The exponential behaviour of nonlinear stochastic functional equations of second order in time

THE EXPONENTIAL BEHAVIOUR OF NONLINEAR STOCHASTIC FUNCTIONAL EQUATIONS OF SECOND ORDER IN TIME Tom´ as Caraballo, Mar´ ıa J. Garrido-Atienza and Jos´ e Real Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain. Abstract Sufficient conditions for exponential mean square stability of solutions to delayed stochastic partial differential equations of second order in time are established. As a consequence of these results, some ones on the pathwise exponential stability of the system are proved. The stability results derived are applied also to partial differential equations without hereditary characteristics. The results are illustrated with several examples. 1 Introduction Stochastic differential delay equations and their asymptotic behaviour have been receiving much attention in the last years (see [1], [2], [4], [8], [6], [7], [9], [11], [12], and the references therein) since these retarded problems often appear in Physics, Biology, Engineering, etc... The delays can enter in the formulations in very different ways, e.g., as a constant or variable delay, as a distributed one, or even some of them can appear in the model at the same time. Also these delay can be bounded (finite) or unbounded (infinite). However, there is a possibility of considering all of them under a unified formulation by using appropriate differential functional equations. In this sense, we will carry out our analysis in a functional framework which will cover a wide variety of situations containing finite delays (see Section 5). 1 On the other hand, a very interesting question is to analyse the long-time behaviour of the solution to a stochastic functional equation. We remark that in some problems the history of the phenomenon has a decisive influence on the future behaviour of the system and, in some cases, not only a short period of the past has to be taken into account, but a large one. This fact motivates the present work. There exists a wide literature concerning pathwise exponential stability of parabolic stochastic evolution equations (with and without delays). We mention here, amongst many others, Caraballo and Liu [3], Liu and Mao [10], Taniguchi [11], Taniguchi et al. [12] and the references therein. However, as far as we know, there are no papers on the asymptotic stability of delay stochastic partial differential equations of second order in time, which is the main aim of this paper. In the case without hereditary characteristics this problem has been considered by Curtain [5], where one can find sufficient conditions for the exponential stability of the expected energy of the system, as well as for the exponential decay of the sample paths, when the main operator generates a strongly continuous contraction semigroup. In this paper we shall develop the theory in a variational framework for non-linear operators in general and under a functional formulation which covers several kinds of delay and, in particular, the non-delay case. In order to motivate our theory let us first study the following example. The simplest model of continuum mechanics is given by the vibrating string, subjected to a constant tension µ, which executes small longitudinal vibrations about the position of stable equilibrium. Our problem is to determine the lateral displacement u(x, t) of a point on the string from its equilibrium position. For a constant linear density %of the string, if we also consider some friction proportional to velocity, ϑv(t), that an external force f(t, u(t), v(t)) acts on the string, and that u0and v0are respectively the initial position and velocity, we then obtain the problem            ∂2u ∂t2−a2∂2u ∂x2+ϑ∂u ∂t + Φ µt, u(t),∂u ∂t (t)¶= 0,in [0,+∞)×[0,1], u(t, 0) = u(t, 1) = 0, t ∈[0,+∞), u(0, x) = u0,∂u ∂t (0, x) = v0(x),in [0,1], where Φ ¡t, u(t),∂u ∂t (t)¢corresponds to 1 %f¡t, u(t),∂u ∂t (t)¢,a=qµ %,and ϑ > 0. This model can be thought to be more realistic if we suppose that Φ contains some random features, for example we can think of Φ ¡t, u(t),∂u ∂t (t)¢=−σ∂u(t) ∂x dW (t) dt , t ≥0,where 2 σ∈R,and W(t) is a one-dimensional Wiener process. Therefore the equation becomes ∂2u ∂t2−a2∂2u ∂x2+ϑ∂u ∂t =σ∂u ∂x(t)dW(t) dt , t ≥0. Curtain proves in [5] that when σ2<4ϑπ2 4π2+ϑ(ϑ+√ϑ2+4π2)and a= 1,the null solution of this problem is stable in mean square, i.e., if the random term is sufficiently small so that this relation is satisfied. But, is it possible to deduce any exponential stability results for the above system when σ2>4ϑπ2 4π2+ϑ(ϑ+√ϑ2+4π2)? As a consequence of the theory we will develop in this paper, we will prove that if σ2<2ϑπ2 ϑ2+2π2the system is exponentially stable both in mean square and pathwise, so these results improve the ones in [5]. On the other hand, if we are interested in some problems concerning the stabilization or controllability of systems, it seems natural to study equations in which some hereditary characteristics can appear. For example, in the above system we can consider the load Φ does not depend just on the present, but on the history of the process, i.e., it is modelled for example by the expression Φ ¡t, ut,∂ut ∂t ¢=−σ∂u(t−τ(t)) ∂x ˙ W(t), t ≥0,where τ(·) is an appropriate delay function (see Example 8, Section 5). The content of the paper is as follows. In Section 2 we present the framework in which our analysis is carried out, and introduce some basic notations and assumptions. Section 3 is devoted to the main result of this work, that is, we establish a sufficient condition ensuring mean square stability for delay stochastic partial differential equations of second order in time in a very general situation. We also indicate how this result can be applied in some particular cases. As a consequence of the mean square stability, in Section 4 we obtain pathwise exponential stability. Moreover, in this section we will be concerned with a more general situation in which our theory can be established. In Section 5 we include some examples to illustrate these results. Finally, some conclusions are included in the last section. 2 Statement of the problem Let Vand Hbe two real separable Hilbert spaces such that V⊂H≡H∗⊂V∗where the injections are continuous and dense. We denote by k·k,|·| and k·k∗the norms in V,Hand V∗respectively; by (·,·) the inner product in H, and by h·,·i the duality product between V∗and V. Let us denote by c > 0 a constant such that |x| ≤ ckxk,∀x∈V. Assume {Ω,F, P}is a complete probability space with a normal filtration {Ft}t≥0, i.e., F0contains the null sets in F, and Ft=∩s>tFs,for all t≥0.Denote Ft=F0for all t≤0. 3 Let us consider a real valued {Ft}−Wiener process {W(t)}t≥0. Given real numbers a < b and a separable Hilbert space H, we will denote by I2(a, b;H) the closed subspace of L2(Ω×(a, b),F⊗B([a, b]) ,dP⊗dt;H) of all stochastic processes which are Ft-adapted for almost every tin (a, b) (in what follows, a.e. t),where B([a, b]) denotes the Borel σ-algebra of subsets in [a, b]. If ϕ∈I2(a, b;H) we will write |ϕ|I2 Hto denote the norm |ϕ|I2(a,b;H). We denote by L2(Ω; C(a, b;H)) the space of processes X∈L2(Ω,F, dP;C(a, b;H)) such that X(t) is Ft-measurable for each tin [a, b], where C(a, b;H) denotes the space of all continuous functions from [a, b] into Hequipped with supremum norm. Let us fix h > 0 and consider T > 0.If we have a function x∈C(−h, T;H),for each t∈[0, T ] we denote by xt∈C(−h, 0; H) the function defined by xt(s) = x(t+s), −h≤s≤0.Moreover, if y∈L2(−h, T ;H) we also denote by yt∈L2(−h, 0; H),a.e. t∈(0, T),the function defined by yt(s) = y(t+s),a.e. s∈(−h, 0). Let A(t) : V→V∗,t≥0,be a family of operators satisfying: (A.1) A(t) is self-adjoint for each t≥0. (A.2) A(t)∈ L(V, V ∗)∀t≥0,and there exists cA>0 such that kA(t)uk∗≤cAkuk, ∀t≥0,∀u∈V. (A.3) ∃α > 0 such that hA(t)u, ui ≥ αkuk2,∀t≥0,∀u∈V. (A.4) hA(·)u, eui ∈ C1(0,+∞), ∀u, eu∈V, and hA0(t)u, ui ≤ 0,∀t≥0,∀u∈V, where hA0(t)u, euidenotes d dthA(t)u, eui. (A.5) there exists a Banach space Xsuch that X⊂ {u∈V;A(t)u∈H, ∀t≥0}, the injection of Xin Vis continuous, and Xis dense in H. Let B(t, ·) : H→Hbe a family of nonlinear operators defined a.e. t≥0 and satisfying: (B.1) ∀v∈H, the map t∈(0,+∞)→B(t, v)∈His Lebesgue measurable. (B.2) the map θ∈R→(B(t, v +θw), z)∈Ris continuous ∀v, w, z ∈H, a.e. t≥0. (B.3) there exists cB>0 such that |B(t, v)| ≤ cB|v|,∀v∈H, a.e. t≥0. (B.4) there exists β > 0 such that (B(t, v)−B(t, ev), v −ev)≥β|v−ev|2,∀v, ev∈H, a.e. t≥0. 4 Let F: [0,+∞)×C(−h, 0; V)×C(−h, 0; H)→Hand G: [0,+∞)×C(−h, 0; V)× C(−h, 0; H)→Hbe two families of nonlinear operators defined a.e. t≥0 such that: (F.1) ∀(ξ, η)∈C(−h, 0; V)×C(−h, 0; H) the map t∈(0,+∞)→F(t, ξ, η)∈His Lebesgue measurable,a.e. t≥0. (F.2) F(t, 0,0) = 0,a.e. t≥0. (F.3) there exist CF,H , CF,V >0 such that ∀ξ, e ξ∈C(−h, 0; V),∀η, eη∈C(−h, 0; H) and a.e. t≥0, |F(t, ξ, η)−F(t, e ξ, eη)|2≤CF,V ||ξ−e ξ||2 C(−h,0;V)+CF,H |η−eη|2 C(−h,0;H). (F.4) there exist m0>0 and constants KF,H =KF,H (m0, h), KF,V =KF,V (m0, h)≥0 such that for all m∈[0, m0],∀x, ex∈C(−h, T;V),∀y, ey∈C(−h, T;H),and ∀t≥0 Zt 0 ems |F(s, xs, ys)−F(s, exs,eys)|2ds ≤KF,V Zt −h ems kx(s)−ex(s)k2ds+KF,H Zt −h ems |y(s)−ey(s)|2ds. (G.1) ∀(ξ, η)∈C(−h, 0; V)×C(−h, 0; H) the map t∈(0,+∞)→G(t, ξ, η)∈His Lebesgue measurable,a.e. t≥0. (G.2) G(t, 0,0) = 0,a.e. t≥0. (G.3) there exist CG,H, CG,V >0 such that ∀ξ, e ξ∈C(−h, 0; V),∀η, eη∈C(−h, 0; H) and a.e. t≥0, |G(t, ξ, η)−G(t, e ξ, eη)|2≤CG,V ||ξ−e ξ||2 C(−h,0;V)+CG,H |η−eη|2 C(−h,0;H). (G.4) there exist m0>0 and constants KG,H =KG,H(m0, h), KG,V =KG,V (m0, h)≥0 such that for all m∈[0, m0],∀x, ex∈C(−h, T;V),∀y, ey∈C(−h, T;H),and ∀t≥0 Zt 0 ems |G(s, xs, ys)−G(s, exs,eys)|2ds ≤KG,V Zt −h ems kx(s)−ex(s)k2ds+KG,H Zt −h ems |y(s)−ey(s)|2ds. Remark 1 Assumptions (F.2),(G.2) are motivated by our interest in analyzing the stability of the zero solution to our problem, but they are not necessary to prove existence of solution. 5 We consider the following problem,                                          u∈I2(−h, T;V)∩L2(Ω; C(0, T;V)),for all T > 0, v∈I2(−h, T;H)∩L2(Ω; C(0, T;H)),for all T > 0, u0(t) = v(t), t ∈[0, T], v(t) + Rt 0A(s)u(s)ds+Rt 0B(s, v(s))ds=v0+Rt 0F(s, us, vs)ds +Rt 0G(s, us, vs)dW(s), t ≥0, u(0) = u0, u(t) = ϕ1(t), v(t) = ϕ2(t),a.e. t∈(−h, 0), (P) where ϕ1∈I2(−h, 0; V), ϕ2∈I2(−h, 0; H), u0∈L2(Ω,F0, P;V) and v0∈L2(Ω,F0, P;H) are given. A similar analysis to that in Remark 1 in [7] shows that, under our assumptions (F.1) − (F.4) and (G.1) −(G.4), all the integrals appearing in problem (P) are well defined, and therefore, the above problem makes sense. On the other hand, several results on the existence and uniqueness of solutions for delay stochastic evolution equations of second order in time can be seen in Garrido-Atienza and Real [7]. In particular, the following one is a consequence of Theorem 4 in [7]: Theorem 2 Assume that hypotheses (A.1) −(A.5),(B.1) −(B.4),(F.1) −(F.4),(G.1) − (G.4) hold. Then, if ϕ1∈I2(−h, 0; V),ϕ2∈I2(−h, 0; H),u0∈L2(Ω,F0, P;V)and v0∈L2(Ω,F0, P;H),there exists a unique solution (u, v)to problem (P), for all T > 0. 3 Exponential stability in mean square In this section we will establish a result about the mean square stability for the solution to problem (P). Theorem 3 Suppose that assumptions (A.1) −(A.5),(B.1) −(B.4),(F.1) −(F.4),(G.1) − (G.4) hold. In addition, assume that there exist some constants ε > 0and δ > 0such that        2α3/2Cδ> cKε, and Cδ³2β−(2K1/2 F,H +ε+KG,H )´>(4α2δ)−1(4αδ +c2(K1/2 F,H +cB)2)Kε, (1) 6 where Cδ= 1 −δ−cK1/2 F,V α−1and Kε=KF,V ε−1+KG,V . Then, the zero solution of problem (P) is exponentially stable in mean square, i.e., there exist m∈(0, m0]and K1= K1(m0, h)>0such that, for all t≥0, E³|v(t)|2+ku(t)k2´≤K1³E|v0|2+Eku0k2+|ϕ2|2 I2 H+kϕ1k2 I2 V´e−mt,(2) for any solution (u, v)of (P). Proof. Let m∈(0, m0]. Applying Itˆo’s formula to the process emt |v(t)|2+ emt hA(t)u(t), u(t)i, we obtain for each t≥0 and P-a.s. emt |v(t)|2+ emt hA(t)u(t), u(t)i =|v0|2+hA(0)u0, u0i+mZt 0 ems(|v(s)|2+hA(s)u(s), u(s)i)ds +Zt 0 ems hA0(s)u(s), u(s)ids−2Zt 0 ems(B(s, v(s)), v(s))ds + 2 Zt 0 ems(F(s, us, vs), v(s))ds+Zt 0 ems |G(s, us, vs)|2ds + 2 Zt 0 ems(G(s, us, vs), v(s))dW(s), and thanks to (A.4) and (B.4), emtE|v(t)|2+ emtEhA(t)u(t), u(t)i(3) ≤E|v0|2+EhA(0)u0, u0i+ (m−2β)Zt 0 emsE|v(s)|2ds +mZt 0 emsEhA(s)u(s), u(s)ids + 2 Zt 0 emsE(F(s, us, vs), v(s))ds+Zt 0 emsE|G(s, us, vs)|2ds. 7 As emt ≤1,∀t∈[−h, 0],from (F.4) and (A.3) we obtain 2Zt 0 emsE(F(s, us, vs), v(s))ds ≤2µZt 0 emsE|v(s)|2ds¶1/2µZt 0 emsE|F(s, us, vs)|2ds¶1/2 ≤2K1/2 F,H Zt −h emsE|v(s)|2ds+ 2 µKF,V Zt −h emsEku(s)k2ds¶1/2µZt 0 emsE|v(s)|2ds¶1/2 ≤³2K1/2 F,H +ε´Zt 0 emsE|v(s)|2ds+KF,V (αε)−1Zt 0 emsEhA(s)u(s), u(s)ids +KF,V ε−1kϕ1k2 I2 V+ 2K1/2 F,H |ϕ2|2 I2 H, for ε > 0,and by (G.4) and (A.3), Zt 0 emsE|G(s, us, vs)|2ds≤KG,H |ϕ2|2 I2 H+KG,H Zt 0 emsE|v(s)|2ds +KG,V kϕ1k2 I2 V+KG,V α−1Zt 0 emsEhA(s)u(s), u(s)ids. Thus, if we substitute these inequalities into (3) we have emtE|v(t)|2+ emtEhA(t)u(t), u(t)i(4) ≤E|v0|2+EhA(0)u0, u0i+Kεkϕ1k2 I2 V+ (2K1/2 F,H +KG,H )|ϕ2|2 I2 H +³m−2β+ 2K1/2 F,H +ε+KG,H ´Zt 0 emsE|v(s)|2ds +¡m+Kεα−1¢Zt 0 emsEhA(s)u(s), u(s)ids. Now, we estimate the last integral on the right hand side of (4). Thanks to d(emt(u(t), v(t))) = memt(u(t), v(t))dt+ emt |v(t)|2dt−emt hA(t)u(t), u(t)idt −emt(B(t, v(t)), u(t))dt+ emt(F(t, ut, vt), u(t)) + emt(G(t, ut, vt), u(t))dW(t), we deduce emtE(u(t), v(t)) = E(u0, v0) + mZt 0 emsE(u(s), v(s))ds +Zt 0 emsE|v(s)|2ds−Zt 0 emsEhA(s)u(s), u(s)ids −Zt 0 emsE(B(s, v(s)), u(s))ds+Zt 0 emsE(F(s, us, vs), u(s))ds. 8 Consequently, Zt 0 emsEhA(s)u(s), u(s)ids ≤E(u0, v0) + Zt 0 emsE|v(s)|2ds +c(m+cB)µZt 0 emsE|v(s)|2ds¶1/2µα−1Zt 0 emsEhA(s)u(s), u(s)ids¶1/2 +µZt 0 emsE|F(s, us, vs)|2ds¶1/2µc2Zt 0 emsEku(s)k2ds¶1/2 +³emtE|v(t)|2´1/2¡c2α−1emtEhA(t)u(t), u(t)i¢1/2 ≤E(u0, v0) + Zt 0 emsE|v(s)|2ds+cK1/2 F,V kϕ1k2 I2 V+c2(m+cB+K1/2 F,H )2(4αδ)−1|ϕ2|2 I2 H +c2(m+cB+K1/2 F,H )2(4αδ)−1Zt 0 emsE|v(s)|2ds +³δ+cK1/2 F,V α−1´Zt 0 emsEhA(s)u(s), u(s)ids +c(2α1/2)−1emtE³|v(t)|2+hA(t)u(t), u(t)i´, and thus CδZt 0 emsEhA(s)u(s), u(s)ids(5) ≤E(u0, v0) + cK1/2 F,V kϕ1k2 I2 V+c2(m+cB+K1/2 F,H )2(4αδ)−1|ϕ2|2 I2 H +c(2α1/2)−1emtE³|v(t)|2+hA(t)u(t), u(t)i´ + (4αδ)−1³4αδ +c2(m+cB+K1/2 F,H )2´Zt 0 emsE|v(s)|2ds. Due to the first condition in (1), the constant Cδis positive. Therefore, we have ¡m+Kεα−1¢Zt 0 emsEhA(s)u(s), u(s)ids ≤¡m+Kεα−1¢C−1 δhE(u0, v0) + c(2α1/2)−1emtE³|v(t)|2+hA(t)u(t), u(t)i´ +cK1/2 F,V kϕ1k2 I2 V+c2(m+cB+K1/2 F,H )2(4αδ)−1|ϕ2|2 I2 H +(4αδ)−1³4αδ +c2(m+cB+K1/2 F,H )2´Zt 0 emsE|v(s)|2ds¸. Thus, if we substitute this into (4) it holds 9 h > r1, r2, r3≥0,are given numbers, f: (−r1,0)→Ris a measurable bounded function such that |f(s)| ≤ Lf,and g:R2→Ris a Lipschitz continuous function such that g(0,0) = 0, with |g(x, y)−g(ex, ey)|2≤L2 g(|x−ex|2+|y−ey|2),∀(x, y),(ex, ey)∈R2. The above problem can be set within our formulation by taking H=L2(0, π), V = H1 0(0, π), A(t)u(t) = −∂2u ∂x2(t),∀u∈V, ∀t≥0, hB(t, v), wi=γZπ 0 dv dx dw dxdx +Zπ 0e k(t, dv dx)dw dxdx, ∀v, w ∈V, ∀t≥0, F(t, ξ, η) = Z0 −r1 f(s)η(s)ds, ∀(t, ξ, η)∈R+×C(−h, 0; V)×C(−h, 0; H), G(t, ξ, η) = gµ∂ξ(−r2) ∂x , η(−r3)¶,∀(t, ξ, η)∈R+×C(−h, 0; V)×C(−h, 0; H). In this situation, operators A,B,Fand Gsatisfy the hypotheses ensuring existence and uniqueness of a solution to the corresponding problem (see Garrido-Atienza [6]). Consequently, for ϕ1∈I2(−h, 0; V), ϕ2∈I2(−h, 0; H), u0∈L2(Ω,F0, P;V) and v0∈L2(Ω,F0, P;H) given, there exists a unique solution u∈I2(−h, T;V)∩L2(Ω; C(0, T;V)),∂u ∂t ∈I2(−h, T;H)∩ L2(Ω; C(0, T;H)). The constants are now cB=ck, β =γ, α =c= 1, KF,H (m0) = r2 1L2 fem0r1, KF,V (m0) = 0, KG,V (m0) = L2 gem0r2, KG,H(m0) = L2 gem0r3. Thanks to (15), the solution is exponentially stable in mean square and almost surely if we impose r1Lf< γ, L2 g<min ½1,2γ−2Lfr1 1 + 2(1 + (ck+Lfr1)2)¾. Example 10 Let us take H=L2(O) and V=H1(O), where O ⊂ Rnis a bounded open set with smooth boundary. Let us consider A(t) = −∆ for all t≥0; B(t, v),where v∈L2(O),the function of L2(O) defined, a.e. x∈ O,by B(t, v)(x) = k(t, v(x)),where k:R+×R→Ris a continuous map such that there exist ck,βk>0 such that |k(t, a)| ≤ ck|a|,(k(t, a)−k(t, ea))(a−ea)≥βk|a−ea|2∀a, ea∈R,∀t≥0. Let us consider two measurable functions f:R+×R→Rand g:R+×R×Rn×R→R, such that f(t, 0) = g(t, 0,0,0) = 0,∀t≥0, and we also suppose that there exist Lf, Lg>0 16 such that |f(t, b)−f(t,eb)| ≤ Lf|b−eb|, |g(t, a, y, b)−g(t, ea, ey,eb)|2≤L2 g(|a−ea|2+|y−ey|2+|b−eb|2), ∀t≥0,∀a, ea, b,eb∈R,∀y, ey∈Rn.Consider also four functions τi∈C1(R+), 1 ≤i≤4, such that 0 ≤τi(t)≤h, ∀t≥0,being τ∗ i= sup 0≤t τ0 i(t)<1.As in Example 8, if we denote θi(t) = t−τi(t),then there exists ki>0 such that θ−1 i(t)≤t+ki,∀t≥τi(0),1≤i≤4. For t∈R+, ξ ∈C(−h, 0; V), η ∈C(−h, 0; H),denote by F(t, ξ, η) and G(t, ξ, η) the families of operators defined, a.e. x∈ O,by F(t, ξ, η)(x) = f(t, η(−τ1(t))(x)), G(t, ξ, η)(x) = g(t, ξ(−τ2(t))(x),∇ξ(−τ3(t))(x), η(−τ4(t))(x)). Under these hypotheses, we can ensure that given ϕ1∈I2(−h, 0; H1(O)), ϕ2∈I2(−h, 0; L2(O)), u0∈L2(Ω,F0, P;H1(O)) and v0∈L2(Ω,F0, P;L2(O)), there exists a unique solution u∈I2(−h, T;H1(O)) ∩L2(Ω; C(0, T ;H1(O)), v ∈I2(−h, T;L2(O)) ∩L2(Ω; C(0, T;L2(O)), to the correspondent system (P), (see Garrido-Atienza [6]). This solution can be seen as a solution to the Neumann problem                              ∂2u ∂t2−∆u+kµt, ∂u ∂t ¶=f(t, u(t−τ1(t))) +gµt, u(t−τ2(t)),∇u(t−τ3(t)),∂u ∂t (t−τ4(t))¶dW(t) dt ,in (0,+∞)×O, ∂u ∂ν = 0,on (0,+∞)×∂O, u(0, x) = u0(x),∂u ∂t (0, x) = v0(x),in O, u(t) = ϕ1(t),∂u(t) ∂t =ϕ2(t), t ∈(−h, 0), where we denote by νthe outward unit normal to ∂O. In this situation, it is not hard to check that β=βk, cB=ck, α =c= 1, KF,H (m0) = L2 f 1−τ∗ 1 em0k1, KF,V (m0) = 0, KG,H(m0) = L2 g 1−τ∗ 4 em0k4, KG,V (m0) = L2 gmax ½em0k2 1−τ∗ 2 ,em0k3 1−τ∗ 3¾. 17 So, using Remark 4, the solution to our problem is exponentially stable in mean square and, therefore, almost sure exponentially stable if we suppose L2 g 1−τ∗ i <1, i = 2,3, L2 g 1−τ∗ 4 <2βk−Ã2 + µck+Lf (1 −τ∗ 1)1/2¶2!L2 g 1−τ∗ i−2Lf (1 −τ∗ 1)1/2, i = 2,3. 6 Conclusions and final remarks Some results on the exponential stability of functional stochastic partial differential equations of second order in time have been proved, which, in the particular case without delay, also improves a stability criterium in [5]. However, another interesting question is, in our opinion, the analysis of the actual decay rate of solutions when we are in a situation in which the stability may not be exponential (which uses to appear when one deals with nonlinear or non-autonomous problems). Only a few works have been done concerning the non-exponential stability of parabolic stochastic systems. It is worth mentioning the paper by Liu [9] on the polynomial stability for semilinear stochastic evolution equations which also covers the delay situation; on the other hand, Caraballo et al. [2] prove some results on the pathwise stability with a general decay function satisfying suitable conditions in both cases. It is our intention to do an investigation in this direction in a future paper. Another point is that, although we have only considered the case of a real Wiener process, the results can be extended to a Hilbert valued situation. However, we have preferred to consider this framework for the sake of clarity. Acknowledgment. This work has been partially supported by Junta de Andaluc´ıa Project FQM314, and by Ministerio de Ciencia y Tecnolog´ıa under the projects HA2001-0075 and BFM2002-03068. References [1] T. Caraballo, Asymptotic exponential stability of stochastic partial differential equations with delay, Stochastics Stochastics Rep. 33 (1990), 27-47. [2] T. Caraballo, M.J. Garrido-Atienza and J. Real, Asymptotic stability for non-linear stochastic evolution equations, Stoch. Anal. Appl. 21 (2003), no. 2 (to appear). 18 [3] T. Caraballo and K. Liu, On exponential stability criteria of stochastic partial differential equations, Stochastic Processes and their Applications 83 (1999), 289-301. [4] T. Caraballo, K. Liu and A. Truman, Stochastic functional partial differential equations: existence, uniqueness and asymptotic decay property, Proc. R. Soc. Lond. A 456 (2000), 1775-1802. [5] R.F. Curtain, Stability of stochastic partial differential equation, J. Math. Anal. Appl. 79 (1981), 352-369. [6] M.J. Garrido-Atienza, “Algunos resultados de existencia, unicidad y estabilidad para EDP funcionales estoc´asticas no lineales”, PhD. Thesis, Universidad de Sevilla, 2002. [7] M.J. 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