Long-time behavior of stochastic reaction–diffusion equation with multiplicative noise

In this paper, we study the dynamical behavior of the solution for the stochastic reaction–diffusion equation with the nonlinearity satisfying the polynomial growth of arbitrary order p≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p\geq2$\end{document} and any space dimension N. Based on the inductive principle, the higher-order integrability of the difference of the solutions near the initial data is established, and then the (norm-to-norm) continuity of solutions with respect to the initial data in H01(U)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H_{0}^{1}(U)$\end{document} is first obtained. As an application, we show the existence of (L2(U),Lp(U))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(L^{2}(U),L^{p}(U))$\end{document} and (L2(U),H01(U))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(L^{2}(U),H_{0}^{1}(U))$\end{document}-pullback random attractors, respectively.

For instance, provided that g ≡ 0 in (1.1), some significant results have been achieved. For instance, Coaraballo and Langa [24] obtained the existence of finite dimensional random attractor in L 2 (U) when f (u) = -βu + u 3 . Li et al. [25] used the quasi-continuity and omega-limit compactness introduced in [15] to obtain the (L 2 (U), L p (U))-random attractor for the problem (1.1), where f (u) is a polynomial of odd degree with a positive leading coefficient. Assuming that f (u) = -βu + u 3 and b = h(t) in (1.1), Fan and Chen [27] gave a new method (without transformations) to study the existence of an L 2 (U)-random attractor. When the nonlinearity f (u) satisfies the polynomial growth of arbitrary order p ≥ 2, Wang and Tang [29] showed the existence of (L 2 (U), H 1 0 (U))-random attractor for the problem (1.1) exploiting the method of the deterministic systems introduced in [33]. When g = 0, Zhao [28] proved the existence of H 1 0 (U)-random attractors for (1.1) by using the quasi-continuity ( [15]) along with the compactness of an omega-limit set.
Inspired by the above papers, we will continue studying the asymptotic behavior for the stochastic reaction-diffusion equation with multiplicative noise. Especially, we are interested in understanding the integrability and continuity of the solutions of Eq. (1.1) with the forcing term g = 0.
On the one hand, we know that obtaining certain higher-order integrability and regularity are significant for better understanding the dynamical systems. When b ≡ 0 and the forcing term g belongs to L 2 (U) or H -1 (U), the solutions of the equation in the deterministic system are at most in H 2 (U) ∩ L 2p-2 (U) or H 1 0 (U) ∩ L p (U) and have no regularities. As regards the stochastic system, if the initial data u 0 and forcing term g belong to L 2 (U), then the solution u with the initial data u(0) = u 0 belongs to L 2 (U) ∩ H 1 0 (U) ∩ L p (U) only and has no higher regularity because of the random noise term. Compared with the case g ≡ 0 mentioned above (from [25]), the case g = 0 is even more complicated. The reason is that the regularity and integrability of the solutions depend not only on the growth exponent p, but also on the regularity and integrability of g. Therefore, a natural question is: can we get some higher integrability when g = 0?
On the other hand, comparing with verifying the (norm-to-norm) continuity and asymptotic compactness, it is easy to check the quasi-continuity and the flattening conditions for most of the dynamical systems, especially in the space H 1 0 (U) and L p (U) (p > 2); see [34,35] for details. For the deterministic autonomous reaction diffusion equations, the authors [36] first proved the continuity of solutions in H 1 0 (U) for any space dimension N and any growth exponent p ≥ 2 by the method of differentiating the equation. However, for the stochastic case, since the Wiener processes W (t) are continuous but are not differentiable functions in R, we cannot use such a method to obtain the continuity in H 1 0 (U). Thus, for any space dimension N and any growth exponent p ≥ 2, we address the question whether or not we can obtain the continuity of solutions in H 1 0 (U) by some new kinds of estimates.
In order to answer the above two problems, we follow the ideas from [21] to obtain our main results, in which the authors investigated the high-order integrability of difference of solutions and existence of random attractors for the reaction-diffusion equations with additive noise.
The remainder article is arranged as follows. In Sect. 2, we first recall some definitions and known results about the pullback random attractors, then we give the well-posedness of a solution and the existence of random attractors in L 2 (U). In Sect. 3, we establish the higher-order integrability of the difference of the solutions near the initial time and get the continuity of solutions in H 1 0 (U). Furthermore, as an application of above continuity and higher-order integrability results of solutions, we show (L 2 (U), L p (U)) and (L 2 (U), H 1 0 (U)) D-pullback random attractors for the problem (1.1).

Preliminaries
Throughout the paper, we denote the norm of a Banach space X by · X . For the sake of convenience, we denote the norm of L r (U) (r ≥ 1, r = 2) by · L r (U) . The inner product and norm of L 2 (U) will be written as (·, ·) and · , respectively.
Next, let (X, · X ) be a separable Banach space with Borel σ -algebra B(X). We use (Ω, F, P) and (X, d) to denote a probability space and a completely separable metric space, respectively. If Y and Z are two nonempty subsets of X, then we use dist X (Y , Z) to denote their Hausdorff semi-distance, i.e., dist X (Y , Z) = sup y∈Y inf z∈Z yz X for any Y ⊂ X, Z ⊂ X. Definition 2.1 Let θ : R × Ω → Ω be a (B(R) × F, F)-measurable mapping. We say (Ω, F, P, (θ t ) t∈R ) is a metric dynamical system if θ 0 is the identity on Ω, θ s+t = θ t • θ s for all t, s ∈ R, and θ t P = P for all t ∈ R. Definition 2.2 Let (Ω, F, P, (θ t ) t∈R ) be a metric dynamical system. If the cocycle mapping Φ : R + × Ω × X → X satisfies the following properties: x ∈ X, ω ∈ Ω, then Φ is called a random dynamical system. Furthermore, Φ is called a continuous random dynamical system if Φ is continuous with respect to x for t ≥ 0 and ω ∈ Ω.
A bounded random set K := {K(ω)} ω∈Ω is said to be tempered (Ω, F, P, (θ t ) t∈R ) if for Pa.e. ω ∈ Ω, lim t→+∞ e -βt diam K(θ -t ω) = 0, for all β > 0. Definition 2.5 Let D be a collection of random sets in X. Then a random set K ∈ D is called a D-pullback absorbing set for a random dynamical system (θ , Φ) if for any random set D ∈ D and P-a.e. ω ∈ Ω, there exists T = T D (ω) > 0 such that Definition 2.6 Let D be a collection of random sets in X. Then Φ is said to be D-pullback asymptotically compact in X if for all P-a.e. ω ∈ Ω, the sequence Definition 2.7 Let D be a collection of some families of nonempty subsets of X. Then A = {A(ω)} ω∈Ω ∈ D is called a D-pullback attractor for a random dynamical system Φ if the following conditions (i)-(iii) are fulfilled: (i) A is a compact random set, that is, ω → dist(x, A(ω)) is measurable for every x ∈ X and A(ω) is nonempty and compact in X for P-a.e. ω ∈ Ω; (ii) A is invariant, that is, Φ(t, ω, A(ω)) = A(θ t ω), for P-a.e. ω ∈ Ω and every t > 0; where dist X is Hausdorff semi-metric given by dist X (Y , Z) = sup y∈Y inf z∈Z yz X for any Y ⊆ X and Z ⊆ X.

Theorem 2.8 ([3])
Let D be an inclusion-closed collection of some families of nonempty subsets of X. Suppose that Φ be a continuous random dynamical system on X over (Ω, F, P, {θ t } t∈R ). If there exists a closed random absorbing set K ∈ D and Φ is D-pullback asymptotically compact in X, then Φ has a unique D-random attractor A ∈ D,

Random attractor in L 2 (U)
In this subsection, we give some estimates of solutions to obtain our main results.

Lemma 2.9
Assume that g ∈ L 2 (U) and (1.2)-(1.4) hold. Let D ∈ D and u 0 ∈ D(ω). Then for P-a.e. ω ∈ Ω, there exists T D (ω) > 0 and the tempered functions Proof Taking the inner product of (2.9) with v in L 2 (U), we find that (2.14) At the same time, applying Hölder's inequality and Young's inequality, we conclude that where λ 1 is the first eigenvalue of -with Dirichlet boundary value in (2.15). Thus, (2.13)-(2.16) imply that Using the Poincaré inequality ∇v 2 ≥ λ 1 v 2 in the above result, we have Substituting ω by θ -t ω for above inequality and using (2 It is obvious that e 2b 0 -t |z(θ τ ω)| dτ is tempered, that is, there exists a random variable r 3 (ω) such that e 2b 0 -t |z(θ τ ω)| dτ ≤ r 3 (ω). In fact, is also tempered. Moreover, it follows from the properties of the Ornstein-Uhlenbeck process that Hence, combining with the above results, we set then (2.10) holds.
Combining the boundedness of solutions in H 1 0 (U) given in Lemma 2.10 with the Sobolev compact embedding H 1 0 (U) → L 2 (U), it is easy to obtain the compactness of solutions in L 2 (U). Thus, by Theorem 2.8 we obtain the following result.

Lemma 2.11
Assume that g ∈ L 2 (U) and (1.2)-(1.4) hold. Then the continuous random dynamical system Φ generated by (1.1) has a unique D-random attractor A, that is, for Pa.e. ω ∈ Ω, A is nonempty, compact, invariant and D-pullback attracting in the topology of L 2 (U).

Uniform estimates of solutions
In this section, the estimates on the higher order integrability for the difference of solutions near initial time will be given. At the same time, we also prove other corresponding results. For the sake of convenience, we choose C as the positive constant which may be different from line to line or in the same line in our paper.
Due to v(t) = α(θ t ω)u(t) with α(θ t ω) = e -bz(θ t ω) , we convert Eq. (3.2) into the equation Our proof will be completed in two steps. We can justify the following estimates by means of the Faedo-Galerkin approximation procedure.
• Assume that (A k ) and (B k ) hold for k ≥ 2. Next, we will prove that (A k+1 ) and (B k+1 ) hold.
Proof The proof is similar to the proof of Theorem 4.5 (from [21]), so we omit it.

L p -Pullback attracting set
In this subsection, we will make uniform estimates for the solutions of Eq. (1.1) so that we prove the existence of a bounded random absorbing set in L p (U) (p ≥ 2). Proof Taking the inner product of (2.9) with |v| p-2 v in L 2 (U), we find that Using (1.3), Hölder's inequality and Young's inequality, we have where c 4 , c 5 are positive constants, and Now, choosing t ≥ T D (ω) (T D (ω) to be the positive number in Lemma 2.9 and integrating (3.39) over (s, t + 1) with respect to t, we obtain Next, integrating (3.40) over (t, t + 1) with respect to s, we have Replacing ω by θ -t-1 ω and using (2.8) and (2.11), we conclude that where ρ 5 (ω) = e pλ 1 2 r p (ω)ρ 4 (ω), that is, for ω ∈ Ω, Therefore, B(ω) is a random absorbing set for Φ in L p (U). Proof Using the interpolation inequality, Theorem 3.1, (2.4), (2.8) and (3.7), we have the following inequality: where r 0 is given by Theorem 3.5. From the above result and Lemma 3.3, it is obvious that the (L 2 (U), L 2 (U)) D-pullback random attractor A ∈ D is compact in L p (U) for P-a.e. ω ∈ Ω.
Thus, we finish the proof of (3.43), which implies that (3.42) holds. Proof Based on Theorem 3.1 and Theorem 3.5, we can utilize the same approach with Theorem 5.5 of [21] and obtain this result. So we omit it.
Combining Theorem 3.5 with Theorem 3.6 and the existence of the absorbing set (Lemma 2.10) in H 1 0 (U), we easily find the existence of a (L 2 , H 1 0 ) D-pullback random attractor.