RANDOM ATTRACTOR OF STOCHASTIC BRUSSELATOR SYSTEM WITH MULTIPLICATIVE NOISE

. Asymptotic dynamics of stochastic Brusselator system with multiplicative noise is investigated in this work. The existence of random attractor is proved via the exponential transformation of Ornstein-Uhlenbeck process and some challenging estimates. The proof of pullback asymptotic compact- ness here is more rigorous through the bootstrap pullback estimations than a non-dynamical substitution of Brownian motion by its backward translation. It is also shown that the random attractor has the attracting regularity to be an ( L 2 × L 2 ,H 1 × H 1 ) random attractor.

{W (t)} t∈R is a one-dimensional, two-sided standard Wiener process (Brownian motion) on a probability space which will be specified later. The terms ρu • dW (t) and ρv • dW (t) indicate that the stochastic PDEs (1.1)-(1.2) are interpreted as the corresponding stochastic integral equations in the Stratonovich sense. The original Brusselator equations were proposed in [11] as a system of ODEs and the diffusive Brusselator equations have been used as a typical mathematical model for morphogenesis and trimolecular autocatalytic reactions in physical chemistry and mathematical biology, cf. [9] and the references in [16].
The concept of random attractor for random dynamical system was first introduced in [7,8] in the study of the asymptotic dynamics of Navier-Stokes equations and other PDEs with multiplicative and additive white noise. The fundamental results on random dynamical systems and related topics have been summarized in [1].
When dealing with random dynamics and the existence of random attractor for stochastic partial differential equations with multiplicative noise, we usually transform the stochastic PDEs into deterministic ones with random parameters and random initial data through the exponential transformation of Brownian motion. In this work, however, we take the approach of the exponential transformation of the Ornstein-Uhlenbeck process. This transformation does change the structure of the original equations and produces the non-autonomous terms, cf. (2.10) and (2.11). It demands more challenging and sophisticated pullback a priori estimates, other than the non-dynamical substitution of ω by θ −t ω as in some other publications.
Another notable aspect is that the Brusselator reaction-diffusion system does not satisfy the usual dissipative condition, cf. [16], and the bootstrap method used in the paper is also different from the decomposition method for the deterministic global attractors in [16]. The result in this work shows that the approach of the Ornstein-Uhlenbeck transformation unifies the treatment of the stochastic PDEs with multiplicative noise and with additive noise in regard to pullback asymptotic dynamics, especially for the reaction-diffusion systems with some sort of weak and hidden dissipativity.
In this paper the initial data or solutions are not restricted to be nonnegative and there are no restrictions on any of the positive parameters in the equations (1.1)- (1.2).
The rest of the paper is organized as follows. In Section 2 we present preliminary concepts on random dynamical system and random attractors. In Section 3 we prove the pullback absorbing property of the Brusselator random dynamical system. In Section 4 we show the pullback asymptotic compactness. In Sections 5 we reach the main results on the existence of a random attractor and its L 2 × L 2 to H 1 × H 1 attracting regularity.
2. Preliminaries and formulation. In this section, we recall the concepts of random dynamical system and random attractor. We refer to [1,3,6,7] for more details. Let (X, · X ) be a real separable Banach space with Borel σ-algebra B(X) and let (Ω, F, P ) be a probability space. R + = [0, ∞).
for all s, t ∈ R and θ t P = P for all t ∈ R on Ω.
Definition 2.4. A random set in X is a set-valued function B(ω) : Ω → 2 X whose graph {(ω, x) : x ∈ B(ω)} ⊂ Ω×X is an element of the product σ-algebra F×B(X). A bounded random set B(ω) ⊂ X means that there is a random variable r(ω) ∈ R + such that ||B(ω)|| := sup x∈B(ω) ||x|| ≤ r(ω) for all ω ∈ Ω. A random set B(ω) is called compact (respectively precompact) if for each ω ∈ Ω the set B(ω) is compact (respectively precompact) in X. A bounded random set is called tempered with respect to (θ t ) t∈R on (Ω, F, P ), if for each ω and for any constant c > 0, In this case, the collection D is called a universe. In the paper, we define D to be the universe of all the tempered random sets in a specified phase space X. Note that all bounded non-random sets are included in D.
Definition 2.5. Let D be a collection of random subsets of X. A random set K ∈ D is called a D-pullback absorbing set with respect to an RDS ϕ over the MDS (Ω, F, P, {θ t } t∈R ), if for any ω ∈ Ω and any bounded set B(ω) ∈ D there exists a finite time t B (ω) > 0 such that Definition 2.6. Let D be a collection of random subsets of X. Then an RDS ϕ is D-pullback asymptotically compact in X if for each ω ∈ Ω, {ϕ(t n , θ −tn ω, x n )} ∞ n=1 has a convergent subsequence in X whenever t n → ∞, and x n ∈ B(θ −tn ω) for any given B ∈ D.
Definition 2.7. Let a universe D of tempered random sets in a Banach space X be given. A random set A ∈ D is called a random attractor in D for a given RDS ϕ on X over the MDS (Ω, F, P, {θ t } t∈R ), if the following conditions are satisfied for each ω ∈ Ω: (i) A is a compact random set.
(ii) A is invariant in the sense that ϕ(t, ω, A(ω)) = A(θ t ω), ∀ t ≥ 0; (iii) A attracts every set B ∈ D in the pullback sense that for every ω ∈ Ω one has lim where the Hausdorff semi-distance is given by dist X (Y, Z) = sup y∈Y inf z∈Z y−z X for subsets Y and Z in X.
We have the following proposition on the existence of random attractor due to Crauel and Flandoli [7,Theorem 3.11].
Proposition 2.8. Given a Banach space X and a collection D of random sets of X, let ϕ be a continuous RDS on X over an MDS (Ω, F, P, {θ t } t∈R ). Suppose that there exists a closed pullback absorbing set {K(ω)} ω∈Ω ∈ D and ϕ is pullback asymptotically compact with respect to D, then the RDS ϕ has a unique random attractor A = {A(ω)} ω∈Ω ∈ D whose basin is D and given by

Define the product Hilbert spaces
The norm and inner-product of H or L 2 (Γ) will be denoted by · and ·, · , respectively. The norm of L p (Γ) or the product space L p (Γ) = [L p (Γ)] 2 will be denoted by · L p if p = 2. By the Poincaré inequality and the homogeneous Dirichlet boundary condition (1.3), there is a constant λ > 0 such that and we take ∇ξ to be the equivalent norm of the space E or the space H 1 0 (Γ). We shall use | · | to denote the Lebesgue measure or a vector norm in a Euclidean space.
The linear sectorial operator is the generator of an analytic C 0 -semigroup on the Hilbert space H, cf. [14]. By Sobolev embedding theorem, H 1 0 (Γ) → L 6 (Γ) is a continuous embedding for n ≤ 3. Thus there is a positive constant ζ > 0 associated with the Sobolev imbedding inequality ϕ L 6 (Γ) ≤ ζ ϕ E = ζ ∇ϕ , for any ϕ ∈ E. (2.3) By Hölder inequality we have for u, v ∈ L 6 (Γ), and the nonlinear mapping is a locally Lipschitz continuous mapping defined on E.
Let {W (t)} t∈R be the standard one-dimensional two-sided Wiener process in the probability space (Ω, F, P ), where the σ-algebra F is generated by the compact-open topology on Ω, and P is the corresponding Wiener measure on F. The shift mapping θ t is defined by Then (Ω, F, P, {θ t } t∈R ) is the canonical MDS and the stochastic process {W (t, ω) = ω(t) : t ∈ R, ω ∈ Ω} is the canonical Wiener process (Brownian motion).
Consider the Ornstein-Uhlenbeck process which solves the linear stochastic differential equation The following proposition is quoted from [3].
Proposition 2.9. Let the metric dynamical system (Ω, F, P, θ t ) and the Ornstein-Uhlenbeck process {z(θ t ω)} t∈R be defined as above. Then there is a θ t -invariant set Ω ∈ Ω of full P -measure such that for every ω ∈Ω, the following statements hold.
In the sequel we consider ω ∈Ω only and will always write Ω forΩ.
As the main approach to investigating the random dynamics of stochastic PDEs, we convert the stochastic Brusselator system (1.1)-(1.2) to a system of pathwise PDEs with the random parameter ω(t) and random initial data. Make the transformation where z(θ t ω) is the Ornstein-Uhlenbeck process in (2.5). Then In view of the equation (2.6), dz+zdt = dW (t), the system (1.1)-(1.2) is transformed by (2.9) to the following random PDE problem: for ω ∈ Ω, x ∈ Γ and t > t 0 , with the homogeneous Dirichlet boundary condition 12) and the initial condition at t = t 0 ∈ R, For every ω ∈ Ω, the problem (2.10)-(2.13) of the pathwise nonautonomous partial differential equations can be written as where g(t, ω; t 0 , g 0 ) = (U (t, ω; t 0 , U 0 ), V (t, ω; t 0 , V 0 )) T and By conducting a priori estimates on the Galerkin approximations of the initial value problem (2.14) and the compactness argument, c.f. [5], but with the extra care on the non-autonomous terms from the random noise, we can prove the local existence and uniqueness of the weak solution g(t, ω; t 0 , g 0 ), t ∈ [t 0 , T (ω, g 0 )] for some T (ω, g 0 ) > t 0 , which depends continuously on the initial data.
By the parabolic regularity [14,Theorem 48.5], every weak solution turns out to be a strong solution for t > t 0 in the existence interval. Similar to Lemma 1.2 in [16], every weak solution g(t, ω; t 0 , g 0 ) of (2.14) on the maximal interval of existence has the property Below we shall study the global existence and the asymptotic dynamics of the weak solutions of the problem (2.14).
Moreover, for terminal time t ∈ [−4, 0] when t 0 ≤ min{T (R, ω), −4}, there exists a random variable M (t, ω) independent of initial data such that the weak solution satisfies Proof. Taking the inner product of (2.11) with V (t, ω), we get It follows that, in the maximal interval of existence [t 0 , T max ), Multiplying the above inequality by e t t 0 (2ρz(θsω)−2λd2)ds and then integrating it over The next step is key to the pullback estimates. We will get rid of the dependence on the initial time and data by the asymptotic decay of the Ornstein-Uhlenbeck process. The arguments go as follows.
in which g(0, ω; −t, g 0 ), t ≥ 0, can be called the pullback quasi-trajectory from g 0 , which is not a trajectory but the terminal values at time t = 0 of the bunch of weak solutions g(0, ω; −t, g 0 ) starting from g 0 more and more backward at time −t. We shall deal with the pullback quasi-trajectories to investigate the pullback asymptotic behavior of the Brusselator random dynamical system ϕ. Proof. This is a direct consequence of Lemma 3.1 and the characterization of the pullback quasi-trajectories of this Brusselator RDS ϕ. Note that M 0 (ω) is a tempered random variable and B 0 ∈ D.
Furthermore we show the pullback absorbing property of the V -component of the random Brusselator system (2.10)-(2.13) in the Banach space L 6 (Γ). This is a key step to pave the way toward the proof of the pullback asymptotic compactness in the next section. Lemma 3.3. For any given initial data (u 0 , v 0 ) ∈ E and terminal time t ∈ [−4, 0], there exists a random time T 6 ( (u 0 , v 0 ) L 6 , ω) ≤ −4 and a positive random variable P (t, ω) such that for any initial time t 0 ≤ T 6 ( (u 0 , v 0 ) L 6 , ω) we have Proof. Taking the inner product of (2.11) with V 3 , we obtain (3.27) Next we use the bootstrap method to take the inner product of (2.11) with V 5 and obtain Then there is a random variable T 6 ( g 0 L 6 , ω) ≤ −4 such that for every ω ∈ Ω, t 0 ≤ T 6 ( g 0 L 6 , ω) and t ∈ [−4, 0], we have Note that the three improper integrals above are convergent by the similar calculations as shown in (3.7) and (3.8). The proof is completed.
The next lemma is instrumental to the proof of pullback asymptotic compactness in the next section.

4.
Pullback asymptotic compactness. In this section, we show that the Brusselator random dynamical system ϕ is pullback asymptotically compact in H by the following uniform Gronwall inequality, cf. [14].
Next we show that the Brusselator random dynamical system ϕ possesses the flattening property in the more regular space E. The first condition of the flattening property is clearly satisfied due to Lemma 4.2. To prove the second condition, we decompose the weak solution g(t, ω; t 0 , g 0 ) of the random Brusselator reactiondiffusion system (2.10)-(2.13) as the high modes and the low modes, g(t, ω) = g 1 (t, ω) + g 2 (t, ω), where g 1 = Q n g, g 2 = (I − Q n )g.