Random attractors for stochastic parabolic equations with additive noise in weighted spaces

In this paper, we establish the existence of random attractors for stochastic parabolic equations driven by additive noise as well as deterministic non-autonomous forcing terms in weighted Lebesgue spaces \begin{document}$ L_{\delta}^r(\mathcal{O})$\end{document} , where \begin{document}$ 1 is the distance from \begin{document}$ x$\end{document} to the boundary. The nonlinearity \begin{document}$ f(x,u)$\end{document} of equation depending on the spatial variable does not have the bound on the derivative in \begin{document}$ u$\end{document} , and then causes critical exponent. In both subcritical and critical cases, we get the well-posedness and dissipativeness of the problem under consideration and, by smoothing property of heat semigroup in weighted space, the asymptotical compactness of random dynamical system corresponding to the original system.


1.
Introduction. Consider the following stochastic parabolic equation with additive noise          du − ∆udt = (f (x, u) + g(x, t))dt + m j=1 h j dω j , t > τ, x ∈ O, u = 0, t > τ, x ∈ ∂O, u(x, τ ) = u τ , τ ∈ R, x ∈ O, where O is the bounded domain in R N with smooth C 2 boundary ∂O, h j ∈ L η δ (O) W 2,r δ (O) (see definition below), η > r as in (3.11), j = 1, · · · , m, and {ω j } m j=1 are independent two-sided real-valued Wiener processes on a probability space which will be specified in Section 3. The nonlinearity f (x, u) and deterministic nonautonomous forcing term g(x, t) satisfy the following assumptions: (P 1 ) There exist ρ > 1 and C > 0 such that (P 2 ) g(x, t) ∈ L γ loc (R; L r δ (O)), i.e., t2 t1 g(x, s) γ L r δ ds < +∞, where 1 < γ, r < ∞, and 1 γ < min{ 1 2 , 1 − 1 ρ }. Based on the wide applications such as describing the random phenomena in physics, chemistry, biology and control theory, the stochastic evolution equations have been extensively investigated in recent decades, see, e.g., [13,19,32]. One of the most important problems in studying stochastic partial differential equation (SPDE) is to understand the asymptotic behavior of them. When the SPDE possesses the random forcing, for example, additive white noise term, the theory on the existence of attractor for deterministic infinite-dimensional dynamical systems, see, e.g., [3,12,20,27,28], can not be used directly. In fact, there is no chance that bounded subsets of the state space remain invariant for SPDE. To overcome this difficulty, H. Cruel and F. Flandoli in [14,15] extended the concept of attractor for deterministic system to random system, and introduced the random attractor (or pullback attractor) which attracts the random set in state space in the sense of pullback. In this situation, one can discover compact invariant sets which are not fixed, but they depend on chance, and they move with time. Afterwards, in order to capture the structure of deterministic non-autonomous dynamical systems, pullback attractor is also used to study the long-time behavior of them, see, e.g., [8,10]. For other related studies, we refer reader to, e.g., [5,7,11,30,31,33].
In this paper, we study the dynamics of SPDE (1.1) in weighted Lebesgue spaces L r δ (O). There are some inspirations in investigating this topic: Firstly, when study SPDE (1.1) in L r δ (O), comparing with the results in [4,7,15,16,29,32,34] we can establish the random attractor for (1.1) in weak topological spaces. Moreover, since δ tends to zero as x close to the boundary of O, the solution of SPDE (1.1) in L r δ (O) can grow more rapidly near the boundary than in normal Lebesgue space L r (O).
Secondly, comparing with nonlinearity f in [4,5,7,15,16,29,30,32,33,34], we remove the assumption on the derivative of f with respect to u, that is, we do not assume its bounded on ∂f (x,u) ∂u : ∂f (x, u) ∂u ≤ c, c > 0. (1.2) The lack of bound on ∂f (x,u) ∂u brings about the critical exponent on f . In the subcritical case, by the smoothing property of the Dirichlet heat semigroup in weighted space, we get the existence of random attractor for (1.1). When nonlinearity f has the critical growth, as it is pointed in the deterministic case in [2,6,9,18,24], the uniform existence time of solutions of (1.1) is only on the compact set of the initial data space. In order to overcome this obstacle, we introduce the almost critical nonlinearity as in [9,23] to get the uniform existence time of solutions on any bounded set of initial data space, and then get the existence of random attractor for (1.1).
Thirdly, for the fixed growth exponents ρ and β (a(x) belongs to L β δ (O)) of f (x, u), there is the new result on the existence of random dynamical system (RDS) generated by SPDE (1.1) in weighted Lebesgue spaces L r δ (O). Under the assumptions (P 1 )-(P 2 ), if r ≥ (ρ − 1)( 2 N − 1 β ) −1 (equivalently, 1 β + ρ−1 r ≤ 2 N ) and r ≥ 1, we know from Remark 4 that the problem (1.1) generates RDS in L r (O); and if ) and r ≥ 1, it follows from Theorem 3.1-3.2 that the problem (1.1) generates RDS in L r δ (O). As it is pointed in Remark 3.2 (a) in [18] . Therefore, there is a "gap" existence interval for r: For other study of elliptic and parabolic problems in L r δ (O), we refer reader to, e.g., [18,26].
Fourthly, by using the smoothing property of the Dirichlet heat semigroup in weighted space L r δ (O), there are some advantages in study SPDE (1.1) with bounded domain. On the one hand, since the nonlinearity f (x, u) of (1.1) depends on the spatial variable, we can get the regularity of solution and existence of random attractor without assumption on the derivative of f with respect to x as in [4,16,29,30,32,33,34]: The reason for this is that we need not take the inner product of corresponding converted equation with −∆v in L 2 (O), see details, e.g., in the proof of Lemma 4.5 in [4]. In this situation, the assumption on f can be relaxed further in some sense. On the other hand, it is convenient to establish random attractor for SPDE (1.1) in general Lebesgue space L r δ (O). Finally, comparing with the previous works, e.g., in [4,16,32,34], we relax the assumptions on nonlinearity f (x, u), that is, f (x, u) does not necessarily satisfy (1.2)-(1.3) and where α 2 is a positive constant, and consider the more general case which only satisfies (P 1 ). Recently, authors in [16] consider the existence of D-uniform attractor for (1.1) in L 2 (O), but the translation compact property for the deterministic non-autonomous forcing term g(x, t) in L 2,w loc (R, L 2 (O)) (translation bounded in L 2 loc (R, L 2 (O))) is needed. Under the assumption (4.3) which g(x, t) does not necessarily be translation compact in L p2,w loc (R, X), we establish the existence of D-random attractor in L r δ (O). Comparing with the known works in obtaining the existence of random attractors for stochastic dynamical systems, we establish the existence of D-random attractor for (1.1) in general initial data space L r δ (O), 1 < r < +∞; and in getting the D-pullback asymptotically compact property for them, we use the difference method of smoothing property of heat semigroups in weighted spaces.
The paper is organized as follows: In the next section, we recall some fundamental results on random attractor for stochastic dynamical systems, and heat semigroup theory on weighted space; in Section 3, we show that (1.1) generates a continuous RDS in L r δ (O); in Section 4 and 5, we establish the existence of random attractor for (1.1) with critical and subcritical cases in L 2 δ (O) and L r δ (O), respectively. For the rest of paper, we denote by C the generic positive constant which may changes its value from line to line or even in the same line.

2.
Preliminaries. In this section, we recall some concepts and existence results on random attractors for stochastic dynamical systems and the working space for (1.1).
2.1. Random dynamical system Let (X, d) be a complete separable metric space with Borel σ-algebra B(X), Ω 1 be a nonempty set, and (Ω 2 , F 2 , P ) be a probability space. For the following concepts, the reader is also referred to, e.g., [4,5,17,29] for more details.
Definition 2.7. Let D be a collection of some families of nonempty subsets of X. Then A = {A(ω 1 , ω 2 ) : ω 1 ∈ Ω 1 , ω 2 ∈ Ω 2 } is called a D-random attractor (or D-pullback attractor) for Φ if the following conditions are satisfied, for all ω 1 ∈ Ω 1 and P -a.e. ω 2 ∈ Ω 2 , (i) A(ω 1 , ω 2 ) is compact, and In order to guarantee D-random attractor A belonging to D and being unique, and obtain a sufficient and necessary condition for the existence of A, we introduce the following concept.
there exists a positive number ε depending on D such that the family The following concept can be used to describe the structure of D-random attractor A. A mapping ψ : R × Ω 1 × Ω 2 → X is called a D-complete orbit of Φ in D if for every τ ∈ R, t ≥ 0, ω 1 ∈ Ω 1 and P -a.e. ω 2 ∈ Ω 2 , there holds Φ(t, θ 1,τ ω 1 , θ 2,τ ω 2 , ψ(τ, ω 1 , ω 2 )) = ψ(t + τ, ω 1 , ω 2 ), and there exists D = {D(ω 1 , ω 2 ) : ω 1 ∈ Ω 1 , ω 2 ∈ Ω 2 } ∈ D such that ψ(t, ω 1 , ω 2 ) belongs to D(θ 1,t ω 1 , θ 2,t ω 2 ) for every t ∈ R, ω 1 ∈ Ω 1 and P -a.e. ω 2 ∈ Ω 2 . The following results on existence and uniqueness of D-random attractor for Φ can be found in [29]. Theorem 2.9. Let D be a neighborhood closed collection of some families of nonempty subsets of X, and Φ be a continuous RDS on X over (Ω 1 , (θ 1,t ) t∈R ) and (Ω 2 , F 2 , P, (θ 2,t ) t∈R ). Then Φ has a D-random attractor A in D if and only if Φ is D-pullback asymptotically compact in X and Φ has a closed measurable D-pullback absorbing set K in D. The D-random attractor A is unique and given by, for each ω 1 ∈ Ω 1 and P -a.e. ω 2 ∈ Ω 2 , When O has a smooth C 2 boundary, it is well-known that there exist constants c 1 , c 2 > 0 such that . Then there exists a constant C > 0 such that 3. Random dynamical system for (1.1). Here we show that there is a continuous random dynamical system on L r δ (O) generated by the SPDE (1.1). Let Ω 1 = R, define a family of operators {θ 1,t } t∈R by for all t, h ∈ R. (3.1) In the following, we consider the probability space (Ω 2 , F 2 , P ), where : ω 2 (0) = 0}, F 2 is the Borel σ-algebra induced by the compact-open topology of Ω 2 , and P is the corresponding Wiener measure on (Ω 2 , F 2 ). Then we can identity ω 2 with Define the time shift by Then (Ω 2 , F 2 , P, (θ 2,t ) t∈R ) is a metric dynamical system. In order to deal with the random forcing term of (1.1), we consider the onedimensional Ornstein-Uhlenbeck equation By Itô integral and its integration by parts formula, see, e.g., [25], one can obtain that the solution of (3.3) is given by Note that the random variable z j (ω j 2 ) is tempered and z j (θ 2,t ω j 2 ) is P -a.e. continuous. Therefore, by Proposition 4.3.3 in [1], one can get that there exists a tempered function q(ω 2 ) > 0 such that Let v(t) = u(t) − z(θ 2,t ω 2 ), where u is the solution of (1.1). Then we can obtain that v satisfies As the proof of Theorem 3.1 in [22], we can get the well-posedness of (3.8)-(3.9) depending on the random parameter: and m(η) > 2. This solution depends continuously on the initial data and is unique in the class Moreover, there exists a constant C > 0 such that Furthermore, for any bounded set (resp. compact set) K in L r δ (O), there is a uniform time T = T (K) such that, for any v τ ∈ K, the solution of (3.8)-(3.9) exists on [τ, T ].
Remark 1. Since we reduce the regularity on deterministic forcing term g(x, t), we can not obtain the higher regularity of solution of (3.8)-(3.9) as in [22].
We know from Theorem 3.1 that in critical case, the uniform existence time of solutions of (3.8)-(3.9) is only on the compact set of initial data spaces. This fact is also pointed in Theorem 1 in [6], Theorem 1 in [2], and Theorem 5 in [18] in studying different types of evolution equations. In order to get the uniform existence time of solutions on any bounded set in initial data spaces for (3.8)-(3.9) with critical nonlinearity, we impose the following assumption on f (x, u) as in [9,23]: Remark 2. The nonlinearity satisfying above growth condition is close to critical case, in this situation, we call it the almost critical nonlinearity.
such that there is a solution v(t) of (3.8)-(3.9) belonging to (3.10). This solution depends continuously on the initial data, is unique in the class (3.12) and satisfies (3.13).
Furthermore, for any bounded set K in L r δ (O), there is a uniform time T = T (K) such that, for any v τ ∈ K, the solution of (3.8)-(3.9) exists on [τ, T ].
Proof. We only give the different ingredient in the proof of Theorem 3.1-3.2 in [22]. For P -a.e. ω 2 ∈ Ω 2 , define (3.14) (3.11), 0 < σ < 1. Without loss of generality, we can assume that 0 It follows from (P 3 ) that for any > 0, there exists C > 0 such that Now, we estimate every term except first one on the right-hand side of (3.17). Let 1 G = 1 β + ρ η . By Theorem 2.10 and (3.5)-(3.6) we have Similarly, Now, by (3.17)-(3.20) we can choose small enough and appropriate T such that We can get the similar estimates as (3.8)-(3.10) in [22], together with above inequality we get that Ψ(X) ⊂ X. The remain proof is similar to that of Theorem 3.1 in [22].

Remark 4.
If the exponents of f (x, u) with a(x) or b(x) belonging to L β (O) satisfy the growth condition 1 β + ρ−1 r ≤ 2 N , as the proof of Theorem 3.1 and 3.2, we can obtain the well-posedness of (3.8)-(3.9) in L r (O).