Weak mean random attractors for non-local random and stochastic reaction-diffusion equations

November 26, 2021 9:41 WSPC/INSTRUCTION FILE 10-NonlocalStochasticMeanRevised Stochastics and Dynamics © World Scientific Publishing Company Weak mean random attractors for non-local random and stochastic reaction-diffusion equations Rub´en Caballero Centro de Investigaci´on Operativa, Universidad Miguel Hern´andez de Elche Avda. Universidad s/n, 03202, Elche (Alicante), Spain ruben.caballer[email protected] Pedro Mar´ın-Rubio Dpto. Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla, C/Tarfia, 41012-Sevilla, Spain [email protected] Jos´e Valero Centro de Investigaci´on Operativa, Universidad Miguel Hern´andez de Elche Avda. Universidad s/n, 03202, Elche (Alicante), Spain jvaler[email protected] Received (Day Month Year) Revised (Day Month Year) In love memory of Prof. Mar´ıa Jos´e Garrido-Atienza, colleague and friend, with deep sorrow In this paper, we prove the existence of weak pullback mean random attractors for a non-local stochastic reaction-diffusion equation with a nonlinear multiplicative noise. The existence and uniqueness of solutions and weak pullback mean random attractors are also established for a deterministic non-local reaction-diffusion equations with random initial data. Keywords: mean random attractor; pullback attractor; stochastic reaction-diffusion equation; non-local equation. AMS Subject Classification: 35B40, 35B41, 35K55, 35K57, 35K59, 37H05, 37H30, 60H15 1. Introduction The theory of global attractors for random dynamical systems in infinitedimensional spaces has been developed intensively over the last 20 years. In particular, the theory of pathwise pullback random attractors [19] has been applied succesfully to stochastic equations of different types (see e.g. [5,10,11,12,18,21,22,24,25,26,32,36,41,42,43] among many others). However, this theory has an important limitation. Namely, it relies on a suitable change of variable which can be applied only to linear stochastic perturbations. For this 1 November 26, 2021 9:41 WSPC/INSTRUCTION FILE 10-NonlocalStochasticMeanRevised 2R. Caballero, P. Mar´ın-Rubio and J. Valero reason, several authors have developed the theory of mean random attractors (see [23,28,37,38,39,40]), which can be applied to equations with much more general nonlinear noises. In this paper we apply the theory of weak mean random attractors developed in [37] to the following stochastic non-local reaction-diffusion problem du = (a(kuk2 V)∆u+f(u) + h(t, x))dt +σ(u)dw (t) in (τ, ∞)× O, u= 0 on (τ, ∞)×∂O, u(τ, x) = uτ(x) for x∈ O, (1.1) where V=H1 0(O) and Ois a bounded open set of Rn.Following [37] under suitable conditions on the domain Oand the functions f,h, σ and awe prove the existence of a weak mean-square random attractor in two situations: 1) The equation is deterministic (that is, σ(u)≡0) but the initial condition uτis a random variable; 2) σ(u)6≡ 0 and w(t) is a two-sided scalar Wiener process. The existence of weak mean random attractors for the local problem (that is, a≡1) was established in [37]. These results were generalized to stochastic reactiondiffusion equations generated by the p-laplacian operator [38]. Reaction-diffusion equations with non-local diffusion of the type ut−a(l(u(t)))∆u=f(t, u), where l:X→Ris a suitable functional and Xis the phase space (usually L2(O) or H1 0(O)), appear in many applications in Physics, Biology and other sciences (see [15,16,17] and the references therein). When lis a linear functional of the type l(u) = ZΩ ξ(x)u(x, t)dx, where ξ( · ) is a given function in L2(O), the existence and properties of global attractors in the autonomous and non-autonomous situations has been established in [6,1,9,8,7]. The drawback of this case is that we cannot obtain a Lypaunov function for the solutions, which makes it difficult to analyze the fine structure of the attractor. However, if we consider a functional lof the type l(u) = kuk2 V, then a Lyapunov function exists [17]. In this situation, the existence and structure of the global attractor in both the single and set-valued frameworks have been studied in the papers [3,4,13,30,33]. In equation (1.1) the functional l(u) = kuk2 Vhelps us to obtain the existence and uniqueness of solutions, because the operator u7→ −a(kuk2 V)∆uis monotone as a map from Vinto its conjugate space V∗. This paper is organized as follows. In Section 2, we recall the main results of the theory of weak pullback mean random attractors developed in [37]. Also, we extend it by proving under suitable assumptions that the mean random attractor can be characterized using complete trajectories. In Section 3, we prove the existence of November 26, 2021 9:41 WSPC/INSTRUCTION FILE 10-NonlocalStochasticMeanRevised Weak mean random attractors for non-local random and stochastic reaction-diffusion eqns. 3 weak pullback mean random attractors for the deterministic equation (1.1) with random initial data. Finally, in Section 4, we obtain the existence of weak pullback mean random attractors for the stochastic equation (1.1). 2. Preliminaries: abstract theory of weak pullback mean random attractors In this section we recall the main results of the theory of pullback mean random attractors developed in [37] and add some new ones about the characterization of the pullback attractor by means of complete trajectories. 2.1. Weak pullback mean random attractors over probability spaces In this subsection we provide the theory of weak pullback mean random attractors for a mean random dynamical system over a probability space. Let Xbe a Banach space with norm k · kXand let (Ω,F, P) be a probability space. For p∈(1,∞) let us consider the Banach space Lp(Ω,F;X) (Lp(Ω; X) for short) of Bochner integrable functions y: Ω →Xsuch that ZΩ kyk2 XdP < ∞. We denote by Da collection of some families D={D(t)}t∈Rof non-empty bounded sets D(t)⊂Lp(Ω; X): D={D={D(t)}t∈R:D(t)∈β(Lp(Ω; X)) satisfy suitable conditions}, where β(Lp(Ω; X)) is the set of all non-empty bounded subsets of Lp(Ω; X). The collection Dis said to be inclusion-closed if D∈ D and B(t)⊂D(t), B(t)∈ β(Lp(Ω; X)), for all t∈R, imply that B={B(t)}t∈R∈ D. A family D={D(t)}t∈Ris said to be compact (weakly compact, bounded, etc.) if each set D(t) is compact (weakly compact, bounded, etc.). A family of maps Φ : R+×R×Lp(Ω; X)→Lp(Ω; X) is called a mean random dynamical system if: •Φ(0, τ, · ) is the identity map for all τ∈R; •Φ(t+s, τ, x) = Φ(t, s +τ, Φ(s, τ, x)) for all t, s ∈R+,τ∈R,x∈Lp(Ω; X). Definition 2.1. A family K={K(t)}t∈R∈ D is called a D-pullback weakly attracting set if for all τ∈R,D∈ D and any weak neighborhood Nw(K(τ)) of K(τ) there exists T(τ, D, Nw(K(τ))) >0 such that Φ(t, τ −t, D(τ−t)) ⊂ Nw(K(τ)) as soon as t≥T. We introduce the main concept of weak D-pullback mean random attractor. November 26, 2021 9:41 WSPC/INSTRUCTION FILE 10-NonlocalStochasticMeanRevised 4R. Caballero, P. Mar´ın-Rubio and J. Valero Definition 2.2. A family A={A(t)}t∈R∈ D is called a weak D-pullback mean random attractor if: (1) Ais weakly compact. (2) Ais D-pullback weakly attracting. (3) Ais minimal, that is, if B∈ D is weakly compact and D-pullback weakly attracting, then A(t)⊂B(t) for all t∈R. It follows from this definition that a weak D-pullback mean random attractor is unique if it exists. Further we need the concept of D-pullback absorbing family. Definition 2.3. A family K={K(t)}t∈R∈ D is called D-pullback absorbing if for all τ∈Rand D∈ D there exists T(τ, D)>0 such that Φ(t, τ −t, D(τ−t)) ⊂K(τ) as soon as t≥T. Theorem 2.1. [37] (p.2183) Let Xbe reflexive. Assume that Dis an inclusionclosed collection. If the mean random dynamical system Φpossesses a weakly compact D-pullback absorbing family K={K(t)}t∈R∈ D, then Φhas a unique weak D-pullback mean random attractor given by A(t) = ∩r≥0∪s≥rΦ(s, t −s, K(t−s))w,t∈R,(2.1) where Cwmeans the closure of Cin the weak topology of Lp(Ω, X). The weak D-pullback mean random attractor is called invariant if A(t) = Φ(τ, t −τ, A(t−τ)) for all τ≥0, t ∈R. The mean random dynamical system Φ is weakly continuous if the map Φ(t, τ) : Lp(Ω, X)→Lp(Ω, X) is weakly continuous for any t≥0, τ∈R. Lemma 2.1. Let the conditions of Theorem 2.1 hold true. Assume that Φis weakly continuous. Then the weak D-pullback mean attractor Ais invariant. Proof. Let y∈ A(t). In view of characterization (2.1) and arguing as in Lemma 3.3 in [27] there exist nets sα,yα∈Φ(sα, t −sα, K(t−sα)) such that sα→+∞ and yα→yweakly in Lp(Ω; X). Hence, yα= Φ(sα, t −sα, zα) for some net zα∈ K(t−sα). Thus, yα= Φ(sα, t −sα, zα) = Φ(τ, t −τ, Φ(sα−τ, t −sα, zα)) = Φ(τ, t −τ, xα). There exists s0such that xα∈K(t−τ) for sα≥s0. Since K(t−τ) is bounded, passing to a subnet we have that xα→xfor some x. The weak continuity of Φ implies then that y= Φ(τ, t −τ, x). Note that xα∈Φ(sα−τ, t −sα, K(t−sα)) = Φ(esα, t −τ−esα, K(t−τ−esα)), November 26, 2021 9:41 WSPC/INSTRUCTION FILE 10-NonlocalStochasticMeanRevised Weak mean random attractors for non-local random and stochastic reaction-diffusion eqns. 5 so x∈ ∩r≥0∪s≥rΦ(s, t −τ−s, K(t−τ−s))w=A(t−τ). We have obtained that A(t)⊂Φ(τ, t −τ, A(t−τ)). Conversely, for any weak neighborhood Nw(A(t)) we have Φ(τ, t −τ, A(t−τ)) ⊂Φ(τ, t −τ, Φ(r, t −τ−r, A(t−τ−r))) = Φ(τ+r, t −τ−r, A(t−τ−r)) ⊂ N w(A(t)), for rlarge enough, so Φ(τ, t −τ, A(t−τ)) ⊂ A(t). The map φ:R→Lp(Ω; X) is a complete trajectory if φ(t) = Φ(t−s, s, φ(s)) for all s<t. We can characterize the attractor in terms of complete trajectories belonging to D. Lemma 2.2. Assume that Dis an inclusion-closed collection. Assume that Φpossesses an invariant weak D-pullback mean attractor A∈D. Then A(t) = {φ(t) : φ∈ D is a complete trajectory}.(2.2) Proof. If φ∈ D is a complete trajectory, then φ(t) = Φ(s, t −s, φ(t−s)) for all s≥0, t ∈R. Hence, for any t∈Rand any weak neighborhood Nw(A(t)) there exists T= T(t, φ, Nw(A(t))) such that φ(t) = Φ(s, t −s, φ(t−s)) ∈ N w(A(t)) for all s≥T, which implies that φ(t)∈ A(t). Conversely, let y∈ A(t), t∈Rbe arbitrary. Since Ais invariant, we have y∈ A(t) = Φ(1, t −1,A(t−1)), so there exists z1∈ A(t−1) such that y= Φ(1, t −1, z1). We put φ1(r) = Φ(r− t+ 1, t −1, z1) for all r≥t−1. By the invariance of the attractor, φ1(r)∈ A(r) for all r≥t−1. Also, φ1(r) = Φ(r−s, s, φ1(s)) for any r≥s≥t−1. By the same argument, there is z2∈ A(t−2) such that z1= Φ(1, t −2, z2). The function φ2(r) = Φ(r−t+2, t−2, z2),for all r≥t−2,satisfies φ2(r)∈ A(r),for all r≥t−2, φ2(r) = Φ(r−s, s, φ2(s)),for any r≥s≥t−2,and φ2(r) = φ1(r) for all r≥t−1. In this way, we construct a sequence of functions φk: [t−k, +∞)→Lp(Ω, X) such that φk(r)∈ A(r),for all r≥t−k, φk(r) = Φ(r−s, s, φk(s)),for any r≥s≥t−k, and φk(r) = φk−1(r) for all r≥t−k+1. Let φ:R→Lp(Ω; X) be the common value of the functions φkat any point t∈R. It is clear that φis a complete trajectory satisfying φ(t)∈ A(r) for all t. Also, φ∈ D because A ∈ D and Dis inclusion closed. A family of sets Dis said to be backwards bounded if there is t0∈Rsuch that ∪t≤t0D(t) is bounded. It is called bounded if ∪t∈RD(t) is bounded. November 26, 2021 9:41 WSPC/INSTRUCTION FILE 10-NonlocalStochasticMeanRevised 6R. Caballero, P. Mar´ın-Rubio and J. Valero Lemma 2.3. Assume that Dis an inclusion-closed collection and that Φpossesses an invariant weak D-pullback mean attractor A ∈ D which is backwards bounded. Then A(t) = {φ(t) : φ∈ D is a backwards bounded complete trajectory}.(2.3) If, moreover, Dcontains any backwards bounded family, then A(t) = {φ(t) : φis a backwards bounded complete trajectory}.(2.4) Proof. The first statement follows directly from (2.2) and the fact that Ais backwards bounded. The second one follows because any backwards bounded complete trajectory has to belong to D, so the two sets defined in (2.3) and (2.4) coincide. In the same way we prove the following. Lemma 2.4. Assume that Φpossesses an invariant weak D-pullback mean attractor A ∈ D which is bounded. Then A(t) = {φ(t) : φ∈ D is a bounded complete trajectory}.(2.5) If, moreover, Dcontains any bounded family, then A(t) = {φ(t) : φis a bounded complete trajectory}.(2.6) 2.2. Weak pullback mean attractors over filtered probability spaces In this subsection we recall the main results of the theory of weak D-pullback mean random attractors for mean random dynamical systems over filtered probability spaces. As usual, we denote by (Ω,F,{Ft}t∈R, P) a complete filtered probability space, where {Ft}t∈Ris an increasing right continuous family of sub-σ-algebras of Fwhich contains all P-null sets. For p∈(1,∞) let Lp(Ω,Ft;X) be the subspace of (class of) functions fof Lp(Ω,F;X) such that fis strongly Ft-measurable. As before, we denote by Da collection of some families D={D(t)}t∈Rof non-empty bounded sets D(t)⊂Lp(Ω,Ft;X): D={D={D(t)}t∈R:D(t)∈β(Lp(Ω,Ft;X)) satisfying suitable conditions}. A family D={D(t)}t∈Ris said to be compact (weakly compact, bounded, etc.) if each set D(t) is compact (weakly compact, bounded, etc.) in Lp(Ω,Ft;X). Definition 2.4. The family of maps Φ(t, τ) : Lp(Ω,Fτ;X)→Lp(Ω,Ft+τ;X), t∈R+,τ∈Ris called a mean random dynamical system on Lp(Ω,F;X) over (Ω,F,{Ft}t∈R, P) if: •Φ(0, τ, · ) is the identity map for all τ∈R; •Φ(t+s, τ) = Φ(t, s +τ, Φ(s, τ, x)) for all t, s ∈R+,τ∈R,x∈Lp(Ω,Fτ;X). November 26, 2021 9:41 WSPC/INSTRUCTION FILE 10-NonlocalStochasticMeanRevised Weak mean random attractors for non-local random and stochastic reaction-diffusion eqns. 7 Definition 2.5. The family K={K(t)}t∈R∈ D is called D-pullback weakly attracting for Φ if for all τ∈R,D∈ D and any weak neighborhood Nw(K(τ)) of K(τ) in Lp(Ω,Fτ;X) there exists T(τ, D, Nw(K(τ))) >0 such that Φ(t, τ −t, D(τ−t)) ⊂ Nw(K(τ)) as soon as t≥T. We introduce the main concept of weak D-pullback mean random attractor. Definition 2.6. A family A={A(t)}t∈R∈ D is called a weak D-pullback mean random attractor for Φ if: (1) A(τ) is weakly compact in Lp(Ω,Fτ;X) for all τ∈R. (2) Ais D-pullback weakly attracting. (3) Ais minimal, that is, if B∈ D satisfies conditions 1-2, then A(t)⊂B(t) for all t∈R. It follows from this definition that a weak D-pullback mean random attractor is unique if it exists. Definition 2.7. The family K={K(t)}t∈R∈ D is called D-pullback absorbing for Φ if for all τ∈Rand D∈ D there exists T(τ, D)>0 such that Φ(t, τ −t, D(τ−t)) ⊂K(τ) as soon as t≥T. Theorem 2.2. [37] (p.2188) Let Xbe reflexive. Assume that Dis an inclusionclosed collection. If the mean random dynamical system Φpossesses a weakly compact D-pullback absorbing family K={K(t)}t∈R∈ D, then Φhas a unique weak D-pullback mean random attractor given by A(t) = ∩r≥0∪s≥rΦ(s, t −s, K(t−s))w,t∈R,(2.7) where the closure is taken with respect to the weak topology of Lp(Ω,Ft;X). As before, the weak D-pullback mean random attractor is called invariant if A(t) = Φ(τ, t −τ, A(t−τ)) for all τ≥0, t ∈R. The mean random dynamical system on Lp(Ω,F;X) over (Ω,F,{Ft}t∈R, P) Φ is weakly continuous if the map Φ(t, τ) : Lp(Ω,Fτ;X)→Lp(Ω,Ft+τ;X) is weakly continuous for any t≥0, τ∈R. Lemma 2.5. Let the conditions of Theorem 2.2 hold true. Assume that Φis weakly continuous. Then the weak D-pullback mean attractor Ais invariant. Proof. Let y∈ A(t). In view of characterization (2.7) and arguing as in Lemma 3.3 in [27] there exist nets sα,yα∈Φ(sα, t −sα, K(t−sα)) such that sα→+∞ November 26, 2021 9:41 WSPC/INSTRUCTION FILE 10-NonlocalStochasticMeanRevised 8R. Caballero, P. Mar´ın-Rubio and J. Valero and yα→yweakly in Lp(Ω,Ft;X). Hence, yα= Φ(sα, t −sα, zα) for some net zα∈K(t−sα). Thus, yα= Φ(sα, t −sα, zα) = Φ(τ, t −τ, Φ(sα−τ, t −sα, zα)) = Φ(τ, t −τ, xα). There exists s0such that xα∈K(t−τ) for sα≥s0. Since K(t−τ) is bounded, passing to a subnet we have that xα→xfor some x. Then the weak continuity of Φ implies that y= Φ(τ, t −τ, x). Note that xα∈Φ(sα−τ, t −sα, K(t−sα)) = Φ(esα, t −τ−esα, K(t−τ−esα)), so x∈ ∩r≥0∪s≥rΦ(s, t −τ−s, K(t−τ−s))w=A(t−τ). We have obtained that A(t)⊂Φ(τ, t −τ, A(t−τ)). Conversely, for any weak neighborhood Nw(A(t)) we have Φ(τ, t −τ, A(t−τ)) ⊂Φ(τ, t −τ, Φ(r, t −τ−r, A(t−τ−r))) = Φ(τ+r, t −τ−r, A(t−τ−r)) ⊂ N w(A(t)), A map φsuch that φ(t)∈Lp(Ω,Ft;X) for all t∈Ris a complete trajectory if φ(t) = Φ(t−s, s, φ(s)) for all s < t. We can characterize the attractor in terms of complete trajectories belonging to D. Lemma 2.6. Assume that Dis an inclusion-closed collection. Assume that Φpossesses an invariant weak D-pullback mean attractor A ∈ D. Then A(t) = {φ(t) : φ∈ D is a complete trajectory}.(2.8) Proof. If φ∈ D is a complete trajectory, then φ(t) = Φ(s, t −s, φ(t−s)) for all s≥0, t ∈R, so for any t∈Rand any weak neighborhood Nw(A(t)) there exists T= T(t, φ, Nw(A(t))) such that φ(t) = Φ(s, t −s, φ(t−s)) ∈ N w(A(t)) for all s≥T, which implies that φ(t)∈ A(t). Conversely, let y∈ A(t), t∈Rbe arbitrary. Since Ais invariant, we have y∈ A(t) = Φ(1, t −1,A(t−1)), so there exists z1∈ A(t−1) such that y= Φ(1, t −1, z1). We put φ1(r) = Φ(r− t+ 1, t −1, z1) for all r≥t−1. By the invariance of the attractor, φ1(r)∈ A(r) for all r≥t−1. Also, φ1(r) = Φ(r−s, s, φ1(s)) for any r≥s≥t−1. By the same argument, there is z2∈ A(t−2) such that z1= Φ(t−1, t −2, z2). The function φ2(r) = Φ(r−t+2, t−2, z1),for all r≥t−2,satisfies φ2(r)∈ A(r),for all r≥t−2, φ2(r) = Φ(r−s, s, φ2(s)),for any r≥s≥t−2,and φ2(r) = φ1(r) for all r≥t−1. In this way, we construct a sequence of functions φksuch that φk(r)∈Lp(Ω,Fr;X), November 26, 2021 9:41 WSPC/INSTRUCTION FILE 10-NonlocalStochasticMeanRevised Weak mean random attractors for non-local random and stochastic reaction-diffusion eqns. 9 φk(r)∈ A(r),for all r≥t−k, φk(r) = Φ(r−s, s, φk(s)),for any r≥s≥t−k, and φk(r) = φk−1(r) for all r≥t−k+1. Let φbe the common value of the functions φk at any point t∈R. It is clear that φis a complete trajectory satisfying φ(t)∈ A(r) for all t. Also, φ∈ D because A∈Dand Dis inclusion closed. Lemma 2.7. Assume that Dis an inclusion-closed collection and that Φpossesses an invariant weak D-pullback mean attractor A ∈ D which is backwards bounded. Then A(t) = {φ(t) : φ∈ D is a backwards bounded complete trajectory}.(2.9) If, moreover, Dcontains any backwards bounded family, then A(t) = {φ(t) : φis a backwards bounded complete trajectory}.(2.10) Proof. The first statement follows directly from (2.3) and the fact that Ais backwards bounded. The second one follows because any backwards bounded complete trajectory has to belong to D, so the two sets defined in (2.9) and (2.10) coincide. In the same way we prove the following. Lemma 2.8. Assume that Φpossesses an invariant weak D-pullback mean attractor A ∈ D which is bounded. Then A(t) = {φ(t) : φ∈ D is a bounded complete trajectory}. If, moreover, Dcontains any bounded family, then A(t) = {φ(t) : φis a bounded complete trajectory}. 3. The mean random attractor for a non-local problem with random initial data We put H=L2(O) with norm k · k(we will use k · kalso for the norm in L2(O)d, d≥1) and V=H1 0(O) with norm kukV=k∇uk.As usual, ( · , · ) is the scalar product in Hd, d ≥1, and also the duality between Lq(O) and Lp(O), where p≥2 and qis its conjugate, that is, 1/p + 1/q = 1. The duality between Vand its dual space V∗is denoted by h · , · iV∗,V . Let us consider the following reaction-diffusion equation      ∂u ∂t −a(kuk2 V)∆u=f(u) + h(t, x) in (τ, ∞)× O, u= 0 on (τ, ∞)×∂O, u(τ, x) = uτ(x) for x∈ O, (3.1) where Ois a bounded open set of Rnwith smooth boundary ∂Oand the functions h,a∈C(R+), f∈C1(R), satisfy the following assumptions: f(r)r≤ −α|r|p+β, (3.2) November 26, 2021 9:41 WSPC/INSTRUCTION FILE 10-NonlocalStochasticMeanRevised 16 R. Caballero, P. Mar´ın-Rubio and J. Valero measurability are equivalent properties when the space is separable (see page 42 in [20]). On the other hand, by (3.14), (3.16) and (3.18) we obtain that ku(t, ω)k2≤ kuτ(ω)k2+ 2TK1+ 2K2(T),∀t∈[τ, τ +T],(3.30) Zτ+T τ ku(s, ω)k2 Vds ≤1 mkuτ(ω)k2+2TK1+ 2K2(T) m,(3.31) Zτ+T τ ku(s, ω)kp Lp(O)ds ≤1 2αkuτ(ω)k2+TK1+K2(T) α,(3.32) ku(t, ω)k2 V≤2TK1+ 2K2(T) + kuτ(ω)k2 mr +K3(T)e2ηr,∀t≥τ+r, (3.33) ZT τ+r k∆un(t, ω)k2dt ≤K4(T, r) m(1 + kuτ(ω)k2) + 1 2m2ZT τ+r kh(t)k2dt +η m2kuτ(ω)k2+ 2TK1+ 2K2(T),(3.34) for any r > 0, τ∈R, T > 0 and ω∈Ω. Since uτ∈L2(Ω; H),we have u∈L∞ loc(τ, ∞;L2(Ω; H)) ∩L2 loc(τ, ∞;L2(Ω; V)) ∩Lp loc(τ, ∞;Lp(Ω; Lp(O))) ∩L∞(τ+ε, τ +T;L2(Ω; V)) ∩L2(τ+ε, τ +T;L2(Ω; D(A))),(3.35) for any 0 < ε < T. For every fixed ω,u(·, ω)∈C([τ, τ +T], H) and in view of (3.30) and the Lebesgue dominated convergence theorem, we have that u∈C([τ, ∞), L2(Ω; H)) obtaining that uis a solution in the sense of Definition 3.1. We will prove that the solution is unique. If u, v are two solutions, then the difference w=u−vsatisfies dw dt −a(kuk2 V)∆u+a(kvk2 V)∆v=f(u1)−f(u2), so multiplying by vand using (3.4) we have 1 2 d dt kvk2+ZO−a(kuk2 V)∆u+a(kvk2 V)∆vwdx ≤ηkvk2. Since (3.7) implies ZO−a(kuk2 V)∆u+a(kvk2 V)∆v(u−v)dx =ZO (a(ku(t)k2 V)|∇u|2−a(ku(t)k2 V)∇u∇v−a(kv(t)k2 V)∇u∇v+a(kv(t)k2 V)|∇v|2)dx ≥a(ku(t)k2 V)ku(t)k2 V−(a(ku(t)k2 V)+a(kv(t)k2 V))ku(t)kVkv(t)kV+a(kv(t)k2 V)kv(t)k2 V =a(ku(t)k2 V)ku(t)kV−a(kv(t)k2 V)kv(t)kV(ku(t)kV− kv(t)kV)≥0,(3.36) November 26, 2021 9:41 WSPC/INSTRUCTION FILE 10-NonlocalStochasticMeanRevised Weak mean random attractors for non-local random and stochastic reaction-diffusion eqns. 17 we obtain by the Gronwall Lemma and taking expectations that Ekw(t)k2≤e2η(t−τ)Eku(τ)−v(τ)k2, which implies uniqueness and the continuity of solutions with respect to the initial datum as well. Finally, equality (3.9) is obtained multiplying the equation by u(t) and taking expectations. Let Φ be the mapping from R+×R×L2(Ω; H) to L2(Ω; H) given by Φ(t, τ, uτ) = u(t+τ) where t≥0, τ ∈R, uτ∈L2(Ω; H),and uis the unique solution to (3.1) with u(τ) = uτ. By the previous result, Φ is a continuous mean random dynamical system on L2(Ω; H). We recall that λ1>0 is the first eigenvalue of −∆ with homogeneous Dirichlet boundary conditions, and assume that there exists µ∈(0,2mλ1) such that Z0 −∞ eµskh(s)k2ds < ∞.(3.37) Remark 3.2. Following Remark 3.1, the function h(t) = h(t) + f(0) satisfies assumption (3.37) as well. According to the previous assumption, let us consider the following universe: denote Dthe class of all families of nonempty bounded subsets of L2(Ω; H), D = {D(τ) : τ∈R},such that lim τ→−∞ eµτ sup v∈D(τ) kvk2 L2(Ω;H)= 0.(3.38) Within this setting we can derive uniform estimates on the solutions to (3.1) that will lead to the existence of D-pullback absorbing family in L2(Ω; H).Namely, we have the following result. Lemma 3.1. Assume that (3.2)-(3.7) and (3.37) hold. Then for any τ∈Rand D={D(t) : t∈R}∈Dthere exists T=T(τ, D)>0such that for all t≥Tand uτ−t∈D(τ−t)we have E(ku(τ)k2)≤M+Me−µτ Zτ −∞ eµskh(s)k2ds, where uis the unique solution satisfying u(t−τ) = ut−τand Mdenotes a positive constant which is dependent on µbut independent of τand D. November 26, 2021 9:41 WSPC/INSTRUCTION FILE 10-NonlocalStochasticMeanRevised 18 R. Caballero, P. Mar´ın-Rubio and J. Valero Proof. From the energy equality (3.9) and assumptions (3.2), (3.3), (3.6) and the Poincar´e inequality we have d dsE(ku(s)k2)+2mλ1E(ku(s)k2) ≤ − 2αE(ku(s)kp Lp(O))+2β|O| +1 2mλ1−µkh(s)k2+ (2mλ1−µ)E(ku(s)k2) for a.a. s > τ. Multiplying by eµs we deduce d ds(eµsE(ku(s)k2)) + 2αeµsE(ku(s)kp Lp(O))≤2β|O|eµs +1 2mλ1−µeµskh(t)k2 for a.a. s > τ. Integrating in [τ−t, τ] E(ku(τ)k2)+2αe−µτ Zτ τ−t eµsE(ku(s)kp Lp(O))ds ≤e−µτ eµ(τ−t)E(kuτ−tk2) + 1 2mλ1−µe−µτ Zτ τ−t eµskh(s)k2ds + 2β|O| µ∀t≥0. Since uτ−t∈D(τ−t),we have that there exists T=T(τ, D) such that e−µτ eµ(τ−t)E(kuτ−tk2)≤1∀t≥T. The proof is complete. Corollary 3.1. Suppose that (3.2)-(3.7) and (3.37) hold. Then, the family K= {K(τ) : τ∈R}with K(τ) = {u∈L2(Ω, L2(O)) : E(kuk2)≤R(τ)},where R(τ) = M+Me−µτ Zτ −∞ eµskh(s)k2ds, belongs to Dand is a weakly compact D-pullback absorbing family for Φ. This allows us to use Theorem 2.1 to conclude the main result of this section. Theorem 3.2. Suppose that (3.2)-(3.7) and (3.37) hold. Then, the continuous mean random dynamical system Φdefined through the solutions to problem (3.1) has the unique weak D-pullback mean random attractor A={A(τ) : τ∈R} ∈ D. In general, the radius R(τ) can be unbounded as τ→ ±∞. However, under an additional assumption on the function h(t) we are able to force it to be bounded in either one or both directions. The following result is straightforward to check. Lemma 3.2. Suppose that (3.2)-(3.7) and (3.37) hold. If, additionally, sup t≤t0 e−µt Zt −∞ eµr kh(r)k2dr < ∞ for some t0∈R, then supτ≤tR(τ)<∞for any t∈R. Hence, the union ∪τ≤tA(τ) is bounded for any t∈R. November 26, 2021 9:41 WSPC/INSTRUCTION FILE 10-NonlocalStochasticMeanRevised Weak mean random attractors for non-local random and stochastic reaction-diffusion eqns. 19 If sup t∈R e−µt Zt −∞ eµr kh(r)k2dr < ∞, then supτ∈RR(τ)<∞. Hence, the union ∪τ∈RA(τ)is bounded. Corollary 3.2. If hdoes not depend on time, that is, h(t)≡h0∈H, then the union ∪τ∈RA(τ)is bounded. 4. The mean random attractor for a stochastic non-local problem We consider now the following stochastic nonlocal reaction-diffusion equation    du = (a(kuk2 V)∆u+f(u) + h(t, x))dt +σ(u)dw (t) in (τ, ∞)× O, u= 0 on (τ, ∞)×∂O, u(τ, x) = uτ(x) for x∈ O, (4.1) where Ois a bounded open set of Rnwith smooth boundary ∂O,(Ω,F,{Ft}t∈R, P) is a complete filtered probability space such that {Ft}t∈Ris a right continuous family of sub-σ-algebras of Fthat contains all P-null sets and w( · ) is a two-sided scalar Wiener process with respect to the filtration {Ft}t∈R. The integral is understood in the Itˆo sense. The functions f∈C1(R), a∈C(R+), σ :R→Rand hsatisfy: f0(r)≤γ1,(4.2) |f(r)| ≤ γ2(1 + |r|),(4.3) f(r)r≤γ3+γ4r2,(4.4) 0< m ≤a(s)≤M, ∀s≥0,(4.5) s7→ a(s2)sis non-decreasing, (4.6) h∈L2 loc(R;H),(4.7) σis globally Lipschitz (with constant Cσ), (4.8) for some γi, m, M > 0 and all s≥0, r ∈R.Additionally, we will need to assume that γ4+C2 σ< mλ1.(4.9) Although some of these conditions are the same as in Section 3, for clarity of exposition we prefer to write them here again. Let e f:H→Hbe the Nemitsky operator given by e f(u) (x) = f(u(x)) for almost all x∈ O. We define the operator B:R×V→V∗given by hB(t, u), viV∗,V =−a(kuk2 V) (∇u, ∇v) + e f(u) + h(t), v. November 26, 2021 9:41 WSPC/INSTRUCTION FILE 10-NonlocalStochasticMeanRevised 20 R. Caballero, P. Mar´ın-Rubio and J. Valero It is straigthforward to see using the Lebesgue theorem that the operator e fis continuous. In the same way, let eσ:H→Hbe the Nemitsky operator given by eσ(u) (x) = σ(u(x)) for almost all x∈ O. It is clear that eσis globally Lipschitz as well. Indeed, keσ(u)−eσ(v)k=ZO (σ(u(x)) −σ(v(x)))2dx1 2 ≤CσZO (u(x)−v(x))2dx1 2 =Cσku−vk.(4.10) Under conditions (4.2)-(4.8) the following lemmas hold. Lemma 4.1. Bis hemicontinuous. Proof. Since a, e fare continuous, for any u, v, z ∈Vwe have that the function λ7→ hB(t, u +λz), viV∗,V =a(ku+λzk2 V) ((∇u, ∇v) + λ(∇z, ∇v)) +e f(u+λz) + h(t), v is continuous. Hence, Bis hemicontinuous. Lemma 4.2. Bis weakly monotone, that is, there exists c > 0such that 2hB(t, u)−B(t, v), u −viV∗,V +kσ(u)−σ(v)k2 H≤cku−vk2∀u, v ∈V. Proof. By (4.2) we obtain that e f(u)−e f(v), u −v=ZO f0(α(x)u(x) + (1 −α(x)) v(x)) (u(x)−v(x))2dx ≤γ1ku−vk2. Hence, the result follows from (3.36) and (4.10). Lemma 4.3. Bis coercive, that is, there are c1≥0, c2>0such that 2hB(t, u), uiV∗,V +keσ(u)k2 H≤c1kuk2−c2kuk2 V+g(t), where g∈L1(τ, T)for all τ < T. Proof. In view of (4.4), (4.5) and (4.8) we have the inequalities −a(kuk2 V) (∇u, ∇u)≤ −mkuk2 V, e f(u), u≤γ3|O| +γ4kuk2, keσ(u)k2 H≤(keσ(0)kH+Cσkuk)2, (h(t), u)≤ kh(t)k kuk ≤ 1 2kh(t)k2+1 2kuk2, November 26, 2021 9:41 WSPC/INSTRUCTION FILE 10-NonlocalStochasticMeanRevised Weak mean random attractors for non-local random and stochastic reaction-diffusion eqns. 21 which imply the result by putting g(t)=2γ3|O| + 2 keσ(0)k2 H+kh(t)k2,c1= 2γ4+ 1 + 2Cσ,c2= 2m. Lemma 4.4. The operator Bis bounded, that is, there exist d1, d2≥0such that kB(t, u)kV∗≤d1+d2kukV+g(t)∀u∈V, where g∈L2(τ, T)for all τ < T. Proof. It follows from (4.3) and (4.5) that kB(t, u)kV∗≤MkukV+  e f(u) + h(t)  V∗ ≤MkukV+ec  e f(u)  +kh(t)k ≤MkukV+ecγ2q2|O| + 2 kuk2+eckh(t)k ≤d1+d2kukV+g(t). We will focus first on the existence and uniqueness of solutions to problem (4.1). Definition 4.1. For τ∈Rand uτ∈L2(Ω,Fτ;H),an H-valued {Ft}t∈R-adapted stochastic process uis called a solution to problem (4.1) if u∈C([τ, +∞), H)∩ L2 loc(τ, +∞;V) P-a.s. and satisfies the equality (u(t), ξ) + Zt τ a(kuk2 V)(∇u, ∇ξ)ds =(uτ, ξ) + Zt τ (e f(u(s)) + h(s), ξ)ds +Zt τ (eσ(u(s)), ξ)dw(s),∀τ < t, P-almost sure for all ξ∈V. From Lemmas 4.1-4.4 and Theorems 4.2.4 and 4.2.5 in [34] we obtain that a unique solution exists and that it satisfies the Itˆo formula. Lemma 4.5. Assume (4.2)-(4.8). Then for any uτ∈L2(Ω,Fτ;H)there exists a unique solution uto problem (4.1) which, moreover, satisfies E sup t∈[τ,τ+T] ku(t)k2!<∞,(4.11) ku(t)k2+ 2 Zt τ a(kuk2 V)k∇uk2ds =ku(τ)k2+ 2 Zt τe f(u(s)) + h(s), u(s)ds +Zt τ keσ(u(s))k2ds + 2 Zt τ (eσ(u(s)), u(s))dw(s)P-a.s., (4.12) November 26, 2021 9:41 WSPC/INSTRUCTION FILE 10-NonlocalStochasticMeanRevised 22 R. Caballero, P. Mar´ın-Rubio and J. Valero for all τ < T. By using (4.11), u∈C([τ, +∞), H) P-a.s. and the Lebesgue theorem we obtain that u∈C([τ, ∞), L2(Ω, H)), so we define the family of maps Φ(t, τ, · ) : Lp(Ω,Fτ;X)→Lp(Ω,Ft+τ;X), t∈R+,τ∈R, by Φ(t, τ, uτ) = u(t+τ), where uis the unique solution to (4.1) with u(τ) = uτ. By the uniqueness of solutions, this family of mappings is a mean random dynamical system on L2(Ω,F;H) over (Ω,F,{Ft}t∈R, P) in the sense of Definition 2.4. We start with an a priori estimate. Lemma 4.6. Assume (4.2)-(4.9). Then there are constants K1, K2>0such that for any uτ∈L2(Ω,Fτ;H)the solution usatisfies the estimate E(ku(t)k2)≤e−ω0(t−τ)E(kuτk2) + K1 ω0 +K2Zt τ e−ω0(t−r)kh(r)k2dr, for all τ < t, for any 0< ω0< mλ1−γ4−C2 σ. Proof. Taking expectations in (4.12) we have Eku(r)k2+ 2 Zr τ Ea(k∇uk2)k∇uk2ds =Eku(τ)k2+2 Zr τ Ee f(u(s)) + h(s), u(s)ds+Zr τ Ekeσ(u(s))k2ds for r≥τ. Thus, for a.a. r > τ, d drEku(r)k2+ 2Ea(k∇uk2)k∇u(r)k2 =2Ee f(u(r)) + h(r), u(r)+Ekeσ(u(r))k2. We estimate each term by using (4.4), (4.7) and (4.8): e f(u(r)), u(r)≤γ3|O| +γ4ku(r)k2, (h(r), u(r)) ≤εmλ1−γ4−C2 σku(r)k2+1 4ε(mλ1−γ4−C2 σ)kh(r)k2, keσ(u(s))k2≤(keσ(0)kH+Cσkuk)2 ≤2keσ(0)k2 H+ 2C2 σkuk2, where ε∈(0,1). Hence, d drEku(r)k2+ 2(1 −ε)(mλ1−γ4−C2 σ)Eku(r)k2 ≤2C2 σkeσ(0)k2 H+1 2ε(mλ1−γ4−C2 σ)kh(r)k2+ 2γ3|O| =K1+K2kh(r)k2. November 26, 2021 9:41 WSPC/INSTRUCTION FILE 10-NonlocalStochasticMeanRevised Weak mean random attractors for non-local random and stochastic reaction-diffusion eqns. 23 Thus, by the Gronwall lemma, E(ku(t)k2)≤e−ω0(t−τ)E(kuτk2) + K1 ω0 +K2Zt τ e−ω0(t−r)kh(r)k2dr, for any 0 < ω0< mλ1−γ4−C2 σ. Further, let us consider the following condition: for some 0 < ω0< mλ1−γ4−C2 σ the function hsatisfies that Z0 −∞ eω0rkh(r)k2dr < ∞.(4.13) Fixing the constant ω0from (4.13) we denote by Dthe collection of all families of non-empty bounded subsets D={D(τ) : τ∈R},D(τ)⊂L2(Ω,Fτ;H), such that lim τ→−∞ eω0τkD(τ)k2 += 0, where kD(τ)k+= supy∈D(τ)kykL2(Ω,Fτ;H). Lemma 4.7. Assume that (4.2)-(4.9) and also condition (4.13) hold true. Then for any t∈Rand D={D(τ) : τ∈R} ∈ D there exists T=T(t, D)such that if s≥T, every solution uwith initial condition at time τ=t−sgiven by ut−s∈D(t−s) satisfies E(ku(t)k2)≤M(1 + e−ω0tZt −∞ eω0rkh(r)k2dr) =: R0(t),(4.14) where M > 0is a constant which depends on ω0. Proof. Since ut−s∈D(t−s), we have e−ω0sE(kut−sk2)≤e−ω0teω0(t−s)kD(t−s)k2 +→0 as s→+∞, so there exists T(t, D) for which e−ω0sE(kut−sk2)≤1 if s≥T. From the estimate in Lemma 4.6 we obtain that E(ku(t)k2)≤1 + K1 ω0 +K2e−ω0tZt −∞ eω0rkh(r)k2dr, for s≥T, proving the result. We define now the family of bounded closed convex sets K0={K0(t) : τ∈R} given by K0(t) = {u∈L2(Ω,Ft;H) : E(kuk2)≤R0(t)}, where R0(t) is the function in (4.14). It is clear that the sets K0(t) are weakly compact. We will prove that this family is D-pullback absorbing and that K0∈ D. November 26, 2021 9:41 WSPC/INSTRUCTION FILE 10-NonlocalStochasticMeanRevised 24 R. Caballero, P. Mar´ın-Rubio and J. Valero Lemma 4.8. Assume the conditions of Lemma 4.7. Then K0is a weakly compact D-pullback absorbing family which belongs to D. Proof. In view of Lemma 4.7, for any t∈Rand D={D(τ) : τ∈R}∈Dthere exists T=T(t, D) such that if s≥T, then Φ(s, t −s, D(t−s)) ⊂K0(t), so K0is a weakly compact D-pullback absorbing family. Finally, we see that eω0τR0(τ) = M(eω0τ+Zτ −∞ eω0rkh(r)k2dr)→0 as τ→ −∞, so K0∈ D. From Lemma 4.8 and Theorem 2.2 we deduce the main result concerning the existence of the weak D-pullback attractor. Theorem 4.1. Assume the conditions of Lemma 4.7. Then the mean random dynamical system Φhas a unique weak D-pullback mean random attractor A0= {A(τ) : τ∈R} ∈ D. As in the previous section, under an additional assumption on the function h( · ) we can prove that the radius R0(t) and the weak D-pullback mean random attractor are bounded in either one or both directions. Lemma 4.9. Under the conditions of Lemma 4.7, if sup t≤t0 e−ω0tZt −∞ eω0rkh(r)k2dr < ∞ for some t0∈R, then supt≤tR0(t)<∞for any t∈R. Hence, the union ∪t≤tA(t) is bounded for any t∈R. If sup t∈R e−ω0tZt −∞ eω0rkh(r)k2dr < ∞, then supt∈RR0(t)<∞. Hence, the union ∪t∈RA(t)is bounded. Corollary 4.1. If hdoes not depend on time, that is, h(t)≡h0∈H, then the union ∪t∈RA(t)is bounded. Acknowledgments This work has been partially supported by the Spanish Ministry of Science, Innovation and Universities, project PGC2018-096540-B-I00, by the Spanish Ministry of Science and Innovation, project PID2019-108654GB-I00, by the Junta de Andaluc´ıa and FEDER, projects US-1254251, P18-FR-4509, and by the Generalitat Valenciana, project PROMETEO/2021/063. 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