Random exponential attractor for a stochastic reaction-diffusion equation in L 2 p ( D )

In this paper, we establish some suﬃcient conditions for the existence of a random exponential attractor for a random dynamical system in a Banach space. As an application, we consider a stochastic reaction-diﬀusion equation with multiplicative noise. We show that the random dynamical system φ ( t , ω ) generated by this stochastic reaction-diﬀusion equation is uniformly Fréchet diﬀerentiable on a positively invariant random set in L 2 p ( D ) and satisﬁes the conditions of the abstract result, then we obtain the existence of a random exponential attractor in L 2 p ( D ), where p is the growth of the nonlinearity satisfying 1 < p ≤ 3.


Introduction
As we know, the random attractor plays a key role in the study of asymptotic behavior of a random dynamical system (RDS).Both the existence and the estimates of Hausdorff and fractal dimensions of random attractors have been studied intensively since Crauel and Flandoli 1994 [1], see, e.g., [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] and the references therein.However, a random attractor is possibly infinite dimensional and sometimes attracts orbits at a slow rate, making it unobservable in practical experiments and numerical simulations.The concept of random exponential attractor was introduced by Shrikyan and Zelik in [17].By definition, a random exponential attractor is a positively invariant finite dimensional set that contains the random attractor and possesses the exponential attraction property.
In [17], Shrikyan and Zelik presented some sufficient conditions for the existence and robustness of random exponential attractors for dissipative RDS.As pointed out there, the main difficulty in constructing a random exponential attractor, in contrast to the deterministic case, is that a typical trajectory of an RDS is unbounded in time.Therefore, some restrictive assumptions were imposed on the global Lipschitz continuity of all nonlinear terms to guarantee the time average of these quantities can be controlled.But the conditions are not easy to verify for some stochastic PDEs.Recently, Zhou [18,19] established a new criterion for the existence of a random exponential attractor for non-autonomous RDS.Their conditions are limited to checking the boundedness of some random variables in the mean.Then they applied the abstract result to a non-autonomous stochastic reaction-diffusion equation in R 3 and the first-order stochastic lattice system.However, in concrete applications, the assumptions rely on the orthogonal projections, so they cannot be directly applied to RDS defined in Banach spaces.
In this article, we mainly consider the existence of a random exponential attractor in Banach space.Motivated by [18][19][20][21], we show that if the cocycle φ(T, ω) is C 1 (in the topology of a Banach space X) on a positively invariant random set χ(ω) for a.s.ω ∈ and some large enough time T (independent of ω), and the Fréchet derivative D v (φ(T, ω)) at every point inside χ(ω) can be split into a compact operator and a contraction (in the mean sense), then we can construct a random exponential attractor for the discrete cocyle φ(nT, θ mT ω) in the Banach space X.Following a similar process as presented in [19], we can get a random exponential attractor for the continuous cocycle φ(t, ω).
As an application of the theory developed in the paper, the following problem in a bounded domain D ⊂ R 3 with smooth boundary ∂D is considered with the initial-boundary value conditions where b is a positive constant, the term u • dW (t) in (1.1) is understood in the sense of Stratonovich interation and W (t) is a two-sided real-valued Wiener process on a probability space specified in Sect.3. The nonlinearity f ∈ C 2 satisfies the following conditions: for some 1 < p ≤ 3, c i > 0 (i = 0, 1, 2, 3), ν > 0 and for all u ∈ R. We also assume that β ∈ L 3p (D) and g ∈ L 6p (D).A typical function in applications is f (x, u) = a|u| q-1 u + h(x) with a > 0 and h(x) ∈ L 3p 2 (D).The above equation (1.1) is known as a reaction-diffusion equation [22] perturbed by a white noise g(x) dt + bu • dW (t).In biology and physics, stochastic equations like (1.1) have been used as models to study the phenomena of stochastic resonance [23][24][25][26][27][28], where g is an input signal and W (t) is a Wiener process used to test the impact of stochastic fluctuations on g.We choose the equation (1.1) since the long-term behavior of solutions for equations like (1.1) has been studied widely for both deterministic and stochastic cases.They are canonical examples to study the existence of global attractors and random attractors.In this respect, we refer the readers to [1, 4, 8-11, 17, 20, 22, 29-32], among others.Until now, as we know, there is no result concerning the existence of random exponential attractors in Banach space for (1.1).We extend the technique presented in [20,21] to stochastic case to get the Fréchet differentiability in the Banach space L 2p (D), and this is nontrival, since the trajectory of an RDS is unbounded in time.Fortunately, some random variables can be controlled in the mean for a large time T in a certain absorbing set and this is sufficient to construct a random exponential attractor in L 2p (D).
Our main tasks in this paper include: (1) Give an abstract result for the existence of a random exponential attractor in general Banach space.(2) Establish the RDS φ(t, ω) generated by equation (1.1)-(1.2) and construct the absorbing set χ(ω).(3) Prove that the RDS generated by (1.1)-(1.5) is uniformly Fréchet differentiable in the topology of L 2p (D).( 4) Check the assumptions in the abstract result presented in Sect. 2 for φ(t, ω) and prove that φ(t, ω) possesses a random exponential attractor in L 2p (D).Our main result in this paper is as follows: This paper is organized as follows.In Sect.2, we recall some basic concepts and present our main result for the existence of a random exponential attractor in a Banach space.In Sect.3, we first prove that the RDS is C 1 on a positively invariant absorbing set in L 2p (D), then apply the abstract result in Sect. 2 to show that the RDS possesses a random exponential attractor in L 2p (D).
Throughout this paper, we denote by • X the norm of Banach space X.The inner product and norm of L 2 (D) are written as (•, •) and • respectively.We also use u r to denote the norm of u ∈ L r (D) (r ≥ 1, r = 2) and |u| to denote the modular of u.The letters c and c i (i = 1, 2, . ..) are generic positive constants and the constant c may change their values from line to line even in the same line.

Preliminaries and abstract results
We first recall some basic concepts and results related to random exponential attractors and then establish a result for the existence of a random exponential attractor in Banach space.
Definition 2.1 Let ( , F, P) be a probability space, ( , F, P, (θ t ) t∈R ) is called a metric dynamical system (MDS) if θ t : R × → is (B(R) × F, F)-measurable, θ 0 is the identity on , θ s+t = θ s • θ t for all s, t ∈ R and θ t P = P for all t ∈ R.

Definition 2.2
The RDS on X over an MDS ( , F, P, An RDS is said to be continuous on X if φ(t, ω) : X → X is continuous for all t ∈ R + and P-a.s.ω ∈ .(2) A random variable r(ω) ≥ 0 is called tempered with respect to (θ t ) t∈R if for P-a.s.ω ∈ , lim t→∞ e -βt r(θ -t ω) = 0 for all β > 0.
In the following, we denote D X and D r the collections of all tempered family of nonempty subsets of X and L r (D) respectively with respect to (θ t ) t∈R .Definition 2. 4 A family E(ω) of subsets of X is called a random exponential attractor in D X for a continuous RDS φ(t, ω) over an MDS ( , F, P, (θ t ) t∈R ) if E(ω) is measurable in ω and there is a set of full measure ˜ ∈ F such that for any ω ∈ ˜ , it holds that (i) Compactness: (iii) Finite-dimensionality: There exists a random variable ζ ω (< +∞) such that and N ε (A) denotes the minimal numbers of balls with radius ε covering A in X; (iv) Exponential attraction: There exist a > 0 (independent of ω), t ω,B ≥ 0 and b ω,B > 0 such that, for any B ∈ D X , where d h (F 1 , F 2 ) denotes the Hausdorff semidistance between F 1 and F 2 .
Remark Here we have borrowed the definition of a random exponential attractor from [18,19].Note that we do not mention the Hölder continuity condition in [17], since the compactness, positive invariance, finite-dimensionality and exponential attraction are intrinsic qualities for the concept of a random exponential attractor.
We denote L(X, Y ) and L(X) the bounded linear maps from X to Y and from X into itself, respectively.For a given λ > 0, we define Let F be a finite dimensional subspace of X.The quotient map L F induced by L is defined by: L F : X → X/F, x → Lx + F and x X/F = inf{ xf X : ∀f ∈ F}.For the quotient map L F , we have the following lemma (see Lemma 2.1 in [21]).Lemma 2.1 For every L ∈ L λ (X) there exist a finite dimensional subspace F ⊂ X such that if L F is the linear map induced by L, then L F < 2λ.
If L ∈ L λ (X), we define ν λ (L) as the minimum integer n such that there exists a subspace F ⊂ X satisfying dim F = n and L F < 2λ.By Lemma 2.1 we see that ν λ (L) is well-defined and finite.We also need a covering result for a linear bounded mapping acts on the balls in X.We give this result in the following lemma, for more details we refer the readers to [21].
for all ε, r > 0, λ > L F .Moreover, the centers of the balls in the covering can be chosen in F.
Finally, by using a similar process presented in [19], we can show that {E(ω)} ω∈ is a random exponential attractor for {φ(t, ω)} t≥0,ω∈ in X, here we omit it.The proof is completed.

The RDS generated by (1.1)∼(1.2) and some useful results
We consider the probability space ( , F, P) where = {ω ∈ C(R, R) : ω(0) = 0}, F is the Borel σ -algebra induced by the compact-open topology of , and P the corresponding Wiener measure on ( , F).The Brownian motion W (t, ω) is identified as ω(t), i.e., For our purpose, we need to convert the stochastic equation (1.1)∼(1.2) into a deterministic equation with a random parameter.We introduce an one-dimensional Ornstein-Uhlenbeck process, which is given by z(θ t ω) := -0 -∞ e τ (θ t ω)(τ ) dτ , t ∈ R, and it solves the Itô equation It is known from [33] that the random variable z(ω) is tempered, and there is a θ t -invariant set ˜ ⊂ of full P measure such that for every ω ∈ ˜ , t → z(θ t ω) is continuous in t and We set α(ω) = e -bz(ω) .From (3.2) we can easily show that α(ω) and α -1 (ω) are tempered.Let v(t) = α(θ t ω)u(t), and we can consider the following evolution equation with random coefficients but without white noise: with Dirichlet boundary condition and initial condition By the normal Faedo-Galerkin methods (see [22]) or a similar result for the deterministic case in [32], one can show that v(t, ω, Then is an RDS generated by (1.1)∼(1.2) and continuous in L 2p (D).To simplify the calculations, we only consider the continuous RDS generated by (3.3)∼(3.5),i.e., and check the conditions presented in Theorem 2.1 for φ(t, ω).With a standard procedure (see [32] for deterministic case and [13,30]) for stochastic case), one can get the existence of a random attractor in L 2p (D) for φ(t, ω).In order to avoid the paper being tediously long, we just give the result below.The above theorem implies that φ(t, ω), defined in (3.6), satisfies the assumption (H0).To prove the Fréchet differentiability and to construct the absorbing subset described in assumptions (H1)∼(H5), we need the following regularity: Then for P-a.s.ω ∈ , there exists T D(ω) > 0 and a tempered random variable M 0 (ω), such that the solution v(t, ω, v 0 (ω)) of (3.3)-(3.5)satisfies, for all t ≥ T D(ω) + 1, For the second term on the left-hand side of (3.7), using (3.10) and (3.13) in [20] and the imbedding theorem, we have and (3.9) So we obtain Applying (1.4), we can estimate the fourth term on the left-hand side of (3.10) as Therefore, applying the Gronwall's inequality, we have We define and M 1 (ω) is a tempered random variable by definition.Thus  Therefore Next, we take the inner product of (3.3) with 6p and a similar procedure of (3.12), we obtain Substituting θ -t-1 ω for ω in the above inequality, we get Then the result holds with M 0 (ω) = M 1 6p 2 (ω).The proof is completed.
for every t ∈ R and a.s.ω ∈ .

Decomposition of v(t, ω)
In this subsection, we consider the linear version in L 2p (D) and (3.42) Clearly, we have In the following, we prove that W 2 (T) is compact in L 2p (D) and W 1 (T) is contractive in the mean in L 2p (D) for some T > 1.The proof of the following lemma is similar to some parts of Lemma 3.1, here we only give the sketch.
(2) there exists a tempered random variable λ t,ω , such that for any t > 0 it holds that (1) As the proof of (3.20), by taking the inner product of (3.40) and (3.41) with |W | 6p-2 W and |W 1 | 6p-2 W 1 respectively we obtain that W (t) and W 1 (t) are bounded from L 2p (D) into L 6p (D) for every t ≥ t * and for some t * = t * (ω) ∈ (0, 1).Thus W 2 (t) = W (t) -W 1 (t) is bounded from L 2p (D) into L 6p (D).similarly, it is also a standard procedure to get that W 2 (t) is bounded from L 2p (D) into H 1 0 (D) for any t > 1.Therefore, by the compact embedding H 1 0 (D) → L 2 (D) and the interpolative inequality

Sketch of Proof
it is easy to prove that W 2 (t) is compact in L 2p (D) for t > 1.
From [18,19], we have the following results: and a family of elements {e j } ∞ j=1 of D(-), which forms an orthogonal basis in both L 2 (D) and H 1 0 (D) such that e j = λ j e j , ∀j ∈ N.

Conclusion
In this paper, we have studied the asymptotic behavior of the RDS φ(t, ω) generated by (3.3)-(3.5).First, an abstract result for the existence of a random exponential attractor is established in general Banach space.Second, a useful asymptotic a priori estimate in L 6p (D) is given.Third, a positively invariant random set χ(ω) in L 2p (D) is constructed and the Fréchet differentiability of φ(T, ω) in χ(ω) is proved for a large time T. Then φ(T, ω) is split into two parts, i. e. W 1 (T, ω) and W 2 (T, ω), and W 1 (T, ω) is proved to be contractive in the mean in L 2p (D) and W 2 (T, ω) to be compact in L 2p (D).Finally, by checking the assumptions (H0)∼(H5) presented in the abstract result for φ(t, ω) and χ(ω), the existence of a random exponential attractor is proved in L 2p (D).
It is worth noticing that our case is different from that of [18].In [18], the author proved the existence of a random exponential attractor for a stochastic non-autonomous reactiondiffusion equation with multiplicative white noise in the entire space R 3 .The author decomposed the solutions into two parts, of whose, one part is finite-dimensional which satisfies the flattening propety [11] and the "tail" part is "quickly decay" for suitable large x ∈ R 3 and large time t.This implies the existence of a finite dimensional random exponential attractor in the Hilbert space L 2 (R 3 ).However, the technique relies on the orthogonal basis {e j } ∞ j=1 in L 2 H 1 0 and the orthogonal projections P n : L 2 (D) → X n , where X n = span {e 1 , . . ., e n }, so that it cannot be applied directly to general Banach space.