Random uniform exponential attractor for stochastic non-autonomous reaction-diffusion equation with multiplicative noise in ℝ3

Abstract We first introduce the concept of the random uniform exponential attractor for a jointly continuous non-autonomous random dynamical system (NRDS) and give a theorem on the existence of the random uniform exponential attractor for a jointly continuous NRDS. Then we study the existence of the random uniform exponential attractor for reaction-diffusion equation with quasi-periodic external force and multiplicative noise in ℝ3.


Introduction
The concept of the exponential attractor was introduced by A. Eden et al., which is a compact positively invariant set with nite fractal dimension and attracts trajectories exponentially fast, see [1]. It can describe the asymptotic behavior of trajectories of autonomous dynamical system or the solutions to dissipative autonomous evolution equations. In contrast to a global attractor, the exponential attractor has nite fractal dimension, if it exists, the asymptotic behavior of in nite dimensional dynamical systems can be characterized by the dynamics on the nite dimensional compact set (i.e., exponential attractor). Besides, exponential attractors are stable under perturbation because of the exponential rate of convergence of trajectories to it. We should note that an exponential attractor is not necessarily unique since it is not invariant, and includes a global attractor in general.
By the notion of pullback attraction, the concept of the exponential attractor can be extended to the case of non-autonomous dynamical system, called pullback exponential attractor, see [2][3][4][5][6] and the references therein. An alternative extension to the case of non-autonomous dynamical system of the concept of the exponential attractor was based on the work [7] of Chepyzhov and Vishik (see also [8,Chapter 4]), in which they introduced an approach to study a family of non-autonomous evolution equations of the form where u ∈ E (Banach space) and Σ is an appropriate compact symbol space. More precisely, they constructed a semigroup (skew-product semi ow) associated with (1) on extended phase space Σ × E and used the theory of semigroup to study the longtime behavior of solutions of non-autonomous evolution equations. Newly extended attractors for non-autonomous dynamical system are called uniform exponential attractors, see [9][10][11][12][13] and the references therein. By its de nition, a uniform exponential attractor is time independent exponentially attracting compact set and has nite fractal dimension. Because we regard extended phase space Σ × E as a whole, symbol space Σ require to be compact and nite dimensional when we use semigroup on Σ × E to study the existence of a uniform exponential attractor of non-autonomous evolution equations, this is why we choose k-dimensional torus T k , k ∈ Z+ (corresponding to the hull of quasi-periodic functions, see [8]) as the symbol space. Our aim in this article is to extend the uniform exponential attractor to the random uniform exponential attractor for a jointly continuous non-autonomous random dynamical system, and give a theorem on the existence of a random uniform exponential attractor for a jointly continuous NRDS. By de nition, a random uniform exponential attractor is a random compact set with nite fractal dimension, which (pullback) attracts uniformly every element of attraction universe D with exponential rate. We emphasize that the attraction universe considered in the de nition is autonomous attraction universe (it contains only autonomous random set, see [14,15]). We also emphasize that the random uniform exponential attractor has no (positive) invariance property along the sample path. According to [16], we have a criterion for the existence of a random exponential attractor for a continuous random dynamical system on a separable Hilbert space. Other results on existence criteria of a random exponential attractor for a random dynamical system can be found in [17,18]. With the help of the concept of skew-product cocycle (i.e., random dynamical system) introduced in [15], we consider the existence of the random exponential attractor for a continuous skew-product cocycle (generated by a jointly continuous NRDS ϕ and a base ow θ) on the extended phase space T k ×E, and project this random exponential attractor onto the phase space E. Then we obtain the random uniform exponential attractor for the jointly continuous NRDS ϕ. Consequently, we formulate Theorem 2.8 on the existence of a random uniform exponential attractor for a jointly continuous NRDS.
As an application of Theorem 2.8, we will consider the following reaction-di usion equation with quasiperiodic external force and multiplicative noise on R , where u = u(x, t) is real-valued functions de ned on R × [ , +∞), the coe cients λ, b are positive constants. W(t) is a two-sided real-valued Brownian motion on a probability space which will be speci ed later. The symbol " • " means that the stochastic integration in system is in the Stratonovich sense. σ(t) = (xt + σ)mod(T k ), where σ ∈ T k , x = (x , . . . , x k ) ∈ R k is a xed vector satisfying that x , . . . , x k are rationally independent. The functions g, f are assumed to satisfy some conditions. There have been many works concerning random attractors for stochastic reaction-di usion equation, see [19,Introduction] for detailed summary. It is worth noting that Zhou [19] considered the existence of random exponential attractor for non-autonomous stochastic reaction-di usion equation with multiplicative noise in R . In the setting of f , g in [19], we here further assume that the external force g is quasi-periodic and prove the existence of a random uniform exponential attractor. This paper is organized as follows. In the next section, we show some preliminaries and give a theorem on the existence of a random uniform exponential attractor for a jointly continuous NRDS. In section , we study the existence of a random uniform exponential attractor for the non-autonomous stochastic reactiondi usion equation (2) de ned on R .

Existence of random uniform exponential attractors
In this section, we present some notations and provide a criterion concerning the existence of a random uniform exponential attractor for a jointly continuous NRDS.
A RDS is said to be continuous if for each t ∈ R + , ω ∈ Ω, the mapping ψ(t, ω, ·) is continuous.
A NRDS is said to be continuous if for each t ∈ R + , ω ∈ Ω and σ ∈ T k , the mapping ϕ(t, ω, σ, ·) is continuous. It is called jointly continuous in T k and X if the mapping ϕ(t, ω, ·, ·) is continuous for each t ∈ R + and ω ∈ Ω. We obtain the general de nition of NRDS by replacing torus T k with general symbol space Σ in De nition 2.2 (see [15, De nition 2.1]).

De nition 2.3. A (autonomous) random set D(·) in X is a multi-valued map D
: Ω → X \ ∅ such that for each x ∈ X the map ω → d X (x, D(ω)) is measurable. It is said that the (autonomous) random set is bounded (resp. closed or compact) if D(ω) is bounded (resp. closed or compact) for a.e. ω ∈ Ω.
We often write D(·) as D or {D(ω)} ω∈Ω . Given two random sets D , Hereafter, we denote by D(X) the collection of all tempered bounded random subset of X. For simplicity, we identify " a.e. ω ∈ Ω" and "ω ∈ Ω" unless otherwise stated.
De ne the extended space X . = T k × X with norm: and Borel σ-algebra B(X).
Then P X is the projection from X to X. Denote by P T k the projection from X to T k .
De nition 2.5 (see [15]). A set-valued mapping B(·) : Ω → X \∅ is called a random set in X if for each X ∈ X the mapping ω → d X (X, B(ω)) is (F, B(R + ))-measurable. If, moreover, B satis es and D X = B : B is a proper random set in X then D ,X ⊂ D X ⊂ D ,X and for any element B ∈ D X , there exist an element B ∈ D ,X such that B ⊆ B . For K-dimensional subspace X K of X (K ∈ N), we de ne the bounded projections P k+K : X → X K = T k ×X K and Q k+K : where P K : X → X K is K-dimensional orthogonal projection from X into X K , Q K = I X −P K and Q k+K = I X −P k+K , where I X is the identity operator on X.
The RDS π is called the skew-product cocycle generated by ϕ and θ. Note that π is continuous, that is, the mapping X → π(·, ·, X) is continuous in X, if and only if ϕ is jointly continuous in T k and X. Now we de ne the random uniform exponential attractor for continuous NRDS {ϕ(t, ω, σ)} t≥ ,ω∈Ω,σ∈T k on a separable Hilbert space X.
De nition 2.6. A random set {M(ω)} ω∈Ω in X is called a D(X)-random uniform exponential attractor for the continuous NRDS {ϕ(t, ω, σ)} t≥ ,ω∈Ω,σ∈T k on X if there is a set of full measureΩ ∈ F such that for every ω ∈Ω, it holds that Remark 2.7. By de nition the random uniform exponential attractor has no (positive) invariance property along the sample path.

Theorem 2.8. Assume that conditions (A )-(A ) hold. Then the continuous skew-product cocycle π acting on X generated by jointly continuous NRDS
Proof. The existence of D X -random exponential attractor E for {π(t, ω)} t≥ ,ω∈Ω follows from Theorem 2.1 in [16]. We claim that , A(ω)) > , ω ∈ I, this is contrary to the exponential attraction of exponential attractor E. In other words, there exist a subset I of Ω such that P(I) > and the exponential attraction of exponential attractor E fails to hold for ω ∈ I, which contradicts the de nition of exponential attractor; The compactness and measurable of P X E follows from E directly.
show the uniform exponential attraction of P X E with respect to D(X). Note that for each x ∈ X, σ ∈ T k , ω ∈ Ω, and t ≥ , we have which indicates the uniform exponential attraction of P X E. The proof is complete.

. Application to stochastic reaction-di usion equation
In this section, we apply Theorem 2.8 to the reaction-di usion equation with quasi-periodic external force and multiplicative noise on R . Namely, we consider the equation (2). The functions g, f are assumed to satisfy the following conditions: (H ) There exist real positive constants c , c , c > and integral functions β ∈ L (R , Hereafter, let (·, ·), · and (·, ·) , · denote the inner products and norms of L (R ) and H (R ), respectively, where

. Setting of the problem
In the sequel, we will use the probability space (Ω, F, P), where F is the Borel σ-algebra on Ω generated by the open compact topology, and P represents the Wiener measure on F. The Brownian motion has a realization then P is ergodic and invariant under ϑ (see [20,21]). It is known that z(ϑ t ω) = − −∞ e s (ϑ t ω)(s)ds (t ∈ R) is a stationary solution of one-dimensional equation dz + zdt = dW(t). From [22], we know that for ω ∈ Ω, t → z(ϑ t ω) is continuous in t and Then (2) is equivalent to the following system with random coe cients We known from [19] for )-measurable in ω and continuous in σ and v . Thus the mapping of solutions generates a NRDS ϕ : , which is continuous both in initial value and symbols. From now on, let D = D(L (R )) be the collection of all tempered bounded random sets of L (R ), i.e., We will prove the existence of D-random uniform exponential attractor of ϕ in this paper.

Lemma 2.9.
For every D ∈ D and ω ∈ Ω, there exist T = T (ω, D) ≥ and a tempered random variable Proof. Taking the inner product of (19) with v(r), we have for r ≥ , (17)) it follows that for r ≥ , where c = λ g + β L . Applying Gronwall inequality to (20) Let hold uniformly for σ ∈ T k .
holds uniformly for σ ∈ T k .
In order to prove the existence of a random uniform exponential attractor for ϕ, we need to prove the existence of a random exponential attractor for π. Consequently, we next present B satisfy (A ), (A ) in Theorem 2.8.

. . Decomposition of solution
It is known that there are a family of eigenfunctions {ẽ m,R } m∈N , which form an orthonormal base of L (U R ) and H (U R ), and a family of eigenvalues {µ m,R } m∈N such that < µ ,R ≤ µ ,R ≤ · · · ≤ µ m,R ≤ · · · , µ m,R → +∞ as m → +∞ .

. . The boundedness of expectation
We next check the boundedness of expectation, we need the following lemma.
Lemma 2.18 (see [24,25] and let ε be small enough, R > be large enough such that Evidently, we can choose ε = ε small enough and R = R ≥ big enough such that For xed ε , R , by µ m+ ,R → ∞, there exists a m = m big enough such that H(ε , λ + µ m + ,R ) ≤ η .