Existence of invariant manifolds for coupled parabolic and hyperbolic stochastic partial differential equations

Existence of invariant manifolds for coupled parabolic and hyperbolic stochastic partial differential equations Tom´as Caraballo†, Igor Chueshov‡, Jos´e A. Langa† †Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain. ‡Department of Mechanics and Mathematics, Kharkov University, 61077 Kharkov, Ukraine. E-mail: [email protected]; [email protected]; [email protected] Abstract. An abstract system of coupled nonlinear parabolic-hyperbolic partial differential equations subjected to additive white noise is considered. The system models temperature dependent or heat generating wave phenomena in a continuum random medium. Under suitable conditions, the existence of an exponentially attracting random invariant manifold for the coupled system is proved, and as a consequence, the system can be reduced to a single stochastic hyperbolic equation with a modified nonlinear term. Finally it is also proved that this random manifold converges to its deterministic counterpart when the intensity of noise tends to zero. AMS classification scheme numbers: 35B40, 37H10, 37L25; Secondary 35M10, 37L55. Submitted to: Nonlinearity 1. Introduction and statement of the problem A description of wave propagation phenomena in random media is usually based on the study of stochastically (or randomly) perturbed hyperbolic partial differential equations (see, e.g., Sobczyk (1984) and the references therein). If these wave phenomena are temperature dependent or heat generating, then the hyperbolic equations are coupled with a stochastic parabolic (heat) equation (see, e.g., Chow (1973) or Hori (1973)). To this respect, the question of how a thermal environment may influence on the long time dynamics of the system arises. In this paper we consider this question and show Invariant manifold for parabolic-hyperbolic SPDE 2 that, under some conditions, temperature field is a slave variable for wave (master) variables. In particular this means that the thermal effects at large time scale can be taken into account by modifying a forcing (nonlinear) term in the corresponding stochastic hyperbolic equation. As a model to present our results we consider a system of stochastic differential equations consisting of the hyperbolic equation vtt +γvt+Lv =F(v, vt, u) + ˙ W1,in X1,(1) and the parabolic one ut+νAu =G(v, vt, u) + K(v, vt) + ˙ W2,in X2,(2) where X1and X2are infinite dimensional separable Hilbert spaces, γand νare positive parameters, and the operators and the noises appearing in (1) and (2) satisfy the following assumptions: (A1) Land Aare positive linear self-adjoint operators in X1and X2respectively with domains D(L) and D(A). (A2) Fand Gare nonlinear mappings, F:D(L1/2)×X1×D(Aα)7→ X1, G:D(L1/2)×X1×D(Aα)7→ X2, where α∈[0,1), and there exist constants MFand MGsuch that kF(v0, v1, u)−F(ˆv0,ˆv1,ˆu)kX1 ≤MF¡kL1/2(v0−ˆv0)k2 X1+kv1−ˆv1k2 X1+kAα(u−ˆu)k2 X2¢1/2(3) and kG(v0, v1, u)−G(ˆv0,ˆv1,ˆu)kX2 ≤MG¡kL1/2(v0−ˆv0)k2 X1+kv1−ˆv1k2 X1+kAα(u−ˆu)k2 X2¢1/2.(4) (A3) The mapping K:D(L1/2)×X17→ [D(Aβ)]0possesses the property kA−β(K(v0, v1)−K(ˆv0,ˆv1)) kX2 ≤MK¡kL1/2(v0−ˆv0)k2 X1+kv1−ˆv1k2 X1¢1/2(5) for some 0 ≤β≤1−α, where MKis a positive constant. Invariant manifold for parabolic-hyperbolic SPDE 3 (A4) For every i= 1,2, Wi(t), t∈R, is a two-sided Xi-valued Wiener process with covariance operator Ki=K∗ i≥0 such that tr K1<∞and tr K2A2(α+ε)−1<∞ for some ε > 0. We assume for simplicity that W1and W2are independent, and denote by (Ω,F,P) the corresponding probability space, and by ˙ Withe generalized derivative with respect to tin (1) and (2). Although it is possible to consider other kinds of randomness to model stochastic wave phenomena, the main reason which justifies the use of additive noise is that it usually models background effects and small effects that have been omitted or neglected in a deterministic modeling procedure. To this respect, from the physical point of view, it is important to know whether qualitative properties of the simplified (deterministic) model are robust enough to perturbations by additive noises. Our result in Section 6 answers this question for system (1)–(2). We also note that system (1)–(2) is an abstract model for a thermoelastic phenomenon in a random medium which can be described by the following equations (see, e.g., Chow (1973)): vtt +γvt−µ∆v−(µ+λ)∇div v=−κ∇θ+e F(v, ∇v) + ˙ W1, t > 0, x ∈ O,(6) θt−ν∆θ=−δ·div vt+e G(θ, ∇θ) + ˙ W2, t > 0, x ∈ O,(7) where Ois a domain in Rd,d= 2,3, v=v(x, t)∈Rddenotes the displacement vector, θ=θ(x, t) the temperature, and µ, λ, κ, ν, δ are positive constants, where µand λare Lam´e moduli. The parameter γ > 0 describes resistance forces, and the functions e F and e Gsatisfy suitable conditions. The white noise processes ˙ W1and ˙ W2(see below for more details) model random fluctuations in external loads ( ˙ W1) and in thermal sources (˙ W2). System (6)-(7) can be easily set in our abstract formulation. To this end, we first need to equip these equations with suitable boundary conditions. For example, we can consider Dirichlet type boundary conditions v= 0, θ = 0 for t > 0, x ∈∂O.(8) If we assume that e F:Rd+d27→ Rdand e G:R1+d7→ Rare globally Lipschitz, then (A1)-(A3) hold for our problem (6), (7) and (8), by setting X1= [L2(O)]d,X2=L2(O), α=β= 1/2, K(v, vt) = −δdiv vt,L=−µ∆−(µ+λ)∇div and A=−∆ with Dirichlet boundary conditions, and finally, Fand Gare defined in the obvious way. We note that the asymptotic behaviour of deterministic thermoelastic models has been receiving increasing attention over the last years (see, e.g., Chandrasekharaiah Invariant manifold for parabolic-hyperbolic SPDE 4 (1998), Chueshov (2004), Jiang and Racke (2000), Munoz Rivera and Barreto (1998), Munoz Rivera and Racke (1995), and the references therein). It is also worth mentioning that, as we do not assume any compactness properties concerning the resolvents of the operators Land A, our problem (6), (7) and (8) on unbounded domains can be also included within the scope of our theory, after an appropriate redetermination of the linear and nonlinear terms in the equations. As far as we know, there are no publications on the dynamics of coupled parabolichyperbolic stochastic partial differential equations, although stochastic parabolic and wave equations have been widely studied by many authors (see, e.g, the monographs Cerrai (2001), Da Prato and Zabczyk (1996) and the references therein for the parabolic case and the papers Barbu and Da Prato (2002), Carmona and Nualart (1993), Dalang and Frangos (1998), Da Prato and Zabczyk (1992), Millet and Morien (2001), Millet and Sanz-Sol´e (2000), Peszat and Zabczyk (2000), Quer-Sardanyons and Sanz-Sol´e (2004) for the wave case). Our main objective in this paper is to prove a reduction principle for the random dynamical system generated by problem (1)–(2) which will allow us to rewrite our coupled system as an equivalent problem for a single stochastic hyperbolic equation with a conveniently modified nonlinear term. To be more precise, we will prove that, for νlarge enough, in the phase space H0=D(L1/2)×X1×X2 of the random dynamical system generated by (1) and (2), there exists an invariant exponentially attracting (random) surface of the form M(ω) = ©(v, ¯v, Φ(ω, v, ¯v)) : (v, ¯v)∈D(L1/2)×X1ª⊂ H0,(9) where Φ : Ω ×D(L1/2)×X17→ X2is a Lipschitz mapping for each ω∈Ω and a stationary process with respect to t(see Theorem 4.1 for more details). Under some additional conditions the existence of this surface Mmakes it possible to prove that the long-time behaviour of the system (1) and (2) can be described by the reduced problem vtt +γvt+Lv =F(v, vt,Φ(θtω, v, vt)) + ˙ W1,in X1.(10) For a similar result in the deterministic framework we refer to Leung (2003) and Chueshov (2004). We also mention that, in contrast with (1), the reduced system (10) contains a random nonlinear term of the form F∗(v, vt, θtω) and, hence, cannot be considered as a perturbation of a deterministic system by an additive white noise process. Invariant manifold for parabolic-hyperbolic SPDE 5 The approach which we adopt in this paper relies on some ideas from the theory of inertial manifolds started by Foias et al. (1988) and developed by many authors (see, e.g., the monographs by Chueshov (1999), Constantin et al. (1989), Temam (1988) for the deterministic case and the papers by Bensoussan and Flandoli (1995), Chueshov (1995), Chueshov and Girya (1995), Chueshov and Scheutzow (2001), Duan et al. (2003) for the stochastic case and also the references therein). To cover our main case α+β= 1 we invoke the idea of the Lyapunov-Perron method (see, e.g., Chow and Lu (1988) and Chow et al. (1992)) in the form presented in Miklavˇciˇc (1991) for the deterministic case. To the best of our knowledge, this idea has not been used earlier in the study of invariance properties of stochastic systems. The paper is organized as follows. In the preliminary Section 2 we represent the problem as a first order stochastic differential equation, for the reader’s convenience recall the basic definitions from the theory of random dynamical systems, and collect several results on stochastic convolutions in a form adapted to our situation. In Section 3 we prove the existence and uniqueness of mild solutions to problem (1) and (2) and show that this problem generates a filtered random dynamical system (RDS). Section 4 contains our main result which is a type of reduction principle (see Theorem 4.1). In Section 5 we establish some properties of the reduced system. In Section 6 we estimate the distance between M(ω) and its deterministic counterpart Mdet in terms of the covariance operators of W1and W2(see Theorem 6.1). In particular, we prove that M(ω) converges to Mdet when the intensity of the noise tends to zero. Some final comments and conclusions are presented in the last section. 2. Basic definitions and auxiliary facts First of all, we will rewrite system (1) and (2) as a first order stochastic partial differential system and will analyze the corresponding Cauchy problem; in other words, problem (1)–(2) is equivalent to dV dt +AV=B(V) + ˙ W, t > s, V |t=s=V0,(11) where s∈R,V=V(t) = (v(t), vt(t), u(t))T,W= (0, W1, W2)Tand A=    0−1 0 L γ 0 0 0 νA    ,B(V) =     0 F(v, vt, u) G(v, vt, u) + K(v, vt)    .(12) We consider now problem (11) in the scale of spaces Hσ=D(L1/2)×X1×D(Aσ), σ ∈R, Invariant manifold for parabolic-hyperbolic SPDE 6 which are equipped with the norms |V|σ=¡kL1/2v0k2 X1+kv1k2 X1+kAσu0k2 X2¢1/2, V = (v0, v1, u0). Recall that if σ < 0, then D(Aσ) is the completion of X2with respect to the norm kAσ· kX2. It is straightforward to check that the operator Agenerates a strongly continuous semigroup e−Atin each space Hσand e−At=ÃTt0 0e−νAt !,(13) where Ttis the strongly continuous group in D(L1/2)×X1generated by the equation vtt +γvt+Lv = 0, t > 0,in X1.(14) Let Pdenote the orthoprojector in Hσonto the first two components, i.e. P(v0, v1, u0) = (v0, v1,0) for (v0, v1, u0)∈ Hσ,(15) and Q=I−P. One can easily establish by a direct calculation (see, e.g., Foias et al. (1998) and also Chueshov (1999) or Temam (1988)) the following dichotomy estimates ¯¯e−AtPV ¯¯σ≤e−γt|PV |σ, t ≤0, V ∈ Hσ,(16) ¯¯e−AtPV ¯¯σ≡ |TtPV |D(L1/2)×X1≤ |PV |σ, t ≥0, V ∈ Hσ,(17) ¯¯e−AtQV ¯¯σ≤h³σ νt´σ+λσ 1ie−νλ1t|QV |0, t > 0, V ∈ Hσ, σ > 0,(18) where λ1>0 is the minimal point in the spectrum of A. We also note (see, e.g., Chueshov (1999, Lemma 5.7.1)) that there exist positive constants C0and γ0such that |Tty|D(L1/2)×X1≤C0e−γ0t|y|D(L1/2)×X1, t ≥0, y ∈D(L1/2)×X1.(19) 2.1. Random dynamical systems We recall now some concepts from the theory of random dynamical systems (see, e.g. Arnold (1998) for more details). As usual, R+denotes the set of all non-negative elements of R. Definition 2.1 Let Xbe a topological space. A random dynamical system (RDS) with time R+and state space Xis a pair (θ, φ) consisting of the following two objects: (i) A metric dynamical system (MDS) θ≡(Ω,F,P,{θt, t ∈R}), i.e. a probability space (Ω,F,P) with a family of measure preserving transformations {θt: Ω 7→ Ω, t ∈R}such that Invariant manifold for parabolic-hyperbolic SPDE 7 (a) θ0= id, θt◦θs=θt+sfor all t, s ∈R; (b) the map (t, ω)7→ θtωis measurable and θtP=Pfor all t∈R. (ii) A (perfect) cocycle φover θof continuous mappings of Xwith one-sided time R+, i.e. a measurable mapping φ:R+×Ω×X7→ X, (t, ω, x)7→ φ(t, ω)x such that the mapping φ(·, ω) : (t, x)7→ φ(t, ω)xis continuous for all ω∈Ω and satisfies the cocycle property: φ(0, ω) = id, φ(t+s, ω) = φ(t, θsω)◦φ(s, ω) for all t, s ≥0 and ω∈Ω. Definition 2.2 Let θbe an MDS, Fthe P-completion of F, and F={Ft, t ∈R}a family of sub-σ-algebras of Fsuch that (i) Fs⊆ Ft,s < t; (ii) Fs=Th>0Fs+h,s∈R, i.e., the filtration Fis right-continuous; (iii) Fscontains all P-null sets in F,s∈R; and (iv) θsis (Ft+s,Ft)-measurable for all s, t ∈R. Then (θ, F) is called a filtered metric dynamical system (FMDS). If, in addition, (θ, φ) is an RDS such that φ(t, ·)x is (Ft,B(X))-measurable for every t≥0 and x∈X, then (θ, F, φ) is called a filtered random dynamical system (FRDS). We note that (θ, F, φ) is an FRDS if and only if (θ, φ) is an RDS, (θ, F) is an FMDS and φ(·,·)xis adapted to Ffor every x∈X. Recall that an X-valued stochastic process Y(t), t ∈T⊆Ris called adapted with respect to the filtration Fif Y(t) is (Ft,B(X))- measurable for every t∈T. 2.2. Stochastic convolution We consider a pair of two-sided independent Wiener processes W1(t) and W2(t), t∈R, with values in X1and X2respectively on the same probability space (Ω,F,P) with covariance operators K1and K2possessing the properties Ki=K∗ i≥0,tr K1<∞,tr K2A2(α+ε)−1<∞, for some ε > 0. For the definitions and properties of such processes see Da Prato and Zabczyk (1992). The property tr K1<∞implies that W1has almost surely strongly continuous trajectories in X1(see Da Prato and Zabczyk (1992, p.119)). In the second case, there exists a Hilbert space X2(containing X2) such that W2 has strongly continuous paths in X2. In addition to the distributional properties of W= (0, W1, W2)T, we will assume that there exists a filtered MDS (θ, F) such that Invariant manifold for parabolic-hyperbolic SPDE 8 (a) W(t)− W(s) is Ft-measurable for t∈R, s ≤tand independent of Fsfor s < t; (b) W(t+s, ω)− W(s, ω) = W(t, θsω), s, t ∈R, ω ∈Ω (helix property). We refer to the monograph Arnold (1998) for the construction of this FMDS and the corresponding Wiener processes. Now, let us consider the following stochastic integral η(t, s) = Zt s e−(t−τ)AdW(τ)≡Zt s e−(t−τ)A    0 dW1(τ) dW2(τ)    , t > s. (20) The integral in (20) exists as an operator stochastic integral (see, e.g., Da Prato and Zabczyk (1992)). The process η(t, s) has the form η(t, s) = (η1(t, s), η2(t, s))T, where η1(t, s) and η2(t, s) are centered (independent) Gaussian processes in D(L1/2)×X1and D(Aα)⊂X2respectively, of the form η1(t, s) = Zt s Tt−τÃ0 dW1(τ)!, η2(t, s) = Zt s e−ν(t−τ)AdW2(τ).(21) One can also prove (see, e.g., Da Prato and Zabczyk (1992)) that E|η(t, s)|2 σ=E|η1(t, s)|2 D(L1/2)×X1+EkAση2(t, s)k2 X2 and E|η1(t, s)|2 D(L1/2)×X1=Zt s tr nTt−τˆ K1T∗ t−τodτ, (22) where ˆ K1= diag {0, K1}is an operator in D(L1/2)×X1, and EkAση2(t, s)k2 X2=1 2νtr ©K2A2σ−1(1 −e−2ν(t−s)A)ª, σ ≤α. (23) We note that (22) and (19) imply E|η1(t, s)|2 D(L1/2)×X1≤C2 0·tr K1·¡1−e−2γ0(t−s)¢(24) and also, since Ttis a strongly continuous contraction semigroup, by Da Prato and Zabczyk (1992, Theorem 6.10) there exists a constant C > 0 independent of K1such that Esup t∈[0,1] |η1(t, 0)|2 D(L1/2)×X1≤CE|η1(1,0)|2 D(L1/2)×X1≤C·tr K1.(25) We will write Π = {(t, s) : −∞ ≤ s≤t < ∞, t > −∞}. By Chueshov and Scheutzow (2001, Proposition 3.1), there exists a (perfect) modification of the processes η1(t, s) and η2(t, s) such that the following properties hold. •Process η1(t, s): Invariant manifold for parabolic-hyperbolic SPDE 9 (i) (t, s)7→ η1(t, s, ω) is continuous from Π into D(L1/2)×X1,ω∈Ω; (ii) (t, s, ω)7→ η1(t, s, ω) is measurable from Π ×Ω into D(L1/2)×X1; (iii) quasi-stationarity: η1(t, s, ω) = η1(t+τ, s +τ, θ−τω),(t, s)∈Π, τ ∈R, ω ∈Ω; (26) (iv) evolution relation: for all −∞ ≤ τ < s ≤t, ω ∈Ω, η1(t, s, ω) = η1(t, τ, ω)−Tt−sη1(s, τ, ω); (27) (v) temperedness: for all β > 0, ω ∈Ω, sup t∈Rn|η1(t, −∞, ω)|D(L1/2)×X1e−β|t|o<∞.(28) •Process η2(t, s): (i) (t, s)7→ η2(t, s, ω) is continuous from Π into D(Aα), ω ∈Ω; (ii) (t, s, ω)7→ η2(t, s, ω) is measurable as a map from Π ×Ω into D(Aα); (iii) quasi-stationarity: η2(t, s, ω) = η2(t+τ, s +τ, θ−τω),(t, s)∈Π, τ ∈R, ω ∈Ω; (29) (iv) evolution relation: for −∞ ≤ τ < s ≤t, ω ∈Ω, η2(t, s, ω) = η2(t, τ, ω)−e−ν(t−s)Aη2(s, τ, ω); (30) (v) temperedness: for all β > 0, ω ∈Ω, sup t∈R©kAαη2(t, −∞, ω)kX2e−β|t|ª<∞.(31) We note that formally Proposition 3.1, as it is stated in Chueshov and Scheutzow (2001), cannot be applied to the process η1. However the arguments given in the proof of this proposition rely only on the fact that the corresponding semigroup (this is Ttin our case) is strongly continuous and exponentially stable and therefore they cover the case of processes like η1(t, s). We also recall (see, e.g., Arnold (1998)) that a random variable v(ω) with values in a Banach space Xis said to be tempered iff sup t∈R©e−β|t|kv(θtω)kXª<∞for all β > 0, ω ∈Ω. By (26) and (29) we have that ηi(t, −∞, ω) = ηi(0,−∞, θtω)≡˜ηi(θtω) for t∈R, ω ∈Ω, and i= 1,2, where, due to (28) and (31), the Gaussian random variables ˜η1(ω) and ˜η2(ω) are tempered in D(L1/2)×X1and X2respectively. 3. Mild solutions and generation of an RDS For a given σ∈R, we denote by C([a, b]; Hσ) the space of strongly continuous functions on the interval [a, b] with values in Hσ, and by L2([a, b]; Hσ) the space of measurable Invariant manifold for parabolic-hyperbolic SPDE 16 Consequently, it is sufficient to prove the existence and uniqueness of solutions to (52) only for the case s= 0. This observation and also the deterministic argument given in Chueshov (2004) make it possible to prove the following assertion. Proposition 4.3 Let s∈Rand γ < µ < νλ1. Then, for every D∈PHαand ω∈Ω the operator BD[·;ω]is continuous from Yα,s into itself and |BD1[V1;ω]−BD2[V2;ω]|Yα,s ≤ |D1−D2|0+κα(ν, µ)· |V1−V2|Yα,s , ω ∈Ω,(57) for every D1, D2∈PHαand V1, V2∈Yα,s, where κα(ν, µ) = MF µ−γ+λα 1MG+λα+β 1MK νλ1−µ.(58) Now we take µ= (γ+νλ1)/2. In this case κα(ν, µ)<1 under the condition ν > ν0, where ν0is given by (46). Thus BD[·;ω] is a contraction in Yα,s and hence Eq. (52) has a unique solution V(·, s)≡V(·, s;ω, D) in the space Yα,s for each ω∈Ω. Using the same (standard) argument as in the deterministic case (see Chueshov (2004)) one can show that this solution V(·, s) possesses the properties V(·)≡V(·, s)∈C((−∞, s],Hσ), σ < min(1 −β, 1/2),(59) and sup t≤s©eµ(t−s)|V(t, s;ω, D1)−V(t, s;ω, D2)|σª≤Cσ|D1−D2|0(60) for any D1, D2∈PHαand ω∈Ω, where Cσis a positive constant. Moreover, it follows directly from (52) that for every r∈(−∞, s) and for almost all t∈[r, s] the function V(·, s) satisfies the relation V(t, s) = e−A(t−r)V(r, s) + Zt r e−A(t−τ)B(V(τ, s))dτ +η(t, r).(61) Now for every s∈Rwe define Φs: Ω ×D(L1/2)×X1→X2as Φs(ω, D) = Zs −inf ty e−νA(s−τ)(G(V(τ, s)) + K(v(τ, s),¯v(τ, s)))dτ +η2(s, −∞),(62) where V(t, s) = (v(t, s),¯v(t, s), u(t, s)) solves the integral equation (52). It is easy to see from (56) that Φs(ω, D) = Φ0(θsω, D)≡Φ(θsω, D), i.e. s7→ Φs(ω, D) is a stationary process. Therefore, it follows from (52) and (61) that the random surface M(ω) given by (49) is positively invariant with respect to the cocycle φ. Moreover, the relation (60) implies the Lipschitz property (47). Invariant manifold for parabolic-hyperbolic SPDE 17 4.2. Tracking properties We will use the method developed in Miklavˇciˇc (1991) for the proof of a tracking property for inertial manifolds in the deterministic case. Let V0= (v0, v1, u0)∈ Hσ, where σsatisfies (48), and let V(t)≡V(t, 0, ω;V0) be a mild solution to (11) for s= 0. We extend V(t) on the semi-axis (−∞,0] by the formula V(t) = (v0, v1,(1 + |t|A)−1u0). It is easy to see that V(·)∈Yα,0∩C((−∞,0],Hσ) and |V|2 Yα,0≡Z0 −∞ e2µt|V(t)|2 αdt ≤C|V0|2 σ.(63) Now we consider the following space Z=½Z(·) : |Z|2 Z≡Z∞ −∞ e2µt|Z(t)|2 αdt < ∞¾ and define the random function Z0(t, ω) =        −V(t) + BPV0[V;ω](t, 0),for t≤0; e−At[−V0+BPV0[V;ω](0,0)] ,for t > 0, (64) where Bis the same as in (52). Below we need the following properties of the random function Z0(t, ω). Lemma 4.4 For every ω∈Ωthe random function Z0(t, ω)belongs to Z. Moreover for every σsatisfying (48) there exist a deterministic constant Cand scalar tempered random variables e R1(ω)and e R2(ω)such that |Z0|Z≤e R1(ω) + C|V0|σand sup t∈R©eµt|Z0(t)|σª≤e R2(ω) + C|V0|σ.(65) Proof. We split Z0(t, ω) into deterministic and stochastic parts, Z0(t, ω) = Zdet 0(t) + Zst 0(t, ω), where Zdet 0(t) =        −V(t) + IPV0[B(V)](t, 0),for t≤0; e−At[−V0+IPV0[B(V)](0,0)] ,for t > 0, and Zst 0(t, ω) =        (−Ttη1(0, t), η2(t, 0))T,for t≤0; ¡0,0, e−νAtη2(0,−∞)¢T,for t > 0, (66) Invariant manifold for parabolic-hyperbolic SPDE 18 Since, by (16) and (18) R∗ 1(ω)≡ |Zst 0(ω)|2 Z≤Z0 −∞ e2(µ−γ)th|η1(0, t)|2 D(L1/2)×X1+kAαη2(t, −∞)k2 X2idt +kAαη2(0,−∞)k2 X2 and R∗ 2(ω)≡sup t∈R©eµt|Zst 0(t, ω)|σª ≤c0sup t∈R©e(µ−γ)t£|η1(0, t)|D(L1/2)×X1+kAση2(t, −∞)kX2¤ª, it follows from (27), (28) and (31) that R∗ 1(ω) and R∗ 2(ω) are tempered random variables. Therefore, estimating the deterministic part Zdet 0(t) by the standard method we arrive at the estimates (65) with e Ri(ω) = C1+C2R∗ i(ω), where C1and C2are deterministic constants. ¤ Now we define an integral operator R:Z 7→ Z by the formula R[Z](t) = Z0(t) + Zt −∞ e−A(t−τ)Q[B(Z(τ) + V(τ)) − B(V(τ))] dτ −Z∞ t e−A(t−τ)P[B(Z(τ) + V(τ)) − B(V(τ))] dτ. (67) Let us prove that Ris a contraction in Z. By (3) and (16) we have that eµt|P(R[Z1](t)−R[Z2](t)) |α≤MFZ∞ t e(µ−γ)(t−τ)·eµτ |Z1(τ)−Z2(τ)|αdτ ≡Z∞ −∞ e(t−τ)f(τ)dτ, (68) where e(t) = 0 for t > 0, e(t) = MFe(µ−γ)tfor t≤0 and f(t) = eµt |Z1(t)−Z2(t)|α, t∈R. Thus, using the Fourier transformation and the Plancherel formula (cf. Lemma 2.2 in Chueshov (2004)) we obtain that |P(R[Z1]−R[Z2]) |Z≤MF µ−γ· |Z1−Z2|Z. Similarly, Q(R[Z1](t)−R[Z2](t)) = (0; 0; Q1(t) + Q2(t)),(69) where Q1(t) = Zt −∞ e−νA(t−τ)[G(Z1(τ) + V(τ)) −G(Z2(τ) + V(τ))] dτ, Q2(t) = Zt −∞ (e−νA(t−τ)AβQ£A−β(K(P[Z1(τ) + V(τ)]) −K(P[Z2(τ) + V(τ)]))¤)dτ. To estimate Q1and Q2we use the following result from Chueshov (2004). Invariant manifold for parabolic-hyperbolic SPDE 19 Lemma 4.5 Let e−Atbe a strongly continuous semigroup in a Hilbert space Hwith a self-adjoint and positive generator A=A∗>0. Let λmin >0be the minimal point in the spectrum of A. The following assertions hold: For any 0≤β≤1and µ≥0the mapping f∈L2(R;H)7→ Iβ[f]∈L2(R;D(A1−β)), where Iβ[f](t) = Zt −∞ e−A(t−τ)(A+µ)βf(τ)dτ, t ∈R, is continuous, and the estimate ZR k(A+µ)αIβ[f](t)k2 Hdt ≤(λmin +µ)2(α+β) λ2 min ZR kf(t)k2 Hdt holds for any 0≤β≤1,−β≤α≤1−βand µ≥0. Applying Lemma 4.5 with A=νA −µwe obtain that |Q1|Z≤λα 1MG νλ1−µ|Z1−Z2|Zand |Q2|Z≤λα+β 1MK νλ1−µ|Z1−Z2|Z. Since µ= (γ+νλ1)/2, we have that |R[Z1]−R[Z2]|Z≤q· |Z1−Z2|Zfor every Z1, Z2∈ Z.(70) Here q=κα(ν, (γ+νλ1)/2) <1 under the condition ν > ν0, where καand ν0are given by (58) and (46). Thus by the contraction principle there exists a unique solution Z∈ Z to the equation Z=R[Z] in Z. Now using the same calculation as in Chueshov (2004) and Miklavˇciˇc (1991) we can conclude that the function e V(t) = Z(t) + V(t), where Z∈ Z solves the equation Z=R[Z], satisfies the relation e V(t) =        BP e V(0)[e V , ω](t, 0),if t≤0; φ(t, 0, ω)e V(0),if t > 0. (71) In particular, e V(0) = BP e V(0)[e V , ω](0,0) and, therefore, by the definition of the operator Bwe obtain that e V(0) = Pe V(0) + Z0 −∞ eAτQB(e V(τ))dτ +Qη(0,−∞). By (62) this implies that e V(0) = ³Pe V(0),Φ(ω, P e V(0))´. Therefore e V(t) = φ(t, 0, ω)e V(0) ∈ M(θtω) for t≥0. Thus to complete the proof we only need to establish (50) and (51). Since e V(t) = Z(t) + V(t) and Z(t) = R[Z](t) = Z0(t) + R[Z](t)−R[0](t),(72) Invariant manifold for parabolic-hyperbolic SPDE 20 from (65) and (70) we obtain the relation |Z|Z≤(1 −q)−1· |Z0|Z≤(1 −q)−1·³e R1(ω) + C|V0|σ´,(73) which implies (50). Now we prove (51). Since |PU|σ=|PU|α, from (68) we have that eµt|P(R[Z](t)−R[0](t)) |σ≤MFZ∞ t e(µ−γ)(t−τ)·eµτ |Z(τ)|αdτ ≤MF·Z∞ t e2(µ−γ)(t−τ)dτ¸1/2 · |Z|Z=MF p2(µ−γ)· |Z|Z. Thus sup t∈R©eµt|P(R[Z](t)−R[0](t)) |σª≤MF p2(µ−γ)cdot |Z|Z.(74) Similarly, using (69) we have that |Q(R[Z](t)−R[0](t)) |σ ≤MGZt −∞ kAσe−νA(t−τ)k·|Z(τ)|αdτ +MKZt −∞ kAσ+βe−νA(t−τ)k·|PZ(τ)|0dτ ≤a1·e−µt · |Z|Z+a2·e−µt ·sup t∈R©eµt|PZ(t)|σª, where a1=MGsup t∈R·Zt −∞ kAσe−(νA−µ)(t−τ)k2dτ¸1/2 <∞, a2=MKsup t∈RZt −∞ kAσ+βe−(νA−µ)(t−τ)kdτ < ∞, (the finiteness of a1and a2follows from (18) by a straightforward computation). From (72) and (74) we have that sup t∈R©eµt|PZ(t)|σª≤sup t∈R©eµt|Z0(t)|σª+MF p2(µ−γ)· |Z|Z. Therefore sup t∈R©eµt|Q(R[Z](t)−R[0](t)) |σª ≤a2sup t∈R©eµt|Z0(t)|σª+"a1+a2MF p2(µ−γ)#· |Z|Z.(75) Consequently, using relations (72), (74) and (65) we obtain that sup t∈R©eµt|Z(t)|σª≤(1 + a2)e R2(ω) + "a1+(1 + a2)MF p2(µ−γ)#· |Z|Z. Invariant manifold for parabolic-hyperbolic SPDE 21 Thus by (73) we have sup t∈R©eµt|Z(t)|σª≤c0(e R1(ω) + e R2(ω)) + c1|V0|σ. with appropriate (deterministic) constants c0and c1. This implies (51) and completes the proof of Theorem 4.1. Remark 4.6 The existence of a positively invariant manifold of the form (49) can be also established in the case γ= 0 under the the condition ν > ν0, where ν0is given by (46) with γ= 0. The point is that we can introduce artificial small damping in problem (1) with γ= 0 by the considering the equation vtt +εvt+Lv =Fε(v, vt, u) + ˙ W1,in X1,(76) where Fε(v, vt, u) = F(v, vt, u) + εvt. We choose εsuch that the relation ν > ν0remains true, where ν0is defined by (46) with γ=εand with the term MF+εinstead of MF. Now we can apply Theorem 4.1 to problem (76) and (2). 5. The reduced system Assume the hypotheses of Theorem 4.1 hold and let Φ ≡Φ0be given by (62) with s= 0. Consider the problem (vtt +γvt+Lv =F(v, vt,Φ(θtω, v, vt)) + ˙ W1, t > s, in X1, v|t=s=v0, vt|t=s=v1,(77) and define its mild solution on the interval [s, T] as a random function D(t, ω)≡D(t, s;ω, v0, v1) = (v(t, ω), vt(t, ω)) ∈C([s, T], D(L1/2)×X1) (78) such that ZT s kAαΦ(θt, v(t, ω), vt(t, ω))k2 X2dt < ∞, ω ∈Ω,(79) and "v(t, ω) vt(t, ω)#=Tt−s"v0 v1#+Zt s Tt−τ"0 F(v(τ), vt(τ),Φ(ω, v(τ), vt(τ))) #dτ +η1(t, s) for almost all t∈[s, T] and ω∈Ω, where Ttis the evolution group generated by (14) and η1(t, s) is given by (21). Proposition 5.1 Let v0∈D(L1/2)and v1∈X1. Then under the conditions of Theorem 4.1 problem (77) has a mild solution on any interval [s, T]. If α < min(1 − β, 1/2), then this solution is unique and any mild solution (ˆv(t),ˆvt(t)) to problem (77) generates a mild solution to problem (1) and (2) by the formula (v(t), vt(t), u(t)) = (ˆv(t),ˆvt(t),Φ(θtω, ˆv(t),ˆvt(t))).(80) Invariant manifold for parabolic-hyperbolic SPDE 22 Moreover, in this case the manifold Mis invariant with respect to the cocycle φgenerated by (1) and (2). Proof. Let V(t) = (v(t), vt(t), u(t)) be a mild solution to problem (11) with the initial data V0= (v0, v1,Φ(ω, v0, v1)). Since Mgiven by (49) is positively invariant, we have that PV (t) = (v(t), vt(t),0) ≡(D(t),0) and QV (t) = (0,0,Φ(θtω, D(t))). Thanks to Theorem 3.3 D(t) possesses the property (78). We also have that ZT s |QV (t)|2 αdt ≤ZT s |V(t)|2 αdt < ∞. Thus (79) holds. Consequently D(t) is a mild solution to (77). If α < min(1 −β, 1/2), then by Theorem 4.1 with σ=αwe have that kAα(Φ(ω, D1)−Φ(ω, D2))kX2≤CkD1−D2kD(L1/2)×X1, for Di∈D(L1/2)×X1.This implies that the function D7→ FΦ(ω, D) := F(v0, v1,Φ(ω, v0, v1)), D = (v0, v1)∈D(L1/2)×X1, is globally Lipschitz, i.e. kFΦ(ω, D1)−FΦ(ω, D2))kX1≤CkD1−D2kD(L1/2)×X1,(81) with Di∈D(L1/2)×X1.Therefore a Gronwall type argument gives us the uniqueness of solutions to (77). Relation (80) easily follows from the uniqueness theorem for (77). Property (81) makes it also possible to solve (77) backwards in time and, hence, one can prove that Mis invariant with respect to the cocycle φ(t, ω). ¤ Observe now that Theorem 4.1 implies that for any mild solution V(t) = (v(t), vt(t), u(t)) to problem (1) and (2) with initial data V0∈ Hσ, where σsatisfies (48), there exists a mild solution D(t) = (ˆv(t),ˆvt(t)) to reduced problem (77) such that kL1/2(v(t)−ˆv(t)k2 X1+kvt(t)−ˆvt(t)k2 X1+kAσ[u(t)−Φ(ω, ˆv(t),ˆvt(t))] k2 X2≤Ce−µ(t−s) for any t≥swith positive constants Cand µ. Thus under the conditions of Theorem 4.1, the long-time behaviour of solutions to (1) and (2) can be described completely by solutions to problem (77). Moreover, under the condition α < min(1 −β, 1/2), due to relation (80), every limiting regime of the reduced system (77) is realized in the coupled system (1) and (2). Invariant manifold for parabolic-hyperbolic SPDE 23 6. Distance between random and deterministic manifolds Theorem 4.1 can be also applied to the deterministic version of problem (1) and (2): vtt +γvt+Lv =F(v, vt, u),in X1,(82) ut+νAu =G(v, vt, u) + K(v, vt),in X2.(83) In this case Theorem 4.1 give us the existence of a (deterministic) invariant exponentially attracting manifold Mdet in the space Hσof the form Mdet =©(v, ¯v, Φdet(v, ¯v)) : (v, ¯v)∈D(L1/2)×X1ª,(84) where Φdet :D(L1/2)×X17→ D(Aσ)⊂X2is a globally Lipschitz mapping and σ satisfies (48). Our goal in this section is to estimate the mean value distance between the deterministic (Mdet) and random (M(ω)) manifolds. Theorem 6.1 There exist a positive constant Csuch that E(sup D∈D(L1/2)×X1 kAσ(Φ(·, D)−Φdet(D))k2 X2)≤C¡tr K1+ tr K2A2α−1¢,(85) where σsatisfies (48). Thus, the random manifold M(ω)is close to its deterministic counterpart when tr K1+ tr K2A2α−1becomes small. Proof. It follows from the definition (see (62)) of the functions Φ and Φdet that Φ(ω, D)−Φdet(D) = Z0 −∞ eνAτ £G(Vst(τ)) −G(Vdet(τ))¤dτ +Z0 −∞ eνAτ £K(vst(τ),¯vst(τ)) −K(vdet(τ),¯vdet(τ))¤dτ +η2(0,−∞), where Vst(t)≡(vst(t),¯vst(t), ust(t)) and Vdet(t)≡(vdet(t),¯vdet(t), udet(t)) are defined on the semi-axis (−∞,0] and solve the equations Vst(t) = ID[B(Vst); ω](t, 0) and Vdet(t) = Idet D[B(Vdet)](t, 0),(86) where IDand Idet Dare defined as in (53). Using the same method as in the proof of relation (75) we can conclude that kAσ(Φ(·, D)−Φdet(D))kX2≤ kAση2(0,−∞)kX2+a1|Vst −Vdet|Yα,0 +a2sup t≤0©eµt|P(Vst(t)−Vdet(t))|0ª,(87) where a1and a2are deterministic constants. Invariant manifold for parabolic-hyperbolic SPDE 24 Now using (86) and the structure of the operators IDand Idet Dwe can write the following estimate |P(Vst(t)−Vdet(t))|0 ≤MFZ0 t e−γ(t−τ)¯¯Vst(τ)−Vdet(τ)¯¯αdτ +eγt |η1(0, t)|D(L1/2)×X1 for t≤0, which implies that sup t≤0©eµt|P(Vst(t)−Vdet(t))|0ª ≤C|Vst −Vdet|Yα,0+ sup t≤0ne(µ−γ)t|η1(0, t)|D(L1/2)×X1o. Therefore from (87) we have that kAσ(Φ(·, D)−Φdet(D))k2 X2 ≤b1|Vst −Vdet|2 Yα,0+b2∆1(ω;η1, η2)+2kAση2(0,−∞)k2 X2,(88) where b1and b2are deterministic constants and ∆1(ω;η1, η2) = sup t≤0ne2(µ−γ)t|η1(0, t)|2 D(L1/2)×X1o.(89) By (86) we have that |Vst −Vdet|Yα,0≤ |BD[Vst;ω](·,0) −BD[Vdet;ω](·,0)|Yα,0+|Σ(0,·)|Yα,0, where BD[V;ω](t, 0) is the same as in (52) and Σ(s, t) is given by (55). Thus by Proposition 4.3 we have that |Vst −Vdet|Yα,0≤(1 −q)−1|Σ(0,·)|Yα,0, where q=κα(ν, (γ+νλ1)/2) <1. Therefore, using (88) we obtain the estimate kAσ(Φ(·, D)−Φdet(D))k2 X2 ≤2kAση2(0,−∞)k2 X2+b1 (1 −q)2|Σ(0,·)|2 Yα,0+b2∆1(ω;η1, η2).(90) It easily follows from relations (23) and (24) and from the definition of Σ(s, t) (see (55)) that EkAση2(0,−∞)k2 X2≤C1·tr K2A2α−1(91) and E|Σ(0,·)|2 Yα,0≤C2¡tr K1+ tr K2A2α−1¢.(92) Invariant manifold for parabolic-hyperbolic SPDE 25 Now we calculate E∆1(ω;η1, η2). From (27) and (17) we obtain that |η1(0, t)|2 D(L1/2)×X1≤2|η1(0,−∞)|2 D(L1/2)×X1+ 2 |η1(t, −∞)|2 D(L1/2)×X1. Hence by (24) we have that E∆1(·;η1, η2)≤Ctr K1+ 2Esup t≤0ne2(µ−γ)t|η1(t, −∞)|2 D(L1/2)×X1o. Since Esup t≤0ne2(µ−γ)t|η1(t, −∞)|2 D(L1/2)×X1o ≤ ∞ X n=1 e−2(µ−γ)(n−1)Esup 0≤t≤1n|η1(−n+t, −∞)|2 D(L1/2)×X1o and, by (26), η1(−n+t, −∞, ω) = η1(t, −∞.θ−nω), we obtain from the invariance of the probability measure with respect to θtthat Esup t≤0ne2(µ−γ)t|η1(t, −∞)|2 D(L1/2)×X1o≤CEsup 0≤t≤1n|η1(t, −∞)|2 D(L1/2)×X1o. Using (27) we have that η1(t, −∞) = η1(t, 0) + Ttη1(0,−∞). Consequently, by (24) and (25) we obtain that Esup t≤0ne2(µ−γ)t|η1(t, −∞)|2 D(L1/2)×X1o≤Ctr K1 and hence E∆1(·;η1, η2)≤Ctr K1. Therefore (85) follows from (90) and (91). ¤ Conclusions We have proved in this paper that the long time behaviour of coupled non-linear parabolic-hyperbolic partial differential equations perturbed by additive noise can be reduced to the analysis of a corresponding hyperbolic random equation with a modified nonlinear term. The main tool is the construction of an invariant random manifold for the coupled system which is given by the graph of a suitable Lipschitz mapping. Of course, one could consider other different expressions for the noisy terms in the equations (multiplicative, cylindrical, etc). We plan to investigate the possibility of doing a similar reduction in the future. Acknowledgments We would like to thank the referees for their useful suggestions and comments. T. Caraballo and J.A. Langa have been partially supported by M.C.Y.T. (Spain) and FEDER (European Community) Project BFM 2002-03068. I. Chueshov has been partially supported by the INTAS Grant 2000-899.