Random attractors for a stochastic nonlocal delayed reaction–diffusion equation on a semi-infinite interval

IMA Journal of Applied Mathematics (2023) 88, 576–601 https://doi.org/10.1093/imamat/hxad025 Advance Access publication on 18 September 2023 Random attractors for a stochastic nonlocal delayed reaction–diffusion equation on a semi-infinite interval Wenjie Hu The MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, China Journal House, Hunan Normal University, Changsha, Hunan 410081, China Quanxin Zhu The MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, China The Key Laboratory of Control and Optimization of Complex Systems, College of Hunan Province, Hunan Normal University, Changsha 410081, China. and Tomás Caraballo∗ Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, c/ Tarfia s/n, E-41012 Sevilla, Spain Department of Mathematics, Wenzhou University, Wenzhou 325035, Zhejiang Province, P. R. China ∗Corresponding author: [email protected] [Received on 5 August 2022; revised on 7 July 2023; accepted on 4 September 2023] The aim of this paper is to prove the existence and qualitative property of random attractors for a stochastic non-local delayed reaction–diffusion equation (SNDRDE) on a semi-infinite interval with a Dirichlet boundary condition at the finite end. This equation models the spatial–temporal evolution of the mature individuals for a two-stage species whose juvenile and adults both diffuse that lives on a semi-infinite domain and subject to random perturbations. By transforming the SNDRDE into a random evolution equation with delay, by means of a stationary conjugate transformation, we first establish the global existence and uniqueness of solutions to the equation, after which we show the solutions generate a random dynamical system. Then, we deduce uniform a priori estimates of the solutions and show the existence of bounded random absorbing sets. Subsequently, we prove the pullback asymptotic compactness of the random dynamical system generated by the SNDRDE with respect to the compact open topology, and hence obtain the existence of random attractors. At last, it is proved that the random attractor is an exponentially attracting stationary solution under appropriate conditions. The theoretical results are illustrated by application to the stochastic non-local delayed Nicholson’s blowfly equation. Keywords:Random attractor; stochastic delayed reaction–diffusion equations; semi-infinite interval; nonlocal; age-structured population model. 1. Introduction When modelling the growth of mature population of a two-stage species (juvenile and adult, with a fixed maturation time τ) whose mature individuals and immature individuals both diffuse, one faces the © The Author(s) 2023. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/ 4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. Downloaded from https://academic.oup.com/imamat/article/88/4/576/7276455 by Sevilla Psych Lib B21 user on 11 March 2024 RANDOM ATTRACTORS FOR AN SNDRDE ON A SEMI-INFINITE INTERVAL 577 following delayed reaction–diffusion equation with spatial non-locality derived in So et al. (2001): ∂u(t,x) ∂t=Δu(t,x)−μu(t,x)+εO Γ(α,x,y)f(u(t−τ,y))dy,(t,x)∈(0, ∞)×D. (1.1) Here, O⊆RNis the spatial domain, and u(t,x)stands for the total mature population at location xand time t. The positive constant μrepresents the death rate of the mature population. εand the immature mobility constant αare defined by ε=e−r 0dI(a)daand α=r 0DI(a)da, where dI(a)and DI(a),a∈ [0, τ], are the age-dependent death rate and diffusion rate of the immature population of the species, respectively. The diffuse kernel Γ(α,x,y)is obtained by integrating along the characteristic based on the general model in Metz & Diekmann (1986), representing the probability of the new-born individuals located at ythat can survive to be matured and moved to location x. Generally speaking, explicit forms of Γ(α,x,y)can only be obtained for some special cases, see Liang et al. (2003). When the spatial domain O is bounded with a Dirichlet boundary value condition (DBVC), existence, uniqueness and attractiveness of the positive steady state and threshold dynamics are important and have been explored by Yi & Zou (2013). When a Neumann boundary value condition (NBVC) is imposed, Zhao (2009) established the global attractiveness of the positive steady state of (1.1) by adopting a fluctuation method. In the real world, there are also species whose individuals live in the whole space, such as the fishes in the infinite ocean. In the situation O=R, the lack of compactness of the infinite domains and the complexity of non-local delayed term cause the global dynamics analysis of (1.1) becomes quite difficult and hence, the existing works mainly focus on the travelling wave solutions. See, for instance, So et al. (2001);Wu & Zou (2001);Yi & Zou (2015). To circumvent this difficulty, Yi et al. (2012) made a first attempt to describe the global dynamics of model (1.1) by adopting the compact open topology combined with delicate analysis of the asymptotic properties of the non-local term and the diffusion operator. Moreover, there are also species whose individuals live in a semi-infinite domain which is neither bounded nor is the whole space. For example, animals living in a big land that has the shore of an ocean or a lake at one side of the land provides such a scenario. For the species whose individuals live in a semi-infinite domain, the kernel function and the spatial domain are neither symmetric nor compact, implying the problem becomes more challenging. Recently, Yi & Zou (2016) derived the kernel Γ(α,x,y)=1 √4παe−(x−y)2 4α−1 √4παe−(x+y)2 4α(1.2) in the scenario O=R+=[0, ∞)with the homogeneous DBVC at the finite end, and investigated global dynamics of Eq. (1.1). The results have been extended to the half plane case, which are more realistic in real world modelings by Hu & Duan (2018),Hu et al. (2018) and Wang (2014). However, the evolution of the mature population is inevitably affected by random perturbations, including the noise generated by the internal self-excitation of the system and the random interference of the external environment. Consequently, for the species living on O=R+that are perturbed by some random effects, a more accurate mathematical model should be the following stochastic non-local delayed reaction–diffusion equation (SNDRDE): ∂u ∂t(t,x)=Δu(t,x)−μu(t,x)+εO Γ(α,x,y)f(u(t−τ,y))dy+ m  j=1 gj(x)dwj dt ,(t,x)∈(0, ∞)×R+, (1.3) which is obtained by adding an additive noise m j=1gj(x)dwj dtto (1.1). Here, Γ(α,x,y)is defined by (1.2), {gj(x)}m j=1are twice continuously differentiable on (0, ∞), standing for the intensity and the shape Downloaded from https://academic.oup.com/imamat/article/88/4/576/7276455 by Sevilla Psych Lib B21 user on 11 March 2024 578 W. HU ET AL. of noise and {wj}m j=1are mutually independent two-sided real-valued Wiener process on an appropriate probability space to be specified later. For presentation simplicity, here we only consider R+=[0, ∞) in the one-dimensional space as earlier works (So et al., 2001;Wu & Zou, 2001;Yi et al., 2012;Yi & Zou, 2015) did. Indeed, the results can also be extended to half plane case or even three dimensions by similar techniques in Hu & Duan (2018),Hu et al. (2018) and Wang (2014). In order to obtain the global complex dynamics and non-local analysis of the qualitative properties of random dynamical systems (RDSs), Crauel and Flandoli proposed the concept of random attractors for infinite dimensional random system in Crauel & Flandoli (1994);Flandoli & Schmalfuss (1996); Crauel (2002), by generalizing the theory of global attractors of infinite-dimensional dissipative systems. Since then, the existence, finite dimensionality and structure of random attractors for various stochastic nonlinear evolution equations or stochastic functional differential equations have been extensively and intensively investigated by adopting the framework in Crauel & Flandoli (1994);Crauel (2002).For example, for the stochastic reaction–diffusion equation without time delay, Caraballo et al. (2000),Gao et al. (2014) and Li & Guo (2008) explored the existence of global attractors on bounded domains. For the stochastic delayed reaction–diffusion equation on bounded domains, the existence of random attractors and their structure have been studied in Caraballo et al. (2007);Bessaih et al. (2014);Chueshov et al. (2014);Wang et al. (2015);Li & Guo (2020). In our recent works Hu & Zhu (2021) and Hu & Zhu (2022), we have obtained the existence, uniqueness and stability of solutions to (1.1) as well as the existence of random attractors when the domain Ois bounded with a DBVC. Therefore, similar questions arise naturally, i.e. under what conditions does (1.3) admit a unique global solution? Under what conditions does (1.3) generate an RDS ? Under what conditions does (1.3) possess random attractors? Moreover, under what conditions is the attractor of (1.3) a random fixed point? In the recent works (Bates et al., 2009), Wang et al. (2018) and Zhou (2017), the authors obtained the existence of global attractors for stochastic reaction–diffusion equations on unbounded domains. The unboundedness of the domain causes the Sobolev embedding to no longer be compact and the asymptotic compactness of solutions cannot be obtained by a standard method. Therefore, in order to overcome the difficulty caused by the unboundedness of the domain, Bates et al. (2009) established uniform estimates on the far-field values of solutions. Nevertheless, it follows from (1.2) that the kernel Γ(α,x,y)is asymmetric and the domain is non-symmetric and non-compact, which together with the time delay imply that the analysis of the long time behaviour of solutions to (1.3) on the semi-infinite interval R+=[0, ∞)is more difficult. This motivates us to establish a new method to analyse the asymptotic behaviour of the following stochastic initial boundary value problem: ⎧ ⎨ ⎩ ∂u ∂t(t,x)=Δu(t,x)−μu(t,x)+εR+Γ(α,x,y)f(u(t−τ,y))dy+m j=1gj(x)dwj dt , u(t,0)=0, t>0, u(t,x)=φ(t,x),(t,x)∈[−τ,0]×R+. (1.4) In the case of deterministic equations, to obtain the global dynamics of (1.1), Yi & Zou (2016) established a priori estimates for non-trivial solutions by exploring the asymptotic properties of the nonlocal delayed effect and the diffusion operator. This method has also been adopted by Hu et al. (Hu & Duan, 2018;Hu et al., 2018) to explore the global dynamics of some non-local delayed differential equations on different half spaces with various boundary conditions. Introducing random factors makes the analysis of the asymptotic behaviour of (1.4) quite different from the deterministic case, since the existence and uniqueness of global solution to SNDRDE (1.4) and whether it generates a random Downloaded from https://academic.oup.com/imamat/article/88/4/576/7276455 by Sevilla Psych Lib B21 user on 11 March 2024 RANDOM ATTRACTORS FOR AN SNDRDE ON A SEMI-INFINITE INTERVAL 579 dynamic system are not so natural as in the deterministic case. Moreover, the framework to deal with random attractors is also quite different from that of the deterministic case. In this paper, we carry out a first attempt to extend the method of exploring the asymptotic properties of the deterministic non-local delayed effect and the diffusion operator to the random case, and prove the existence and qualitative property of random attractors for the SNDRDE (1.4) on the unbounded domain R+. Unlike the previous works (Caraballo et al., 2007;Bessaih et al., 2014;Chueshov et al., 2014;Wang et al., 2015;Li & Guo, 2020), where the phase space is a Hilbert space, we need to work here with a Banach space as natural phase space. Due to the lack of an inner product, we prove the existence of a global solution and obtain uniform a priori estimates of the solution by using the semigroup approach together with a careful analysis of the diffusion operator instead of taking inner product. Moreover, to overcome the difficulty caused by the non-compactness of the spatial domain, we also adopt the compact open topology to describe the asymptotic behaviour. It is clear that our method can be used for a variety of other equations on half spaces, as it was done for the deterministic case (Hu et al., 2018). The remaining part of this paper is structured as follows. In Section 2, we recall some basic results from the theory of RDSs and random attractors as well as some notation and preliminary lemmas needed for the proof of our main results. In Section 3, by means of the Ornstein–Uhlenbeck (O-U) process, we first transform SNDRDE (1.4) into a random partial differential equation with delay, and we then show that SNDRDE (1.4) has global solutions by the Banach fixed point theorem together with the properties of the semigroup generated by the linear part of (1.4). Furthermore, we show that solutions to (1.4) generate RDSs. To prove the existence of random attractors for SNDRDE (1.4), we first establish uniform a priori estimates of the solutions in Section 4, and we then show the asymptotic compactness of RDSs generated by (1.2) with respect to the compact open topology, implying the existence of random attractors by the results in Crauel & Flandoli (1994);Crauel (2002). In Section 5, we derive sufficient conditions ensuring the random attractor becomes an exponentially attracting stationary solution. In Section 6, the theoretical results are applied to the stochastic non-local delayed Nicholson’s blowfly equation. At last, we summarize the paper by pointing out some potential directions deserving further research. 2. Preliminaries We first recall some notation to be used throughout this paper, and then we introduce the theory of RDSs as well as random attractors. We denote by BUC R+,Rthe set of all bounded and uniformly continuous functions from R+to R, and by C=C([−τ,0],X)the set of all continuously functions from [−τ,0]to Xequipped with the usual supremum norm ϕC=sup{ϕ(ξ)X:ξ∈[−τ,0]}for any ϕ∈C. For any real interval J⊆R,setJ+[−τ,0] ={t+ξ:t∈Iand ξ∈[−τ,0]}. For any u:(J+[−τ,0])→X and t∈J, we define ut(·)∈Cby ut(ξ) =u(t+ξ) for all ξ∈[−τ,0]. In the sequel, we introduce the concept of random attractor and RDS following Arnold (1998) and Crauel & Flandoli (1994);Flandoli & Schmalfuss (1996);Crauel (2002). Definition 2.1. Let θt:Ω→Ω,t∈Rbe a family of measure preserving transformations such that (t,ω) → θtωis measurable and θ0=id, θt+s=θtθs, for all s,t∈R.Theflowθttogether with the probability space Ω,F,P,θtt∈Ris called a metric dynamical system. For a given complete separable metric space (X,d), denote by B(X)the Borel-algebra of open subsets in X. Downloaded from https://academic.oup.com/imamat/article/88/4/576/7276455 by Sevilla Psych Lib B21 user on 11 March 2024 580 W. HU ET AL. Definition 2.2. A mapping Φ:R+×Ω×X→Xis said to be an RDS on a complete separable metric space (X,d)with Borel σ-algebra B(X)over the metric dynamical system Ω,F,P,θtt∈Rif (i) Φ(·,·,·):R+×Ω×X→Xis (B(R+)×F×B(X),B(X))-measurable; (ii) Φ(0, ω,·)is the identity on Xfor P-a.e. ω∈Ω; (iii) Φ(t+s,ω,·)=Φ(t,θsω,·)◦Φ(s,ω,·), for all t,s∈R+and P-a.e. ω∈Ω. An RDS Φis continuous or differentiable if Φ(t,ω,·):X→Xis continuous or differentiable for all t∈R+and P-a.e. ω∈Ω. Definition 2.3. A set-valued map Ωω→ D(ω) ∈2Xis said to be a random set in Xif the mapping ω→ d(x,D(ω)) is (F,B(R))-measurable for any x∈X, where d(x,D(ω)) infy∈D(ω) d(x,y)is the distance in Xbetween the element xand the set D(ω) ⊂X. Definition 2.4. A random set {D(ω)}ω∈Ωof Xis called tempered with respect to {θt}t∈Rif for P-a.e. ω∈Ω, lim t→∞e−βtdDθ−tω=0, for all β>0, where d(D)=supx∈DxX. Definition 2.5. Let D={D(ω) ⊂X,ω∈Ω}be a family of random set. A random set K(ω) ∈D is said to be a D-pullback absorbing set for Φif for P-a.e. ω∈Ωand for every B∈D, there exists T=T(B,ω) > 0 such that Φt,θ−tω,Bθ−tω⊆K(ω) for all t≥T. If, in addition, for all ω∈Ω,K(ω) is a closed non-empty subset of Xand K(ω) is measurable in Ωwith respect to F, then we say Kis a closed measurable D-pullback absorbing set for Φ. Definition 2.6. An RDS Φis said to be D-pullback asymptotically compact in Xif for P-a.e. ω∈Ω, Φtn,θ−tnω,xnn≥1has a convergent subsequence in Xwhenever tn→∞and xn∈Dθ−tnωfor any given D∈D. Definition 2.7. A compact random set A(ω) is said to be a D-pullback random attractor associated with the RDS Φif it satisfies the invariance property Φ(t,ω)A(ω) =Aθtω, for all t≥0, and the pullback attracting property lim t→∞dist Φt,θ−tωDθ−tω,A(ω)=0, for all t≥0, D∈D,P−a.e.ω∈Ω, where dist(·,·)denotes the Hausdorff semidistance dist(A,B)=sup x∈A inf y∈Bd(x,y),A,B⊂X. Downloaded from https://academic.oup.com/imamat/article/88/4/576/7276455 by Sevilla Psych Lib B21 user on 11 March 2024 RANDOM ATTRACTORS FOR AN SNDRDE ON A SEMI-INFINITE INTERVAL 581 Lemma 2.1. Let (θ,Φ) be a continuous RDS. Suppose that Φis D-pullback asymptotically compact and has a closed pullback D-absorbing set K={K(ω)}ω∈Ω∈D. Then it possesses a random attractor {A(ω)}ω∈Ω, where A(ω) =∩ τ≥0∪t≥τΦt,θ−tω,Kθ−tω. For convenience, we introduce the following Gronwall inequality in Bessaih et al. (2014) that will be frequently used in our subsequent proofs. Lemma 2.2. Let T>0 and u,α,fand gbe non-negative continuous functions defined on [0, T] such that u(t)≤α(t)+f(t)t 0 g(r)u(r)dr,fort∈[0, T]. Then u(t)≤α(t)+f(t)t 0 g(r)α(r)et rf(τ)g(τ )dτdr,fort∈[0, T]. 3. Global solutions and RDSs In this section, we will prove the existence of global solutions to SNDRDE (1.4) under the given initial condition, and then show that the solutions generate an RDS. By the Fourier sine transform defined by Eq. (10.5.39) in Haberman (2004), we can obtain that the semigroup S(t)generated by the linear system ⎧ ⎨ ⎩ ∂u ∂t=Δu−μu,t>0 u(t,0)=0, t≥0 u(0, x)=φ(x),x∈R+ (3.1) is S(0)[φ](x)=φ(x), S(t)[φ](x)=exp(−μt) √4πt∞ 0φ(y)exp −(x−y)2 4t−exp −(x+y)2 4tdy, t>0, (3.2) for (x,φ) ∈R+×X.LetZ=BUC(R,R)be the set of all bounded and uniformly continuous functions from Rto Requipped with the usual supremum norm ·Z. Then, the Fourier transformation method indicates that the semigroup U(t):Z→Zgenerated by Δ−μIis defined as U(0)[φ](x)=φ(x), U(t)[φ](x)=exp(−μt) √4πt∞ −∞ φ(y)exp −(x−y)2 4tdy for all t∈(0, ∞),(3.3) for (x,φ) ∈R×Z, which is analytic and strongly continuous on Z. Downloaded from https://academic.oup.com/imamat/article/88/4/576/7276455 by Sevilla Psych Lib B21 user on 11 March 2024 582 W. HU ET AL. We introduce the following results concerning the properties of semigroup S(t), which is frequently used thorough the whole paper. The details of the proof can be found in Yi & Zou (2016) Lemma 2.1. Lemma 3.1. Let S(t)and U(t)be defined in (3.2) and (3.3), respectively, then we have the following results. (i) S(t)[φ](x)=e−μtU(t)[˜ φ](x)for all φ∈X,t∈R+and x∈R+, where ˜ φrepresents the odd extension of φ. (ii) S(t)is an analytic and strongly continuous semigroup on X. (iii) For all t∈(0, ∞)and (x,φ) ∈(0, ∞)×X, there hold S(t)[φ]≤e−μtφ, ∂S(t)[φ](x) ∂t≤(1+μt)exp(−μt)φ t,  ∂S(t)[φ](x) ∂x≤exp(−μt)φ √πt, ∂2S(t)[φ](x) ∂x2≤exp(−μt)φ t. (iv) For any t1,t2∈(0, ∞),x1,x2∈R+and φ∈X, there holds St1[φ]x1−St2[φ]x2≤1+μmin t1,t2exp −μmin t1,t2||φ|| min t1,t2t2−t1 +exp −μmin t1,t2||φ|| πmin t1,t2x2−x1. For the purpose of later use, we prove the following property on the non-local diffusion operator of (1.4). Lemma 3.2. Define K:X→Xby K(φ)(·)=R+ Γ(α,·,y)φ(y)dy for all φ∈X. Then, Ksup{K(φ) φ:φ = 0}≤1. Proof. For any x∈R+,wehave |K(φ)(x)|=R+ 1 √4παe−(x−y)2 4αφ(y)dy−R+ 1 √4παe−(x+y)2 4αφ(y)dy =x −∞ 1 √4παe−u2 4αφ(x−u)du−∞ x 1 √4παe−u2 4αφ(u−x)du ≤φ∞ −∞ 1 √4παe−u2 4αdu=φ. (3.4) Therefore, we have Ksup φ=0 K(φ) φ≤1.  In the sequel, we always impose the following assumptions on the nonlinear drift term f. Downloaded from https://academic.oup.com/imamat/article/88/4/576/7276455 by Sevilla Psych Lib B21 user on 11 March 2024 RANDOM ATTRACTORS FOR AN SNDRDE ON A SEMI-INFINITE INTERVAL 583 (H)f(·):R→Ris continuously differentiable, f(0)=0, and  df (s) ds ≤Lf. Here, Lfis a positive constant, representing the bound of the derivation. Hence, it is clear that for all s1,s2∈R fs1−fs2≤Lfs1−s2,(3.5) In this paper, we consider the canonical probability space (Ω,F,P)with Ω=ω=ω1,ω2,...,ωm∈CR;Rm:ωi(0)=0 and Fis the Borel σ-algebra induced by the compact open topology of Ω, while Pis the corresponding Wiener measure on (Ω,F). Then, we identify Wwith W(t,ω) =w1(t),w2(t),...,wm(t)for t∈R. Moreover, we define the time shift by θtω(·)=ω(·+t)−ω(t),t∈R. Then, Ω,F,P,θtt∈Ris a metric dynamical system. In order to construct the conjugate transformation, we consider the stochastic stationary solution of the one dimensional Ornstein–Uhlenbeck equation dzj+μzjdt=dwj(t),j=1, ...,m. (3.6) Thesolutionto(3.6) is given by zj(t)zjθtωj=−μ0 −∞ eμsθtωj(s)ds,t∈R, (3.7) By Definition 2.4, one can see that the random variable zjωjis tempered and zjθtωjis P-a.e. ω continuous. Therefore, Proposition 4.3.3 in Arnold (1998) implies that there exists a tempered function 0<r(ω) < ∞such that m  j=1zjωj 2≤r(ω), (3.8) where r(ω) satisfies, for P-a.e. ω∈Ω, rθtω≤eμ 2|t|r(ω),t∈R. (3.9) Combining (3.8) with (3.9), we obtain that for P-a.e. ω∈Ω, m  j=1zjθtωj 2≤eμ 2|t|r(ω),t∈R(3.10) Downloaded from https://academic.oup.com/imamat/article/88/4/576/7276455 by Sevilla Psych Lib B21 user on 11 March 2024 584 W. HU ET AL. Putting zθtω(x)=m j=1gj(x)zjθtωj,by( 3.6), we have dz+μzdt= m  j=1 gj(x)dwj. To prove that (1.4) possesses a global solution that generates an RDS, we consider the transformation v(t)=u(t)−zθtω, where uis a solution of (1.4), and show that vis a global solution of the transformed equation and generates an RDS. Then, we show that (1.4) also has a global solution and generates a conjugated RDS due to the inverse transformation. This method has also been adopted by Hu & Zhu (2022),Li & Guo (2020) and Wang et al. (2015) to deal with random attractors as well as Duan et al. (2003, 2004);Lu & Schmalfuß(2007) and Lu & Schmalfuß(2008) to tackle invariant manifolds of stochastic partial differential equations with or without delay. Apparently, vsatisfies ∂v(t,x) ∂t=Δv(t,x)−μv(t,x)+Fvt+zθt+·ω(x)+Δzθtω(x),t>0, x∈(0, ∞)(3.11) with boundary condition v(t,0)=0, for t∈(0, ∞), (3.12) and initial condition v(ξ,x,ω) =ψ(ξ,x,ω) φ(x,ξ)−zθξω(x)for (x,ξ) ∈R+×[−τ, 0]. (3.13) Here, F:C→Xis defined by F(ϕt+zθt+·ω)(x)=εR+ Γ(α,x,y)fϕ(t−τ,y)+zθt−τω(y)dy =εK[fϕ(t−τ,·)+zθt−τω,·](x), for any ϕ∈C. We now show that the pathwise deterministic problem (3.11)–(3.13) has a global mild solution under assumption (H). We aim at solving the following integral equation: v(t,ω,ψ) =S(t)ψ(0)+t 0S(t−r)Fvr+zθr+·ωdr +t 0S(t−r)Δzθrωdr, ψ(t),t∈[−τ,0], (3.14) for the initial data ψ∈C. We have the following results. Theorem 3.1. Assume that fsatisfies (H). Then, for any ψ∈Cand for P-a.e. ω∈Ω, there exists a global mild solution to (3.11)–(3.13). Moreover, if f:C→Xis globally bounded, i.e. there exists M>0 such that f(ϕ)≤Mfor all ϕ∈C, then the solution is pullback bounded, i.e., there exists C(ω) > 0 such that v(t,θ−tω,ψ)≤C(ω) for P-a.e. ω∈Ω. Proof. We first prove that (3.11)–(3.13) has a local mild solution and then show it can be extended to a global one by an argument of steps. For any ψ∈Cand P-a.e. ω∈Ω, we show in the following that Downloaded from https://academic.oup.com/imamat/article/88/4/576/7276455 by Sevilla Psych Lib B21 user on 11 March 2024 RANDOM ATTRACTORS FOR AN SNDRDE ON A SEMI-INFINITE INTERVAL 591 for P-a.e. ω∈Ω. Keep in mind that  vt  C co =sup v(t+ζ)X co :ζ∈[−τ,0] . Hence, we can obtain  vt(·,ω,ψ)  C co ≤eμτ [e−μtψC co +εLft 0 eμ(s−t) vs(·,ω,ψ)  C co ds +t 0 eμ(s−t)(ΔzθsωX co +εLfzθs−τωX co)ds. (4.5) By replacing ωby θ−tω, we derive from (4.5) that, for all t≥τ,  vt·,θ−tω,ψθ−tω  C co ≤eμτ [e−μt ψθ−tω  C co +t 0 eμ(s−t)Δzθs−tωX co +εLfzθs−t−τωX cods +εLft 0 eμ(s−t) vs·,θ−tω,ψθ−tω  C co ds]. (4.6) Since gjare twice continuously differentiable and z(ω)(x)=m j=1gj(x)zjωj, there exists constant c such that p1(ω) Δz(ω)X co +εLfzθ−τωX co ≤cm j=1zjωj 2. Therefore, it follows from (3.9) and (3.10) that t 0 eμ(s−t)p1θs−tωds≤ct 0 eμ 2(s−t)r(ω)ds≤cr(ω).(4.7) Incorporating (4.7)into(4.6) gives rise to  vt·,θ−tω,ψθ−tω  C co ≤eμτ [e−μt ψθ−tω  C co +εLft 0 eμ(s−t) vs·,θ−sω,ψθ−sω  C co ds+cr(ω)]. (4.8) Multiply the both sides of (4.8)byeμtand adopt the Grönwall inequality, we have eμt vt·,θ−tω,ψθ−tω  C co ≤eμτ  ψθ−tω  C co +ceμtr(ω)) +εeμτ Lft 0 ( ψθ−tω  C co +ceμsr(ω))eεLfeμτ (t−s)ds ≤ceμτ eμtr(ω) +eμτ  ψθ−tω  C co +εeμτ Lf ψθ−tω  C co t 0 eεLfeμτ (t−s)ds +cεeμτ Lfr(ω) t 0 eεLfeμτ (t−s)eμsds. (4.9) Downloaded from https://academic.oup.com/imamat/article/88/4/576/7276455 by Sevilla Psych Lib B21 user on 11 March 2024 592 W. HU ET AL. Therefore, we have  vt·,θ−tω,ψθ−tω  C co ≤ceμτ r(ω) +e−μteμτ  ψθ−tω  C co + ψθ−tω  C co (e(εLfeμτ −μ)t−1) +cεeμτ Lf μ−εeμτ Lf [e−εeμτ Lf−e−μt]r(ω). (4.10) Note that ψ(ω) =φ−z(θt+·ω). The above estimate (5.1) implies that, for all t≥τ,  ut·,θ−tω,φ  C co ≤ vt·,θ−tω,ψθ−tω  C co +z(θ−tθt+·ω)C co ≤ceμτ r(ω) +e−μteμτ  ψθ−tω  C co + ψθ−tω  C co (e(εLfeμτ −μ)t−1) +cεeμτ Lf μ−εeμτ Lf [e−εeμτ Lf−e−μt]r(ω) +ceμτ 2r(ω). (4.11) Therefore, if φ∈Dθ−tωand εLfeμτ −μ<0, then there exists a TD>τsuch that, for all t≥TD(ω), e−μteμτ  ψθ−tω  C co + ψθ−tω  C co (e(εLfeμτ −μ)t−1)−cεeμτ Lf μ−εeμτ Lf e−μtr(ω) ≤c1(ω),(4.12) which, along with (4.11), shows that, for all t≥TD(ω)  ut·,θ−tω,φ  C co ≤2ceμτ r(ω) +cεeμτ Lf μ−εeμτ Lf e−εeμτ Lfr(ω) +c1(ω).(4.13) Given ω∈Ω, define K(ω) ={ϕ∈C:ϕC co ≤2ceμτ r(ω) +cεeμτ Lf μ−εeμτ Lf e−εeμτ Lfr(ω) +c1(ω)}.(4.14) Then, K={K(ω)}ω∈Ω∈D. Furthermore, (4.13)impliesthatKis a random absorbing set for the RDS Φin D. In the sequel, we first show that Ψis a continuous random semiflow with respect to ·C co. Lemma 4.3. Assume (H) is satisfied and fis globally bounded, then Φt,θ−tω,φis a continuous random semiflow with respect to the compact open topology induced by the norm ·C co. Proof. It suffices to prove that for any given φn∈Dθ−tnωand φ∈Dθ−tωsuch that tn→t,φn→φ then the sequence Ψtn,θ−tnω,φnconvergent to Ψt,θ−tω,φwith respect to · C co. Here, for convenience, we assume that tn≥tsince the case tn<tcan be proved similarly. To prove the continuity, define P:R+×Ω×C→Xco by P(t,θ−tω,φ)(x)=Ψ(t,θ−tω,φ)(0)(x) for all (t,φ) ∈R+×C. By Theorem 3.2 and the cocycle property of z, we only need to prove that P(t,θ−tω,φ)(x)is a random semiflow. Take tn,φnn∈N⊂R+×Dθ−tnωsuch Downloaded from https://academic.oup.com/imamat/article/88/4/576/7276455 by Sevilla Psych Lib B21 user on 11 March 2024 RANDOM ATTRACTORS FOR AN SNDRDE ON A SEMI-INFINITE INTERVAL 593 that limn→∞ tn−t=0 and limn→∞ drφn,φ=0. Denote by Gn|P(tn,θ−tnω,φn)(x)− P(t,θ−tω,φ)(x)|. For any given a bounded closed interval Iand any x∈I,wehave Gn=Ψ(tn,θ−tnω,φn)(0)(x)−Ψt,θ−tω,φ(0)(x) ≤Φ(tn,θ−tnω,ψn)(0)(x)−Φt,θ−tω,ψ(0)(x)+|z(θtω)(x)−z(θtnω)(x)| ≤|S(tn)ψn(0, x)−S(t)ψ(0, x)|+|tn 0 S(tn−r)Fvn r+zθ−tnθr+·ω(x) −t 0 S(t−r)Fvr+zθ−tθr+·ω(x)dr|+|tn 0 S(tn−r)Δzθ−tnθrω(x)dr −t 0 S(t−r)Δzθ−tθrω(x)dr|+|z(θtω)(x)−z(θtnω)(x)| I1+I2+I3+I4, (4.15) for P-a.e. ω∈Ω. Now, we estimate each term on the right-hand side of (4.15). I1|[S(tn)−S(t)]ψn(0, x)|+|S(t)[ψn(0, x)−ψ(0, x)]| ≤1+tnexp −μtn|ψn(0, x)| tnt−tn+e−μt|ψn(0, x)−ψ(0, x)| I11 +I12. (4.16) It follows from limn→∞ drφn,φ=0, the continuity of z(θtω) with respect to tand limn→∞ tn−t= 0 that limn→∞ I12 =0. Clearly, I11 →0 because of tn→t. I2|tn 0 S(tn−r)Fvn r+zθ−tnθr+·ω(x)dr −t 0 S(tn−r)Fvn r+zθ−tnθr+·ω(x)dr| +|t 0 S(tn−r)Fvn r+zθ−tnθr+·ω(x)dr −t 0 S(tn−r)Fvr+zθ−tnθr+·ω(x)dr| +|t 0 S(tn−r)Fvr+zθ−tnθr+·ω(x)dr −t 0 S(t−r)Fvr+zθ−tnθr+·ω(x)dr| +|t 0 S(tn−r)Fvr+zθ−tnθr+·ω(x)dr −t 0 S(t−r)Fvr+zθ−tθr+·ω(x)dr| I21 +I22 +I23 +I24. (4.17) By Lemmas 3.1 and 3.2 and the boundedness of f, we have for any x∈Iand P-a.e. ω∈Ω I21 ≤εtn t e−μ(tn−r)fvn r+zθr+·ω(x)dr ≤εMtn t e−μ(tn−r)dr.(4.18) Downloaded from https://academic.oup.com/imamat/article/88/4/576/7276455 by Sevilla Psych Lib B21 user on 11 March 2024 594 W. HU ET AL. Since tn→t, we can see I21 →0. I22 ≤εLft 0 e−μ(tn−r)vn r−vrCco dr ≤εLf|vr(·,θ−tnω,ψn)(x)−vr(·,θ−tnω,ψ)(x)|tn t e−μ(tn−r)dr. (4.19) By similar procedure as the proof of Theorem 2.8-(i) in Yi et al. (2012), we have I22 →0 provided ψn→ψ. I23 ≤t 0 [S(tn−r)−S(t−r)]Fvr+zθr+·ωdr =tn−√δ 0 [S(tn−r)−S(t−r)]Fvr+zθr+·ωdr +t tn−√δ [S(tn−r)−S(t−r)]Fvr+zθr+·ωdr ≤εMtn−√δ 01+μ(tn−r)exp −μ(tn−r) tn−r(tn−t)dr +2M|t−tn+√δ| ≤εMt−tn1 √δ+μ+2M|t−tn+√δ|, (4.20) where δ∈(0, 1)and t<tn<t+δwith δ→0. Hence, we have I23 →0 when tn→t. Therefore, limn→∞ I2=0, as tn→t,φn→φ. In the following, we estimate I3. I3|tn 0 S(tn−r)Δzθ−tnθrω(x)−t 0 S(t−r)Δzθ−tnθrω(x)dr| |tn 0 S(tn−r)Δzθ−tnθrω(x)dr −t 0 S(tn−r)Δzθ−tnθrω(x)dr| +|t 0 S(tn−r)Δzθ−tnθrω(x)dr −t 0 S(t−r)Δzθ−tnθrω(x)dr| +|t 0 S(t−r)Δzθ−tnθrω(x)dr −t 0 S(t−r)Δzθ−tθrω(x)dr|I31 +I32 +I33. (4.21) I31 ≤c|Δzθrω(x)|tn t e−μ(tn−r)eμ(tn−r)/2r(ω)dr.(4.22) Since {gj(x)}m j=1are twice continuously differentiable, there exists M>0 such that for any x∈Iand P-a.e. ω,|Δzθrω(x)|⩽M. Thus, limn→∞ I31 =0 and limn→∞ I31 =0 in the case tn→t.It follows from Δzθtωis continuous with respect to tthat limn→∞ I33 =0. By the same arguments as the estimation of I23 in (4.20), we have limn→∞ I32 =0, indicating that limn→∞ I3=0. Moreover, since zjθtωjis P-a.e. ωcontinuous, we have that limn→∞ I4=0. Summing up the above computation Downloaded from https://academic.oup.com/imamat/article/88/4/576/7276455 by Sevilla Psych Lib B21 user on 11 March 2024 RANDOM ATTRACTORS FOR AN SNDRDE ON A SEMI-INFINITE INTERVAL 595 together with Lemma 4.1, we can see that Φt,θ−tω,φis a continuous random semiflow with respect to the compact open topology induced by the norm ·co. Now, we are in the position to prove the D-pullback asymptotically compact in Dwith respect to ·co C. Lemma 4.4. Assume that (H)holds and fis bounded. Then, the RDS Ψgenerated by SNDRDE (1.4)isD-pullback asymptotically compact in Cco for t>τ, i.e., for P-a.e. ω∈Ω, the sequence {Ψ(tn,θ−tnω,φnθ−tnω)}has a convergent subsequence in Cco provided tn→∞,B={B(ω)}ω∈Ω∈D and φnθ−tnω∈Bθ−tnω. Proof. Take an arbitrary random set {B(ω)}ω∈Ω∈D, a sequence tn→+∞and φn∈Bθ−tnω.We have to prove that Ψtn,θ−tnω,φnis precompact. Since {K(ω)}is a random absorbing for Ψ, then there exists T>0 such that, for all ω∈Ω, Ψt,θ−tωBθ−tω⊂K(ω) (4.23) for all t≥T. Because tn→+∞, we can choose n1≥1 such that tn1−1≥T. Applying (4.23)for t=tn1−1 and ω=θ−1ω,wefindthat η1Ψtn1−1, θ−tn1ω,φn1∈Kθ−1ω. (4.24) Similarly, we can choose a subsequence nkof {n}such that n1<n2<···<nk→+∞such that ηkΨtnk−k,θ−tnkω,φnk∈Kθ−kω. (4.25) Hence, by the assumption we conclude that the sequence Ψk,θ−kω,ηk (4.26) is precompact. On the other hand by (4.25), we have Ψ(k,θ−kω,ηk)=Ψ(k,θ−kω,Ψ(tnk−k,θ−tnkω,φnk)=Ψtnk,θ−tnkω,φnk, (4.27) for all k≥1. Combining (4.26) and (4.27), we obtain that the sequence Ψtnk,θ−tnkω,φnkis precompact. Therefore, Ψtn,θtnω,φnkis precompact, which completes the proof.  Lemma (4.2) says that the continuous RDS Ψhas a random absorbing set while Lemma (4.1) tells us that (θ,Ψ) is pullback asymptotically compact in Cco.Thus,itfollowsfromLemma2.1 that the continuous RDS (θ,Ψ)possesses a random attractor. Namely, we obtain the following result. Theorem 4.1. Assume that (H)holds, εLfeμτ −μ<0 and fis bounded, then the continuous RDS Ψ generated by (1.4) admits a unique D-pullback attractor in Cco belonging to the class D. Downloaded from https://academic.oup.com/imamat/article/88/4/576/7276455 by Sevilla Psych Lib B21 user on 11 March 2024 596 W. HU ET AL. 5. Existence of exponentially attracting stationary solutions In this section, we are devoted to deriving sufficient conditions that guarantee the random attractor being an exponentially attracting random fixed point ξ∗by adopting the general Banach fixed point theorem. We first introduce the general Banach fixed point theorem, which was established in Schmalfus (1998) and extended in Duan et al. (2003) to infinite case in the following. Lemma 5.1. Let (Y,dY)be a complete metric space with bounded metric. Suppose that Φ(t,ω,Y)⊂Y for ω∈Ω,t≥0, and that x→Φ(t,ω,x)is continuous. In addition, we assume the contraction condition: There exists a constant k<0 such that, for ω∈Ω, sup x=y∈Y log dY(Φ(1, ω,x),Φ(1, ω,y)) dY(x,y)≤k Then Φhas a unique generalized fixed point γ∗in Y. Moreover, the following convergence property holds: lim t→∞Φt,θ−tω,x=γ∗(ω) for any ω∈Ωand x∈Y. Theorem 5.1. Assume that fis bounded and satisfies (H). Moreover, assume that 0 <τ<1 and μ>max{εLf 1−τ,εLfeμτ }. Then the RDS Ψgenerated by SNDRDE (1.4) possess a tempered random fixed point ξ∗, which is unique under all tempered random variables in Cco and attracts exponentially fast every random variable in Cco. Proof. If μ>εLfeμτ , then the conditions of Theorem 4.1 hold and hence (1.4) possess random attractors in Cco. We will prove that (1.4) admits a unique globally exponentially attracting random stationary solution in Cco, which immediately implies the random attractor in Cco obtained in Theorem 4.1 is the random fixed point. If suffices to prove that the RDS Φgenerated by (3.11) has a unique exponentially attracting generalized fixed point χ∗. Since the transformation v(t)=u(t)−zθtωand u(t)=v(t)+zθtωare conjugation, one can see that ξ∗=χ∗+zθtωis a unique exponentially attracting generalized fixed point of (1.4) by conjugation technique. By (4.5) and the Grönwall inequality, one can see that for any ψ∈Cco  vt(·,ω,ψ(ω))  C co ≤ceμτ r(ω) +e−μteμτ ψ(ω)C co +ψ(ω)C co (e(εLfeμτ −μ)t−1) +cεeμτ Lf μ−εeμτ Lf [e−εeμτ Lf−e−μt]r(ω), (5.1) which implies that for any ψ∈Cco,Φ(t,ω,ψ) ∈Cco, i.e. Cco is invariant under the random semiflow Φ. Moreover, it follows from Lemma 4.3 that Φis continuous in C. Therefore, we only need to prove Downloaded from https://academic.oup.com/imamat/article/88/4/576/7276455 by Sevilla Psych Lib B21 user on 11 March 2024 RANDOM ATTRACTORS FOR AN SNDRDE ON A SEMI-INFINITE INTERVAL 597 the contraction property. That is, there exists k<0 such that sup ϕ=ψ∈Cco Φ(1, ω,ϕ) −Φ(1, ω,ψ)C co ≤ekϕ−ψC co.(5.2) Hence, it suffices to prove that for any ϕ,ψ∈C Φ(1, ω,ϕ) −Φ(1, ω,ψ)C co =v1(·,ω,ϕ) −v1(·,ω,ψ)C co ≤ekϕ(ζ,x)−ψ(ζ,x)C co.(5.3) By Eq. (3.14), we have for any ϕ,ψ∈C v1(·,ω,ϕ) −v1(·,ω,ψ)C co ≤S(1)[ϕ(0)−ψ(0)] +sup ζ∈[−τ,0] 1+ζ 0 S(1+ζ−r)Fvϕ r+zθr+·ω−Fvφ r+zθr+·ωdr ≤eμτ e−μφ−ψC co +εLf1 0 e−μ(1−r) vr(·,ω,φ) −vr(·,ω,ψ)  C co dr . (5.4) Multiply both sides of (6.3)byeμleads to euv1(·,ω,ϕ) −v1(·,ω,ψ)C co ≤eμτ [ϕ−ψC co +εLf1 0 eμr vr(·,ω,ϕ) −vr(·,ω,ψ)  C co dr]. (5.5) Again, the Grönwall inequality gives rise to euv1(·,ω,ϕ) −v1(·,ω,ψ)C co ≤eμτ+εLfϕ−ψC co,(5.6) implying that vϕ−vψC co ≤eμ(τ−1)+εLfφ−ψC co. Thus, μ> εLf 1−τmeans that μ(τ −1)+εLf<0, indicating that (6.1) satisfies. Therefore, Φ(t,ω,·)admits a random exponentially attracting generalized fixed point χ∗and ξ∗=χ∗+zθtωis a unique exponentially attracting generalized fixed point of (1.4). This completes the proof.  6. Applications In this section, we will apply our main results to the stochastic non-local delayed Nicholson’s blowfly equation. As pointed before, we consider the ideal case, i.e. the one-dimensional domain. The results can be extended to half planes similarly to the deterministic case Wang (2014);Hu & Duan (2018);Hu et al. (2018). We consider equation (1.4) with the birth function being Ricker’s function b(u)=pue−qu, which was first employed by Gurney et al. (1980) to fit the Nicholson’s blowfly experiment data and has been widely used as a prototype of birth function for many species such as blowfly and fish. Here, pis the maximum per capita egg production, τis the maturation time rate and 1/qis the size at which the Downloaded from https://academic.oup.com/imamat/article/88/4/576/7276455 by Sevilla Psych Lib B21 user on 11 March 2024 598 W. HU ET AL. population reproduces at its maximum rate. Thus, (1.4) can be written as ⎧ ⎨ ⎩ ∂u ∂t(t,x)=Δu(t,x)−μu(t,x)+εR+Γ(α,x,y)pu(t−τ,y)e−qu(t−τ,y)dy+m j=1gj(x)dwj dt , u(t,0)=0, t>0, u(t,x)=φ(t,x),(t,x)∈[−τ,0]×R+. (6.1) Theorem 6.1. For (6.1), the following statements are true: (i) If q>e2, and μ>εpqe−2eμτ , then (6.1) admits a random attractor. Furthermore, if the delay τsatisfies 0 <τ<1 and μ>max{εpqe−2 1−τ,εpqe−2eμτ }, then the random attractor is a random exponentially attracting generalized fixed point. (ii) If 0 <q≤e2and μ>εpeμτ , then (6.1) admits a random attractor. Furthermore, if the delay τ satisfies 0 <τ<1 and μ>max{εp 1−τ,εpeμτ }, then the random attractor is a random exponentially attracting generalized fixed point. Proof. Differentiating ftwice gives f(u)=p(1−qu)e−qu, (6.2) f(u)=−qp(2−qu)e−qu. (6.3) Hence, f(u)≤f(1 q)=p qe−1, i.e. fis bounded by p qe−1. Moreover, by (6.3), we can see f(u)is decreasing on [0, 2 q)and increasing on (2 q,+∞). Moreover, f(u)<0on(1 q,+∞)and f(u)>0on [0, 1 q). Therefore, we have |f(u)|≤max{f(0),|f(2 q)|} = max{p,pqe−2}. In the case q>e2,wehave |f(u)−f(v)|≤pqe−2|u−v|, (6.4) i.e. we can take Lf=pqe−2. Thus, by Theorem 4.1,wehaveif μ>εpqe−2eμτ , (6.5) then (6.1) admits a random attractor. Furthermore, if the delay τsatisfies 0 <τ<1 and μ> max{εpqe−2 1−τ,pqεe−2eμτ }, then the random attractor is a random exponentially attracting generalized fixed point. In the scenario q≤e2, we can take Lf=p. Thus, by Theorem 4.1,wehaveif μ>εpeμτ , (6.6) then (6.1) admits a random attractor. Furthermore, if the delay τsatisfies 0 <τ<1 and μ> max{εp 1−τ,εpeμτ }, then the random attractor is a random exponentially attracting generalized fixed point.  In the following text, we give some comments of Theorem 6.1 from the biological point of view. Remark 6.1. Since 1/qis the size at which the population reproduces at its maximum rate, Theorem 6.1 says that if the mature population arrives its maximum reproduction rate at a small size, say 0 <u<1 e2, Downloaded from https://academic.oup.com/imamat/article/88/4/576/7276455 by Sevilla Psych Lib B21 user on 11 March 2024 RANDOM ATTRACTORS FOR AN SNDRDE ON A SEMI-INFINITE INTERVAL 599 the parameter qaffects the existence and the structure of random attractors, while in the case the mature population arrives its maximum reproduction rate at a large size, say more than 1 e2, then the existence and the structure of random attractors do not depend on the parameter q. Remark 6.2. For simplicity, we use Theorem 6.1 (ii) to explain the influence of other parameters on the existence and the structure of random attractors since it is the same for (i). If we fix the maximum birth rate p, the death rate μof mature population and εof immature population such that μ>max{εp 1−τ,εpeμτ }, then the increase of maturation time τmay destroy the existence of attractors and the attractors being a random exponentially attracting fixed point, i.e. the long maturation time may make the population unstable. It is clear that if we fix other parameters such that the inequalities hold, then the increase of p,εand τwill all destroy the random attractor. At last, if p,εand τare fixed, the left of μ>εpeμτ increases at a linear rate, while the right increases at an exponential rate and hence with the increase of μ, the right will at last be larger than the left, which destroys the condition. In summary, when we fix other parameters, each single parameter in (6.1) may destabilize the system when it becomes large enough. 7. Summary In this paper, we have obtained the existence and qualitative property of random attractors for (1.4)on a semi-infinite interval R+. We show that under certain conditions, the random attractor is a globally exponentially attracting random stationary solution. From dynamical system theory, the conditions for the attractors being fixed point are so strong that they could be hardly met in the real world applications. Indeed, from the dissipative system theory, if some estimates on the dimension of random attractors can be given, it will benefit the researchers a lot in studying the structure of the random attractor. Nevertheless, the lack of inner product of the phase space and the asymmetry as well as non-compactness of spatial domain made this problem quite challenging, requiring further studies. Furthermore, in order to obtain the global complex dynamics and non-local analysis of the qualitative properties of the system, existence and structure of the associated invariant manifolds of the stationary solutions, and the existence of connecting orbits (including the heteroclinic orbits or homoclinic orbits) are all of great significance and deserve much attention. Acknowledgement The authors would like to thank the anonymous referees for their valuable comments, especially for suggesting an application part, which led to improvement in our presentation. This work was jointly supported by the National Natural Science Foundation of China (62173139), China Postdoctoral Science Foundation (2019TQ0089), Hunan Provincial Natural Science Foundation of China (2020JJ5344), the Science and Technology Innovation Program of Hunan Province (2021RC4030) and the Scientific Research Fund of Hunan Provincial Education Department (20B353). The research of T. Caraballo has been partially supported by Spanish Ministerio de Ciencia e Innovación (MCI), Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER), under the project PID2021-122991NB-C21. This work was completed when Wenjie Hu was on a visit to the Universidad de Sevilla as a visiting scholar, and he would like to thank the staff in the Facultad de Matemáticas for their hospitality and also thank the university for its excellent facilities and support during his stay. Downloaded from https://academic.oup.com/imamat/article/88/4/576/7276455 by Sevilla Psych Lib B21 user on 11 March 2024 600 W. HU ET AL. References Arnold, L. (1998) Random Dynamical System. New York, Berlin: Springer-Verlag. Bates, P. W., Lu, K. & Wang, B. (2009) Random attractors for stochastic reaction-diffusion equations on unbounded domains. J. Differ. Equ., 246, 845–869. Bessaih, H., Garrido-Atienza, M. J. & Schmalfuß, B. 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