# Upper Semicontinuity of Random Attractors for Nonautonomous Stochastic Reversible Selkov System with Multiplicative Noise

Authors: Chunxiao Guo; Yanfeng Guo; Xiaohan Li
Journal: Advances in Mathematical Physics
 (2019)

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## Abstract

In this paper, the existence of random attractors for nonautonomous stochastic reversible Selkov system with multiplicative noise has been proved through Ornstein-Uhlenbeck transformation. Furthermore, the upper semicontinuity of random attractors is discussed when the intensity of noise approaches zero. The main difficulty is to prove the asymptotic compactness for establishing the existence of tempered pullback random attractor.


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## Body

## 1. Introduction

Selkov equations [1] were a system of ODEs originally proposed by Selkov [2, 3] as a simplified model of a biochemical process called glycolysis, through which living cells get energy from breaking down sugar. The existence and robustness of global attractors of the reversible Selkov system have been investigated in the deterministic case (i.e.,σ=0) by You in [4]. Li proved the existence of random attractor of the stochastic reversible Selkov system on infinite lattice with additive noise in [5]. In this paper, we consider the upper semicontinuity of random attractors for nonautonomous stochastic reversible Selkov system with multiplicative noise(1)du=d1△u+ρ-au+u2v-Gu3+f1t,xdt+σu∘dWt,(2)dv=d2△v+β-bv-u2v+Gu3+f2t,xdt+σv∘dWt,where Γ⊂Rn(n≤3) is a bounded smooth domain. There are boundary conditions(3)ut,x=vt,x=0,x∈∂Γ,and initial conditions(4)uτ,x=uτx,vτ,x=vτx.All the coefficients d1,d2,ρ,β,a,b, and G are arbitrarily given positive constants; fi(i=1,2) denote the time-dependent external forces. {W(t)}t∈R is a one-dimensional two-sided standard Wiener process on a probability space. The terms σu∘dW(t), σv∘dW(t) indicate that (1)-(2) are interpreted in the sense of the Stratonovich integration.In the past decades, great progress has been made in various aspects of random dynamical systems; see [6–22]. The long-time behavior of solutions for nonautonomous three-component reversible Gray-Scott system was discussed in [23] and the existence and upper semicontinuity of random attractor for stochastic three-component reversible Gray-Scott system was proved in [24]. In [25], Tu and You introduced the pullback asymptotic compactness of stochastic Brusselator system with multiplicative noise. Notice that (1) and (2) are nonautonomous stochastic equations with time-dependent external terms f1 and f2. For such a nonautonomous stochastic system, Wang established an efficacious theory about the existence and upper semicontinuity of the random attractor by introducing two parametric spaces [26–29].In this paper, we first obtain the existence of random attractors for the stochastic perturbed reversible Selkov system (1)–(4) defined on Γ⊂Rn(n≤3). When σ→0, we will consider the limit behaviors of random attractors and prove the upper semicontinuity of these perturbed random attractors. As usual, we must prove the pullback asymptotic compactness of the solution operators in L2(Γ). The main difficulty for proving such compactness lines is the uniform estimates of the solutions in [L6(Γ)]2 when we need to obtain some estimates in [H1(Γ)]2. We obtain the important L6(Γ)-norm estimates by using the Mean Value Theorem.The outline of the paper is as follows. In Section2, some basic concepts related to the nonautonomous random dynamical system and upper semicontinuity of random attractors are introduced. Section 3 is devoted to the pullback asymptotic compactness and the existence of random attractors. In Section 4, we establish the upper semicontinuity of random attractors when the coefficient σ approaches zero.


## 2. Preliminaries

In this section, we use some concepts of nonautonomous random dynamical system and random attractor. Let(X,·X) be a real separable Banach space with Borel σ-algebra B(X) and (Ω,F,P,(θt)t∈R) be an ergodic metric dynamical system. Some concepts and definitions of continuous random dynamical system have been seen in [6, 8, 17, 30] for more details. The following theorem is used to obtain the existence of random attractors for the continuous random dynamical system.Theorem 1 (see [6, 8]).
Letφ be a continuous 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 φ and φ is asymptotically compact in X, then A(τ,ω) is a random attractor of φ, where (5)Aτ,ω=⋂r≥0⋃t≥rφt,τ-t,θ-tω,Bτ-t,θ-tω¯,τ∈R,ω∈Ω. Moreover, A(τ,ω) is the random attractor of φ.Now, we define the product Hilbert spaces(6)H=L2Γ2,E=H01Γ2.The norm and inner-product in H or L2(Γ) will be denoted by · and (·,·), respectively. The norm in Lp(Γ) or [Lp(Γ)]2 will be denoted by ·Lp, if p≠2. The norm in E will be denoted by ·E.Let{W(t)}t∈R be the standard one-dimensional, two-sided Wiener process in the complete probability space (Ω,F,P), where(7)Ω=ω∈CR,R:ω0=0,the σ-algebra F is generated by the compact-open topology on Ω, and P is the corresponding Wiener measure on F; see [6, 8]. The Wiener shift θt is defined by(8)θtω·=ω·+t-ωt,ω∈Ω,t∈R. Consider the Ornstein-Uhlenbeck process(9)zθtω≔-∫-∞0esθtωsdω=-∫-∞0esωt+sds+ωt,which solves the Itô equation(10)dz+zdt=dWt.Proposition 2 (see [6, 8]).
Let the metric dynamical system(Ω,F,P,(θt)t∈R) and the Ornstein-Uhlenbeck process {z(θtω)}t∈R be defined as above. There is a θt-invariant set Ω~∈Ω of full P-measure such that for every ω∈Ω~, the following statements hold:
(1) The Ornstein-Uhlenbeck process{z(θtω)}t∈R has the asymptotically sublinear growth property; i.e.,(11)limt→±∞zθtωt=0.
(2)z(θtω) is continuous in t and, for any fixed t0∈R,(12)limt→±∞1t-t0∫t0tzθsωds=0.In the sequel, we consider ω∈Ω~ only and will always write Ω for Ω~.Proposition 3 ((upper semicontinuity of random attractors) (see [8, 28])).
Suppose the following conditions are satisfied:
(1) Givenσ>0, we suppose that φσ is a random dynamical system over a metric system (Ω,F,P,(θt)t∈R). For P-a.e. τ∈R, ω∈Ω, t≥0, σn→0, and xn,x∈X with xn→x, there holds(13)limn→±∞φσt,τ,ω,xn=φt,τx.
(2) Assume thatφσ has a random attractor Aσ(τ,ω)={Aσ(τ,ω):τ∈R,ω∈Ω}∈D and a random absorbing set Kσ(τ,ω)={Kσ(τ,ω):τ∈R,ω∈Ω}∈D such that for some deterministic positive constant c and for P-a.e. τ∈R,ω∈Ω(14)limsupσ→0Kστ,ωX≤c,where Kστ,ωX=supx∈Kσ(τ,ω)xX.
(3) There exists aσ0>0 such that for P-a.e. τ∈R,ω∈Ω(15)⋃0<σ≤σ0Aστ,ωisprecompactinX.Then, for P-a.e. τ∈R,ω∈Ω,(16)distAστ,ω,A0τ,ω→0,asσ→0.To investigate the random dynamics of stochastic PDEs, we convert the stochastic evolutionary system to a system of pathwise PDEs with the random parameter. Let(17)Ut,τ,ω,Uτ=e-σzθtωut,τ,ω,Uτ,Vt,τ,ω,Vτ=e-σzθtωvt,τ,ω,vτ,where z(θtω) is the Ornstein-Uhlenbeck process. Then(18)dU=e-σzθtωdu-σe-σzθtωu∘dzθtω,dV=e-σzθtωdv-σe-σzθtωv∘dzθtω.So we can consider the following equations with random coefficients, but without white noise:(19)dUdt=d1△U+e-σzθtωρ-aU+e2σzθtωU2V-Ge2σzθtωU3+e-σzθtωf1t,x+σUzθtω,(20)dVdt=d2△V+e-σzθtωβ-bV-e2σzθtωU2V+Ge2σzθtωU3+e-σzθtωf2t,x+σVzθtω,for x∈Γ and t≥τ, with the homogeneous Dirichlet boundary conditions(21)Ut,x=Vt,x=0,t≥τ∈R,x∈∂Γ,and the initial value conditions(22)Uτ,x=Uτx=e-σzθtωuτx,Vτ,x=Vτx=e-σzθtωvτx.For every ω∈Ω, τ∈R, the initial-boundary value problem (19)–(22) is formulated into the following reversible Selkov equation:(23)dgdt=Ag+Jg+ft,x,(24)gτ,x=gτx=Uτx,VτxT,where g(t,τ,ω,gτ)=(U(t,τ,ω,Uτ),V(t,τ,ω,Vτ))T, and (25)A=d1△00d2△,and(26)Jg,θtω=e-σzθtωρ-aU+e2σzθtωU2V-Ge2σzθtωU3+e-σzθtωf1t,x+σUzθtωe-σzθtωβ-bV-e2σzθtωU2V+Ge2σzθtωU3+e-σzθtωf2t,x+σVzθtω,for any t≥τ with initial data(27)gτx=Uτx,VτxT=e-σzθτωuτ,e-σzθτωvτT.By the Galerkin method, we can prove the existence and uniqueness of the weak solution, which is continuously depending on the initial data and(28)gt,τ,ω,gτ∈Cτ,+∞;H∩L2τ,+∞;E.This is similar to the autonomous case studied in [4]. Define the continuous random dynamical system φ associated with the problem (23)-(24) as follows:(29)φ:R+×R×Ω×H→H,φt,τ,ω,gτ=gt+τ,τ,θ-τω,gτ=Ut+τ,τ,θ-τω,Uτ,Vt+τ,τ,θ-τω,Vτfor all (t,τ,ω,gτ)∈R+×R×Ω×H. The continuous dynamical system φ and a continuous random dynamical system associated with the solutions of (1)–(4) are equivalent. Thus, we only discuss the dynamical system φ induced by (19)–(22).Assume that(30)X=fismeasurable:∫-∞0eγ/2s(fs+τ,·Lppds<+∞,p=2,4,6.Sometimes, we also assume that f1 and f2 are tempered in the following sense. For every c>0,(31)limλ→-∞ecλ∫-∞0eγ/2sf1s-λ,·Lpp+f2s-λ,·Lppds=0,p=2,4,6.


## 3. Existence of Random Attractors

In this section, we derive uniform estimates of solution for the stochastic reversible Selkov equations; these estimates are necessary for proving the existence of bounded absorbing sets and the asymptotic compactness of the random dynamical system. We will useD to denote the collection of all tempered random subsets of H from now on.For brevity, we writeU(t,τ,ω,Uτ),V(t,τ,ω,Vτ) as U(t,ω),V(t,ω) or simply as U,V; similarly, we write weak solution g(t,τ,ω,gτ) as g(t,ω) or g.Lemma 4.
Suppose thatfi(t,x)∈X,i=1,2. Then, for every τ∈R,ω∈Ω, and B={B(τ,ω):τ∈R,ω∈Ω}∈D, there exist a number T0(τ,ω,B,σ)≥0 and a tempered random variable M0(τ,ω)(32)M0τ,ω=1+C1∫-∞0e-2σzθsω-∫0s2σzθrωdr+γsds+2Ga∫-∞0e-2σzθsω-∫0s2σzθrωdr+γsf1s+τ2ds+2b∫-∞0e-2σzθsω-∫0s2σzθrωdr+γsf2s+τ2ds,such that for all t≥T0, the solution g of (19)–(22) with gτ-t∈B(τ-t,θ-tω) satisfies(33)gτ,τ-t,θ-τω,gτ-t≤M0τ,ω,∀t≥T0τ,ω,B,σ.Proof.
Multiplying (19) by GU and (20) by V, we get (34)12ddtGU2+Gd1∇U2=e-σzθtω∫ΓGUρdx-aGU2+e2σzθtω∫ΓGU3Vdx-e2σzθtω∫ΓG2U4dx+∫Γe-σzθtωf1t,xGUdx+σzθtωGU2,and(35)12ddtV2+d2∇V2=e-σzθtω∫ΓVβdx-bV2-e2σzθtω∫ΓU2V2dx+e2σzθtω∫ΓGU3Vdx+∫Γe-σzθtωf2t,xVdx+σzθtωV2.Adding up the above two equations, we obtain (36)ddtGU2+V2+2Gd1∇U2+2d2∇V2+2aGU2+bV2-2σzθtωGU2+V2=2e-σzθtω∫ΓGUf1t,x+Vf2t,xdx+2e-σzθtω∫ΓGUρ+Vβdx-2e2σzθtω∫ΓGU2-UV2dx.By the ϵ-Young inequality, we have(37)e-σzθtω∫ΓGUρ+Vβdx≤aG4U2+b4V2+e-2σzθtωρ2ΓGa+e-2σzθtωβ2Γb,(38)e-σzθtω∫ΓGUf1t,x+Vf2t,xdx≤aG4U2+b4V2+e-2σzθtωGaf12+e-2σzθtωbf22.It follows that(39)ddtGU2+V2+2Gd1∇U2+2d2∇V2+2aGU2+bV2-2σzθtωGU2+V2≤aGU2+bV2+e-2σzθtω2ρ2ΓGa+2β2Γb+e-2σzθtω2Gaf12+2bf22.Let γ=min{a,b}, C1=2ρ2|Γ|G/a+2β2|Γ|/b. It is easy to see that(40)ddtGU2+V2+γ-2σzθtωGU2+V2+2Gd1∇U2+2d2∇V2≤C1e-2σzθtω+e-2σzθtω2Gaf12+2bf22.By Gronwall’s lemma on the interval [τ-t,r](r≥τ-t) for above inequality and substituting ω by θ-τω, we can deduce(41)GUr,τ-t,θ-τω,Uτ-t2+Vr,τ-t,θ-τω,Vτ-t2≤e∫-tr-τ2σzθsωds-γr+t-τGUτ-t2+Vτ-t2+C1∫-tr-τe-2σzθsω+∫sr-τ2σzθrωdr+γs+τ-rds+2Ga∫-tr-τe-2σzθsω+∫sr-τ2σzθrωdr+γs+τ-rf1s+τ2ds+2b∫-tr-τe-2σzθsω+∫sr-τ2σzθrωdr+γs+τ-rf2s+τ2ds.Note that(42)2Ga∫-tr-τe-2σzθsω+∫sr-τ2σzθrωdr+γs+τ-rf1s+τ2ds=2Gaeγτ-re-∫r-τ02σzθrωdr∫-tr-τeγse-∫0s2σzθrωdr-2σzθsωf1s+τ2ds.By (11) and (12), we find that there exists a T1=T1(τ,ω,B,σ0)(0<σ<σ0) such that for ∀t≥T1>0, we have (43)zθ-tω≤γt8σ0,(44)∫0-tzθrωdr≤γt8σ0.We get for all t≥T1,(45)∫-t-T1eγse-∫0s2σzθrωdr-2σzθsωf1s+τ2ds≤∫-t-T1eγ/2sf1s+τ2ds.For the term on the right-hand side of (45), since f1,f2∈X, we have(46)∫-t-T1eγ/2sf1s+τ2+f2s+τ2ds<+∞.Thus, we get(47)2Ga∫-tr-τe-2σzθsω-∫r-τs2σzθrωdr+γs+τ-rf1s+τ2ds≤2Ga∫-∞r-τe-2σzθsω-∫r-τs2σzθrωdr+γs+τ-rf1s+τ2ds,where the integral is convergent due to (46). Similarly, we also obtain(48)2b∫-tr-τe-2σzθsω-∫r-τs2σzθrωdr+γs+τ-rf2s+τ2ds≤2b∫-∞r-τe-2σzθsω-∫r-τs2σzθrωdr+γs+τ-rf2s+τ2ds.Thus, we yield(49)GUr,τ-t,θ-τω,Uτ-t2+Vr,τ-t,θ-τω,Vτ-t2≤e∫-tr-τ2σzθsωds-γr+t-τGUτ-t2+Vτ-t2+C1∫-tr-τe-2σzθsω+∫sr-τ2σzθrωdr+γs+τ-rds+2Ga∫-∞r-τe-2σzθsω+∫sr-τ2σzθrωdr+γs+τ-rf1s+τ2ds+2b∫-∞r-τe-2σzθsω+∫sr-τ2σzθrωdr+γs+τ-rf2s+τ2ds.By (49), we get that(50)GUτ,τ-t,θ-τω,Uτ-t2+Vτ,τ-t,θ-τω,Vτ-t2≤e∫-t02σzθsωds-γtGUτ-t2+Vτ-t2+C1∫-∞0e-2σzθsω+∫s02σzθrωdr+γsds+2Ga∫-∞0e-2σzθsω+∫s02σzθrωdr+γsf1s+τ2ds+2b∫-∞0e-2σzθsω+∫s02σzθrωdr+γsf2s+τ2ds.According to the properties of the Ornstein-Uhlenbeck process,(51)∫-∞0e-2σzθsω+∫s02σzθrωdr+γsds<+∞.Noticing that {B(τ,ω):τ∈R,ω∈Ω}∈D is tempered, then for any (Uτ-t,Vτ-t)∈B(τ-t,θ-tω), we have(52)limt→+∞e∫-t02σzθsω-γdsGUτ-t2+Vτ-t2=0.Then, M0(τ,ω) is tempered. By (49), the proof is completed.Lemma 5.
Suppose thatfi(t,x)∈X,i=1,2. Then, for every τ∈R,ω∈Ω, and B={B(τ,ω):τ∈R,ω∈Ω}∈D, there exist a number T2(τ,ω,B,σ)≥0 and tempered random variables M1(τ,ω), M2(τ,ω)(53)M1τ,ω=1+C12e∫-102σzθsωds+γ∫-∞0e-2σzθsω+∫s02σzθrωdr+γsds+Gae∫-102σzθsωds+γ∫-∞0e-2σzθsω+∫s02σzθrωdr+γsf1s+τ2ds+1be∫-102σzθsωds+γ∫-∞0e-2σzθsω+∫s02σzθrωdr+γsf2s+τ2ds,M2τ,ω=1+C12e∫-2-12σzθsωds+γ+1∫-∞-1e-2σzθsω+∫s-12σzθrωdr+γs+1ds+Gae∫-2-12σzθsωds+γ+1∫-∞-1e-2σzθsω+∫s-12σzθrωdr+γs+1f1s+τ2ds+1be∫-2-12σzθsωds+γ+1∫-∞-1e-2σzθsω+∫s-12σzθrωdr+γs+1f2s+τ2ds.such that for all t≥T2, the solution g of (19)–(22) with gτ-t∈B(τ-t,θ-tω) satisfies(54)∫τ-1τ∇Us,τ-t,θ-τω,Uτ-t2ds+∫τ-1τ∇Vs,τ-t,θ-τω,Vτ-t2ds≤M1τ,ω,∫τ-2τ-1∇Us,τ-t,θ-τω,Uτ-t2ds+∫τ-2τ-1∇Vs,τ-t,θ-τω,Vτ-t2ds≤M2τ,ω,Proof.
Substitutingr for τ-1 in (41), for t≥1, we yield(55)GUτ-1,τ-t,θ-τω,Uτ-t2+Vτ-1,τ-t,θ-τω,Vτ-t2≤e∫-t-12σzθsωds-γt-1GUτ-t2+Vτ-t2+C1∫-t-1e-2σzθsω+∫s-12σzθrωdr+γ1+sds+2Ga∫-t-1e-2σzθsω+∫s-12σzθrωdr+γ1+sf1s+τ2ds+2b∫-t-1e-2σzθsω+∫s-12σzθrωdr+γ1+sf2s+τ2ds.Applying Gronwall’s lemma to (40) over [τ-1,τ] and by (55), we obtain(56)d1G∫τ-1τe∫sτ2σzθs-τωds-γτ-s∇Us,τ-t,θ-τω,Uτ-t2ds+d2∫τ-1τe∫sτ2σzθs-τωds-γτ-s∇Vs,τ-t,θ-τω,Vτ-t2ds≤12e∫-t02σzθsωds-γtGUτ-tθ-τω2+Vτ-tθ-τω2+C12∫-∞0e-2σzθsω+∫s02σzθrωdr+γsds+Ga∫-∞0e-2σzθsω+∫s02σzθrωdr+γsf1s+τ2ds+1b∫-∞0e-2σzθsω+∫s02σzθrωdr+γsf2s+τ2ds.It is easy to show that(57)∫τ-1τe∫sτ2σzθs-τωds-γτ-s∇Us,τ-t,θ-τω,Uτ-t2ds≥e-∫-102σzθsωds-γ∫τ-1τ∇Us,τ-t,θ-τω,Uτ-t2ds,and(58)∫τ-1τe∫sτ2σzθs-τωds-γτ-s∇Vs,τ-t,θ-τω,Vτ-t2ds≥e-∫-102σzθsωds-γ∫τ-1τ∇Vs,τ-t,θ-τω,Vτ-t2ds.
Now letd=min{d1,d2,d1G}. According to (56), (57), and (58), there exists a T3(τ,ω,B,σ) such that for all t≥T3(τ,ω,B,σ), we can get(59)∫τ-1τ∇Us,τ-t,θ-τω,Uτ-t2ds+∫τ-1τ∇Vs,τ-t,θ-τω,Vτ-t2ds≤1de∫-102σzθsωds+γ∫τ-1τe∫sτ2σzθs-τωds-γτ-s∇Us,τ-t,θ-τω,Uτ-t2ds+∫τ-1τe∫sτ2σzθs-τωds-γτ-s∇Vs,τ-t,θ-τω,Vτ-t2ds≤1+C12e∫-102σzθsωds+γ∫-∞0e-2σzθsω+∫s02σzθrωdr+γsds+Gae∫-102σzθsωds+γ∫-∞0e-2σzθsω+∫s02σzθrωdr+γsf1s+τ2ds+1be∫-102σzθsωds+γ∫-∞0e-2σzθsω+∫s02σzθrωdr+γsf2s+τ2ds≐M1τ,ω.Similarly, substituting r for τ-2 in (41), applying Gronwall’s lemma to (40) over [τ-2,τ-1], and combining the estimate (49), there exists a T4(τ,ω,B,σ) such that for all t≥T4(τ,ω,B,σ), we get(60)∫τ-2τ-1∇Us,τ-t,θ-τω,Uτ-t2ds+∫τ-2τ-1∇Vs,τ-t,θ-τω,Vτ-t2ds≤1+C12e∫-2-12σzθsωds+γ+1∫-∞-1e-2σzθsω+∫s-12σzθrωdr+γs+1ds+Gae∫-2-12σzθsωds+γ+1∫-∞-1e-2σzθsω+∫s-12σzθrωdr+γs+1f1s+τ2ds+1be∫-2-12σzθsωds+γ+1∫-∞-1e-2σzθsω+∫s-12σzθrωdr+γs+1f2s+τ2ds≐M2τ,ω.It is easy to check that M1(τ,ω) and M2(τ,ω) are tempered. Let T2=max{T3,T4}. This completes the proof.Lemma 6.
Letfi(t,x)∈X,i=1,2, q∈[τ-1,τ]. If there exists a t0∈(τ-t,q) such that g(t0,τ-t,θ-τω,gτ-t)∈E with gt0,τ-t,θ-τω,gτ-tE≤R(τ,ω), where R(τ,ω)>0 is any given random variable. Then for every τ∈R,ω∈Ω, and B(τ,ω)∈D, there exist a number T5(τ,ω,B,σ)≥0 and two tempered random variables M3(τ,ω) and M4(τ,ω)(61)M3τ,ω=1+C2∫-∞q-τe-4σzθsω+∫sq-τ4σzθsωds+γs-q+τds+8a3∫-∞q-τe-4σzθsω+∫sq-τ4σzθsωds+γs-q+τf1s+τL44ds+8b3G3∫-∞q-τe-4σzθsω+∫sq-τ4σzθsωds+γs-q+τf2s+τL44ds,and(62)M4τ,ω=1+C3∫-∞q-τe-6σzθsω+∫sq-τ6σzθsωds+γs-q+τds+32a5∫-∞q-τe-6σzθsω+∫sq-τ6σzθsωds+γs-q+τf1s+τL66ds+32b5G5∫-∞q-τe-6σzθsω+∫s06σzθsωds+γs-q+τf2s+τL66ds,such that for all t≥T5, the solution g of (19)–(22) with gτ-t∈B(τ-t,θ-tω) satisfies(63)gq,τ-t,θ-τω,gτ-tL44≤M3τ,ω,gq,τ-t,θ-τω,gτ-tL66≤M4τ,ω.Proof.
Here we invoked Sobolev’s embedding theorem to assert thatH01(Γ)↪L6(Γ)↪L4(Γ) is a continuous embedding for n≤3. For g(t0,τ-t,θ-τω,gτ-t)∈E with gt0,τ-t,θ-τω,gτ-tE≤R(τ,ω), it follows that(64)gt0,τ-t,θ-τω,gτ-tL6≤ηRτ,ω,gt0,τ-t,θ-τω,gτ-tL4≤λRτ,ω,where η and λ are positive constants.
LetU~=U, V~=V/G. Then, the transformed system (19)-(20) becomes a rescaled system(65)dU~dt=d1△U~+e-σzθsωρ-aU~+e2σzθsωGU~2V~-Ge2σzθsωU~3+e-σzθsωf1t,x+σU~zθsω,(66)dV~dt=d2△V~+e-σzθsωβG-bV~-e2σzθsωU~2V~+e2σzθsωU~3+e-σzθsωf2t,xG+σV~zθsω.Multiplying (65) by U~3, we obtain(67)14ddtU~L44+3d1U~∇U~2=∫Γe-σzθsωρU~3+f1U~3dx-aU~4L4+∫Γe2σzθsωGU~5V~-GU~6dx+σzθsωU~L44.Multiplying (66) by GV~3, we get(68)14ddtGV~L44+3Gd2V~∇V~2=∫Γe-σzθsωβV~3+f2V~3dx-bGV~4L4+∫Γe2σzθsω-GU~2V~4+GU~3V~3dx+σGzθsωV~L44.Summing up the resulting equalities, we have(69)14ddtU~L44+GV~L44+3dU~∇U~2+GV~∇V~2≤e-σzθsω∫ΓρU~3+βV~3+f1U~3+f2V~3dx-aU~4L4-bGV~4L4+e2σzθsωG∫Γ-U~6+U~5V~+U~3V~3-U~2V~4dx+σzθsωU~L44+GV~L44=e-σzθsω∫ΓρU~3+βV~3+f1U~3+f2V~3dx-aU~4L4-bGV~4L4+e2σzθsωG∫ΓU~2-U~4+U~3V~+U~V~3-V~4dx+σzθsωU~L44+GV~L44.Using the Young inequality, we have(70)-U~4+U~3V~+U~V~3-V~4≤-U~4+34U~4+14V~4+14U~4+34V~4-V~4≤0,and(71)e-σzθsω∫ΓρU~3+βV~3+f1U~3+f2V~3dx≤e-σzθsωρΓ1/4U~L43+βΓ1/4V~L43+e-σzθsωf1L4U~L43+f2L4V~L43≤2a3e-4σzθsωρ4Γ+3a8U~L44+3bG8V~L44+2b3G3e-4σzθsωβ4Γ+2a3e-4σzθsωf1L44+3a8U~L44+2b3G3e-4σzθsωf2L44+3bG8V~L44=34aU~L44+bGV~L44+e-4σzθsω2ρ4Γa3+2β4Γb3G3+e-4σzθsω2f1L44a3+2f2L44b3G3.Recalling that γ=min{a,b}, it follows that(72)ddtU~L44+GV~L44+γ-4σzU~L44+GV~L44≤e-4σzθsωC2+e-4σzθsω8f1L44a3+8f2L44b3G3,where C2=8ρ4|Γ|/a3+8β4|Γ|/b3G3. By Gronwall’s lemma on the interval [t0,q], q∈[τ-1,τ], for the above inequality, and replacing ω by θ-τω, we get(73)U~q,τ-t,θ-τω,U~τ-tL44+GV~q,τ-t,θ-τω,V~τ-tL44≤e∫t0-τq-τ4σzθsωds-γq-t0U~t0,τ-t,θ-τω,U~τ-tL44+GV~t0,τ-t,θ-τω,V~τ-tL44+C2∫-∞q-τe-4σzθsω+∫sq-τ4σzθsωds+γs-q+τds+8a3∫-∞q-τe-4σzθsω+∫sq-τ4σzθsωds+γs-q+τf1s+τL44ds+8b3G3∫-∞q-τe-4σzθsω+∫sq-τ4σzθsωds+γs-q+τf2s+τL44ds.Recalling that U~=U, V~=V/G, we have(74)Uq,τ-t,θ-τω,Uτ-tL44+G-3Vq,τ-t,θ-τω,Vτ-tL44≤1+C2∫-∞q-τe-4σzθsω+∫sq-τ4σzθsωds+γs-q+τds+8a3∫-∞q-τe-4σzθsω+∫sq-τ4σzθsωds+γs-q+τf1s+τL44ds+8b3G3∫-∞q-τe-4σzθsω+∫sq-τ4σzθsωds+γs-q+τf2s+τL44ds≐M3τ,ω.Similarly, multiplying (65) by U~5 and (66) by GV~5, we get(75)ddtU~L66+GV~L66+γ-6σzθsωU~L66+GV~L66≤e-6σzθsω32ρ6Γa5+32β6Γb5G5+e-6σzθsω32f1L66a5+32f2L66b5G5.Put C3=32ρ6|Γ|/a5+32β6|Γ|/b5G5. Applying Gronwall’s inequality over [t0,q] and replacing ω by θ-τω, we have(76)U~q,τ-t,θ-τω,U~τ-tL66+GV~q,τ-t,θ-τω,V~τ-tL66≤e∫t0q6σzθs-τωds-γq-t0U~t0,τ-t,θ-τω,U~τ-tL66+GV~t0,τ-t,θ-τω,V~τ-tL66+C3∫t0qe-6σzθs-τω+∫sq6σzθs-τωds-γq-sds+32a5∫t0qe-6σzθs-τω+∫sq6σzθs-τωds-γq-sf1sL66ds+32b5G5∫t0qe-6σzθs-τω+∫sq6σzθs-τωds-γq-sf2sL66ds.Recalling that U~=U, V~=V/G, we get(77)Uq,τ-t,θ-τω,Uτ-tL66+G-5Vq,τ-t,θ-τω,Vτ-tL66≤1+C3∫-∞q-τe-6σzθsω+∫sq-τ6σzθsωds+γs-q+τds+32a5∫-∞q-τe-6σzθsω+∫sq-τ6σzθsωds+γs-q+τf1s+τL66ds+32b5G5∫-∞q-τe-6σzθsω+∫sq-τ6σzθsωds+γs-q+τf2s+τL66ds≐M4τ,ω.The proof is completed.Lemma 7.
Suppose thatfi(t,x)∈X,i=1,2. Then, for every τ∈R,ω∈Ω, and B={B(τ,ω):τ∈R,ω∈Ω}∈D, there exist a number T6(τ,ω,B,σ)≥0 and a tempered random variable M5(τ,ω)(78)M5τ,ω=1+2σmax-1≤s≤0zθsω-2γM1+M4∫-104e4σzθsω3d2+3G2ds+∫-10e-2σzθsωΓρ2+β2dds+∫-10e-2σzθsωf1s+τ2+f2s+τ2dds.such that for all t≥T6, the solution g of (19)–(22) with gτ-t∈B(τ-t,θ-tω) satisfies(79)∇Uτ,τ-t,θ-τω,Uτ-tθ-τω2+∇Vτ,τ-t,θ-τω,Vτ-tω2≤M5τ,ω,where gτ-t=(Uτ-t,Vτ-t)∈B(τ-t,θ-tω).Proof.
Multiplying (19) by -△U and (20) by -△V, we have(80)12ddt∇U2=-d1△U2+ρe-σzθtω,-△U-a∇U2-e2σzθtωU2V,△U+Ge2σzθtωU3,△U+e-σzθtωf1,-ΔU+σzθtω∇U2,(81)12ddt∇V2=-d2△V2+βe-σzθtω,-△V-b∇V2+e2σzθtωU2V,△V-Ge2σzθtωU3,△V+e-σzθtωf2,-ΔV+σzθtω∇V2.Then summing up the resulting equalities, we have(82)12ddt∇U2+∇V2+d1△U2+d2△V2+a∇U2+b∇V2-σzθtω∇U2+∇V2=e-σzθtω∫Γ-ρ△U-β△Vdx+e-σzθtω∫Γ-f1ΔU-f2ΔV+e2σzθtω∫Γ-U2V△U+GU3△Udx+e2σzθtω∫ΓU2V△V-GU3△Vdx.Due to Hölder’s inequality and ϵ-Young inequality, we obtain (83)e-σzθtω∫Γ-ρ△U-β△Vdx≤e-2σzθtωρ2Ωd1+β2Ωd2+d14△U2+d24△V2.e-σzθtω∫Γ-f1ΔU-f2ΔV≤e-2σzθtωf12d1+f22d2+d14△U2+d24△V2.For the third term on the right-hand side of (82), we have (84)e2σzθtω∫Γ-U2V△U+GU3△Udx≤e4σzθtωd1∫ΓU4V2dx+e4σzθtωd1∫ΓG2U6dx+d14△U2+d14△U2≤d12△U2+e4σzθtω3d1∫Γ3G2U6dx+∫ΓU6+U6+V6dx≤d12△U2+e4σzθtω3d12+3G2UL66+VL66.For the last term in (82), we get (85)e2σzθtω∫ΓU2V△V-GU3△Vdx≤e4σzθtωd2∫ΓU4V2dx+e4σzθtωd2∫ΓG2U6dx+d24△V2+d24△V2≤d22△V2+e4σzθtω3d2∫Γ3G2U6dx+∫ΓU6+U6+V6dx≤d22△V2+e4σzθtω3d22+3G2UL66+VL66.Recalling that d=min{d1,d2} and γ=min{a,b}, it follows that(86)ddt∇U2+∇V2+2γ-2σzθtω∇U2+∇V2≤2e4σzθtω31d1+1d22+3G2UL66+VL66+2e-2σzθtωΓρ2d1+β2d2+2e-2σzθtωf12d1+f22d2≤4e4σzθtω3d2+3G2UL66+VL66+2e-2σzθtωΓρ2+β2d+2e-2σzθtωf12+f22d.For t≥T5(τ,ω,B)≥1 and s∈[τ-1,τ], we integrate (86) over [s,τ] to get(87)∇Uτ,τ-t,θ-τω,Uτ-t2+∇Vτ,τ-t,θ-τω,Vτ-t2≤∇Us,τ-t,θ-τω,Uτ-t2+∇Vs,τ-t,θ-τω,Vτ-t2+∫τ-1τ2σzθs-τω-γ∇Us,τ-t,θ-τω,Uτ-t2+∇Vs,τ-t,θ-τω,Vτ-t2ds+∫τ-1τ4e4σzθs-τω3d2+3G2UL66+VL66ds+∫τ-1τ2e-2σzθs-τωΓρ2+β2dds+∫τ-1τ2e-2σzθs-τωf12+f22dds.Now integrating the above with respect to s over (τ-1,τ), we find that for t≥T6(τ,ω,B,σ), it follows that(88)∇Uτ,τ-t,θ-τω,Uτ-t2+∇Vτ,τ-t,θ-τω,Vτ-t2≤1+2σmax-1≤s≤0zθsω-2γ∫τ-1τ∇Us,τ-t,θ-τω,Uτ-t2+∇Vs,τ-t,θ-τω,Vτ-t2ds+∫τ-1τ4e4σzθs-τω3d2+3G2UsL66+VsL66ds+∫τ-1τ2e-2σzθs-τωΓρ2+β2dds+∫τ-1τ2e-2σzθs-τωf1s2+f2s2dds.By the Mean Value Theorem in (60), there is a time t0∈[τ-2,τ-1] such that(89)∇Ut0,τ-t,θ-τω,Uτ-t2+∇Vt0,τ-t,θ-τω,Vτ-t2≤M2τ,ω.Then by Lemma 6, for q∈[τ-1,τ], there is a random variable M4(τ,ω)>0 such that(90)Uq,τ-t,θ-τω,Uτ-tL66+Vq,τ-t,θ-τω,Uτ-tL66≤M4τ,ω.By Lemma 5, it follows from (77) that, for all t≥T5(τ,ω,B,σ),(91)∇Uτ,τ-t,θ-τω,Uτ-t2+∇Vτ,τ-t,θ-τω,Vτ-t2≤1+2σmax-1≤s≤0zθsω-2γM1+M4∫-104e4σzθsω3d2+3G2ds+∫-10e-2σzθsωΓρ2+β2dds+∫-10e-2σzθsωf1s+τ2+f2s+τ2dds≐M5τ,ω.It is easy to see that M5(τ,ω) is tempered. This completes the proof.Lemma 8.
Suppose thatfi(t,x)∈X(i=1,2). Then for each τ∈R and ω∈Ω, the continuous cocycle φ associated with problem (19)–(22) has a closed measurable D-pullback absorbing set Bσ(τ,ω)∈D, which is given by (92)Bστ,ω=U,V∈H:U,V2≤Rστ,ω,where Rσ(τ,ω) is defined as in (32).Theorem 9.
Suppose thatfi(t,x)∈X(i=1,2). Then the continuous cocycle φ associated with problem (19)–(22) has a unique D-pullback attractor Aσ={Aσ(τ,ω):τ∈R,ω∈Ω} in H.Proof.
By Lemma8, the continuous cocycle φ has a D-pullback absorbing set Bτ(σ,ω). By the compact embedding E↪H, the continuous cocycle φ is pullback asymptotically compact in H. According to Theorem 1, there exists a unique random attractor Aσ in H for the continuous cocycle φ.


## 4. Upper Semicontinuity of Random Attractors

In this section, we prove the upper semicontinuity of random pullback attractors for the nonautonomous stochastic reversible Selkov system when the intensity of noiseσ→0. Given σ∈[0,1], we write the cocycle of problem (19)–(22) as φσ to indicate its dependence on σ.The deterministic system corresponding to (19)–(22) is given by(93)dudt=d1△u+ρ-au+u2v-Gu3+f1t,x,(94)dvdt=d2△v+β-bv-u2v+Gu3+f2t,x,with the homogeneous Dirichlet boundary conditions(95)ut,x=vt,x=0,t>τ,x∈∂Γ,and the initial conditions(96)uτ,x=uτx,vτ,x=vτx,x∈Γ.Similarly, we can prove that problem (93)–(96) generates a continuous cocycle φ0 in H. Moreover, φ0 has a unique pullback attractor A0={A0(τ),τ∈R} in H and has a pullback absorbing set B0(τ)={B0(τ),τ∈R}, where B0(τ) is given by(97)B0τ=u,v∈H:u,v2≤R0τand(98)R0τ=1+C1∫-∞0eγsds+Ga∫-∞0eγsf1s+τ2ds+1b∫-∞0eγsf2s+τ2ds.The next lemma shows the convergence of solutions of (19)–(22) as σ→0.Lemma 10.
Suppose thatfi(t,x)∈X(i=1,2). Let gσ=(U,V) and g=(u,v) be the solution of (19)–(22) and (93)–(96) with initial conditions gτ,σ and gτ, respectively. If limσ→0gσ,τ=gτ, then, for ω∈Ω,t>τ and ε∈[0,1],(99)limσ→0gσt,τ,ω,gτ,σ=gt,τ,gτ.Proof.
LetU and V be the solution of (19)–(22), and u~=U-u, v~=V-v, and k=(u~,v~)=(U-u,V-v)=gσ-g. Then from (19)-(20) and (93)-(94), we get (100)du~dt=d1△u~+e-σzθtω-1ρ-au~+e2σzθtωU2V-u2v-Ge2σzθtωU3+Gu3+e-σzθtωf1t,x-f1t,x+σUzθtω,dv~dt=d2△v~+e-σzθtω-1β-bv~-e2σzθtωU2V+u2v+Ge2σzθtωU3-Gu3+e-σzθtωf2t,x-f2t,x+σVzθtω,which implies(101)12ddtu~2+d1∇u~2+au~2=∫Γe-σzθtω-1ρu~dx+∫Γe-σzθtω-1f1u~dx+∫Γe2σzθtωU2V-u2v-Ge2σzθtωU3+Gu3u~dx+∫ΓσzθtωUu~dx,12ddtv~2+d2∇v~2+bv~2=∫Γe-σzθtω-1βv~dx+∫Γe-σzθtω-1f2v~dx-∫Γe2σzθtωU2V-u2v-Ge2σzθtωU3+Gu3v~dx+∫ΓσzθtωVv~dx.Summing up the two equalities together, it yields that(102)ddtu~2+v~2+2d∇u~2+∇v~2+2au~2+2bv~2=2∫Γe-σzθtω-1ρu~+βv~dx+2∫Γe-σzθtω-1f1u~+f2v~dx+2∫ΓσzθtωUu~+Vv~dx+2∫Γe2σzθtωU2V-u2v-Ge2σzθtωU3+Gu3u~-v~dx.Note that(103)∫Γe2σzθtωU2V-u2v-Ge2σzθtωU3+Gu3u~-v~dx=∫Γe2σzθtωU2V-GU3u~-v~dx+∫ΓGu3-u2vu~-v~dx=∫Γe2σzθtω-1U2V-GU3u~-v~dx+∫ΓU2V-GU3+Gu3-u2vu~-v~dx=∫Γe2σzθtω-1U2V-GU3u~-v~dx+∫ΓU2V-u2v-GU3-Gu3u~-v~dx.Given that τ∈R, ω∈Ω, T>0, and ε∈[0,1), since ω is continuous on R, we obtain that there exists a σ1=σ1(ω,T,ε)>0 such that for all σ∈(0,σ1) and t∈[τ,τ+T],(104)e-σzθtω-1+eσzθtω-1+e2σzθtω-1+e-2σzθtω-1<ε.Thus, we obtain(105)ddtu~2+v~2+2d∇u~2+∇v~2+2au~2+2bv~2≤au~2+bv~2+ε2ρ2Γa+ε2β2Γb+au~2+bv~2+ε2f12a+ε2f22b+2σzθtω∫Uu~+Vv~dx+2ε∫ΓU2V+GU3u~+v~dx+2∫ΓU2V-u2v+GU3-Gu3u~+v~dx.Lemma 6 is true even if q∈[τ,t]. By Hölder’s inequality and Gagliardo-Nirenberg inequality, we have(106)∫ΓU2V-u2v+GU3-Gu3u~+v~dx≤d∇u~+∇v~+c1′g+gL42+gL63+gσ+gσL42+gσL63u~2+v~2,where c1′ is a constant depending on d, G and the Gagliardo-Nirenberg inequality constant. It follows from (105) and (106) that there exists a σ2=σ2(ω,T,ε)>0 such that for all σ∈(0,σ2) and t∈[τ,τ+T],(107)ddtu~2+v~2≤ε2ρ2Γa+ε2β2Γb+ε2f12a+ε2f22b+2c2′g+gL42+gL63+gσ+gσL42+gσL63u~2+v~2≤c4ε2+ε2f12a+ε2f22b+2c3′,where c4=2ρ2|Γ|/a+2β2|Γ|/b, c3′=2c2′(g+gL42+gL63+gσ+gσL42+gσL63). For all σ∈(0,σ2) and t∈[τ,τ+T], due to Gronwall’s lemma, we get(108)kt2≤ec3′t-τ)u~τ2+v~τ2+ε2∫τte-c3′t-sc4+f1r2a+f2r2bdr.From (108), recalling that k=(u~,v~)=(U-u,V-v)=gσ-g, we find that(109)gσt,τ,ω,gτ,σ-gt,τ,gτ2≤ec3′t-τgτ,σ-gτ2+ε2∫τte-c3′t-sc4+f1r2a+f2r2bdr,which finishes the proof.Theorem 11.
Provided thatfi(t,x)∈X(i=1,2). Then for every τ∈R and ω∈Ω,(110)limσ→0dHAστ,ω,A0τ=0.Proof.
To achieve the result, it suffices to verify conditions (1), (2), and (3) in Proposition3. For condition (1), by taking a sequence σn→0 and gτ,n→gτ, from Lemma 10 we have(111)φσnt,τ,ω,gτ,n→φt,τ,gτinH.By Lemma 8 and Theorem 9, φσ has a tempered pullback absorbing set Bσ and a unique D-pullback random attractor Aσ∈D. Here,(112)Bστ,ω=U,V∈H:U,V2≤Rστ,ω,where Rσ(τ,ω) is defined as in (32). It follows from (97) and (112) that for all τ∈R and ω∈Ω, we have(113)limsupσ→0Bστ,ω≤B0τ,which deduces condition (2) immediately.
Now consider condition (3). By Lemma7, we know that {Kσ(τ,ω)}∈D is also a closed and tempered absorbing set for φσ in H, where (114)Kστ,ω=U,V∈H:U,VE2≤M5τ,ω.Let (115)M~5τ,ω=1+2max-1≤s≤0zθsω-2γM1+M4∫-104e4zθsω3d2+3G2ds+∫-10e2zθsωΓρ2+β2dds+∫-10e2zθsωf1s+τ2+f2s+τ2dds,and (116)Kτ,ω=U,V∈H:U,VE2≤M~5τ,ω.We find that for all τ∈R and ω∈Ω,(117)⋃0<σ≤1Aστ,ω⊆⋃0<σ≤1Kστ,ω⊆Kτ,ω.By the compactness of embedding H↪E, we know that K(τ,ω) is compact in H. By (117), it follows that ⋃0<σ≤1Aσ(τ,ω) is precompact in H. Hence, condition (3) is clear and the proof is finished.



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*Source: 2763245-2019-06-27.xml*