Dynamics of a stochastic fractional nonlocal reaction-diffusion model driven by additive noise

Dynamics of a stochastic fractional nonlocal reaction-diffusion model driven by additive noise Lingyu Li∗1, Zhang Chen†1and Tomás Caraballo‡§2 1School of Mathematics, Shandong University, Jinan, Shandong 250100, People’s Republic of China 2Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Tarfia s/n, 41012-Sevilla, Spain October 7, 2022 Abstract. In this paper, we are concerned with the long-time behavior of stochastic fractional nonlocal reaction-diffusion equations driven by additive noise. We use the techniques of random dynamical systems to transform the stochastic model into a random one. To deal with the new nonlocal term appeared in the transformed equation, we first use a generalization of Peano’s theorem to prove the existence of local solutions, and then adopt the Galerkin method to prove existence and uniqueness of weak solutions. Next, the existence of pullback attractors for the equation and its associated Wong-Zakai approximation equation driven by colored noise are shown, respectively. Furthermore, we establish the upper semi-continuity of random attractors of the Wong-Zakai approximation equation as δ→0+. Keywords and phrases: Stochastic fractional nonlocal reaction-diffusion equation, additive noise, random attractor, colored noise, upper semi-continuity. 1 Introduction Nonlocal problems modeled by partial differential equations have been extensively studied in [1,2,6,7, 8,14,17,22,23,38,45] and the references therein. Chipot et al. [13] studied the behavior of a large class of nonlocal nonlinear elliptic problems as follows        −aZD u∆u+λu =f, in D, ∂nu+γZD0 u= 0,on ∂D, ∗E-mail address: [email protected] †E-mail address: zc[email protected] ‡E-mail address: [email protected] §Corresponding author. 1 where Dis a bounded open subset of RN, the boundary ∂D is Lipschitz, D0⊂D,λ > 0, the functions a and γbelong to CR;R+,f∈L2(D)and ∂nuis the normal derivative of u. Instead of considering the nonlocal term aZD u, Chipot et al. [15,16] extended this term to a more general nonlocal operator a(l(u)) where the functional l∈ L L2(D); R, i.e. for some g∈L2(D) l(u) = lg(u) = ZD g(x)u(x)dx. Then the following nonlocal parabolic equation ∂u ∂t −a(l(u))∆u=f received much attention. More precisely, the case in which fis independent of uwas considered in [11,12], where existence and uniqueness of solutions, as well as the long time dynamics, are established under some suitable assumptions. Furthermore, as the nonlocal operator is not acting globally in the whole domain in [36,37] but contained in a ball centered on each position point, radial solutions, bifurcation analysis, branch of solutions and their stability are studied. When fdepends on uin a semilinear form, making use of a fixed point theorem, the existence and uniqueness of a weak solution were proved in [33] for a semilinear problem with nonlocal diffusion where the domain has smooth boundary and the term fis Lipschitz continuous. In addition, the existence and uniqueness of periodic solutions were also analyzed. Later, the existence and regularity of pullback attractors were studied for a deterministic non-autonomous parabolic equation with nonlocal diffusion and additional non-autonomous terms (see [8]). Based on [8], time delays were taken into account in non-autonomous nonlocal partial differential equations where the time-dependent term h(t, ut)involves delays, one can see [45] for more details. It is worth noting that random attractors play an important role in the study of dynamical behavior of stochastic partial differential equations. One can refer to [3,4,5,9,10,18,19,27] and the references therein. Applying the theory of random dynamical systems, stochastic nonlocal partial differential equations with linear noise (multiplicative noise and additive noise) were investigated in [42], where the well-posedness and long-time behavior of solutions were exploited. Moreover, the well-posedness and asymptotic behavior of a class of stochastic nonlocal partial differential equations driven by nonlinear white noise were studied in [44], as well as the random non-autonomous problem driven by colored noise. As special cases, Wong-Zakai approximation models with additive and multiplicative noise possess random attractors which converge upper-semicontinuously to the respective random attractors of the stochastic equations driven by standard Brownian motions. We also remark that more results can be found in [21,24,26,39] for colored noise and [32,34,40,51] for Wong-Zakai approximations. Notice that the Laplacian operator in these aforementioned papers is the standard one. When γ∈(0,1), the operator (−∆)γis called the fractional Laplacian. The relationship between the standard Laplacian and the fractional Laplacian follows from the properties that lim γ→1−(−∆)γu=−∆uand lim γ→0+(−∆)γu=u(see [35, Proposition 4.4]). As far as we are aware, great attention has been devoted to the study of fractional partial differential equations, see, e.g., [25,28,30,31,41,46,49,50] and the references therein. In fact, there are different definitions about fractional Laplacian operator, one being called integral fractional operator and the other spectral fractional operator. More details on fractional Laplacian can be seen in [35]. Very recently, [43] dealt with stochastic nonlocal reaction-diffusion equations driven by multiplicative noise which focused on the integral fractional operator. Since the 2 fractional Laplacian (−∆)γleads to not enough dissipativity, the regularity of solutions is proved by a priori estimates instead of obtaining energy estimates directly. Motivated by these papers above, in this work we consider the following non-autonomous stochastic fractional nonlocal reaction-diffusion equation driven by additive white noise,          ∂u ∂t +a(l(u))(−∆)γu=f(u) + h(t) + φdW dt ,in O × (τ, +∞), u= 0,on ∂O × (τ, +∞), u(x, τ) = uτ(x),in O, (1.1) where Ois a smooth bounded domain of Rn, τ ∈R,l∈ L L2(O); R,h∈L2 loc R;L2(O),φ∈Hγ(O)∩ H2(O)and a∈CR;R+satisfies that there exist two positive constants mand Msuch that m≤a(s)≤M, ∀s∈R,(1.2) and its corresponding approximation by colored noise given by          ∂uδ ∂t +a(l(uδ))(−∆)γuδ=f(uδ) + h(t) + φζδ(θtω),in O × (τ, +∞), uδ= 0,on ∂O × (τ, +∞), uδ(x, τ) = uδ,τ (x),in O, (1.3) where the process ζδ(θtω)is the so-called colored noise (see Section 5 for more details). Our main purposes are to prove the existence and uniqueness of weak solutions of both problems, as well as the existence and the upper semi-continuity of random attractors of problem (1.3) to (1.1) when δgoes to zero. Unlike the case of stochastic fractional nonlocal reaction-diffusion equation driven by multiplicative white noise, there appears a new nonlocal term in the process of transforming the stochastic equation (1.1) into a random one by means of the Ornstein-Uhlenbeck transformation. Thus, the existence and uniqueness of solutions become much more difficult to be proved. In this case, we present a more detailed proof of the well-posedness of the solutions. We first need to apply a generalization of Peano’s theorem to prove the existence of local solutions, and then use the well-known Galerkin method to prove existence and uniqueness of weak solutions. In what follows, we establish the existence of pullback attractors of problem (1.1) and its associated problem driven by colored noise (1.3), respectively. Finally, we give the upper semi-continuity of attractors. The structure of the paper is as follows. In Section 2, we first recall the basic concepts on random dynamical systems. Next, we present some ergodic properties of the Ornstein-Uhlenbeck process and the fractional Laplacian operator, respectively. And then we transform the fractional stochastic nonlocal problem into a random one via the Ornstein-Uhlenbeck process. Section 3 is devoted to proving the well-posedness of problem (1.1). In Section 4, we show the existence of pullback attractors of problem (1.1) by obtaining the existence of random absorbing set and the asymptotic compactness of the random dynamical system which is generated by solutions to problem (1.1). Similar to Section 4, the existence of pullback attractors of problem driven by colored noise associated with (1.1) is proved in Section 5. In the last section, the convergence of solutions and attractors of problem (1.1) are discussed as δ→0+. 2 Preliminaries In this section, we will recall some basic concepts related to non-autonomous random dynamical systems and properties of Ornstein-Uhlenbeck processes which will be used throughout this paper. One can refer 3 to [3,48] for more details. Besides, the concept and some properties of integral fractional Laplacian operator are shown as follows. 2.1 Non-autonomous random dynamical systems Let (X, k·kX)be a complete separable metric space, and (Ω,F,P,(θt)t∈R)an ergodic metric dynamical system, where Ω = C0(R;R) = {ω∈C(R;R) : ω(0) = 0},Fis the Borel-σ-algebra induced by the compact-open topology of Ω,Pis the Wiener measure on (Ω,F)and θ={θt}t∈Ris a measurable flow on Ω, which is defined by θ: (R×Ω,F ⊗ B(R)) →(Ω,F), θ0= idΩ, θt+s=θt◦θs, t, s ∈R.(2.1) The Wiener shift operators which form the flow θ θtω(·) = ω(·+t)−ω(t), t ∈R, ω ∈Ω leave the Wiener measure Pinvariant. More precisely, Pis ergodic with respect to θ. Since the above probability space is canonical we have for a Wiener process and its shift operator W(t, ω) = ω(t), W (t, θsω) = ω(t+s)−ω(s) = W(t+s, ω)−W(s, ω). It is important to note that the measurability in (2.1) is not true if we replace Fby its completion, see Appendix A3 of [3]. Definition 2.1. A continuous non-autonomous random dynamical system on X over (Ω,F,P, (θt)t∈R)is a BR+× F × B(X),B(X)-measurable mapping: ϕ:R+×R×Ω×X→X, (·, τ, ·,·)7→ ϕ(·, τ, ·,·) such that ϕsatisfies: (1) ϕ(0, τ, ω, ·)is identity on X; (2) ϕ(t+s, τ, ω, ·) = ϕ(t, τ +s, θsω, ϕ(s, τ, ω, ·)) for all s, t ≥0; (3) ϕ(t, τ, ω, ·) : X→Xis continuous for all t≥0. Definition 2.2. A set-valued mapping D(τ, ·):Ω→2X,ω7→ D(τ, ω)is said to be a random set, if for every τ∈R, the mapping ω7→ d(u, D(τ, ω)) is measurable for any u∈X, moreover, {D(τ, ω) : τ∈ R, ω ∈Ω}is also closed for each ω∈Ω. Definition 2.3. A random set {D(τ, ω) : τ∈R, ω ∈Ω}is called tempered, if for every τ∈R, ω ∈Ω and β > 0 lim t→+∞e−βtd(D(τ−t, θ−tω)) = 0, where d(D) = sup {kbkX:b∈D}. Definition 2.4. A random set {B(τ, ω) : τ∈R, ω ∈Ω}is said to be a random pullback absorbing set if for any tempered random set {D(τ, ω) : τ∈R, ω ∈Ω}, every τ∈R, ω ∈Ω, there exists t0such that for each t≥t0, ϕ(t, τ −t, θ−tω, D (τ−t, θ−tω)) ⊂B(τ, ω). 4 Definition 2.5. Let Dbe a collection of random subset of X,ϕis said to be D-pullback asymptotically compact in X, if for all τ∈R,ω∈Ω,{ϕ(tn, τ −tn, θ−tnω, xn)}+∞ n=1 has a convergent subsequence in X whenever tn→+∞,and xn∈B(τ−tn, θ−tnω)with {B(τ, ω) : τ∈R, ω ∈Ω}∈D. Definition 2.6. A family A={A(τ, ω) : τ∈R, ω ∈Ω} ∈ D is called a D-pullback attractor for ϕif for every τ∈R, ω ∈Ω, (1) A(τ, ω)is compact in Xand for every τ∈R,ω7→ d(X, A(τ, ω)) is measurable; (2) Ais invariant: ϕ(t, τ, ω, A(τ, ω)) = A(t+τ, θtω),for every τ∈R, ω ∈Ω; (3) Aattracts all member of D: for every B∈ D, τ ∈Rand ω∈Ω,we have lim t→∞ dX(ϕ(t, τ −t, θ−tω, B (τ−t, θ−tω)) ,A(τ, ω)) = 0, where dXis Hausdorff semi-distance in X. The following result is a sufficient condition ensuring existence and uniqueness of random attractors for non-autonomous random dynamical system. Theorem 2.1. ([48, Theorem 2.23]) Let ϕbe a continuous non-autonomous random dynamical system on X over (Ω,F,P,(θt)t∈R), if there exists a closed random tempered absorbing set {B(τ, ω) : τ∈R, ω ∈Ω} of ϕand ϕis asymptotically compact in X, then the D-pullback attractor Ais unique and is given by, A(τ, ω) = \ r≥0[ t≥r ϕ(t, τ −t, θ−tω, B (τ−t, θ−tω)), τ ∈R, ω ∈Ω. 2.2 Ornstein-Uhlenbeck process Consider the following linear one-dimensional stochastic differential equation dZ +Zdt =dW (2.2) with initial value Z(t0) = Z0. Then Z(t, t0, Z0) =e−(t−t0)Z0+Zt t0 e−(t−s)dW(s) =e−(t−t0)Z0+W(t)−e−(t−t0)W(t0)−Zt t0 e−(t−s)W(s)ds is the unique solution of (2.2). Let z(θtω) = lim t0→−∞ Z(t, t0, Z0) = W(t)−Zt −∞ e−(t−s)W(s, ω)ds, then Z(t, ω) = z(θtω)is a stationary solution of (2.2). Moreover, the random variable z(ω)satisfies the following properties: Lemma 2.1. (See [9]) (1) There exists a{θt}t∈R-invariant set Ω1of full measure such that lim t→±∞ |z(θtω)| |t|= 0, lim t→±∞ 1 tZt 0 z(θrω)dr = 0, lim t→±∞ 1 tZt 0 |z(θrω)|dr =E|z|<+∞. (2) The mapping t7→ z(θtω)is continuous. 5 2.3 Fractional settings Let Sbe the Schwartz space of rapidly decaying C∞functions in Rnand 0< γ < 1. For every u∈ S, the fractional Laplacian operator (−∆)γat the point xis defined by (−∆)γu(x) = C(n, γ)P.V. ZRn u(x)−u(y) |x−y|n+2γdy, x ∈Rn,(2.3) where P.V. is a commonly used abbreviation for “in the principal value sense” and C(n, γ) = ZRn 1−cos ξ1 |ξ|n+2γdξ−1with ξ={ξ1, ξ2,· · · , ξn} ∈ Rn. By a standard change of variable, (2.3) is equivalent to (−∆)γu(x) = −C(n, γ) 2ZRn u(x+y) + u(x−y)−2u(x) |y|n+2γdy, x ∈Rn. For any 0< γ < 1, the fractional Sobolev space Wγ,2(Rn) := Hγ(Rn)is defined by Hγ(Rn) = u∈L2(Rn) : ZRnZRn |u(x)−u(y)|2 |x−y|n+2γdxdy < +∞, equipped with the norm kukHγ(Rn)=ZRn |u(x)|2dx +ZRnZRn |u(x)−u(y)|2 |x−y|n+2γdxdy1 2 and inner product (u, v)Hγ(Rn)=ZRn u(x)v(x)dx +ZRnZRn (u(x)−u(y))(v(x)−v(y)) |x−y|n+2γdxdy, ∀u, v ∈Hγ(Rn). From now on, we denote by k · k and (·,·)the norm and the inner product of L2(Rn), respectively. Moreover, we denote by k·k˙ Hγ(Rn)the Gagliardo semi-norm of Hγ(Rn)which is defined by kuk2 ˙ Hγ(Rn)=ZRnZRn |u(x)−u(y)|2 |x−y|n+2γdxdy, u ∈Hγ(Rn). Thus, kuk2 Hγ(Rn)=kuk2+kuk2 ˙ Hγ(Rn)=kuk2+2 C(n, γ)k(−∆)γuk2for all u∈Hγ(Rn). Since (−∆)γis a nonlocal operator, we here see the homogeneous Dirichlet boundary (1.1) as u= 0 on Rn\O instead of u= 0 only on ∂O. From this perspective, we define two spaces V={u∈Hγ(Rn) : u= 0 a.e. on Rn\O} and H=u∈L2(Rn) : u= 0 a.e. on Rn\O}. Furthermore, let b:V×V→Rbe a bilinear form given by, for v1, v2∈V, b(v1, v2) = µ(v1, v2) + C(n, γ) 2ZRnZRn (v1(x)−v1(y)) (v2(x)−v2(y)) |x−y|n+2γdxdy, where µis a constant chosen later. For convenience, we associate an operator A:V→V∗with bsuch that (A(v1), v2)V∗,V =b(v1, v2),for all v1, v2∈V, where V∗is the dual space of Vand (·,·)V∗,V is the duality paring of V∗and V. Since Ais injective and surjective, the inverse A−1:V∗→Vis well-defined. On the other hand, the embedding V ,→His compact and H=H∗⊂V∗yield that A−1:H→V⊂His a symmetric 6 compact operator. Then, by means of the Hilbert-Schmidt theorem, Ahas a family of eigenfunctions {wj}+∞ j=1 which forms an orthonormal basis of H. Moreover, if λjdenotes the eigenvalue of operator A corresponding to wj, i.e., Awj=λjwj, j = 1,2,· · · , then λjsatisfies 0< µ < λ1≤λ2≤ · · · ≤ λj→+∞as j→+∞. 2.4 Setting of the problem Let f∈C(R)and assume that there exist positive constants Cf, κ, β1and β2>1such that (f(s)−f(r))(s−r)≤Cf(s−r)2,∀s, r ∈R,(2.4) and −κ−β1|s|2≤f(s)s≤κ−β2|s|2,∀s∈R.(2.5) Now we consider its associated Nemytskii operator ˜ f:L2(O)→L2(O)which is defined as ˜ f(u)(x) = f(u(x)) for all u∈L2(O)and x∈ O. By simplicity, we identify fwith ˜ fwhen no confusion is possible. Next, we fix a number µsuch that 0<µ<min β1 m,β2 M,C(n, γ) 2,and define the operator F: L2(O)→L2(O)by F(u) = a(l(u))µu +f(u), which is given by F(u)(x) := a(l(u))µu(x) + f(u(x)), x ∈ O. It is easy to check that Fis continuous in L2(O). In what follows, we will write F(u(x)) instead of F(u)(x), although this notation does not match the typical one for the Nemytskii operator associated to a real valued function. By (2.5) and the Young inequality, we have, for all u∈L2(O)and x∈ O, −κ−(β1−mµ)|u(x)|2≤F(u(x))u(x)≤κ−(β2−Mµ)|u(x)|2.(2.6) From (2.6), we deduce that there exists α > 0such that |F(u(x))| ≤ α(|u(x)|+ 1) .(2.7) Then, problem (1.1) can be written as          ∂u ∂t +a(l(u))(−∆)γu+a(l(u))µu =F(u) + h(t) + φdW dt , x ∈ O, t > τ, u(x, t)=0, x ∈Rn\O, t > τ, u(x, τ) = uτ(x), x ∈ O. (2.8) To study the dynamics of problem (2.8), we first transform the stochastic fractional nonlocal differential equation (2.8) into a random one by doing a change of variable. To start off, we denote by u(·) := u(·, τ, ω, uτ)the solution to problem (2.8), we do the change of variable v(t) = u(t)−φz (θtω)with 7 v(τ) = u(τ)−φz (θτω), then we obtain                  ∂v ∂t +a(l(v) + z(θtω)l(φ))((−∆)γ+µ)v+a(l(v) + z(θtω)l(φ))z(θtω) ((−∆)γ+µ)φ =F(v+φz (θtω)) + h(t) + φz (θtω), x ∈ O, t > τ, v(x, t)=0, x ∈Rn\O, t > τ, v(x, τ) = uτ(x)−φz (θτω) := vτ(x), x ∈ O. (2.9) Notice that this change of variable is, in reality, a homeomorphism and therefore a conjugation (see [3] for more details). 3 Well-posedness of problem (2.8) In this section, we are interested in proving the well-posedness of weak solution of problem (2.8). To this end, we first introduce the definition of a weak solution of problem (2.9), and then prove the existence and uniqueness of weak solution of problem (2.9). For that, we need the following assumption to obtain the existence of local solution of problem (2.9). (H1)For any t∈[τ, T], the mapping t→h(t)is measurable. Definition 3.7. Given φ∈Vand the initial value vτ∈H. A weak solution to equation (2.9)is a function v(·) := v(·, τ, ω, vτ)that belongs to L2([τ, +∞); V)∩L∞([τ, +∞); H), such that for every ζ∈V, d dt(v, ζ) + a(l(v) + z(θtω)l(φ)) C(n, γ) 2ZRnZRn (v(x)−v(y))(ζ(x)−ζ(y)) |x−y|n+2γdxdy +µ(v, ζ) +a(l(v) + z(θtω)l(φ)) z(θtω)C(n, γ) 2ZRnZRn (φ(x)−φ(y))(ζ(x)−ζ(y)) |x−y|n+2γdxdy +µ(φ, ζ) =ZO F(v+φz (θtω)) ζ(x)dx +ZO h(t)ζ(x)dx +ZO z(θtω)φ(x)ζ(x)dx, (3.1) where the above equation must be understood in the sense of distribution on (τ, +∞). Theorem 3.1. Suppose (H1)holds, a∈CR;R+is locally Lipschitz and satisfies (1.2),f∈C(R) fulfills (2.4)-(2.5), which implies F:L2(Rn)→L2(Rn)satisfies (2.6)-(2.7). In addition, h∈L2 loc (R;H) and l∈ L L2(O); R. Then, for every τ∈R, ω ∈Ωand each initial datum vτ∈H, there exists a unique weak solution v(t, τ, ω, vτ)to problem (2.9)in the sense of Definition 3.7. Proof. Step 1: Existence of local solution. Using spectral theory, there exists a sequence {wi}i≥1 which is a Hilbert basis of Hcomposed by the eigenfunctions of A. Firstly, we consider the function vn(t) := vn(t, τ, ω, vτ) = n X j=1 ϕnj(t)wjfor all n≥1, which satisfies the following system          d dt (vn(t), wj) + a(l(vn(t)) + z(θtω)l(φ)) (Avn(t), wj)V∗,V +a(l(vn(t)) + z(θtω)l(φ)) z(θtω) (Aφ, wj)V∗,V = (F(vn(t) + φz (θtω)) , wj)+(h(t), wj) + z(θtω) (φ, wj), (vn(τ), wj) = (vτ, wj), j = 1,2,· · · , n. (3.2) Notice that the above equation is a Cauchy problem for the following ordinary differential system in Rn, ϕ0 nj(t) + λja(l(vn(t)) + z(θtω)l(φ)) ϕnj(t) + a(l(vn(t)) + z(θtω)l(φ)) z(θtω) (Aφ, wj)V∗,V = (F(vn(t) + φz (θtω)) , wj)+(h(t), wj) + z(θtω) (φ, wj), j = 1,2,· · · , n, (3.3) 8 where t≥τ,λjis the eigenvalue associated to the eigenfunction wj, and the vector (ϕn1, ϕn2,· · · , ϕnn) is the unknown variable. Define g:T → Rn (t, x)7→ −λ1ξ(x)x1−ξ(x)z(θtω) (Aφ, w1)V∗,V + F n X i=1 xiwi+φz (θtω)!, w1!+ (h(t), w1) + z(θtω) (φ, w1), . . . , −λnξ(x)xn−ξ(x)z(θtω) (Aφ, wn)V∗,V + F n X i=1 xiwi+φz (θtω)!, wn!+ (h(t), wn) + z(θtω) (φ, wn)!, where T=n(t, x)∈[τ, T ]×Rn:τ≤t≤T, x−(vτ, w1),...,(vτ, wn)≤do for any fixed d∈R+and x= (x1, . . . , xn)7→ ξ(x) = a l n X i=1 xiwi!+z(θtω)l(φ)!. Next, we need to prove that gis a Caratheodory function. Firstly, let xbe fixed. Following the continuity of zand f, as well as assumption (H1), the function gj(·, x) = −λjξ(x)xj−ξ(x)z(θ·ω) (Aφ, wj)V∗,V + F n X i=1 xiwi+φz (θ·ω)!, wj!+(h(·), wj)+z(θ·ω) (φ, wj) is measurable. Therefore, the function g(·, x)is measurable. Secondly, we need to verify that the function g(t, ·)is continuous a.e. t∈[τ, T ]. In fact, the functions ξand x∈Rn7→ F n X i=1 xiwi+φz (θtω)!, wj!are continuous, then we can obtain gj(t, x) = −λjξ(x)xj−ξ(x)z(θtω) (Aφ, wj)V∗,V + F n X i=1 xiwi+φz (θtω)!, wj!+(h(t), wj)+z(θtω) (φ, wj) is a continuous function with respect to x. Thirdly, we are going to prove that for each (t, x)∈ T , there exist a neighborhood V(t, x)and a Lebesgue measurable function Q(·)∈L1(τ, T )such that for any (s, y)∈V(t, x), |g(s, y)| ≤ Q(s). From the definition of T, there exists a constant CTsuch that |x| ≤ d+|((vτ, w1),...,(vτ, wn))| ≤ CT. By (2.7), we have  F n X i=1 xiwi+φz (θtω)!, wj!≤αCT n X i=1 kwik!kwjk+α|O|1/2+|z(θtω)|kφkkwjk.(3.4) Observe that taking into account the expression of ξand the continuity of the function ain the compact interval I0:= "−klkCT n X i=1 kwik−|z(θtω)|klkkφk,klkCT n X i=1 kwik+|z(θtω)|klkkφk#, there exists a constant M0>0such that ξ(x)≤M0,∀x∈Rn:|x| ≤ CT. 9 Replacing ωby θ−τωand then changing variables, we obtain the following result kv(σ, τ −t, θ−τω, vτ−t)k2+mC(n, γ)Zσ τ−t eµm 2(s−σ)kv(s, τ −t, θ−τω, vτ−t)k2 ˙ Hγ(Rn)ds ≤eµm 2(τ−t−σ)kvτ−tk2+C1Zσ−τ −t eµm 2(s+τ−σ)|z(θsω)|2kφk2ds +4µM2 mZσ−τ −t eµm 2(s+τ−σ)|z(θsω)|2k(−∆)γφk2ds +2 µm Zσ−τ −t eµm 2(s+τ−σ)kh(s+τ)k2ds +2C2 µm (1 −eµm 2(τ−t−σ)). Since v(t, τ −t, θ−τω, vτ−t) = u(t, τ −t, θ−τω, uτ−t)−φz (θt−τω)with vτ−t=uτ−t−φz (θ−tω), we have ku(σ, τ −t, θ−τω, uτ−t)k2 ≤2eµm 2(τ−t−σ)kuτ−t−φz (θ−tω)k2+ 2C1Zσ−τ −t eµm 2(s+τ−σ)|z(θsω)|2kφk2ds +8µM2 mZσ−τ −t eµm 2(s+τ−σ)|z(θsω)|2k(−∆)γφk2ds +4 µm Zσ−τ −t eµm 2(s+τ−σ)kh(s+τ)k2ds +4C2 µm (1 −eµm 2(τ−t−σ))+2|z(θσ−τω)|2kφk2 ≤4eµm 2(τ−t−σ)kuτ−tk2+ 4eµm 2(τ−t−σ)kφk2|z(θ−tω)|2 + 2C1Zσ−τ −t eµm 2(s+τ−σ)|z(θsω)|2kφk2ds +8µM2 mZσ−τ −t eµm 2(s+τ−σ)|z(θsω)|2k(−∆)γφk2ds +4 µm Zσ−τ −t eµm 2(s+τ−σ)kh(s+τ)k2ds +4C2 µm (1 −eµm 2(τ−t−σ))+2|z(θσ−τω)|2kφk2. (4.9) Note that uτ−t∈B(τ−t, θ−tω)and B={B(τ, ω) : τ∈R, ω ∈Ω}is tempered, there exists T1= T1(τ, ω, σ, B, m, µ)>0such that for all t≥T1, 4eµm 2(τ−t−σ)kuτ−tk2≤1 2.(4.10) By Lemma 2.1, there exists T2=T2(ω, m, µ)≥T1such that for all t≥T2, 4eµm 2(τ−t−σ)kφk2|z(θ−tω)|2≤1 2.(4.11) Since φ∈V, we deduce that 8µM2 mZσ−τ −t eµm 2(s+τ−σ)|z(θsω)|2k(−∆)γφk2ds < +∞.(4.12) On the other hand, (4.1) and the continuity of z(θ·ω)indicate that 2C1Zσ−τ −t eµm 2(s+τ−σ)|z(θsω)|2kφk2ds +4 µm Zσ−τ −t eµm 2(s+τ−σ)kh(s+τ)k2ds < +∞.(4.13) 16 By (4.9)-(4.13), we have for all t≥T2, ku(σ, τ −t, θ−τω, uτ−t)k2 ≤1+2C1Zσ−τ −t eµm 2(s+τ−σ)|z(θsω)|2kφk2ds +8µM2 mZσ−τ −t eµm 2(s+τ−σ)|z(θsω)|2k(−∆)γφk2ds +4 µm Zσ−τ −t eµm 2(s+τ−σ)kh(s+τ)k2ds +4C2 µm + 2|z(θσ−τω)|2kφk2. Therefore, the proof is complete. Based on Lemma 4.1, we will prove that (2.8) has a D-pullback absorbing set in H. Lemma 4.2. Suppose the conditions of Theorem 3.1 and (H2)hold. Then the continuous cocycle Φhas aD-pullback absorbing set B={B(τ, ω) : τ∈R, ω ∈Ω}∈D, which is given by B(τ, ω) = u∈H:kuk2≤R(τ, ω), where R(τ, ω) =1 + 2C1Z0 −∞ eµm 2s|z(θsω)|2kφk2ds +8µM2 mZ0 −∞ eµm 2s|z(θsω)|2k(−∆)γφk2ds +4 µm Z0 −∞ eµm 2skh(s+τ)k2ds +4C2 µm + 2|z(ω)|2kφk2. Proof. As a special case of Lemma 4.1 with σ=τ, there exists T=T(τ, ω, B, m, µ, α)>0such that for all t≥T, u(τ, τ −t, θ−τω, uτ−t)∈B(τ, ω). Now we only need to check that Ris tempered, i.e., for any β > 0, lim t→+∞e−βtR(τ−t, θ−tω)=0.(4.14) In fact, e−βtR(τ−t, θ−tω) =e−βt + 2C1e−βt Z0 −∞ eµm 2s|z(θs−tω)|2kφk2ds +8µM2 me−βt Z0 −∞ eµm 2s|z(θs−tω)|2k(−∆)γφk2ds +4 µme−βt Z0 −∞ eµm 2skh(s+τ)k2ds +4C2 µm e−βt + 2e−βt|z(θ−tω)|2kφk2. Combining Lemma 2.1 with (4.2), one can easily check that e−βtR(τ−t, θ−tω)→0as t→+∞. Lemma 4.3. Suppose the conditions of Lemma 4.1 hold. Then the continuous cocycle Φis asymptotically compact in H. Proof. By Lemma 4.1, there exist T=T(τ, ω, B, m, µ, α)>0and C=C(τ, ω, m, µ, α)>0such that, for all t≥Tand uτ∈B(τ−t, θ−tω), ku(τ−1, τ −t, θ−τω, uτ)k ≤ C. (4.15) 17 When tk→+∞(k→+∞)and uτ,k ∈B(τ−tk, θ−tkω), there is K1=K1(τ, ω, B)>0such that for all k≥K1, ku(τ−1, τ −tk, θ−τω, uτ,k)k ≤ C. This shows that {u(τ−1, τ −tk, θ−τω, uτ,k)}is bounded in H. (4.16) Follows from [43, Lemma 3.3], there exists bu∈L2(τ−1, τ;H)such that, up to a subsequence, and for all n∈Nas k→+∞ u(·, τ −tk, θ−τω, uτ,k) =u(·, τ −1, θ−τω, u(τ−1, τ −tk, θ−τω, uτ,k)) →bu(·)in L2([τ−1, τ]; H). (4.17) By choosing a further subsequence (which we do not relabel), it follows from (4.17) that u(s, τ −tk, θ−τω, uτ,k)→bu(s)in Hfor almost all s∈[τ−1, τ],(4.18) which means the sequence {u(s, τ −tk, θ−τω, uτ,k)}has a convergent subsequence in H. The proof is complete. By Lemma 4.2, Lemma 4.3 and Theorem 2.1, we deduce the existence of D-pullback attractors for Φ as stated in the following result. Theorem 4.1. Suppose the conditions of Lemma 4.1 hold. Then the continuous cocycle Φgenerated by problem (2.8)has a unique D-pullback random attractor A={A(τ, ω) : τ∈R, ω ∈Ω}∈Din H. 5 Existence of random attractors to fractional random nonlocal PDEs driven by colored noise In this section, we discuss the following equation driven by colored noise as an approximation to fractional stochastic nonlocal differential equation (1.1),          ∂uδ ∂t +a(l(uδ))(−∆)γuδ=f(uδ) + h(t) + φζδ(θtω),in O × (τ, +∞), uδ= 0,on ∂O × (τ, +∞), uδ(x, τ) = uδ,τ (x),in O, (5.1) where the process ζδ(θtω)is the so-called colored noise. For convenience, we rewrite (5.1) in the following form, similarly as we did with (2.8):          ∂uδ ∂t +a(l(uδ))(−∆)γuδ+a(l(uδ))µuδ=F(uδ) + h(t) + φζδ(θtω), x ∈ O, t > τ, uδ(x, t)=0, x ∈Rn\O, t > τ, uδ(x, τ) = uδ,τ (x), x ∈ O. (5.2) Let us recall some properties of the colored noise which will be useful in our subsequent analysis. Although it can be found in several published works, we prefer to present them here to make our paper more readable. For every ω∈Ωand δ > 0,yδsatisfies dyδ dt =−yδ+ζδ(θtω), 18 which possesses a special solution given by yδ(θtω) = e−tZt −∞ esζδ(θsω)ds. And it has the following properties: Lemma 5.1. [20, Lemma 3.2] (1) The mapping t7→ yδ(θtω)is continuous. For every ω∈Ωand 0< δ ≤1 2, lim t→±∞ |yδ(θtω)| |t|= 0, and lim t→±∞ 1 tZt 0 yδ(θsω) ds= 0. (2) Let τ∈R, ω ∈Ωand T > 0. Then for every ε > 0,there exists δ0=δ0(τ, ω, T, ε)>0such that for all 0< δ < δ0and t∈[τ, τ +T], |yδ(θtω)−z(θtω)|< ε. Performing the change of variable vδ(t) = uδ(t)−φyδ(θtω)with vδ(τ) = uδ(τ)−φyδ(θτω), we obtain                  ∂vδ ∂t +a(l(vδ) + yδ(θtω)l(φ))((−∆)γ+µ)vδ+a(l(vδ) + yδ(θtω)l(φ))yδ(θtω) ((−∆)γ+µ)φ =F(vδ+φyδ(θtω)) + h(t) + φyδ(θtω), x ∈ O, t > τ, vδ(x, t) = 0, x ∈Rn\O, t > τ, vδ(x, τ) = uδ,τ (x)−φyδ(θτω) := vδ,τ (x), x ∈ O. (5.3) Since (5.3) is a random equation, similar to Theorem 3.1 in the previous section, one can prove that problem (5.3) has a unique solution vδ(·) := vδ(·, τ, ω, vτ)∈L2([τ, +∞); V)∩L∞([τ, +∞); H). We now define a cocycle Φδ:R+×R×Ω×H→Hsuch that, for all t∈R+, τ ∈R, ω ∈Ωand uδ,τ ∈H, Φδ(t, τ, ω, uδ,τ ) = uδ(t+τ, τ, θ−τω, uδ,τ ) = vδ(t+τ, τ, θ−τω, vδ,τ ) + φyδ(θtω). Lemma 5.2. Suppose the conditions of Theorem 3.1 and (H2)hold. Then, for every σ∈R,τ∈R, ω∈Ω, the continuous cocycle Φδhas a D-pullback absorbing set Bδ={Bδ(τ, ω) : τ∈R, ω ∈Ω} ∈ D, which is given by Bδ(τ, ω) = uδ∈H:kuδk2≤Rδ(τ, ω), where Rδ(τ, ω) =1 + 2C1Z0 −∞ eµm 2s|yδ(θsω)|2kφk2ds +8µM2 mZ0 −∞ eµm 2s|yδ(θsω)|2k(−∆)γφk2ds +4 µm Z0 −∞ eµm 2skh(s+τ)k2ds +4C2 µm + 2|yδ(ω)|2kφk2. Proof. Taking the inner product of (5.3) with vδ(t) = vδ(t, τ, ω, vδ,τ ), and following (1.2), (2.7) and the Young inequality, we obtain d dt kvδ(t)k2+µm 2kvδ(t)k2+mC(n, γ)kvδ(t)k2 ˙ Hγ(Rn) ≤C1|yδ(θtω)|2kφk2+4µM2 m|yδ(θtω)|2k(−∆)γφk2+2 µmkh(t)k2+C2, (5.4) 19 where C1and C2are showed as Lemma 4.1. Multiplying (5.4) by eµm 2tand integrating over (τ−t, σ)with σ≥τ−t, we have for every ω∈Ω, kvδ(σ, τ −t, ω, vδ,τ−t)k2+mC(n, γ)Zσ τ−t e(µm 2−2α)(s−σ)kvδ(s, τ −t, ω, vδ,τ−t)k2 ˙ Hγ(Rn)ds ≤eµm 2(τ−t−σ)kvδ,τ−tk2+C1Zσ τ−t eµm 2(s−σ)|yδ(θsω)|2kφk2ds +4µM2 mZσ τ−t eµm 2(s−σ)|yδ(θsω)|2k(−∆)γφk2ds +2 µm Zσ τ−t eµm 2(s−σ)kh(s)k2ds +2C2 µm (1 −eµm 2(τ−t−σ)). Replacing ωby θ−τωand then changing variables appropriately, we obtain the following result kvδ(σ, τ −t, θ−τω, vδ,τ−t)k2+mC(n, γ)Zσ τ−t eµm 2(s−σ)kvδ(s, τ −t, θ−τω, vδ,τ−t)k2 ˙ Hγ(Rn)ds ≤eµm 2(τ−t−σ)kvδ,τ−tk2+C1Zσ−τ −t eµm 2(s+τ−σ)|yδ(θsω)|2kφk2ds +4µM2 mZσ−τ −t eµm 2(s+τ−σ)|yδ(θsω)|2k(−∆)γφk2ds +2 µm Zσ−τ −t eµm 2(s+τ−σ)kh(s+τ)k2ds +2C2 µm (1 −eµm 2(τ−t−σ)). Since vδ(t, τ −t, θ−τω, vδ,τ−t) = uδ(t, τ −t, θ−τω, uδ,τ−t)−φyδ(θt−τω)with vδ,τ−t=uδ,τ−t−φyδ(θ−tω), we have kuδ(σ, τ −t, θ−τω, uδ,τ−t)k2 ≤2eµm 2(τ−t−σ)kuδ,τ−t−φyδ(θ−tω)k2+ 2C1Zσ−τ −t eµm 2(s+τ−σ)|yδ(θsω)|2kφk2ds +8µM2 mZσ−τ −t eµm 2(s+τ−σ)|yδ(θsω)|2k(−∆)γφk2ds +4 µm Zσ−τ −t eµm 2(s+τ−σ)kh(s+τ)k2ds +4C2 µm (1 −eµm 2(τ−t−σ))+2kφyδ(θσ−τω)k2 ≤4eµm 2(τ−t−σ)kuδ,τ−tk2+ 4eµm 2(τ−t−σ)kφk2|yδ(θ−tω)|2 + 2C1Zσ−τ −t eµm 2(s+τ−σ)|yδ(θsω)|2kφk2ds +8µM2 mZσ−τ −t eµm 2(s+τ−σ)|yδ(θsω)|2k(−∆)γφk2ds +4 µm Zσ−τ −t eµm 2(s+τ−σ)kh(s+τ)k2ds +4C2 µm (1 −eµm 2(τ−t−σ))+2kφyδ(θσ−τω)k2. Lemma 5.1 and inequality (4.1) imply that there exists T=T(τ, ω, B, σ, m, µ)≥0such that for all t≥T, kuδ(σ, τ −t, θ−τω, uδ,τ−t)k2 ≤1+2C1Zσ−τ −t eµm 2(s+τ−σ)|yδ(θsω)|2kφk2ds +8µM2 mZσ−τ −t eµm 2(s+τ−σ)|yδ(θsω)|2k(−∆)γφk2ds +4 µm Zσ−τ −t eµm 2(s+τ−σ)kh(s+τ)k2ds +4C2 µm + 2kφyδ(θσ−τω)k2. 20 On the other hand, combining Lemma 5.1 with inequality (4.2), one can easily check that Rδis tempered (we omit the details here). Similar to Lemma 4.3, we have the following result on Φδ. Lemma 5.3. Suppose the conditions of Lemma 5.2 hold. Then the continuous cocycle Φδis asymptotically compact in H. By Lemma 5.2 and Lemma 5.3, we deduce the following result about existence of D-pullback attractors for Φδ. Theorem 5.1. Suppose the conditions of Lemma 5.2 hold. Then the continuous cocycle Φδof problem (5.2)has a unique D-pullback random attractor Aδ={Aδ(τ, ω) : τ∈R, ω ∈Ω}∈Din H. 6 Upper semi-continuity of random attractors In this section, we will establish the convergence of solutions and the upper semi-continuity of random attractors of (5.2) as δ→0+. Lemma 6.1. Suppose the conditions of Theorem 3.1 hold. If uδand uare the solutions of problems (5.2)and (2.8), respectively, then for all τ∈R, ω ∈Ω,ε > 0and T > 0, there exist δ0=δ0(τ, ω, T, ε) and C=C(τ, ω, T, µ, m, M, Cf)such that for all 0< δ < δ0and t∈[τ, τ +T], kuδ(t, τ, ω, uδ,τ )−u(t, τ, ω, uτ)k2≤eC(t−τ)kuδ,τ −uτk2+Cε. (6.1) Proof. By (5.3) and (2.9), taking the inner product with vδ(t)−v(t)we have 1 2 d dt kvδ(t)−v(t)k2+a(l(v(t)) + z(θtω)l(φ)) bvδ(t)−v(t), vδ(t)−v(t) ≤[a(l(vδ(t)) + yδ(θtω)l(φ)) −a(l(v(t)) + z(θtω)l(φ))] Avδ(t), vδ(t)−v(t) +[a(l(vδ(t) + yδ(θtω)l(φ))) −a(l(v(t)) + z(θtω)l(φ))] Aφ, vδ(t)−v(t) +F(vδ(t) + φyδ(θtω)) −F(v(t) + φz (θtω)) , vδ(t)−v(t) +yδ(θtω)−z(θtω)φ, vδ(t)−v(t). (6.2) Since ais locally Lipschitz and l∈ L(L2(O); R), the first term on the right-hand side of (6.2) can be bounded by [a(l(vδ(t)) + yδ(θtω)l(φ)) −a(l(v(t)) + z(θtω)l(φ))] Avδ(t), vδ(t)−v(t) ≤La(R)klkhkvδ(t)−v(t)k+kφk|yδ(θtω)−z(θtω)|i(−∆)γvδ(t) + µvδ(t), vδ(t)−v(t) ≤La(R)klkhkvδ(t)−v(t)k+kφk|yδ(θtω)−z(θtω)|i ×C(n, γ) 2kvδ(t)k˙ Hγ(Rn)kvδ(t)−v(t)k+µkvδ(t)kkvδ(t)−v(t)k ≤C(n, γ) 2L2 a(R)klk2kvδ(t)k˙ Hγ(Rn)+kvδ(t)k ×kvδ(t)−v(t)k2+kvδ(t)−v(t)kkφk|yδ(θtω)−z(θtω)| ≤C(n, γ) 2L2 a(R)klk2kvδ(t)k˙ Hγ(Rn)+kvδ(t)k3 2kvδ(t)−v(t)k2+1 2kφk2|yδ(θtω)−z(θtω)|2. (6.3) 21 Similarly, for the second term on the right-hand side of (6.2), we have |([a(l(vδ(t) + yδ(θtω)l(φ))) −a(l(v(t)) + z(θtω)l(φ))] Aφ, vδ(t)−v(t))| ≤La(R)klkhkvδ(t)−v(t)k+kφk|yδ(θtω)−z(θtω)|i(−∆)γφ+µφ, vδ(t)−v(t) ≤La(R)klkhkvδ(t)−v(t)k+kφk|yδ(θtω)−z(θtω)|i ×C(n, γ) 2kφk˙ Hγ(Rn)kvδ(t)−v(t)k+µkφkkvδ(t)−v(t)k ≤C(n, γ) 2L2 a(R)klk2kφk˙ Hγ(Rn)+kφk ×kvδ(t)−v(t)k2+kvδ(t)−v(t)kkφk|yδ(θtω)−z(θtω)| ≤C(n, γ) 2L2 a(R)klk2kφk˙ Hγ(Rn)+kφk3 2kvδ(t)−v(t)k2+1 2kφk2|yδ(θtω)−z(θtω)|2. (6.4) In terms of f∈C(R), the locally Lipschitz continuity of aand l∈ L(L2(O); R), for the third term on the right-hand side of (6.2), we obtain F(vδ(t) + φyδ(θtω)) −F(v(t) + φz (θtω)) , vδ(t)−v(t) =F(vδ(t) + φyδ(θtω)) −F(v(t) + φyδ(θtω)) , vδ(t)−v(t) +F(v(t) + φyδ(θtω)) −F(v(t) + φz (θtω)) , vδ(t)−v(t) ≤µMkvδ(t)−v(t)k2+µLa(R)klk(kvkC([τ,τ+T];H)+|yδ(θtω)|kφkL∞(O))kvδ(t)−v(t)k2 +µM|yδ(θtω)−z(θtω)|kφkkvδ(t)−v(t)k +µLa(R)klk|yδ(θtω)−z(θtω)|kφkh|kvkC([τ,τ+T];H)+|z(θtω)|kφkL∞(O)ikvδ(t)−v(t)k +Cfkvδ(t)−v(t)k2+Cf|yδ(θtω)−z(θtω)|kφkkvδ(t)−v(t)k ≤µM +µLa(R)klk(kvkC([τ,τ+T];H)+|yδ(θtω)|kφkL∞(O)) + 3 2+Cfkvδ(t)−v(t)k2 +1 2|yδ(θtω)−z(θtω)|2kφk2µ2M2+µ2L2 a(R)klk2h|kvkC([τ,τ+T];H)+|z(θtω)|kφkL∞(O)i2. (6.5) Applying the Young inequality to the last term on the right-hand side of (6.2), we have yδ(θtω)−z(θtω)φ, vδ(t)−v(t)≤|yδ(θtω)−z(θtω)|kφkkvδ(t)−v(t)k ≤1 2kvδ(t)−v(t)k2+1 2|yδ(θtω)−z(θtω)|2kφk2. (6.6) Combining (6.2)-(6.6), we have the following estimation d dt kvδ(t)−v(t)k2 ≤3C(n, γ) 2L2 a(R)klk2kvδ(t)k˙ Hγ(Rn)+kvδ(t)k+kφk˙ Hγ(Rn)+kφk + 2µM +µLa(R)klk(kvkC([τ,τ+T];H)+|yδ(θtω)|kφkL∞(O)) + 3 2+Cf+ 1kvδ(t)−v(t)k2 +|yδ(θtω)−z(θtω)|2kφk2C(n, γ) 2L2 a(R)klk2kvδ(t)k˙ Hγ(Rn)+kvδ(t)k+kφk˙ Hγ(Rn)+kφk +µ2M2+µ2L2 a(R)klk2h|kvkC([τ,τ+T];H)+|z(θtω)|kφkL∞(O)i2+ 1. (6.7) By Lemma 5.1, for every ε > 0,there exists δ0=δ0(τ, ω, T, ε)>0such that, for all 0< δ < δ0and t∈[τ, τ +T], |yδ(θtω)−z(θtω)|< ε. 22 Note that v∈L2([τ, τ +T]; V)∩C([τ, τ +T]; H)and the mapping t7→ z(θtω)is continuous, we derive d dt kvδ(t)−v(t)k2≤Ckvδ(t)−v(t)k2+Cε. The Gronwall lemma implies that kvδ(t, τ, ω, vδ,τ )−v(t, τ, ω, vτ)k2≤eC(t−τ)kvδ,τ −vτk2+Cε. By the definition of u, it follows that kuδ(t, τ, ω, uδ,τ )−u(t, τ, ω, uτ)k2≤eC(t−τ)kuδ,τ −uτk2+|yδ(θτω)−z(θτω)|2+|yδ(θtω)−z(θtω)|2kφk2+Cε, which together with Lemma 5.1 implies that kuδ(t, τ, ω, uδ,τ )−u(t, τ, ω, uτ)k2≤eC(t−τ)kuδ,τ −uτk2+Cε. As a straightforward consequence of Lemma 6.1, we obtain the following convergence of solution. Lemma 6.2. Suppose the conditions of Lemma 6.1 hold and δn→0+. Let uδnand ube the solutions of problem (5.2)and (2.8)with initial data uδn,τ and uτ, respectively. If uδn,τ →uτin Has δn→0+, then for every τ∈R, ω ∈Ωand t>τ, uδn(t, τ, ω, uδn,τ )→u(t, τ, ω, uτ)in Has δn→0+. For the attractor Aδof Φδ,we have the following compactness. Lemma 6.3. Suppose the conditions of Theorem 3.1 and (H2)hold. Then for every τ∈R, ω ∈Ω, if δn→0+and uδn∈ Aδn(τ, ω), the sequence {uδn}+∞ n=1 is precompact in H. Proof. Since uδn∈ Aδn(τ, ω), by the invariance of Aδn, there exists ˜uδn∈ Aδn(τ−1, θ−1ω)such that uδn= Φδn(1, τ −1, θ−1ω, ˜uδn) = uδn(1, τ −1, θ−τω, ˜uδn).(6.8) By Lemma 5.2, there exists N1=N1(τ, ω, α)⩾1such that, for all n⩾N, Rδn(τ−1, θ−1ω) =1 + 2C1Z0 −∞ eµm 2s|yδ(θs−1ω)|2kφk2ds +8µM2 mZ0 −∞ eµm 2s|yδ(θs−1ω)|2k(−∆)γφk2ds +4 µm Z0 −∞ eµm 2skh(s+τ−1)k2ds +4C2 µm + 2|yδ(θ−1ω)|2kφk2, which together with the fact ˜uδn∈ Aδn(τ−1, θ−1ω)⊆Bδn(τ−1, θ−1ω)implies that for all n≥N1, k˜uδnk2≤1+2C1Z0 −∞ eµm 2s|yδ(θs−1ω)|2kφk2ds +8µM2 mZ0 −∞ eµm 2s|yδ(θs−1ω)|2k(−∆)γφk2ds +4 µm Z0 −∞ eµm 2skh(s+τ−1)k2ds +4C2 µm + 2|yδ(θ−1ω)|2kφk2. By definition, we know that {˜vδn}∞ n=1 is bounded in H. Therefore, there exists ¯v∈L2([τ−1, τ]; H)such that, up to a subsequence, vδn(·, τ −1, θ−τω, ˜vδn)→¯v(·)in L2([τ−1, τ]; H), 23 which implies, up to a subsequence, vδn(s, τ −1, θ−τω, ˜vn)→¯v(s)in H a.e. for s∈(τ−1, τ). We then have uδn(s, τ, θ−τω, ˜uδn)→¯v(s) + φyδ(θsω)in H a.e. for s∈(τ−1, τ).(6.9) Since δn→0, by Lemma 6.2 and (6.9), we have uδn(τ, s, θ−τω, uδn(s, τ −1, θ−τω, ˜uδn)) →u(τ, s, θ−τω, ¯v(s) + φyδ(θsω)) in H. Note that uδn(τ, s, θ−τω, uδn(s, τ −1, θ−τω, ˜uδn)) = uδn(τ, τ −1, θ−τω, ˜uδn). Thus, we have uδn(τ, τ −1, θ−τω, ˜uδn)→u(τ, s, θ−τω, ¯v(s) + φyδ(θsω)) in H, which along with (6.8) completes the proof. Now we present the upper semi-continuity of random attractors as δ→0+. Theorem 6.1. Suppose the conditions of Theorem 3.1 and (H2)hold. Then, for all τ∈R, ω ∈Ω, lim δ→0+dH(Aδ(τ, ω),A(τ, ω)) = 0. Proof. Let δn→0and uδn,τ →uτin H. We derive from Lemma 5.1 that, for all τ∈R, ω ∈Ω, lim δ→0kBδ(τ, ω)k=kB(τ, ω)k.(6.10) Then by (6.10), Lemmas 6.2 and 6.3, we prove this theorem from Theorem 3.1 in [47]. 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