Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise

In this paper, we first establish some sufficient conditions for the existence and construction of a random exponential attractor for a continuous cocycle on a separable Banach space. Then we mainly consider the random attractor and random exponential attractor for stochastic non-autonomous damped wave equation driven by linear multiplicative white noise with small coefficient when the nonlinearity is cubic. First step, we prove the existence of a random attractor for the cocycle associated with the considered system by carefully decomposing the solutions of system in two different modes and estimating the bounds of solutions. Second step, we consider an upper semicontinuity of random attractors as the coefficient of random term tends zero. Third step, we show the regularity of random attractor in a higher regular space through a recurrence method. Fourth step, we prove the existence of a random exponential attractor for the considered system, which implies the finiteness of fractal dimension of random attractor. Finally we remark that the stochastic non-autonomous damped cubic wave equation driven by additive white noise also has a random exponential attractor.


(Communicated by Nikolay Tzvetkov)
Abstract. In this paper, we first establish some sufficient conditions for the existence and construction of a random exponential attractor for a continuous cocycle on a separable Banach space. Then we mainly consider the random attractor and random exponential attractor for stochastic non-autonomous damped wave equation driven by linear multiplicative white noise with small coefficient when the nonlinearity is cubic. First step, we prove the existence of a random attractor for the cocycle associated with the considered system by carefully decomposing the solutions of system in two different modes and estimating the bounds of solutions. Second step, we consider an upper semicontinuity of random attractors as the coefficient of random term tends zero. Third step, we show the regularity of random attractor in a higher regular space through a recurrence method. Fourth step, we prove the existence of a random exponential attractor for the considered system, which implies the finiteness of fractal dimension of random attractor. Finally we remark that the stochastic non-autonomous damped cubic wave equation driven by additive white noise also has a random exponential attractor.

1.
Introduction. It is known that the existence of attractor and the estimate of its dimension are two main topics in studying the asymptotic behavior of infinitedimensional dynamical systems [2,3,9,26,32,39,41,43,50]. The finite fractal dimension of attractor implies that the attractor can be mapped onto a compact subset of a finite dimensional Euclidean space and hence the random attractor can be described by finite independent parameters [25]. But just knowing the boundedness of Hausdorff dimension for a set, we can not have such an available finite parametrization (see [34,47]).
However, there is an intrinsic drawback that random attractor sometimes is infinite dimensional and attracts orbits at a relatively slow rate so that it takes an unexpected long time to reach it. Moreover, the attractor is possible sensitive to perturbations which makes it unobservable in experiments and numerical simulations. To overcome these drawback, Shirikyan and Zelik in [49] introduced the concept of random exponential attractor and present some sufficient conditions for constructing a random exponential attractor for an autonomous random dynamical system, and gave an application to parabolic partial differential equations with random noise. By definition, a random exponential attractor for a random dynamical system is a compact random set, which has finite fractal dimension and attracts exponentially any trajectory and is positive invariant, then it contains random attractor and becomes an appropriate alternative to study the asymptotic behavior of random dynamical systems. Moreover, the existence of a random exponential attractor implies the existence of a random attractor with finite fractal dimension.
We notice that the evolution mode of states in a random system is, in some sense, similar to the deterministic non-autonomous one and there were several construction methods to obtain a pullback exponential attractor for a (deterministic) process, see [14,15,16,17,20,35,61]. We also notice that a trajectory of a random system is often unbounded in time along the path of sample point with probability 1 which is different from deterministic one [14,15,16,17,20,35,61]. Thus, in general, a simple straightforward extension from deterministic system to random system does not work. Fortunately, some time averages of those quantities bounding the trajectories of a random system in large times can be finite and possibly controlled, which provides a useful way for constructing an exponential attractor for a random system. It is observable that the existing only conditions given in [49] for constructing a random exponential attractor of random systems are not easy to be verified for some stochastic partial differential equations driven by white noises, including our considered system (1) below.
In this article, motivated by the ideas of [20,49,61], we first establish a new criterion (some sufficient conditions) for the existence and construction of a random exponential attractor for a continuous cocycle on a separable Banach space. It is worth mentioning that our conditions just need to check the boundedness of some random variables in the mean and can be easily verified for some stochastic evolution equations.
For the stochastic non-autonomous evolution equations with a time-dependent external term g, Wang established an useful theory about the existence and upper semi-continuity of random attractors for the corresponding cocycle by introducing two parametric spaces, see [52,53].
The random attractor and the bounds of its Hausdorff and fractal dimensions for the stochastic wave equations with additive noise (i.e., the random term in (1) is "adW (t)" independent of u) have been studied by many authors, see [12,13,8,18,21,38,54,57,63,65]. For the stochastic system (1) with linear multiplicative noise "au • dW (t)" (depending on the state variable u) and sufficient small coefficient |a| of random term, when the nonlinear function f has a subcubic growth exponent (i.e., f 1 ≡ 0 in (A1)), the existence and the boundedness of fractal dimension of random attractor were studied, see [22,36,52,66], of those, Zhou and Zhao in [66] gave some sufficient conditions to bound the fractal dimension of a random invariant set for a cocycle and applied these conditions to get an upper bound of fractal dimension of the random attractor of system (1).
But as we know, when f has a cubic growth exponent (i.e., f 1 = 0 in (A1)), there is no results about the existence and dimension of random attractor in both cases of stochastic autonomous and non-autonomous wave equations (1). In this case, there are two main essential difficulties. The first difficulty arises in showing the asymptotic compactness of system that is the key step to prove the existence of a random attractor, which is caused by the cubic growth condition (2) of f and can not be overcome by decomposing the solutions of system just one time like in the deterministic case [27,58,62] or the subcubic growth exponential case [36,52,65,66]. The second difficulty occurs in the possible unboundedness of the expectation of bound of random attractor in a "higher regularity" space here, which is a basic requiring condition in known existing methods to show the boundedness of the fractal dimension of a random attractor [34,56,65,66]. To solve these problems, we have to use some new different technique.
We will do the following objects for the random attractor of system (1).
(I) Study the existence of random attractor for (1) when f satisfies (A1) and the coefficient |a| of random term is sufficient small. To prove the asymptotic pullback compactness of the cocycle Φ associated with (1), we decompose carefully the solutions of system and estimate the bounds of solutions through two different modes, which is different from the known publications [38,57,58]. Then we construct a compact measurable tempered attracting set and prove the existence of a random attractor for Φ in phase space H 1 0 (U ) × L 2 (U ). (II) Establish the upper semicontinuity of the random attractors for (1) with respect to the coefficient a of white noise term as a → 0. We will show that the random attractors of (1) tends to the pullback attractor of deterministic nonautonomous system (1) a=0 as a → 0 in the sense of Hausdorff semi-distance between two subsets of phase space.
(III) Study the regularity of random attractor by constructing a compact measurable tempered attracting set through a recurrence method and hence prove the boundedness of random attractor in higher regular space [H 2 (U ) ∩ H 1 0 (U )] × H 1 0 (U ) for the cocycle Φ.
(IV) Prove the existence of a random exponential attractor for a continuous cocycle Φ in H 1 0 (U ) × L 2 (U ) when f satisfies ( A1)-(A2) by applying our new criterion, which implies the finiteness of fractal dimension of random attractor for (1). This paper is organized as follows. In section 2, we establish some sufficient conditions for the existence of a random exponential attractor for a continuous cocycle on a separable Banach space. In section 3, we first transfer the stochastic differential system (1) into an equivalent random differential system, then we show that the mapping of solutions for this random system generates a continuous cocycle, finally we prove the ultimately pullback boundedness of solutions of random equation. In section 4, we carefully decompose the solutions of random differential equation into a sum of two components in two ways: one component decays exponentially and another component is bounded in a higher regular space. In section 5, we construct a compact pullback attracting set and obtain the existence of random attractor. In section 6, we consider an upper semicontinuity of random attractors as a → 0. In section 7, we prove the regularity of random attractor basing on a recurrence mode. In section 8, we prove the existence of a random exponential attractor for (1) when f satisfies (A1)-(A2). In section 9, we consider a special case of (1) where f (u, x) is a 3 order polynomial of u with a positive leading coefficient. At last, we point out that the following non-autonomous stochastic damped wave equation with additive white noise and initial-boundary conditions: has a random exponential attractor under conditions (A1)-(A2).
2. Random exponential attractor for continuous cocycles. In this section, we establish some sufficient conditions for the existence and construction of a random exponential attractor for a continuous cocycle on a separable Banach space.
(iii) Positive invariance: (iv) Exponential attraction: there exists a constantã > 0 such that for any Remark 2.1. By Definition 2.1 and definition of random attractor (see section 5), it is easy to see that the existence of a random exponential attractor for every τ ∈ R and ω ∈Ω. Here "ω ∈Ω" means "a.e. ω ∈ Ω".
In the following of this article, for simplicity, we identify "a.e. ω ∈ Ω" and "ω ∈ Ω". In fact, the property holds for a.e. ω ∈ Ω throughout the article.
then system (1) can be written as the following equivalent random system in E: In the following, we always assume that condition (A1) holds. It follows from Galerkin approximation method or [9,42,48,52,54,64] and Lemma 3.1 below that for every fixed τ ∈ R, ω ∈ Ω and ϕ τ ∈ E, the problem (47) has a unique globally weak solution ϕ(·, τ, ω, ϕ τ ) ∈ C([τ, +∞); E), which defines a continuous cocycle Φ: Obviously, the dynamics of solution (u, u t ) of (1) is same to that of cocycle Φ associated with (47) in E. From now, we consider the existence, upper semicontinuity, regularity of random attractor and the existence of random exponential attractor for Φ in E.
Let D = D(E) denote the collection of all tempered families of nonempty subsets of E and the inner and norm of L 2 (U ) are denoted as (·, ·) and || · ||. We remark that all the numbers c i (i ∈ N) below are positive constants independent of (ω, τ, t). and Then there exist a random variable M 0 (ω) and a tempered measurable D(E)-pullback absorbing set for Φ In particular, there exists a T B0 (ω) ≥ 0 (independent of τ ) such that Proof. For every fixed τ ∈ R and ω ∈ Ω, let ϕ [22,63], (48) and ε < √ Taking the inner product (·, ·) E of (47) with ϕ(r), we find that for r ≥ τ − t, thus, where By Gronwall's inequality to (58) By (47), (48) and (59), Thus, by (60), we have Since {θ t } t∈R is ergodic measure-preserving on (Ω, F, P), by Birkhoff ergodic Theorem [51], it holds that for every ω ∈ Ω (in fact a.e.ω ∈ Ω), which implies that there exists a T 0 (ω) > 0 such that and Thus (62) and (65) imply (54). By (60) and the cocycle property of ϕ, (55) holds. For any γ > 0, Remark 3.1. By (45), (52) and (51), it follows that for we have and for it holds that 4. Decomposition of solutions. In this section, we split the solutions of (47) into a sum of two components: one component decays "exponentially" and another one is ultimately pullback bounded in a "higher regular" space. To do this, we have to decompose the solutions of (47) in two modes with different initial data.
For the first component ϕ 1 , we have its "exponentially" decay as follows.
Then there exist positive constants K 0,l andK l such that for given τ ∈ R, ω ∈ Ω, t ≥ 0 and T > 0, the solution ϕ(r) of (47) has a decomposition: and ν is as in (80).

ZHAOJUAN WANG AND SHENGFAN ZHOU
By taking the expectation, we have Similar to (78), we have The proof is completed. Based on Lemmas 4.1-4.4, we have the following ultimately pullback boundedness of component ϕ N in a "higher regular" space E ν .

ZHAOJUAN WANG AND SHENGFAN ZHOU
Then by applying the Gronwall inequality to (159), we have that for r ≥ τ, and by (69), The proof is completed.
Remark 7.1. From the expression (171) of b 1 (ω) (the obtained bound of random attractor in the norm of E 1 ), it is seen that the expectation of b 2 1 (ω) is possible unbounded, which is different from the case of subcubic growth exponent in [65,66]. 8. Existence of random exponential attractor. In this section, we assume that conditions (A1)-(A2) hold. Now let us prove the existence of a random exponential attractor for Φ in E according to Theorem 2.1. For this aim, in the following, we will check that Φ satisfies the conditions (H1)-(H4) in Theorem 2.1.
Here we notice that for the system (1), we require that the coefficient a of the random term is small (see (196)), but for the system (213), we don't need such a condition, this is because that the multiplicative noise term depends on the state variable u but the additive noise term is independent of u.