# Pullback𝒟-Attractor of Nonautonomous Three-Component Reversible Gray-Scott System on Unbounded Domains

Authors: Anhui Gu
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

The long time behavior of solutions of the nonautonomous three-components reversible Gray-Scott system defined on the entire spaceℝn is studied when the external forcing terms are unbounded in a phase space. The existence of a pullback global attractor for the equation is established in L2ℝn3 and H1ℝn3, respectively. The pullback asymptotic compactness of solutions is proved by using uniform estimates on the tails of solutions on unbounded domains.


---

## Body

## 1. Introduction

In this paper, we consider the dynamical behavior of the nonautonomous three-components reversible Gray-Scott system(1)∂u∂t=d1Δu-(F+k)u+u2v-Gu3+Nw+f1(t,x),∂v∂t=d2Δv-Fv-u2v+Gu3+f2(t,x),∂w∂t=d3Δw-(F+N)w+ku+f3(t,x),
on [τ,∞)×𝒪 with initial data
(2)u(τ,x)=u0(x),v(τ,x)=v0(x),w(τ,x)=w0(x)on𝒪,
where 𝒪=ℝn×ℝn×ℝn; all the parameters are arbitrarily given positive constants; fi(x,t) for t∈ℝ and x∈ℝn is an external forcing term which is locally square integrable in time. That is, fi∈Lloc2(ℝ,L2(ℝn)), i=1,2,3.Historically, whenw=0, G=0, and the external forces f1=f3=0, f2=F, system (1) reduces to the two-component Gray-Scott system which signified one of the Brussell school led by the renowned physical chemist and Nobel Prize laureate (1977), Ilya Prigogine, which originated from describing an isothermal, cubic autocatalytic, continuously fed, and diffusive reactions of two chemicals (see, e.g., [1, 2]). The three-component reversible Gray-Scott model was firstly introduced by Mahara et al., which is based on the scheme of two reversible chemical or biochemical reactions [3]. Then in [4], You takes some nondimensional transformations, and the three-component reversible system is reduced to the system (1) without external forces. In [4], You considers the existence of global attractor for the system with Neumann boundary condition on a bounded domain of space dimension n≤3 by the method of the rescaling and grouping estimation. For more recent dynamical behaviors of the nonautonomous three-component reversible Gray-Scott system, we can refer to [5, 6] for the existence of random attractors of the stochastic cases on bounded and unbounded domains and [7] for the existence of uniform attractor of the deterministic case on a bounded domain.As pointed in [8], we can discuss the same or similar coupled reaction-diffusion systems on a higher dimensional domain with the space dimension n>3 and on an unbounded domain, to work with various different phase spaces. Here, we intend to investigate the dynamical behavior of the nonautonomous three-component reversible Gray-Scott system on unbounded domains. It is worth mentioning that the Sobolev embeddings are not compact on domains of infinite volume. This introduces a major obstacle for proving the existence of attractors for PDEs on unbounded domains. For some deterministic equations, the difficulty caused by the unboundedness of domains can be overcome by the energy equation approach which is developed by Ball in [9] and used by many authors (see, e.g., [10, 11]). In this paper, we will use the uniform estimates on the tails of solutions to circumvent the difficulty caused by the unboundedness of the domain. This idea was developed in [12] to prove the asymptotic compactness of solutions for autonomous parabolic equations on ℝn and later extended to stochastic equations in, for example, [6, 13–15]. Here, we will use the method of tail estimates to investigate the asymptotic behavior of system (1) with initial data (2). We first establish the pullback asymptotic compactness of solutions of system (1) and prove the existence of a pullback global attractor in ℍ. Then we extend this result and show the existence of a pullback global attractor in 𝔼.The paper is organized as follows. In the next section, we recall the fundamental concepts and results for pullback attractors for nonautonomous dynamical systems. In Section3, we define a cocycle for the nonautonomous three-component reversible Gray-Scott system on ℝn. Section 4 is devoted to deriving the uniform estimates of solutions for large space and time variables. In the last section, we prove the existence of a pullback global attractor for the equation in ℍ and 𝔼.The following notations will be used throughout the paper. We denote by∥·∥ and (·,·) the norm and inner product in L2(ℝn) or ℍ=[L2(ℝn)]3 and use ∥·∥6, ∥·∥4 to denote the norm in L6(ℝn) or 𝕍=[L6(ℝn)]3, L4(ℝn) or 𝕌=[L4(ℝn)]3; 𝕍′,𝕌 the dual of 𝕍,𝕌; 𝔼=[H1(ℝn)]3. The letters M is a generic positive constant which may change its value from line to line or even in the same line.


## 2. Preliminaries

In this section, we recall some basic concepts related to pullback attractors for nonautonomous dynamical systems. It is worth noticing that these concepts are quite similar to those of random attractors for stochastic systems. We can refer to [16–19] for more details.LetΩ be a nonempty set and X a metric space with distance d(·,·).Definition 1.
A family of mappings{θt}t∈ℝ from Ω to itself is called a family of shift operators on Ω if {θt}t∈ℝ satisfies the group properties:(i)
θ0ω=ω, for all ω∈Ω,(ii)
θt(θτω)=θt+τω, for all ω∈Ω and t,τ∈ℝ.Definition 2.
Let{θt}t∈ℝ be a family of shift operators on Ω. Then a continuous θ-cocycle ϕ on X is a mapping
(3)ϕ:ℝ+×Ω×X⟶X,(t,ω,x)↦ϕ(t,ω,x),
which satisfies, for all ω∈Ω and t,τ∈ℝ, the following:(i)
ϕ(0,ω,·) is the identity on X,(ii)
ϕ(t+τ,ω,·)=ϕ(t,θτω,·)∘ϕ(τ,ω,·),(iii)
ϕ(t,ω,·):X→X is continuous.Hereafter, we always assume thatϕ is a continuous θ-cocycle on X and 𝒟 is a collection of families of subsets of X:
(4)𝒟={D={D(ω)}ω∈Ω:D(ω)⊆Xforeveryω∈Ω}.Definition 3.
Let𝒟 be a collection of families of subsets of X. Then 𝒟 is called inclusion closed if D={D(ω)}ω∈Ω∈𝒟 and D~={D~(ω)⊆X:ω∈Ω} with D~(ω)⊆D(ω) for all ω∈Ω imply that D~∈𝒟.Definition 4.
Let𝒟 be a collection of families of subsets of X, and {K(ω)}ω∈Ω is called a pullback absorbing set for ϕ in 𝒟 if, for every B∈𝒟 and ω∈Ω, there exists t(ω,B)>0 such that
(5)ϕ(t,θ-tω,B(θ-tω))⊆K(ω),∀t≥t(ω,B).Definition 5.
Let𝒟 be a collection of families of subsets of X. Then ϕ is said to be 𝒟-pullback asymptotically compact in X if, for every ω∈Ω, {ϕ(tn,θ-tnω,xn)}n=1∞ has a convergent subsequence in X whenever tn→∞ and xn∈B(θ-tnω) with {B(ω)}ω∈Ω∈𝒟.Definition 6.
Let𝒟 be a collection of families of subsets of X and {𝒜(ω)}ω∈Ω∈𝒟. Then {𝒜(ω)}ω∈Ω∈𝒟 is called a 𝒟-pullback global attractor for ϕ if the following conditions are satisfied, for every ω∈Ω:(i)
𝒜(ω) is compact,(ii)
{𝒜(ω)}ω∈Ω is invariant, that is,
(6)ϕ(t,ω,𝒜(ω))=𝒜(θtω),∀t≥0,(iii)
{𝒜(ω)}ω∈Ω attracts every set in 𝒟, that is, for every B={B(ω)}ω∈Ω∈𝒟,
(7)limt→∞d(ϕ(t,θ-tω,B(θ-tω)),𝒜(ω))=0,where d is the Hausdorff semimetric given by d(Y,Z)=supy∈Yinfz∈Z∥y-z∥X for any Y⊆X and Z⊆X.Theorem 7.
Let𝒟 be an inclusion-closed collection of families of subsets of X and ϕ a continuous θ-cocycle on X. Suppose that {K(ω)}ω∈Ω∈𝒟 is a closed absorbing set for ϕ in 𝒟 and ϕ is 𝒟-pullback asymptotically compact in X. Then ϕ has a unique 𝒟-pullback global attractor {𝒜(ω)}ω∈Ω∈𝒟 which is given by
(8)𝒜(ω)=⋂τ≥0‍⋃t≥τ‍ϕ(t,θ-tω,K(θ-tω))¯.


## 3. Cocycle Related to Three-Component Reversible Gray-Scott System

In this section, we constructed aθ-cocycle for the nonautonomous three-component reversible Gray-Scott system defined on ℝn. For every τ∈ℝ and t>τ, system (1) with initial data (2) can be rewritten as
(9)∂g∂t-Ag+H(g)=f(x,t),x∈𝒪,
with initial condition
(10)g(x,τ)=gτ(x),x∈ℝn,
where
(11)g=(u,v,w)T,f(x,t)=(f1(x,t),f2(x,t),f3(x,t))T,x∈𝒪,A=(d1Δ000d2Δ000d3Δ),H(g)=(-(F+k)u+u2v-Gu3+Nw-Fv-u2v+Gu3ku-(F+N)w),
and here T denotes the transposition.As in the case of bounded domains, by conducting a priori estimate on the Galerkin approximate solutions of system (9)-(10) similar to the autonomous system studied in [7], we can prove that if f∈Lloc2(ℝ,ℍ)), then problem (9)-(10) is well defined in ℍ. That is, for any gτ∈ℍ and τ∈ℝ, (9) possesses a unique solution g satisfying
(12)g∈C([τ,∞);ℍ)∩Lloc2([τ,∞);𝔼),
which continuously depends on the initial data gτ∈ℍ. To construct a cocycle ϕ for problem (9)-(10), we denote by Ω=ℝ and define a shift operator θt on Ω for every t∈ℝ by
(13)θt(τ)=t+τ,∀τ∈ℝ.
Let ϕ be a mapping from ℝ+×Ω×ℍ to ℍ given by
(14)ϕ(t,τ,gτ)=g(t+τ,τ,gτ),
where t≥0, τ∈ℝ, gτ∈ℍ, and g is the solution of problem (9)-(10). By the uniqueness of solutions, we find that, for every t,s≥0, τ∈ℝ and gτ∈ℍ,
(15)ϕ(t+s,τ,gτ)=g(t,s+τ,ϕ(s,τ,gτ)).
Then it follows that ϕ is a continuous θ-cocycle on ℍ. The purpose of this paper is to study the existence of pullback attractors for ϕ in an appropriate phase space.Letℍ- be a subset of ℍ and denote
(16)∥ℍ-∥=supx∈ℍ-∥x∥ℍ.
Let D={D(t)}t∈ℝ be a family of subsets of ℍ. That is, D(t)∈ℍ for every t∈ℝ and satisfying
(17)limt→-∞eFt∥D(t)∥2=0,
where F is a positive number given in (1). Hereafter, we use 𝒟F to denote the collection of all families of subsets of ℍ satisfying (17), that is,
(18)𝒟F={D={D(t)}t∈ℝ:Dsatisfies(17)}.
Throughout this paper, we assume the following conditions for the external term:
(19)∫-∞τ‍eFξ∥f(ξ)∥2dξ<∞,∀τ∈ℝ,(20)∫-∞τ‍eFξ∥f(ξ)∥𝕍′6dξ<∞,∫-∞τ‍eFξ∥f(ξ)∥𝕌′4dξ<∞,∀τ∈ℝ,(21)limsupK→∞∫-∞τ‍∫|x|≥K‍eFξ|f(x,ξ)|2dxdξ=0,∀τ∈ℝ.It is useful to note that condition (21) implies that, for every τ∈ℝ and β>0, there is K=K(τ,β)>0 such that
(22)∫-∞τ‍∫|x|≥K‍eFξ|fi(x,ξ)|2dxdξ≤βeFτ,i=1,2,3.
As we will see later, inequality (22) is crucial for deriving uniform estimates on the tails of solutions, and these estimates are necessary for proving the asymptotic compactness of solutions.


## 4. Uniform Estimates of Solutions

In this section, we derive uniform estimates of solutions of problem (9)-(10) defined on ℝn when t→∞. We start with the estimates in ℍ.Lemma 8.
Suppose that (19) holds. Then for every τ∈ℝ and D={D(t)}t∈ℝ∈𝒟F, there exists T=T(τ,D)>0 such that for all t≥T,
(23)∥g(τ,τ-t,g0(τ-t))∥2≤Me-Fτ∫-∞τ‍eFξ∥f(ξ)∥2dξ,∫τ-tτ‍eFξ∥∇g(ξ,τ-t,g0(τ-t))∥2ds≤M∫-∞τ‍eFξ∥f(ξ)∥2dξ,
where g0(τ-t)∈D(τ-t) and M depends on G, F, N, and k.Proof.
Define(24)W(t,x)=Nkw(t,x),μ=kN,
then the system (1) becomes
(25)∂u∂t=d1Δu-(F+k)u+u2v-Gu3+kW+f1(t,x),(26)∂v∂t=d2Δv-Fv-u2v+Gu3+f2(t,x),(27)μ∂W∂t=μd3ΔW+ku-(μF+k)W+f3(t,x).
Taking the inner products (∂u/∂t,Gu(t)), (∂v/∂t,v(t)), and (μ(∂W/∂t),GW(t)) and  then suming up the resulting equalities, we get
(28)12ddt(G∥u∥2+∥v∥2+μG∥W∥2)+d1G∥∇u∥2+d2∥∇v∥2+μGd3∥∇W∥2=2kG∫ℝn‍uWdx-(G(F+k)∥u∥2-F∥v∥2-G(μF+k)∥W∥2)-∫ℝn‍(Gu2-uv)2dx+G∫ℝn‍uf1dx+∫ℝn‍vf2dx+G∫ℝn‍Wf3dx≤G∫ℝn‍uf1dx+∫ℝn‍vf2dx+G∫ℝn‍Wf3dx-F(G∥u∥2+∥v∥2+μG∥W∥2)≤-12F(G∥u∥2+∥v∥2+μG∥W∥2)+G2F|f1|2+12F|f2|2+G2μF|f3|2.
Setting
(29)d=min{d1,d2,d3},C1=max{1,G,G/μ}Fmin{1,G,G/μ},
then (28) becomes
(30)ddt∥g(t)∥2+2d∥∇g(t)∥2+F∥g(t)∥2≤C1∥f∥2.
Multiplying (30) by eFt and then integrating the resulting inequality on (τ-t,τ) with t≥0, we find that
(31)∥g(τ,τ-t,g0(τ-t))∥2+2de-Fτ∫τ-tτ‍eFξ∥∇g(ξ)∥2dξ≤e-FτeF(τ-t)∥g0(τ-t)∥2+C1e-Fτ∫τ-tτ‍eFξ∥f(ξ)∥2dξ≤e-FτeF(τ-t)∥g0(τ-t)∥2+C1e-Fτ∫-∞τ‍eFξ∥f(ξ)∥2dξ.
Notice that g0(τ-t)∈D(τ-t) and D={D(t)}t∈ℝ∈𝒟F. We find that for every τ∈ℝ, there exists T=T(τ,D) such that for all t≥T,
(32)eF(τ-t)∥g0(τ-t)∥2≤C1∫-∞τ‍eFξ∥f(ξ)∥2dξ.
By (31) and (32), we get that, for all t≥T,
(33)∥g(τ,τ-t,g0(τ-t))∥2+2de-Fτ∫τ-tτ‍eFξ∥∇g(ξ)∥2dξ≤2C1e-Fτ∫-∞τ‍eFξ∥f(ξ)∥2dξ,
which completes the proof.The following lemma is useful for deriving uniform estimates of solutions in𝔼.Lemma 9.
Suppose that (19) holds. Then for every τ∈ℝ and D={D(t)}t∈ℝ∈𝒟F, there exists T=T(τ,D)>2 such that for all t≥T,
(34)∫τ-2τ‍eFξ∥g(ξ,τ-t,g0(τ-t))∥2dξ≤M∫-∞τ‍eFξ∥f(ξ)∥2dξ,∫τ-2τ‍eFξ∥∇g(ξ,τ-t,g0(τ-t))∥2dξ≤M∫-∞τ‍eFξ∥f(ξ)∥2dξ,
where g0(τ-t)∈D(τ-t) and M relies on G, F, N, and k.Proof.
By (30) we find that
(35)ddt∥g(t)∥2+F∥g(t)∥2≤C1∥f∥2.
Let s∈[τ-2,τ] and τ>2. Multiplying the above by eFt and integrating over (τ-t,s), we get
(36)eFs∥g(s,τ-t,g0(τ-t))∥2≤eF(τ-t)∥g0(τ-t)∥2+C1∫τ-ts‍eFξ∥f(ξ)∥2dξ≤eF(τ-t)∥g0(τ-t)∥2+C1∫-∞τ‍eFξ∥f(ξ)∥2dξ.
Therefore, there exists T=T(τ,D)>2 such that for all t≥T and s∈[τ-2,τ],
(37)eFs∥g(s,τ-t,g0(τ-t))∥2≤2C1∫-∞τ‍eFξ∥f(ξ)∥2dξ.
Integrate the above with respect to s on (τ-2,τ) to obtain
(38)∫τ-2τ‍eFs∥g(s,τ-t,g0(τ-t))∥2ds≤4C1∫-∞τ‍eFξ∥f(ξ)∥2dξ.
On the other hand, for s=τ-2, (37) implies that
(39)eF(τ-2)∥g(τ-2,τ-t,g0(τ-t))∥2≤2C1∫-∞τ‍eFξ∥f(ξ)∥2dξ.
Multiplying (30) by eFτ and then integrating over (τ-2,τ), by (38) we get that, for all t≥T,
(40)eFτ∥g(τ,τ-t,g0(τ-t))∥2+2d∫τ-2τ‍eFξ∥∇g(ξ)∥2dξ≤eF(τ-2)∥g(τ-2,τ-t,g0(τ-t))∥2+C1∫τ-tτ‍eFξ∥f(ξ)∥2dξ≤3C1∫-∞τ‍eFξ∥f(ξ)∥2dξ.
which along with (39) completes the proof.Note thateλξ≥eλτ-2λ for any ξ≥τ-2. So as an immediate consequence of Lemma 9 we have the following estimates.Corollary 10.
Suppose that (19) holds. Then for every τ∈ℝ and D={D(t)}t∈ℝ∈𝒟F, there exists T=T(τ,D)>2 such that for all t≥T,
(41)∫τ-2τ‍∥g(ξ,τ-t,g0(τ-t))∥2ds≤Me-Fτ∫-∞τ‍eFξ∥f(ξ)∥2dξ,∫τ-2τ‍∥∇g(ξ,τ-t,g0(τ-t))∥2ds≤Me-Fτ∫-∞τ‍eFξ∥f(ξ)∥2dξ,
where g0(τ-t)∈D(τ-t) and M depends on G, F, N, and k.Before the derivation of uniform estimates of solutions in𝔼, we firstly give two propositions which will frequently be used in the next results.Proposition 11.
Suppose that (20) holds. Then for every τ∈ℝ and D={D(t)}t∈ℝ∈𝒟F, there exists T=T(τ,D)>0 such that for all t≥T,
(42)∥g(τ,τ-t,g0(τ-t))∥66≤Me-Fτ∫-∞τ‍eFξ∥f(ξ)∥𝕍′6dξ,(43)∫τ-2τ‍eFξ∥g(ξ,τ-t,g0(τ-t))∥66dξ≤M∫-∞τ‍eFξ∥f(ξ)∥𝕍′6dξ,
where g0(τ-t)∈D(τ-t) and M depends on G, F, N, and k.Proof.
LetV(t,x)=(v(t,x))/G, then (25)–(27) become
(44)∂u∂t=d1Δu-(F+k)u+Gu2V-Gu3+kW+h1(t,x),∂V∂t=d2ΔV-FV-u2V+u3+h2(t,x),μ∂W∂t=μd3ΔW+ku-(μF+k)W+h3(t,x).
Taking the inner products (∂u/∂t,u5(t)), (∂V/∂t,GV5(t)) and (μ(∂W/∂t),W5(t)) and then suming up the resulting equalities, we get
(45)16ddt(∥u∥66+G∥V∥66+μ∥W∥66)+5(d1∥u2∇u∥2+d2G∥V2∇V∥2+μd3G∥W2∇W∥2)≤-(F+k)∫ℝn‍u6ds+F∫ℝn‍V5dx-GF∫ℝn‍V6dx-(μF+k)∫ℝn‍W6dx+k∫ℝn‍u5Wdx+k∫ℝn‍uW5dx-G∫ℝn‍(u8-u7V-u3V5+u2V6)dx+∫ℝn‍u5f1dx+G∫ℝn‍V5f2dx+∫ℝn‍W5f3dx.
By Young's inequality,
(46)-G∫ℝn‍(u8-u7V-u3V5+u2V6)dx≤0,G∫ℝn‍V5f2dx≤5FG6∥V∥66+G∥f2∥𝕍′66F5,∫ℝn‍u5f1dx≤5F6∥u∥66+∥f1∥𝕍′66F5,∫ℝn‍W5h3dx≤5μF6∥W∥66+∥f3∥𝕍′66(μF)5.
From (46) then (45) yields
(47)ddt(∥u∥66+G∥V∥66+μ∥W∥66)+F(∥u∥66+G∥V∥66+μ∥W∥66)≤∥f1∥𝕍′6F5+G∥f2∥𝕍′6F5+∥f3∥𝕍′6(μF)5.
Denote
(48)C4=max{1,G,1/μ5}F5min{1,1/G5,1/μ5},
then from (47) implies that
(49)ddt∥g(t)∥66+F∥g(t)∥66≤C4∥f(t)∥𝕍′6.
Multiplying (49) by eFt and then integrating the resulting inequality on (τ-t,τ) with t≥0, we find that(50)∥g(τ,τ-t,g0(τ-t))∥66≤e-FτeF(τ-t)∥g0(τ-t)∥66+C4e-Fτ∫τ-tτ‍eFξ∥f(ξ)∥𝕍′6dξ≤e-FτeF(τ-t)∥g0(τ-t)∥66+C4e-Fτ∫-∞τ‍eFξ∥f(ξ)∥𝕍′6dξ.
Notice thatg0(τ-t)∈D(τ-t) and D={D(t)}t∈ℝ∈𝒟F. We find that, for every τ∈ℝ, there exists T=T(τ,D) such that for all t≥T,
(51)eF(τ-t)∥g0(τ-t)∥66≤C4∫-∞τ‍eFξ∥f(ξ)∥𝕍′6dξ.
By (50) and (51), we get that, for all t≥T,
(52)∥g(τ,τ-t,g0(τ-t))∥66≤2C4e-Fτ∫-∞τ‍eFξ∥f(ξ)∥𝕍′6dξ,
which completes the proof.Similarly, we have the following.Proposition 12.
Suppose that (20) holds. Then for every τ∈ℝ and D={D(t)}t∈ℝ∈𝒟F, there exists T=T(τ,D)>0 such that for all t≥T,
(53)∥g(τ,τ-t,g0(τ-t))∥44≤Me-Fτ∫-∞τ‍eFξ∥f(ξ)∥𝕌′4dξ,∫τ-2τ‍eFξ∥g(ξ,τ-t,g0(τ-t))∥44dξ≤M∫-∞τ‍eFξ∥f(ξ)∥𝕌′4dξ,
where g0(τ-t)∈D(τ-t) and M depends on G, F, N, and k.Proof.
The proof is similar to Proposition11 except for few trivial details, and thus we omit it here.Lemma 13.
Suppose that (19) and (20) hold. Then for every τ∈ℝ and D={D(t)}t∈ℝ∈𝒟F, there exists T=T(τ,D)>2 such that for all t≥T,
(54)∥∇g(τ,τ-t,g0(τ-t))∥2≤Me-Fτ(∫-∞τ‍eFξ∥f(ξ)∥2dξ+∫-∞τ‍eFξ∥f(ξ)∥𝕍′6dξ),∫τ-1t‍∥gξ(ξ,τ-t,g0(τ-t))∥2dξ≤Me-Fτ(∫-∞τ‍eFξ∥f(ξ)∥2dξ+∫-∞τ‍eFξ∥f(ξ)∥𝕍′6dξ),
where g0(τ-t)∈D(τ-t) and M depends on d, G, F, N, and k.Proof.
Taking the inner product of the first equation of system (1) with ut, the second equation with vt, and the third equation with wt, respectively, in L2(ℝn) and then replacing t by ξ, we obtain
(55)∥uξ(ξ,τ-t,u0(τ-t))∥2+ddξ(d12∥∇u(ξ,τ-t,u0(τ-t))∥2+(F+k)2∥u(ξ,τ-t,u0(τ-t))∥2)≤12∥uξ(ξ,τ-t,u0(τ-t))∥2+(43+2G2)∥u(ξ,τ-t,u0(τ-t))∥66+23∥v(ξ,τ-t,v0(τ-t))∥66+2N2∥w(ξ,τ-t,w0(τ-t))∥2+2∥f1(ξ)∥2,(56)∥vξ(ξ,τ-t,v0(τ-t))∥2+ddξ(d22∥∇v(ξ,τ-t,u0(τ-t))∥2+F2∥v(ξ,τ-t,v0(τ-t))∥2d22)≤12∥vξ(ξ,τ-t,v0(τ-t))∥2+(1+32G2)∥u(ξ,τ-t,u0(τ-t))∥66+12∥v(ξ,τ-t,v0(τ-t))∥66+32∥f2(ξ)∥2,(57)∥wξ(ξ,τ-t,w0(τ-t))∥2+ddξ(d32∥∇w(ξ,τ-t,w0(τ-t))∥2+(F+N)2∥w(ξ,τ-t,w0(τ-t))∥2d32)≤12∥wξ(ξ,τ-t,w0(τ-t))∥2+k2∥u(ξ,τ-t,u0(τ-t))∥2+∥f3(ξ)∥2,
where (55) and (56) are partly due to the Young inequality ∥u4v2∥≤(2/3)∥u∥66+(1/3)∥v∥66. Denote u0(τ-t), v0(τ-t), w0(τ-t), and g0(τ-t) with u0, v0, w0, and g0, respectively. Adding the three inequalities (55)–(57) together, we have
(58)∥uξ(ξ,τ-t,u0)∥2+∥vξ(ξ,τ-t,v0)∥2+∥wξ(ξ,τ-t,w0)∥2+ddξ(d1∥∇u(ξ,τ-t,u0)∥2+d2∥∇v(ξ,τ-t,v0)∥2+d3∥∇w(ξ,τ-t,w0)∥2+(F+k)∥u(ξ,τ-t,u0)∥2+F∥v(ξ,τ-t,v0)∥2+(F+N)∥w(ξ,τ-t,w0)∥2)≤4(1+32G2)∥u(ξ,τ-t,u0)∥66+2∥v(ξ,τ-t,v0)∥66+4N2∥w(ξ,τ-t,w0)∥2+2k2∥u(ξ,τ-t,u0)∥2+4∥f1(ξ)∥2+3∥f2(ξ)∥2+2∥f3(ξ)∥2.
Let
(59)C2=4+6G2,C3=2max{2N2,k2}.
Then (58) yields
(60)∥gξ(ξ,τ-t,g0)∥2+ddξ(d∥∇g(ξ,τ-t,g0)∥2+F∥g(ξ,τ-t,g0)∥2)≤C2∥g(ξ,τ-t,g0)∥66+C3∥g(ξ,τ-t,g0)∥2+4∥f(ξ)∥2.
That is,
(61)ddξ(d∥∇g(ξ,τ-t,g0)∥2+F∥g(ξ,τ-t,g0)∥2)≤C2∥g(ξ,τ-t,g0)∥66+C3∥g(ξ,τ-t,g0)∥2+4∥f(ξ)∥2.
Let s≤τ and t≥2. By integrating (61) over (s,τ), we get
(62)d∥∇g(τ,τ-t,g0)∥2+F∥g(τ,τ-t,g0)∥2≤d∥∇g(s,τ-t,g0)∥2+F∥g(s,τ-t,g0)∥2+C2∫sτ∥g(ξ,τ-t,g0)∥66dξ+C3∫sτ∥g(ξ,τ-t,g0)∥2dξ+4∫sτ∥f(ξ)∥2dξ.
Now integrating the above with respect to s on (τ-1,τ), we find
(63)d∥∇g(τ,τ-t,g0)∥2+F∥g(τ,τ-t,g0)∥2≤d∫τ-1τ∥∇g(s,τ-t,g0)∥2ds+F∫τ-1τ∥g(s,τ-t,g0)∥2ds+C2∫τ-1τ∥g(ξ,τ-t,g0)∥66dξ+C3∫τ-1τ∥g(ξ,τ-t,g0)∥2dξ+4∫τ-1τ∥f(ξ)∥2dξ,
which along with Corollary 10 and Proposition 11 implies that there exists T=T(τ,D)>2 such that for all t≥T,
(64)d∥∇g(τ,τ-t,g0)∥2+F∥g(τ,τ-t,g0)∥2≤(d+F+C3+4eF)Me-Fτ∫-∞τeFξ∥f(ξ)∥2dξ+C2eFMe-Fτ∫-∞τ∥f(ξ)∥𝕍′6dξ≤Me-Fτ(∫-∞τeFξ∥f(ξ)∥2dξ+∫-∞τeFξ∥f(ξ)∥𝕍′6dξ).
Similarly, first integrating (61) with respect to ξ on (s,τ-1) and then integrating with respect to s on (τ-2,τ-1), by using Corollary 10 and Proposition 11, we can get that for all t≥T,
(65)d∥∇g(τ-1,τ-t,g0)∥2+F∥g(τ-1,τ-t,g0)∥2≤(d+F+C3+4eF)Me-Fτ∫-∞τeFξ∥f(ξ)∥2dξ+C2eFMe-Fτ∫-∞τ∥f(ξ)∥𝕍′6dξ≤Me-Fτ(∫-∞τeFξ∥f(ξ)∥2dξ+∫-∞τeFξ∥f(ξ)∥𝕍′6dξ).
Now integrating (60) over (τ-1,τ), we obtain
(66)∫τ-1τ∥gξ(ξ,τ-t,g0)∥2dξ+d∥∇g(τ,τ-t,g0)∥2+F∥g(τ,τ-t,g0)∥2≤d∥∇g(τ-1,τ-t,g0)∥2+F∥g(τ-1,τ-t,g0)∥2+C2∫τ-1τ∥g(ξ,τ-t,g0)∥66dξ+C3∫τ-1τ∥g(ξ,τ-t,g0)∥2dξ+4∫τ-1τ∥f(ξ)∥2dξ,
which along with (65) shows that, for all t≥T,
(67)∫τ-1τ∥gξ(ξ,τ-t,g0)∥2dξ≤Me-Fτ(∫-∞τeFξ∥f(ξ)∥2dξ+∫-∞τeFξ∥f(ξ)∥𝕍′6dξ).
Then Lemma 13 follows from (64) and (67) which completes the proof.Lemma 14.
Suppose that (19) and (20) hold, and let
(68)dfdt∈[Lloc2(ℝ,L2(ℝn))]3.
Then for every τ∈ℝ and D={D(t)}t∈ℝ∈𝒟F, there exists T=T(τ,D)>2 such that for all t≥T,
(69)∥gτ(τ,τ-t,g0(τ-t))∥2≤Me-FτeMe-Fτ∫-∞τ‍eFξ∥f(ξ)∥𝕌′4dξ×(∫-∞τeFξ∥f(ξ)∥2dξ+∫-∞τeFξ∥f(ξ)∥𝕍′6dξ)+M∫τ-1τ∥fξ(ξ)∥2dξ,
where g0(τ-t)∈D(τ-t) and M depends on d, G, F, N, k, and ϱ (a positive constant in the Gagliardo-Nirenberg inequality).Proof.
Letut=u~, vt=v~,  and Wt=w~ and differentiate system (1) with respect to t to get that
(70)∂u~∂t=d1Δu~-(F+k)u~+2uvu~+u2v~-3Gu2u~+kw~+∂f1∂t(t,x),∂v~∂t=d2Δv~-Fv~-2uvu~-u2v~+3Gu2u~+∂f2∂t(t,x),∂w~∂t=d3Δw~+ku~-(F+N)w~+∂f3∂t(t,x).
Taking the inner products (∂u~/∂t,u~), (∂v~/∂t,v~), and (∂w~/∂t,w~) in L2(ℝn) and then putting the three equalities together, we have
(71)12ddt(∥u~∥2+∥v~∥2+∥w~∥2)+d1∥∇u~∥2+d2∥∇v~∥2+d3∥∇w~∥2+(F+k)∥u~∥2+F∥v~∥2+(N+k)∥w~∥2≤(2+32G)(∥u∥42+∥v∥42)(∥u~∥42+∥v~∥42)+N∥w~∥42+k∥u~∥42+N4∥u~∥42+k4∥w~∥42+F2(∥u~∥2+∥v~∥2+∥w~∥2)+12F(∥f1t(t)∥2+∥f2t(t)∥2+∥f3t(t)∥2).
That is,
(72)ddt∥g~∥2+2d∥∇g~∥2+F∥g~∥2≤(4+3G)∥g∥42∥g~∥42+12max{N,k}∥g~∥2+1F∥ft(t)∥2.
Due to the Hölder inequality and Gagliardo-Nirenberg inequality,
(73)ddt∥g~∥2+2d∥∇g~∥2+F∥g~∥2≤(4+3G)∥g∥42∥g~∥42+12max{N,k}∥g~∥2+1F∥ft(t)∥2≤(4+3G)ϱ∥g∥42∥g~∥∥∇g~∥+12max{N,k}∥g~∥2+1F∥ft(t)∥2≤2d∥∇g~∥2+(4+3G)2ϱ28d∥g∥44∥g~∥2+δ∥g~∥2+1F∥ft(t)∥2,
where δ=(1/2)max{N,k}. Then (73) implies that
(74)ddt∥g~∥2≤((4+3G)2ϱ28d∥g∥44+δ)∥g~∥2+1F∥ft(t)∥2.
By the Gronwall lemma, letting s∈[τ-1,τ] and t≥1 and integrating on (s,τ), by g~=gt we get
(75)∥gτ(τ,τ-t,g0(τ-t))∥2≤∥gτ(s,τ-t,g0(τ-t))∥2×e((4+3G)2ϱ2/8d)∫sτ‍∥g(ξ,τ-t,g0(τ-t))∥44dξ+δ(τ-s)+1F∫sτ∥fξ(ξ)∥2dξ≤∥gτ(τ,τ-t,g0(τ-t))∥2×e((4+3G)2ϱ2/8d)∫sτ‍∥g(ξ,τ-t,g0(τ-t))∥44dξ+δ(τ-s)+1F∫sτ∥fξ(ξ)∥2dξ.
Now integrating (75) with respect to s on (τ-1,τ), we find
(76)∥gτ(τ,τ-t,g0(τ-t))∥2≤e((4+3G)2ϱ2/8d)e-Fτ∫τ-1τ‍e-Fξ∥g(ξ,τ-t,g0(τ-t))∥44dξ+δ×∫τ-1τ∥gτ(s,τ-t,g0(τ-t))∥2ds+1F∫τ-1τ∥fξ(ξ)∥2dξ≤eMe-Fτ∫-∞τ‍eFξ∥f(ξ)∥𝕌′4dξ+δ×∫τ-1τ∥gτ(s,τ-t,g0(τ-t))∥2ds+1F∫τ-1τ∥fξ(ξ)∥2dξ,
which along with Lemma 13 and Proposition 12 shows that there exists T=(τ,D)>2 such that for all t≥T,
(77)∥gτ(τ,τ-t,g0(τ-t))∥2≤Me-FτeMe-Fτ∫-∞τ‍eFξ∥f(ξ)∥𝕌′4dξ×(∫-∞τeFξ∥f(ξ)∥2dξ+∫-∞τeFξ∥f(ξ)∥𝕍′6dξ)+M∫τ-1τ∥fξ(ξ)∥2dξ,
which completes the proof.We now establish uniform estimates on the tails of solutions whent→∞. We show that the tails of solutions are uniformly small for large space and time variables. These uniform estimates are crucial for proving the pullback asymptotic compactness of the cocycle ϕ.Lemma 15.
Suppose that (19) and (20) hold. Then for every ε>0, τ∈ℝ, and D={D(t)}t∈ℝ∈𝒟F, there exist T=T(τ,D,ε)>2 and K=K(τ,ε)>0 such that for all t≥T and N≥K,
(78)∫|x|≥K|g(x,τ,τ-t,g0(τ-t))|2dx≤ε,
where g0(τ-t)∈D(τ-t), K depends on τ and ε, and T depends on τ, D, and ε.Proof.
Choose a smooth cut-off function satisfying0≤ρ(s)≤1 for s∈ℝ+, ρ(s)=0 for 0≤s≤1, and ρ(s)=1 for s≥2. Suppose that there exists a constant c such that |ρ′(s)|≤c for s∈ℝ+.
Taking the inner product of (25), (26), and (27) with Gρ(|x|2/K2)u, ρ(|x|2/K2)v, and Gρ(|x|2/K2)W in L2(ℝn), respectively, we get
(79)G2ddt∫ℝnρ(|x|2K2)|u|2-d1G∫ℝnρ(|x|2K2)uΔu+G(F+k)∫ℝnρ(|x|2K2)|u|2=G∫ℝnρ(|x|2K2)u3vdx-G2∫ℝnρ(|x|2K2)u4dx+kG∫ℝnρ(|x|2K2)uWdx+G∫ℝnρ(|x|2K2)f1udx,12ddt∫ℝnρ(|x|2K2)|v|2-d2∫ℝnρ(|x|2K2)vΔvdx+F∫ℝnρ(|x|2K2)|v|2dx=-∫ℝnρ(|x|2K2)u2v2dx+G∫ℝnρ(|x|2K2)vu3dx+∫ℝnρ(|x|2K2)f2vdx,μG2ddt∫ℝnρ(|x|2K2)|W|2-μGd3∫ℝnρ(|x|2K2)WΔW+G(μF+k)∫ℝnρ(|x|2K2)|W|2=kG∫ℝnρ(|x|2K2)Wudx+G∫ℝnρ(|x|2K2)Wf3dx.
Add up the three equalities. Then we have
(80)ddt(G2∫ℝnρ(|x|2K2)|u|2+12∫ℝnρ(|x|2K2)|v|2+μG2∫ℝnρ(|x|2K2)|W|2)-(d1G∫ℝnρ(|x|2K2)uΔu+d2∫ℝnρ(|x|2K2)vΔv+μGd3∫ℝnρ(|x|2K2)WΔW)+G(F+k)∫ℝnρ(|x|2K2)|u|2+F∫ℝnρ(|x|2K2)|v|2dx+G(μF+k)∫ℝnρ(|x|2K2)|W|2≤-∫ℝnρ(|x|2K2)(Gu2-uv)2dx+2kG∫ℝnρ(|x|2K2)uWdx+G∫ℝnρ(|x|2K2)f1udx+∫ℝnρ(|x|2K2)f2vdx+G∫ℝnρ(|x|2K2)Wf3dx≤Gk∫ℝnρ(|x|2K2)|u|2+Gk∫ℝnρ(|x|2K2)|W|2+GF2∫ℝnρ(|x|2K2)|u|2+F2∫ℝnρ(|x|2K2)|v|2dx+μGF2∫ℝnρ(|x|2K2)|W|2dx+G2F∫|x|≥K|f1(x,t)|2dx+G2μF∫|x|≥K|f3(x,t)|2dx+12F∫|x|≥K|f2(x,t)|2dx.
That is,
(81)ddt(G∫ℝnρ(|x|2K2)|u|2+∫ℝnρ(|x|2K2)|v|2+μG∫ℝnρ(|x|2K2)|W|2)+GF∫ℝnρ(|x|2K2)|u|2+F∫ℝnρ(|x|2K2)|v|2+GμF∫ℝnρ(|x|2K2)|W|2≤2cK(d1G(∥u∥2+∥∇u∥2)+d2(∥v∥2+∥∇v∥2)+d3Gμ(∥W∥2+∥∇W∥2))+GF∫|x|≥K|f1(x,t)|2dx+GμF∫|x|≥K|f3(x,t)|2dx+1F∫|x|≥K|f2(x,t)|2dx.
Here, we use the integration by parts. We have
(82)∫ℝnρ(|x|2K2)uΔudx≤cK(∥u∥2+∥∇u∥2).
Denote do=max{d1,d2,d3}, then we can deduce from (81) that
(83)ddt∫ℝnρ(|x|2K2)|g|2dx+F∫ℝnρ(|x|2K2)|g|2≤2cdoK(∥g∥2+∥∇g∥2)+C1∫|x|≥K|f(x,t)|2dx.
Multiplying (83) by eFt and then integrating over (τ-t,τ) with t≥0, we get
(84)∫ℝnρ(|x|2K2)|g(x,τ,τ-t,g0(τ-t))|2dx≤e-FτeF(τ-t)∫ℝnρ(|x|2K2)|g0(x,τ-t)|2dx+2cdoKe-Fτ∫τ-tτeFξ∥g(ξ,τ-t,g0(τ-t))∥2+∥∇g(ξ,τ-t,g0(τ-t))∥2dξ+C1e-Fτ∫τ-tτ∫|x|≥K|f(x,ξ)|2dxdξ≤e-FτeF(τ-t)∫ℝn|g0(x,τ-t)|2dx+C1e-Fτ∫-∞τ∫|x|≥K|f(x,ξ)|2dxdξ+2cdoKe-Fτ∫τ-tτeFξ∥g(ξ,τ-t,g0(τ-t))∥2+∥∇g(ξ,τ-t,g0(τ-t))∥2dξ.
Note that for given ε>0, there is T1=T1(τ,D,ε)>0 such that for all t≥T1,
(85)e-FτeF(τ-t)∫ℝn|g0(x,τ-t)|2dx≤ε.
By (22), there is K1=K1(τ,ε)>0 such that for all N≥K1,
(86)C1e-Fτ∫-∞τ∫|x|≥K|f(x,ξ)|2dxdξ≤ε.
For the last term on the right-hand side of (84), it follows from Lemma 8 that there is T2=T2(τ,D) such that for all t≥T2,
(87)2cdoKe-Fτ∫τ-tτeFξ∥g(ξ,τ-t,g0(τ-t))∥2+∥∇g(ξ,τ-t,g0(τ-t))∥2dξ≤2cdoKMe-Fτ∫-∞τeFξ∥f(ξ)∥2dξ.
Therefore, there is K2=K2(τ,ε)>K1 such that for all K≥K2 and t≥T2,
(88)2cdoKe-Fτ∫τ-tτeFξ∥g(ξ,τ-t,g0(τ-t))∥2+∥∇g(ξ,τ-t,g0(τ-t))∥2dξ≤ε.
Let T=max{T1,T2} and then by (85)–(88), we find that, for all K≥K2 and t≥T,
(89)∫ℝnρ(|x|2K2)|g(x,τ,τ-t,g0(τ-t))|2dx≤3ε,
and hence for all K≥K2 and t≥T,
(90)∫|x|≥2Nρ(|x|2K2)|g(x,τ,τ-t,g0(τ-t))|2dx≤∫ℝnρ(|x|2K2)|g(x,τ,τ-t,g0(τ-t))|2dx≤3ε,
which completes the proof.


## 5. Existence of Pullback Attractors

In this section, we prove the existence of a𝒟F-pullback global attractor for the nonautonomous three-component reversible Gray-Scott system on ℝn. We first establish the 𝒟F-pullback asymptotic compactness of solutions and prove the existence of a pullback attractor in ℍ. Then we show that this attractor is actually a 𝒟F-pullback attractor in 𝔼.Lemma 16.
Suppose that (19) and (20) hold. Then ϕ is 𝒟F-pullback asymptotically compact in ℍ. That is, for every τ∈ℝ, D={D(t)}t∈ℝ∈𝒟F, tn→∞, and g0,n∈D(τ-tn), the sequence ϕ(tn,τ-tn,g0,n) has a convergent subsequence in ℍ.Proof.
The proof is a slightly modification of Lemma 5.1 in [20] and thus is omitted here.Theorem 17.
Suppose that (19) and (20) hold. Then problem (9)-(10) has a unique 𝒟F-pullback global attractor {𝒜(τ)}τ∈ℝ∈𝒟F in ℍ. That is, for every τ∈ℝ,(i)
𝒜(τ) is compact in ℍ,(ii)
{𝒜(τ)}τ∈ℝ is invariant. That is,
(91)ϕ(t,τ,𝒜(τ))=𝒜(t+τ),∀t≥0,(iii)
{𝒜(τ)}τ∈ℝ attracts every set in 𝒟F with respect to the norm of ℍ. That is, for every B={B(τ)}τ∈ℝ∈𝒟F,
(92)limt→∞dH(ϕ(t,τ-t,B(τ-t),𝒜(τ)))=0,where for any X,Y⊆ℍ, dℍ(X,Y)=supx∈Xinfy∈Y∥x-y∥ℍ.Proof.
Forτ∈ℝ, denote
(93)B(τ)={g:∥g∥2≤Me-Fτ∫-∞τ‍eFξ∥f(ξ)∥2dξ}.
Note that B={B(τ)}τ∈ℝ∈𝒟F is a 𝒟F-pullback absorbing for ϕ in ℍ by Lemma 8. In addition, ϕ is 𝒟F-pullback asymptotically compact by Lemma 16. Thus, the existence of a 𝒟F-pullback global attractor for ϕ in ℍ follows from Theorem 7.In what follows, we strengthen Theorem17 and show that the global attractor {𝒜(τ)}τ∈ℝ is actually a 𝒟F-pullback global attractor in 𝔼. As a necessary step towards this goal, we first prove the asymptotic compactness of solutions in 𝔼.Lemma 18.
Suppose that (19) and (20) hold. Let
(94)dfdt∈[Lloc2(ℝ,L2(ℝn))]3.
Then ϕ is 𝒟F-pullback asymptotically compact in 𝔼. That is, for every τ∈ℝ,  D={D(t)}t∈ℝ∈𝒟F, tn→∞, and g0,n∈D(τ-tn), the sequence ϕ(tn,τ-tn,g0,n) has a convergent subsequence in 𝔼.Proof.
By Lemma16, the sequence ϕ(tn,τ-tn,g0,n)=g(τ,τ-t,g0,n) has a convergent subsequence in ℍ, and hence there exists g-∈ℍ such that, up to a subsequence,
(95)g(τ,τ-t,g0,n)⟶g-inℍ.
This shows that ϕ(tn,τ-tn,g0,n) is a Cauchy sequence in ℍ. Next, we prove that the sequence is actually a Cauchy sequence in 𝔼. For any n,m≥1, it follows from (9) that
(96)gτ(τ,τ-tn,g0,n)-gτ(τ,τ-tm,g0,m)=A(g(τ,τ-tn,g0,n)-g(τ,τ-tm,g0,m))-H(g(τ,τ-tn,g0,n)-g(τ,τ-tm,g0,m)).
That is,
(97)∂(un-um)∂τ=d1Δ(un-um)-(F+k)(un-um)+(un2vn-um2vm)-G(un3-um3)+N(wn-wm),∂(vn-vm)∂τ=d2Δ(vn-vm)-F(vn-vm)-(un2vn-um2vm)+G(un3-um3),∂(wn-wm)∂τ=d3Δ(wn-wm)+k(un-um)-(F+N)(wn-wm),
where un, um denote u(τ,τ-tn,u0,n), u(τ,τ-tm,u0,m) and so do as vn, vm, wn, wm, and gn, gm. Taking the inner products (∂(un-um)/∂τ,un-um), (∂(vn-vm)/∂τ,vn-vm) and (∂(wn-wm)/∂τ,wn-wm), respectively, and summing up the three equalities, we get
(98)d(∥∇(un-um)∥2+∥∇(vn-vm)∥2+∥∇(wn-wm)∥2)+(F+k)∥un-um∥2+F∥vn-vm∥2+(N+k)∥wn-wm∥2≤(N+k)(wn-wm,un-um)+(un2vn-um2vm,un-um-vn+vm)-G(un3-um3,un-um-vn+vm)-∥∂∂τ(un-um+vn-vm+wn-wm)∥×∥un-um+vn-vm+wn-wm∥.
Because of the Hölder inequality and Gagliardo-Nirenberg inequality, we have
(99)(un2vn-um2vm,un-um-vn+vm)-G(un3-um3,un-um-vn+vm)≤d2∥∇(gn-gm)∥2+C2ϱ22d(∥gn∥2+∥gn∥44+∥gn∥66)∥gn-gm∥2.
From (98) and (99), it yields
(100)d∥∇(un-um+vn-vm+wn-wm)∥2+2F∥un-um+vn-vm+wn-wm∥2≤-2∥∂∂τ(un-um+vn-vm+wn-wm)∥×∥un-um+vn-vm+wn-wm∥+(C2ϱ2d(∥gn∥2+∥gn∥44+∥gn∥66)+2δ)×∥un-um+vn-vm+wn-wm∥2.
By Lemma 14 we find that, for every τ∈ℝ, there exists T=T(τ,D) such that for all t≥T,
(101)∥gτ(τ,τ-t,g0(τ-t))∥≤M.
Since tn→∞, there exists N-=N-(τ,D) such that tn≥T for all n≥T. Thus, we obtain that, for all n≥N-,
(102)∥gτ(τ,τ-t,g0,n)∥≤M,
which along with (100) shows that, for all n,m≥N-,
(103)d∥∇(un-um+vn-vm+wn-wm)∥2+2F∥un-um+vn-vm+wn-wm∥2≤2M∥un-um+vn-vm+wn-wm∥+(C2ϱ2d(∥gn∥2+∥gn∥44+∥gn∥66)+2δ)×∥un-um+vn-vm+wn-wm∥2.
That is,
(104)∥∇(g(τ,τ-t,g0,n)-g(τ,τ-t,g0,m))∥2≤2M∥g(τ,τ-t,g0,n)-g(τ,τ-t,g0,m)∥+(C2ϱ2d(∥gn∥2+∥gn∥44+∥gn∥66)+2δ)×∥g(τ,τ-t,g0,n)-g(τ,τ-t,g0,m)∥2.
By Lemma 8 and Propositions 11 and 12, and combining with the fact that g(τ,τ-t,g0,n) is a Cauchy sequence in ℍ, we complete the proof.We are now ready to prove the existence of a global attractor for problem (9)-(10) in 𝔼.Theorem 19.
Suppose that (19) and (20) hold. Let
(105)dfdt∈[Lloc2(ℝ,L2(ℝn))]3.
Then problem (9)-(10) has a unique 𝒟F-pullback global attractor {𝒜(τ)}τ∈ℝ∈𝒟F in 𝔼. That is, for every τ∈ℝ,(i)
𝒜(τ) is compact in 𝔼,(ii)
{𝒜(τ)}τ∈ℝ is invariant. That is,
(106)ϕ(t,τ,𝒜(τ))=𝒜(t+τ),∀t≥0,(iii)
{𝒜(τ)}τ∈ℝ attracts every set in 𝒟F with respect to the norm of 𝔼. That is, for every B={B(τ)}τ∈ℝ∈𝒟F,
(107)limt→∞d𝔼(ϕ(t,τ-t,B(τ-t),𝒜(τ)))=0,where, for any X,Y⊆ℍ, dℍ(X,Y)=supx∈Xinfy∈Y∥x-y∥𝔼.Proof.
The proof is a slightly modification of Theorem 5.4 in [20] and thus is omitted here.Remark 20.
In this paper, we study the nonautonomous three-component reversible Gray-Scott system, which is probably similar to the system considered in [20]. But, in fact, the asymptotically dissipative condition of the system cannot be satisfied (see, e.g., [4, 8] for more details), which is different from the system considered in [20]. Here, we have to use the method of the rescaling and grouping estimation to deduce the uniform estimates of solutions.Remark 21.
In the original three-component reversible Gray-Scott system (see [4]), the first constant F that appears in the second variable v-section does not depend on the space variable x. Here we have to affiliate the constant F to f2(x), or else, it will give an obstacle to establish the uniform estimates of solutions when t→∞ (see also [6, Remark 5.3]).



---

*Source: 719063-2013-09-23.xml*