPONENTIAL STABILITY FOR NSDDE 17 6. An example. Although we could consider several applications to show how our theory can be applied, we have preferred to treat a simple situation in which a usual coercivity condition (which is easier to check than the integral condition (14)) ensures the asymptotic behaviour of the model. Indeed, let us consider the problem d[X(t)−k(t, X(t−r))] = [A(t)X(t) + f(t, X(t−r))] dt +g(t, X(t))dW(t) X(t) = ϕ(s),−r≤t≤0, where Aand ksatisfy the assumptions in the previous sections, f:R+×H→V∗ and g:H→ L2 0(K, H) are Lipschitz continuous and it holds 2hu−k(t, v), A(t)u+f(t, v)i+||g(t, u)||2 L2 0(24) ≤ −α1||u||2+α2|u|2+β1||v||2+β2|v|2+ξ1(t), for u, v ∈Vwhere α1> β1≥0 and α2, β2≥0.If we assume that (α1−β1)λ1> α2+β2,then (24) implies (14) since for any u∈L2(−r, T;V)∩C(−r, T ;H) it follows Zt t0 ems h2hu(s)−k(s, u(s−r)), A(s)u(s) + f(s, u(s−r))i+||g(s, u(s))||2 L2 0ids ≤Zt t0 ems −α1||u(s)||2+α2|u(s)|2+β1||u(s−r)||2+β2|u(s−r)|2+ξ1(s)ds ≤Zt t0 ems −α1||u(s)||2+α2|u(s)|2+ξ1(s)ds +Zt t0 ems β1||u(s−r)||2+β2|u(s−r)|2ds ≤Zt t0 ems −α1||u(s)||2+α2λ−1 1||u(s)||2+ξ1(s)ds +emr Zt−r t0−r ems β1||u(s)||2+β2|u(s)|2ds ≤Zt t0 ems −α1+α2λ−1 1||u(s)||2+ξ1(s)ds +emr Zt−r t0−r ems β1+β2λ−1 1||u(s)||2ds ≤Zt t0 ems −α1+α2λ−1 1+emr β1+β2λ−1 1||u(s)||2+ξ1(s)ds +emr Zt0 t0−r ems β1+β2λ−1 1||u(s)||2ds ≤Zt t0 ems −α1+α2λ−1 1+emr β1+β2λ−1 1||u(s)||2ds +Zt t0 emsξ1(s)ds +β1+β2λ−1 1emr Zt0 t0−r ems||u(s)||2ds. As (α1−β1)λ1> α2+β2, there exists m0>0 such that for all m∈[0, m0) we have that −α1+α2λ−1 1+emr β1+β2λ−1 1=−(α1−emrβ1−λ−1 1(α2+emrβ2)) = −α1(m)<0. 18 T. CARABALLO, J. REAL, & T. TANIGUCHI On the other hand, setting α2(m) = β1+β2λ−1 1emr and ξ(t, m) = Zt 0 emsξ1(s)ds we have the desired conclusion. Now, if −α1+α2λ−1 1+β1+β2λ−1 1<0,(25) it is obvious that there exists m0>0 such that for all m∈[0, m0] it follows that α1(m)>0 and our theory can be applied to ensure exponential stability of our problem. Finally, we will especialise this situation to a more explicit one. Indeed, let H=L2(0, π), V =H1 0(0, π) and V∗=H−1(0, π),with the usual norms in the spaces Hand Vdefined as |ξ|=Rπ 0ξ2(x)dx1/2for ξ∈H, and ||u|| =Rπ 0(∂u ∂x )2dx1/2 for u∈V, respectively. Let us define A(t) = ∂ ∂x a(t, x)∂ ∂x, where a(t, x) is measurable in (0,+∞)×(0, π),and satisfies 0< ν ≤a(t, x)≤αin (0,+∞)×(0, π). The family of operators A(t) satisfies A(·)∈L∞(0, T ;L(V, V ∗)) for all T > 0, and it is known that hA(t)u, ui ≤ −νkuk2, u ∈V, and λ1= 1. Consider the following neutral stochastic delay differential equation.            d(X(t, x)−cX(t−r, x)) =∂ ∂x a(t, x)∂X(t, x) ∂x +bX(t−r, x) + g(t, X(t, x))dW (t), X(t, 0) = X(t, π) = 0, t ≥0, X(t, x) = ϕ(t, x), t ∈[−r, 0], (26) where W(t) is a cylindrical Wiener process with values in L2(0, π), ϕ(t) is an adequate continuous square integrable process and b, c ∈R.We assume that ||g(t, u)||2 L2 0(L2(0,π)) ≤cg|u|2+g1(t), cg>0, t ≥ −r, |c|<1 y 2ν > |b|(2 + |c|) + cg+α|c|,(27) with 0 ≤g1(t)≤g0e−ρt, g0>0, ρ > 0 for any t≥0.Let k(t, v) = cv and f(t, v) = bv. Then we have 2hu−k(t−r, v), A(t)u+f(t−r, v)i+||g(t, u)||2 L0 2 = 2 hu−cv, A(t)u+bvi+||g(t, u)||2 L0 2 ≤ −2ν||u||2+ 2|b||u||v|+α|c|||u||||v||+|bc||v|2+cg|u|2+g1(t) ≤ −(2ν−α|c| 2)||u||2+ (|b|+cg)|u|2+ (|b|+|bc|)|v|2+α|c| 2||v||2+g1(t). 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Taniguchi, Almost sure exponential stability for stochastic partial functional differential equations, Stochastic Analysis and Applications 16(1998), 965-975. [24] J. Wu, Theory and Applications of Partial Functional Differential Equations, Applied Mathematical Sciences Volume 119, Springer-Verlag, New York, 1996 E-mail address, Tom´as Caraballo: [email protected] E-mail address, Jos´e Real: [email protected] E-mail address, Takeshi Taniguchi: takeshi [email protected]