Effects of additive and multiplicative noise on the dynamics of a parabolic equation

Appl. Math. Inf. Sci. Vol?, No. No?, 1-9 (2015?) 1 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.12785/amis/152AMISfinalversionwithphoto Effects of additive and multiplicative noise on the dynamics of a parabolic equation Tom´ as Caraballo1and Renato Colucci2 1Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain 2Department of Mathematical Sciences, Xi’an Jiaotong-Liverpool University, 111 Ren’ai Road, Suzhou, P. R. China, 215123 Received: ..., Revised: ..., Accepted: ... Published online: ... Abstract: We consider the effects of additive and multiplicative noise on the asymptotic behavior of a fourth order parabolic equation arising in the study of phase transitions. On account that the deterministic model presents three different time scales, in this paper we have established some conditions under which the third time scale, which encounter finite dimensional behavior of the system, is preserved under both additive and multiplicative linear noise. In particular we have proved the existence of a random attractor in both cases, and observed that the order of magnitude of the third time scale is also preserved. Keywords: Random attractor, Phase Transitions, White Noise 1 Introduction The study of phase transitions has been an important subject of research over the last decades (see [1]). Several models have been introduced, among the others, the Cahn-Hilliard (see for example [2]) equations have been intensively studied. In [3] the authors introduced a model related to that of Cahn and Hilliard (C-H) in the sense that if uis the solution of the equation    ut=− ε 2uxxxx +1 2[W′(ux)]x u=uxx =0,on ∂ I, u(0,t) = u0, (1) then uxsolve the C-H equation. In (1), Iis an open interval, the function Wis the so called double well potential, that is, W(p) = (p2−1)2and ε is a small parameter. The equation (1) represents the L2-gradient dynamics associated to the energy functional (see [10] and [14]): F ε (u) = 1 2ZIu2 xxdx+1 2ZIW(ux)dx. The global dynamics of (1) have been studied in [3] by numerical experiments and, in particular, the existence of three time scales with different dynamical behavior has been pointed out. In a first time scale of order O( ε 2), the energy of the initial datum is drastically reduced and a microstructure appears in the region in which the gradients of the initial data are in the set in which the potential Wis non convex, that is (−√3 3,√3 3)(see [10] for an analysis of the first time scale using variational techniques). In the second time scale of order O(1), the region without microstructure evolves in a heat equation like behavior, while the region with microstructure remains almost stationary. In the last time scale of order O( ε −2), equation (1) exhibits a finite dimensional behavior, the solution is approximately the union of consecutive segments with slops ±1. In the papers [11], [12], [13], and [4], the third time scale has been studied. In [11] the authors prove the existence of an exponential attractor M ε whose dimension is of order O( ε −10). In particular the time for which the solutions enter the absorbing set is of order ( ε −2). In [12] the authors found an estimate of the dimension, of order O( ε −1)(in accordance with the numerical experiments presented in [3]), of the global attractor A ε by the volume elements evolution method and they proved the existence of an inertial manifold whose dimension is of order O( ε −19). Moreover, some estimates on the regularity estimates and on the embedding dimension are ∗Corresponding author e-mail: [email protected] c 2015? NSP Natural Sciences Publishing Cor. 2 T. Caraballo & R. Colucci: Effects noise on the dynamics of a parabolic equation performed. Since natural systems are subjected to random perturbation we consider interesting to reconsider the problem (1) in the context of stochastic processes. Namely, when some multiplicative and additive noise may appear in the formulation. Thus, in Section 2 below we state the problem in the multiplicative case, state some necessary preliminary definitions on the theory of random attractors, and prove that our problem generates a random dynamical system. In Section 3 we prove the existence of a random attractor. Finally, in Section 4 we consider the case in which the noise appears as an additive one. We do not intend to establish all the results in a formal and detailed way, but only wish to provide the main estimates in order to justify the existence of a random attractor, highlighting the main differences with the multiplicative case. This additive case requires of some additional technicalities which make the analysis more involved. 2 Preliminary definitions We consider the following stochastic version of the system (1) with multiplicative noise    dX = (− ε 2△2X+1 2W′′(∇X)△X)dt + σ X◦dw, X=Xxx =0,on ∂ I, X(0,t) = X0, (2) where w(t)is a two-sided standard Wiener process defined on a probability basis ( Ω ,F,{F}t∈R,P), where {F}t∈Ris a filtration that is an increasing collection of sigma-algebras on Ω and F0contains all the null sets. In order to study the dynamics of equation (2) we first recall some basic definitions of the theory of random dynamical systems. Let { θ t: Ω → Ω ,t∈R}be a family of measure preserving transformations such that (t, ω )→ θ t ω is measurable, θ 0=id and θ t+s= θ t θ sfor all s,t∈R. The flow θ ttogether with the corresponding probability space ( Ω ,F,P,( θ t)t∈R)is called a measurable dynamical system. Finally we suppose that θ tis ergodic. A continuous random dynamical system is a measurable map ϕ :R+× Ω ×X→X, such that P-a.s. (i) ϕ (0, ω ) = id on X, (ii) ϕ (t+s, ω ) = ϕ (t, θ s ω ) ϕ (s, ω ),∀t,s∈R+, where (X,d)is a Polish space with Borel σ -algebra B over θ on ( Ω ,F,P). In [16] the authors introduced, in the context of random dynamical systems, the concept of attractor for stochastic partial differential equations in order to study the qualitative dynamical behavior of the solutions. Definition 1. A random compact set {K( ω )} ω ∈ Ω is a family of compact sets indexed by ω such that for every x∈X the mapping ω →d(x,K( ω )) is measurable with respect to F. Definition 2. A random set K( ω )⊂X is said to absorb the set B if for almost all ω ∈ Ω there exists t(B, ω )such that for all t ≥t(B, ω ) ϕ (t, θ −t ω )B⊂K( ω ). Definition 3. A random set A( ω )is said to be a random attractor associated to the random dynamical system ϕ if P-a.s. (i)A( ω )is a random compact set. (ii) ϕ (t, ω )A( ω ) = A( θ t ω ),∀t≥0, (iii)attracts all B ⊂X bounded and non-random, that is lim t→∞dist( ϕ (t, θ −t ω )B,A( ω )) = 0, where dist denotes the usual Hausdorff semi-distance. In the sequel we will use the fact that ϕ (t, θ −t ω )can be interpreted as the position at t=0 of the trajectory which was in xat time −tand we consider the attraction property at tgoes to −∞. This is the so called pullback convergence which means to look at the position of the solutions at present time when the initial ones go to −∞. In the next section we will prove the existence of a random attractor for equation (2) using the following theorem (see [16]): Theorem 1. Assume that there exists a compact set D( ω ) absorbing every boundednon-randomset B ⊂X.Then,the set A( ω ) = [ B⊂X Λ B( ω ) is a random attractor for ϕ , where the union is taken over all B ⊂X bounded, and Λ B( ω )is the omega-limit set of B given by Λ B( ω ) = \ n≥0[ t≥n ϕ (t, θ −t ω )B. Moreover in [16] the author proved that random attractors are unique and, by the ergodicity of θ t, there exists a compact set K⊂Xsuch that P-a.s. the random attractor is the omega limit set of K, that is: A( ω ) = \ n≥0[ t≥n ϕ (t, θ −t ω )K. In order to prove the existence of the random attractor, the idea is to transform the stochastic evolution equation (2) containing a noise term into an evolution equation without noise but with a random coefficient, that is a real function which takes random values. In this case all the techniques and tools for the study of standard evolutions equations are available. For this purpose we consider the following change of variable ψ =T−1( ω )X,(3) c 2015? NSP Natural Sciences Publishing Cor. Appl. Math. Inf. Sci. Vol?, No. No?, 1-9 (2015?) / www.naturalspublishing.com/Journals.asp 3 where T( ω ) = e σ z∗( ω ), and where z∗is the stationary solution of the equation dz =−zdt +dwt, ω (t) = wt( ω ),(4) which is the so called Ornstein-Uhlenbeck process. In particular, if ϕ is a random dynamical system and T and T−1are measurable then (t, ω ,x)→T−1( θ t ω , ϕ (t, ω ,T( ω ,x))) := ψ (t, ω ,x) is a random dynamical systems (see [6]). Before proceeding with the analysis of the random system we recall some important properties of z∗( ω )(see [6] for more details): Proposition 1. There exists a { θ t}t∈R-invariant subset ¯ Ω ∈Fof Ω =C0(R,R)of full measure such that lim t→±∞| ω (t)| t=0,for ω ∈¯ Ω , and for such ω the random variable is given by z∗( ω ):=−Z0 −∞e τω ( τ )d τ and is well defined. Moreover, for ω ∈¯ Ω , the mapping (t, ω )→z∗( θ t ω ) = −Z0 −∞e τθ t( ω ( τ ))d τ =−Z0 −∞e τ ( ω (t+ τ )− ω ( τ ))d τ is a stationary solution of equation (4)with continuous trajectories. In addition, for ω ∈¯ Ω : lim t→±∞|z∗( θ t ω )| |t|=0,lim t→±∞ 1 tZt 0z∗( θτω )d τ =0, and E|z∗|=lim t→±∞ 1 tZt 0|z∗( θτω )|d τ <∞. By the change of variable (3) the equation (2) becomes: ψ t=− ε 2 ψ xxxx +T−1( θ t ω ) 2[W′(T( θ t ω ) ψ x)]x(5) + σ z∗( θ t ω ) ψ , ψ (0) = ψ 0,(6) The existence and uniqueness of solution follows from classical methods such as Galerkin approximations (see for example [26] Theorem 3.1.1), and we will omit the details. In particular, we can prove the following result: Theorem 2. For P−a.s. ω ∈ Ω and all T >0, the problem (5)admits a unique solution ψ such that if ψ 0∈H=L2(I)then ψ ∈C0([0,T],H)∩L2(0,T;V); if ψ 0∈V=H2(I)then ψ ∈C([0,T],V)∩L2(0,T;D(A)); where D(A) = { ψ ∈H4(I): ψ = ψ xx =0on ∂ I}is the domain of the differential operator A =− ∂ 4 ∂ x4. The continuity of the cocycle is guarranteed by the following stronger result: Proposition 2. The solution of (5)is Lipschitz continuous with respect to the initial data. Proof. Let ψ i,i=1,2 be two solutions associated respectively to the initial data ψ 0,i,i=1,2 and set u= ψ 1− ψ 2. Then the equation fulfilled by uis the following: ut=− ε 2uxxxx + σ z∗( θ t ω )u +T−1( θ t ω ) 2W′(T( θ t ω ) ψ 1x)−W′(T( θ t ω ) ψ 2x)x, Multiplying the previous equation by uin Hyields: 1 2 d dtkuk2+ ε 2kuxxk2 +T−1( θ t ω ) 2ZIW′(T( θ t ω ) ψ 1x)−W′(T( θ t ω ) ψ 2x)uxdx + σ z∗( θ t ω )kuk2=0, from which 1 2 d dt kuk2+ ε 2kuxxk2≤2kuxk2+2T2ZI¯uu2 xdx+ σ |z∗|kuxk2, where ¯u= ψ 1x ψ 2x. In the next section we will prove the absorbing properties in several spaces, in particular, we will find the existence of random variables ri( ω )such that k ψ (0)k≤r0( ω ),k ψ x(0)k≤r2( ω ),k ψ xx(0)k≤r3( ω ), where again we intend the aboveestimates in the following sense S(t, θ −t ω ) ψ 0= ψ (0). Here we make use of that in order to conclude the proof (see section below for the complete details). In particular we use: k¯uk∞≤ k ψ 1xk∞k ψ 2xk∞ ≤ k ψ 1kk ψ 2kk ψ 1xxkk ψ 2xxk ≤r2 0( ω )r2 3( ω ):=K( ω ) Then, we can easily obtain 1 2 d dt kuk2+ ε 2kuxxk2≤2(1+T2K( ω )+ 1 2 σ |z∗|)kuxk2 :=2h( ω ,t)kuxk2 ≤ ε 2kuxxk2+h2( ω ,t) ε 2kuk2, and integrating we arrive at kS(t, θ −t ω ) ψ 1,0−S(t, θ −t ω ) ψ 2,0k =k ψ 1(0, ω ,−t, ψ 1,0)− ψ 2(0, ω ,−t, ψ 2,0)k =k ψ 1(0)− ψ 2(0)k ≤e1 ε 2R0 −th2( ω ,t)dtk ψ 0,1− ψ 0,2k. c 2015? NSP Natural Sciences Publishing Cor. 4 T. Caraballo & R. Colucci: Effects noise on the dynamics of a parabolic equation 3 Existence of the random attractor In order to use Theorem 1to prove the existence of the random attractor, we prove the existence of a compact random absorbing set. We start by proving the existence of an absorbing ball in H. Proposition 3. There exists a random variable r0( ω )such that for any ρ >0there exists t( ω , ρ )<−1satisfying k ψ (0, ω ;t0, ψ 0)k≤r0( ω ), for all t0≤t( ω , ρ )and k ψ 0k≤ ρ . Proof. If we multiply the equation (5) by ψ we obtain the following inequality: 1 2 d dt k ψ k2+ ε 2k ψ xxk2+2[T( θ t ω )]2k ψ xk4 4 =2k ψ xk2+ σ z∗( θ t ω )k ψ k2 ≤(2+ σ z∗( θ t ω ))k ψ xk2 ≤(2+ σ z∗( θ t ω ))k ψ kk ψ xxk, from which 1 2 d dt k ψ k2+ ε 2 2k ψ xxk2≤(2+ σ z∗( θ t ω ))2k ψ k ≤(2+ σ z∗( θ t ω ))4 ε 2+ ε 2 4k ψ xxk2. Then d dtk ψ k2+ ε 2 2k ψ xxk2(7) ≤2[2+ σ z∗( θ t ω )]4 ε 2:=F( θ t ω ). Integrating the previousinequality in [t0,−1]witht0≤−1: k ψ (−1)k2≤ k ψ (t0)k2e− ε 2 2(−1−t0) +Z−1 t0 e− ε 2 2(−1−s)F( θ s ω )ds, from which k ψ (−1)k2≤e ε 2 2k ψ (t0)k2e ε 2 2t0+Z−1 t0 e ε 2 2sF( θ s ω )ds ≤e ε 2 2T−2( θ t0 ω )kX(t0)k2e ε 2 2t0 +Z−1 −∞e ε 2 2sF( θ s ω )ds. Now, if we fix the initial datum X0in B(0, ρ )⊂H, then there exists a time t( ω , ρ )such that for all t0≤t( ω , ρ ) and for all X0∈B(0, ρ )we have: k ψ (−1)k2≤r2 1( ω ), where r2 1( ω ) = e ε 2 21+Z−1 −∞e ε 2 2sF( θ s ω )ds. In detail, it is sufficient to chose t( ω , ρ )such that: T−2( θ t0 ω ) ρ 2e ε 2 2t0≤1, and this is possible since P-a.s. T−2( θ t0 ω )e ε 2 2t0→0 as t0→−∞. Using again (7), for all t∈[−1,0]we have that k ψ (t)k2≤k ψ (−1)k2e− ε 2 2(t+1)+Zt −1e− ε 2 2(t−s)F( θ s ω )ds, and then k ψ (0)k2≤r2 1( ω )e− ε 2 2+Z0 −1e ε 2 2sF( θ s ω )ds :=r2 0( ω ). We remark that, from Proposition 1, it follows that Z−1 −∞e2 ε 2sF( θ s ω )ds <∞. Remark. For any fixed ω ∈ Ω the time for which the absorbing property is fulfilled is of order O( ε −2), and this is in perfect accordance with the numerical experiments in [3] and with the autonomous case (see [11] and [12]). Integrating inequality (7), we deduce the following estimate that will be useful later Z0 −1k ψ k2dt ≤Z0 −1k ψ xk2dt ≤Z0 −1k ψ xxk2dt ≤1 2 ε 2r2 1( ω )+Z0 −1 F( θ t ω )dt:=R( ω ), while integrating in (t0,0)and letting t0→−∞we derive Z0 −∞k ψ xxk2dt ≤1 ε 2 ρ 2 2+Z0 −∞ F( θ t ω )dt. From the previous proposition we obtain the absorbing property in Hand we can pass to the proof of the absorbing property in H1 0(I):={v∈L2(I):vx∈L2(I)and v=0 on ∂ I}. Proposition 4. There exists a random variable r2( ω )such that for any ρ >0there exists t( ω , ρ )<−1satisfying k ψ (0, ω ;t0, ψ 0)kH1 0(I)≤r2( ω ), for all t0≤t( ω , ρ )and k ψ 0k≤ ρ . c 2015? NSP Natural Sciences Publishing Cor. Appl. Math. Inf. Sci. Vol?, No. No?, 1-9 (2015?) / www.naturalspublishing.com/Journals.asp 5 Proof. If we multiply equation (5) by ψ xx we obtain 1 2 d dt k ψ xk2+ ε 2k ψ xxxk2+6[T( θ t ω )]2ZI ψ 2 x ψ 2 xx =2k ψ xxk2+ σ z∗( θ t ω )k ψ xk2 ≤(2+ σ z∗( θ t ω ))k ψ xxk2,(8) from which d dtk ψ xk2≤2 σ z∗( θ t ω )+ 1 ε 2k ψ xk2,(9) and by the uniform Gronwall lemma k ψ x(0)k2≤R( ω )exp2Z0 −1 σ z∗( θ s ω )+ 1 ε 2ds :=r2 2( ω ). Before proving the absorbing property for the norm k ψ xxk, we perform some important estimates. We consider the following inequality Zt t+t0k ψ xk2ds ≤Zt t+t0k ψ xxk2ds ≤1 ε 2 ρ 2 2+Zt t+t0 F( θ s ω )ds, and apply the uniform Gronwall lemma in (t+t0,t)with t,t0≤0 : k ψ x(t)k2≤   1 ε 2n ρ 2 2+Rt t+t0F( θ s ω )dso −t0  × ×exp2Zt t+t0 σ z∗( θ s ω )+ 1 ε 2ds. From the Birkhof ergodic theorem (see Da Prato and Zabczyk [17, Chapter 1] and the stationarity of the Ornstein-Uhlenbeck process we have that lim t0→−∞−1 t0Zt t+t0|z∗( θ p ω )|dp =E(|z∗|). Then, there exists T0( ω )such that for all t0≤T0( ω )we have −1 t0Zt t+t0|z∗( θ p ω )|dp ≤2E(|z∗|). Moreover, by the same argument, there exists T1( ω )≤ T0( ω )such that for all t0≤T1( ω ): −1 t0Zt t+t0 F( θ s ω )ds =−1 t0Zt t+t0 2 ε 2(2+ σ z∗( θ s ω ))4ds ≤−1 t0Zt t+t0 16 ε 2[24+( σ z∗( θ s ω ))4]ds ≤28 ε 2+16 σ 4 ε 21 −t0Zt t+t0 (z∗( θ s ω ))4ds ≤28 ε 2+16 σ 4 ε 22E([z∗]4):=f( ω ). Then, k ψ x(t)k2≤ ρ 2 2 ε 2(−T1( ω )) +f( ω )e4 σ E(|z|)+ 2 ε 2T1( ω ) :=˜ R2( ω ). Additionally, integrating equation (8) in (−1,0), we have Z0 −1k ψ xxxk2dt ≤2 ε 2r2 1( ω )+2 σ M( ω )+ 2 ε 2R( ω ) :=R′( ω ), where M( ω ):=max t∈[−1,0][T( θ t ω )]4. Proposition 5. There exists a random variable r3( ω )such that, for any ρ >0, there exists a constant T( ω , ρ ):=min{t( ω , ρ ),T1( ω )}<−1satisfying k ψ xx(0, ω ;t0, ψ 0)k≤r3( ω ), for all t0≤T( ω , ρ )and k ψ 0k≤ ρ . Proof. In order to prove the absorbing property in Vwe multiply equation (5) by ψ xxxx: 1 2 d dtk ψ xxk2+ ε 2k ψ xxxxk2 =6[T( θ t ω )]2ZI ψ 2 x ψ xx ψ xxxxdx +2k ψ xxxk2+ σ z∗( θ tw)k ψ xxk2 ≤6[T( θ t ω )]2k ψ xk2 ∞k ψ xxkk ψ xxxxk+ ε 2 2k ψ xxxxk2 +2 ε 2k ψ xxk2+ σ z∗( θ tw)k ψ xxk2 ≤6[T( θ t ω )]2˜ Rk ψ xxk2k ψ xxxxk+ ε 2 2k ψ xxxxk2 +( 2 ε 2+ σ z∗( θ t ω ))k ψ xxk2, from which: d dtk ψ xxk2≤18 ε 2[T( θ t ω )]4k ψ xxk4(10) +2 ε 2+2 σ z∗( θ t( ω ))k ψ xxk2. If we integrate equation (10) in (s,0)for s∈(−1,0)we have k ψ xx(0)k2≤ k ψ xx(s)k2+18 ε 2Z0 s[T( θ t ω )]4k ψ xxk4dt +Z0 s2 ε 2+2 σ z∗( θ t( ω ))k ψ xxk2dt. c 2015? NSP Natural Sciences Publishing Cor. 6 T. Caraballo & R. Colucci: Effects noise on the dynamics of a parabolic equation Integrating again with respect to sin (−1,0)we obtain k ψ xx(0)k2≤Z0 −1k ψ xx(s)k2dsx +18 ε 2Z0 −1[T( θ s ω )]4k ψ xxk4ds +Z0 −12 ε 2+2 σ z∗( θ s(w))k ψ xxk2ds ≤R( ω )+ 18M( ω ) ε 2˜ R2( ω )R′( ω ) +2 ε 2+2 σ max s∈(−1,0)|z∗( θ s ω )|R( ω ) :=r2 3( ω ), where we have used Z0 −1k ψ xxk4ds ≤Z0 −1k ψ xk2k ψ xxxk2ds ≤˜ R2( ω )Z0 −1k ψ xxxk2ds ≤R2( ω )R′( ω ). From the above propositions we get that there exists a random ball B( ω )in L2(I)∩H1 0(I)∩H2(I)which absorbs any bounded non random subset of Hat time 0 for any t0≤min{T1( ω ),T0( ω ),t( ω , ρ )}. The compactness of B( ω )in L2(I)follows from the compact embedding of H1 0(I)in L2(I). Then, by Theorem 1, we conclude that Theorem 3. The equation (5)admits a random attractor A ε ( ω )in H. 4 Additive noise In this final section we consider some remarks on the stochastic version of (1) with additive noise and compare the results with that of multiplicative noise. The system takes the form:    dX = (− ε 2△2X+1 2W′′(∇X)△X)dt + σφ (x)dw(t), X=Xxx =0,on ∂ I, X(t0,x) = X0(x),(11) where σ >0 and φ ∈Hsatisfies φ xxxx ∈L∞(I). As in the previous section we consider a transformation involving the Orstein-Uhlenbeck process: v(t) = X(t)− σ z∗( θ t ω ) φ . Then the equation can be rewritten as vt+ ε 2vxxxx =1 2W′(vx+ σφ xz∗( θ t ω ))x + σ z∗( θ t ω )g(x),(12) where for simplicity we have set: g(x):=g(x, φ , ε , σ ) = φ (x)− ε 2 φ xxxx(x). Once at this point, it is possible to establish a result ensuring the existence and uniqueness of solution of the initial value problems associated to (12), as well as the continuity with respect to the initial data. However, we will only comment on the main differences between this case and the multiplicative noise one. Thus, we will show how to obtain inequalities concerning kvk,kvxkand kvxxk similar to those of the previous sections. We note that obtaining the estimates requires more technicalities in our computations. We start with the absorption in L2(I). Multiplying equation (12) by vin L2(I)we obtain 1 2 d dtkvk2+ ε 2kvxxk2 =1 2ZIW′′(vx+ σ z∗( θ t ω ) φ x)(vxx + σφ xxz∗( θ t ω ))vdx + σ z∗( θ t ω )ZIgvdx, and, in more details, 1 2 d dtkvk2+ ε 2kvxxk2+2kvxk4 4 =6 σ YZIv2 xv φ xxdx+6 σ 2Y2ZIvxxv φ 2 xdx +6 σ 3Y3ZIv φ 2 x φ xxdx +12 σ YZIvxxvxv φ xdx +12 σ 2Y2ZIvxv φ xdx+ σ YZIgvdx, where we have set, for notational simplicity,Y:=z∗( θ t ω ). We observe that using integration by parts we have: 6 σ YZIv2 xv φ xxdx+12 σ YZIvxxvxv φ xdx =−6 σ YZIv3 x φ xdx ≤6 σ |Y|k φ xk∞kvxk3 3 ≤kvxk4 4+9 σ 2Y2k φ xk2kvxk2 ≤kvxk4 4+9 σ 2Y2k φ xk2kvkkvxxk ≤kvxk4 4+ ε 2 4kvxxk2+ σ 4Y4 ε 2k φ xk4 ∞kvk2. Then, 1 2 d dtkvk2+ ε 23 4kvxxk2+kvxk4 4 ≤ σ 4Y4 ε 2k φ xk4 ∞kvk2+6 σ 2Y2kvxxkkvkk φ xk2 ∞ +6 σ 3|Y3|k φ xk2 ∞k φ xxkkvk +12 σ 2Y2kvkkvxkk φ xk∞+ σ |Y|kgkkvk, c 2015? NSP Natural Sciences Publishing Cor. Appl. Math. Inf. Sci. Vol?, No. No?, 1-9 (2015?) / www.naturalspublishing.com/Journals.asp 7 from which 1 2 d dt kvk2+ ε 2 2kvxxk2+kvxk4 4≤f1(t)kvxk2+f2(t), where f1(t):=2+12 σ 2Y2k φ xk∞+37 σ 4Y4 ε 2k φ xk4 ∞, and f2(t):= σ 2Y4 4kgk2+9 σ 6Y6k φ xk4 ∞k φ xxk. Finally, using kvxk2≤|I|1 2kvxk2 4, we conclude that d dtkvk2+ ε 2kvxxk2≤h(t), where h(t):=2|I| 4f2 1(t)+ f2(t). Thus, we have obtained a similar inequality to the one in the previous section and, for this reason, we do not repeat the details while we pass to obtain the inequality for kvxk. By multiplying equation (12) by vxx in L2(I)we arrive at 1 2 d dt kvxk2+ ε 2kvxxxk2 +1 2ZIW′′(vx+ σφ xz∗( θ t ω ))(vxx + σ z∗( θ t ω ) φ xx)vxxdx + σ z∗( θ t ω )ZIgvxxdx =0. In details 1 2ZIW′′(vx+ σφ xz∗( θ t ω ))(vxx + σ z∗( θ t ω ) φ xx)vxxdx =6kvxvxxk2+6 σ z∗( θ t ω )ZIv2 xvxx φ xxdx +6 σ 2[z∗( θ t ω )]2ZI φ 2 xv2 xxdx +6 σ 3[z∗( θ t ω )]3ZI φ 2 x φ xxvxxdx +12 σ [z∗( θ t ω )]ZIvxv2 xx φ xdx +12 σ 2[z∗( θ t ω )]2ZIvxvxx φ x φ xxdx −2kvxxk2−2 σ z∗( θ t ω )ZI φ xxvxxdx. Then 1 2 d dt kvxk2+ ε 2kvxxxk2+6kvxvxxk2 ≤ σ |Y|kgkkvxxk+6 σ |Y|kk φ xxk∞kvxkkvxvxxk +6 σ 3|Y3|k φ xk2 ∞k φ xxkkvxxk +12 σ |Y|k φ xk∞kvxxkkvxvxxk +12 σ 2Y2k φ xk∞k φ xxk∞kvxkkvxxk +2kvxxk2+2 σ |Y|k φ xxkkvxxk. Simply using 2ab ≤a2+b2we obtain 1 2 d dtkvxk2+ ε 2kvxxxk2≤f3(t)kvxxk2+f4(t), where f3(t):=5+9 σ 2Y2k φ xxk2 ∞+36 σ 2Y2k φ xk2 ∞ +12 σ 2Y2k φ xk∞k φ xxk∞ and f4(t):= σ 2Y2 4kgk2+ σ 2Y2k φ xxk2+9 σ 6Y6k φ xk4 ∞k φ xxk2. Thus we conclude d dtkvxk2≤2f2 3(t) ε 2kvxk2+f4(t). We note that the use of the uniform Gronwall lemma is possible using similar arguments to those of the previous section such as the ergodic theorem. We conclude with the inequality involving kvxxk. Multiplying equation (12) by vxxxx in H, 1 2 d dtkvxxk2+ ε 2kvxxxxk2 =1 2ZIW′′(vx+ σ z∗( θ t ω ) φ x)(vxx + σ z∗( θ t ω ) φ xx)vxxxxdx + σ z∗( θ t ω )ZIgxxvxxdx. The term involving the potential can be written in details: 6ZIv2 xvxxvxxxxdx+6 σ z∗( θ t ω )ZIv2 x φ xx φ xxxxdx +6 σ 2[z∗( θ t ω )]2ZI φ 2 xvxxvxxxxdx +6 σ 3[z∗( θ t ω )]3ZI φ 2 x φ xxvxxxxdx +12 σ z∗( θ t ω )ZI φ xvxvxxvxxxxdx +12 σ 2[z∗( θ t ω )]2ZI φ x φ xxvxvxxxxdx −2ZIvxxvxxxxdx−2 σ z∗( θ t ω )ZI φ xxvxxxxdx, and can be estimated by 6kvxk2 ∞kvxxkkvxxxxk+6 σ Yk φ xxkk φ xxxxk∞kvxxk2 +6 σ 2Y2k φ xk2 ∞kvxxkkvxxxxk +6 σ 3Y3k φ xk2 ∞k φ xxkkvxxxxk +12 σ Yk φ xk∞kvxk∞kvxxkkvxxxxk +12 σ 2Y2k φ xk∞k φ xxk∞kvxxkkvxxxxxk +2kvxxxk2+2 σ Yk φ xxkkvxxxxk. Then, by using the estimate in H1 0(I), that is kvxk≤Rx, the interpolating inequalities kuxk∞≤kuxk1/2kuxxk1/2,kvxxxk2≤kvxxkkvxxxxk, c 2015? NSP Natural Sciences Publishing Cor. 8 T. Caraballo & R. Colucci: Effects noise on the dynamics of a parabolic equation and 2ab ≤a2+b2, we have d dtkvxxk2≤f5(t)kvxxk4+f6(t)kvxxk2+f7(t), where f5(t) = 144R2 x ε 21+4 σ 2Y2k φ xk2 ∞, f6(t) = 21+6 σ Yk φ xxkk φ xxxxk∞+72 ε 2 σ 4Y4k φ xk4 ∞ +288 ε 2 σ 4Y4k φ xk2 ∞k φ xxk2 ∞+8 ε 2, and f7(t) = 2 σ 2Y2 4kgxxk2+72 ε 2 σ 6Y6k φ xk4 ∞k φ xxk2 +8 σ 2Y2 ε 2k φ xxk2. Again we have obtained a similar inequality to that of the case of multiplicative noise. Then, it is not difficult to obtain the existence of the random attractor also in this additive case. Acknowledgements. We thank the referee for the interesting and helpful suggestions which allowed us to improve the presentation of this paper. This work has been partially supported by FEDER and the Spanish Ministerio de Econom´ıa y Competitividad project MTM2011-22411 and the Consejer´ıa de Innovaci´on, Ciencia y Empresa (Junta de Andaluc´ıa) under grant 2010/FQM314 and Proyecto de Excelencia P12-FQM-1492 References [1] G. Alberti, S. 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Schmalfuß, Backward cocycle and attractors of stochastic differential equations, in V. Reitmann, T. Redrich and N. J. Kosch (eds.), International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behaviour (1992), 185-192. c 2015? NSP Natural Sciences Publishing Cor. Appl. Math. Inf. Sci. Vol?, No. No?, 1-9 (2015?) / www.naturalspublishing.com/Journals.asp 9 [25] M. Slemrod, Dynamics of measured valued solutions to a backward-forward heat equation, Journal of Dynamics and Differential Equations 3 (1991), 1–28. [26] R. Temam, Infinite Dimensonal Dynamical Systems in Mechanics and Physics. Springer; 2nd edition, 1997. Tom´ as Caraballo is Professor of Mathematical Analysis at the Universidad de Sevilla (Spain). He leads the research group Stochastic Analysis of Differential Systems and his main research interests are in the area of dynamical systems, mainly in non-autonomous and stochastic dynamics, including aspects of stochastic partial differential equations (e.g. existence, uniqueness, stability and related topics), asymptotic behaviour of dynamical systems perturbed by random terms and multivalued dynamical systems. Renato Colucci is Lecturer at the department of Mathematical Sciences of Xi’an Jiatong-Liverpool University of Suzhou (China). His research interests are in the area of Dynamical systems, Calculus of variations and Differential Equations. He is particularly interested in applications arising from natural, social sciences and arts. c 2015? NSP Natural Sciences Publishing Cor.