Pullback 𝒟-Attractor of Nonautonomous Three-Component Reversible Gray-Scott System on Unbounded Domains

Abstract

The long time behavior of solutions of the nonautonomous three-components reversible Gray-Scott system defined on the entire space ℝ^𝑛 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 [𝐿²(ℝ^𝑛)]³ and [𝐻¹(ℝ^𝑛)]³, respectively. The pullback asymptotic compactness of solutions is proved by using uniform estimates on the tails of solutions on unbounded domains.

1. Introduction

In this paper, we consider the dynamical behavior of the nonautonomous three-components reversible Gray-Scott system 𝜕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), (1) on [𝜏,∞)×𝒪 with initial data 𝑢(𝜏,𝑥)=𝑢₀(𝑥),𝑣(𝜏,𝑥)=𝑣₀(𝑥),𝑤(𝜏,𝑥)=𝑤₀(𝑥)on𝒪, (2) where 𝒪=ℝ^𝑛×ℝ^𝑛×ℝ^𝑛; all the parameters are arbitrarily given positive constants; 𝑓_𝑖(𝑥,𝑡) for 𝑡∈ℝ and 𝑥∈ℝ^𝑛 is an external forcing term which is locally square integrable in time. That is, 𝑓_𝑖∈𝐿²_loc(ℝ,𝐿²(ℝ^𝑛)), 𝑖=1,2,3.

Historically, when 𝑤=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 ℝ^𝑛 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 Section 3, we define a cocycle for the nonautonomous three-component reversible Gray-Scott system on ℝ^𝑛. 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(ℝ^𝑛) or ℍ=[𝐿²(ℝ^𝑛)]³ and use ‖⋅‖₆, ‖⋅‖₄ to denote the norm in 𝐿⁶(ℝ^𝑛) or 𝕍=[𝐿⁶(ℝ^𝑛)]³, 𝐿⁴(ℝ^𝑛) or 𝕌=[𝐿⁴(ℝ^𝑛)]³; 𝕍^′,𝕌 the dual of 𝕍,𝕌; 𝔼=[𝐻¹(ℝ^𝑛)]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.

Let Ω be a nonempty set and 𝑋 a metric space with distance 𝑑(⋅,⋅).

Definition 1. A family of mappings {𝜃_𝑡}_𝑡∈ℝ from Ω to itself is called a family of shift operators on Ω if {𝜃_𝑡}_𝑡∈ℝ satisfies the group properties:(i)𝜃₀𝜔=𝜔, for all 𝜔∈Ω,(ii)𝜃_𝑡(𝜃_𝜏𝜔)=𝜃_𝑡+𝜏𝜔, for all 𝜔∈Ω and 𝑡,𝜏∈ℝ.

Definition 2. Let {𝜃_𝑡}_𝑡∈ℝ be a family of shift operators on Ω. Then a continuous 𝜃-cocycle 𝜙 on 𝑋 is a mapping 𝜙∶ℝ⁺×Ω×𝑋⟶𝑋,(𝑡,𝜔,𝑥)↦𝜙(𝑡,𝜔,𝑥), (3) 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,𝜃_𝜏𝜔,⋅)∘𝜙(𝜏,𝜔,⋅),(iii)𝜙(𝑡,𝜔,⋅)∶𝑋→𝑋 is continuous.

Hereafter, we always assume that 𝜙 is a continuous 𝜃-cocycle on 𝑋 and 𝒟 is a collection of families of subsets of 𝑋: 𝒟=𝐷={𝐷(𝜔)}_𝜔∈Ω∶𝐷(𝜔)⊆𝑋forevery𝜔∈Ω. (4)

Definition 3. Let 𝒟 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. Let 𝒟 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 𝜙𝑡,𝜃_−𝑡𝜔,𝐵𝜃_−𝑡𝜔⊆𝐾(𝜔),∀𝑡≥𝑡(𝜔,𝐵). (5)

Definition 5. Let 𝒟 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(𝑡_𝑛,𝜃_−𝑡𝑛𝜔,𝑥_𝑛)}^∞_𝑛=1 has a convergent subsequence in 𝑋 whenever 𝑡_𝑛→∞ and 𝑥_𝑛∈𝐵(𝜃_−𝑡𝑛𝜔) with {𝐵(𝜔)}_𝜔∈Ω∈𝒟.

Definition 6. Let 𝒟 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, 𝜙(𝑡,𝜔,𝒜(𝜔))=𝒜𝜃_𝑡𝜔,∀𝑡≥0, (6)(iii){𝒜(𝜔)}_𝜔∈Ω attracts every set in 𝒟, that is, for every 𝐵={𝐵(𝜔)}_𝜔∈Ω∈𝒟, lim_𝑡→∞𝑑𝜙𝑡,𝜃_−𝑡𝜔,𝐵𝜃_−𝑡𝜔,𝒜(𝜔)=0, (7)where 𝑑 is the Hausdorff semimetric given by 𝑑(𝑌,𝑍)=sup_𝑦∈𝑌inf_𝑧∈𝑍‖𝑦−𝑧‖_𝑋 for any 𝑌⊆𝑋 and 𝑍⊆𝑋.

Theorem 7. Let 𝒟 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 𝒜(𝜔)=_{𝜏≥0𝑡≥𝜏}𝜙𝑡,𝜃_−𝑡𝜔,𝐾𝜃_−𝑡𝜔. (8)

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

In this section, we constructed a 𝜃-cocycle for the nonautonomous three-component reversible Gray-Scott system defined on ℝ^𝑛. For every 𝜏∈ℝ and 𝑡>𝜏, system (1) with initial data (2) can be rewritten as 𝜕𝑔𝑔()𝜕𝑡−𝐴𝑔+𝐻=𝑓𝑥,𝑡,𝑥∈𝒪, (9) with initial condition 𝑔(𝑥,𝜏)=𝑔_𝜏(𝑥),𝑥∈ℝ^𝑛, (10) where 𝑔=(𝑢,𝑣,𝑤)^𝑇,𝑓(𝑥,𝑡)=(𝑓₁(𝑥,𝑡),𝑓₂(𝑥,𝑡),𝑓₃(𝑥,𝑡))^𝑇𝑑,𝑥∈𝒪,𝐴=¹Δ000𝑑₂Δ000𝑑₃Δ,𝐻𝑔=−(𝐹+𝑘)𝑢+𝑢²𝑣−𝐺𝑢³+𝑁𝑤−𝐹𝑣−𝑢²𝑣+𝐺𝑢³()𝑤,𝑘𝑢−𝐹+𝑁 (11) 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∈𝐿²_loc(ℝ,ℍ)), then problem (9)-(10) is well defined in ℍ. That is, for any 𝑔_𝜏∈ℍ and 𝜏∈ℝ, (9) possesses a unique solution 𝑔 satisfying 𝑔∈𝐶([𝜏,∞);ℍ)∩𝐿²_loc([𝜏,∞);𝔼), (12) which continuously depends on the initial data 𝑔_𝜏∈ℍ. To construct a cocycle 𝜙 for problem (9)-(10), we denote by Ω=ℝ and define a shift operator 𝜃_𝑡 on Ω for every 𝑡∈ℝ by 𝜃_𝑡(𝜏)=𝑡+𝜏,∀𝜏∈ℝ. (13) Let 𝜙 be a mapping from ℝ⁺×Ω×ℍ to ℍ given by 𝜙𝑡,𝜏,𝑔_𝜏=𝑔𝑡+𝜏,𝜏,𝑔_𝜏, (14) where 𝑡≥0, 𝜏∈ℝ, 𝑔_𝜏∈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 𝑔_𝜏∈ℍ, 𝜙𝑡+𝑠,𝜏,𝑔_𝜏=𝑔𝑡,𝑠+𝜏,𝜙𝑠,𝜏,𝑔_𝜏. (15) 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.

Let ℍ be a subset of ℍ and denote ℍ=sup_𝑥∈ℍ‖𝑥‖_ℍ. (16) Let 𝐷={𝐷(𝑡)}_𝑡∈ℝ be a family of subsets of ℍ. That is, 𝐷(𝑡)∈ℍ for every 𝑡∈ℝ and satisfying lim_{𝑡→−∞}𝑒^𝐹𝑡‖𝐷(𝑡)‖²=0, (17) where 𝐹 is a positive number given in (1). Hereafter, we use 𝒟_𝐹 to denote the collection of all families of subsets of ℍ satisfying (17), that is, 𝒟_𝐹=𝐷={𝐷(𝑡)}_𝑡∈ℝ∶𝐷satisfies(17). (18) Throughout this paper, we assume the following conditions for the external term: ^𝜏_−∞𝑒^𝐹𝜉𝑓(𝜉)²𝑑𝜉<∞,∀𝜏∈ℝ,_𝜏−∞𝑒^𝐹𝜉𝑓(𝜉)⁶_𝕍′𝑑𝜉<∞,_𝜏−∞𝑒^𝐹𝜉𝑓(𝜉)⁴_𝕌′𝑑𝜉<∞,∀𝜏∈ℝ,limsup_𝐾→∞^𝜏_{−∞|𝑥|≥𝐾}𝑒^𝐹𝜉𝑓(𝑥,𝜉)²𝑑𝑥𝑑𝜉=0,∀𝜏∈ℝ. (19)(20)(21)

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 ^𝜏_{−∞|𝑥|≥𝐾}𝑒^𝐹𝜉𝑓_𝑖(𝑥,𝜉)²𝑑𝑥𝑑𝜉≤𝛽𝑒^𝐹𝜏,𝑖=1,2,3. (22) 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 ℝ^𝑛 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∈𝒟_𝐹, there exists 𝑇=𝑇(𝜏,𝐷)>0 such that for all 𝑡≥𝑇, 𝑔𝜏,𝜏−𝑡,𝑔₀(𝜏−𝑡)²≤𝑀𝑒^{−𝐹𝜏𝜏}_−∞𝑒^𝐹𝜉𝑓(𝜉)²𝑑𝜉,_{𝜏𝜏−𝑡}𝑒^𝐹𝜉∇𝑔𝜉,𝜏−𝑡,𝑔₀(𝜏−𝑡)²𝑑𝑠≤𝑀^𝜏_−∞𝑒^𝐹𝜉𝑓(𝜉)²𝑑𝜉, (23) where 𝑔₀(𝜏−𝑡)∈𝐷(𝜏−𝑡) and 𝑀 depends on 𝐺, 𝐹, 𝑁, and 𝑘.

Proof. Define 𝑊()=𝑁𝑡,𝑥𝑘𝑤()𝑘𝑡,𝑥,𝜇=𝑁, (24) then the system (1) becomes 𝜕𝑢𝜕𝑡=𝑑₁Δ𝑢−(𝐹+𝑘)𝑢+𝑢²𝑣−𝐺𝑢³+𝑘𝑊+𝑓₁(𝑡,𝑥),𝜕𝑣𝜕𝑡=𝑑₂Δ𝑣−𝐹𝑣−𝑢²𝑣+𝐺𝑢³+𝑓₂(𝑡,𝑥),𝜇𝜕𝑊𝜕𝑡=𝜇𝑑₃Δ𝑊+𝑘𝑢−𝜇𝐹+𝑘𝑊+𝑓₃(𝑡,𝑥). (25)(26)(27) 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 12𝑑𝐺‖𝑢‖𝑑𝑡²+‖𝑣‖²+𝜇𝐺‖𝑊‖²+𝑑₁𝐺‖∇𝑢‖²+𝑑₂‖∇𝑣‖²+𝜇𝐺𝑑₃‖∇𝑊‖²=2𝑘𝐺_ℝ𝑛−𝐺(𝑢‖𝑢𝑊𝑑𝑥𝐹+𝑘)‖²−𝐹‖𝑣‖²−𝐺𝜇𝐹+𝑘‖𝑊‖²−_ℝ𝑛𝐺𝑢²−𝑢𝑣²𝑑𝑥+𝐺_ℝ𝑛𝑢𝑓₁+𝑑𝑥_ℝ𝑛𝑣𝑓₂𝑑𝑥+𝐺_ℝ𝑛𝑊𝑓₃𝑑𝑥≤𝐺_ℝ𝑛𝑢𝑓₁𝑑𝑥+_ℝ𝑛𝑣𝑓₂𝑑𝑥+𝐺_ℝ𝑛𝑊𝑓₃𝐺‖𝑢‖𝑑𝑥−𝐹²+‖𝑣‖²+𝜇𝐺‖𝑊‖²1≤−2𝐹𝐺‖𝑢‖²+‖𝑣‖²+𝜇𝐺‖𝑊‖²+𝐺2𝐹|𝑓₁|²+12𝐹|𝑓₂|²+𝐺2𝜇𝐹|𝑓₃|². (28) Setting 𝑑=min𝑑₁,𝑑₂,𝑑₃,𝐶₁=max1,𝐺,𝐺/𝜇,𝐹min1,𝐺,𝐺/𝜇 (29) then (28) becomes 𝑑𝑔(𝑡𝑑𝑡)²+2𝑑∇𝑔(𝑡)²+𝐹𝑔(𝑡)²≤𝐶₁𝑓². (30) Multiplying (30) by 𝑒^𝐹𝑡 and then integrating the resulting inequality on (𝜏−𝑡,𝜏) with 𝑡≥0, we find that 𝑔𝜏,𝜏−𝑡,𝑔₀(𝜏−𝑡)²+2𝑑𝑒^{−𝐹𝜏𝜏}_𝜏−𝑡𝑒^𝐹𝜉∇𝑔(𝜉)²𝑑𝜉≤𝑒^−𝐹𝜏𝑒^{𝐹(𝜏−𝑡)}𝑔₀(𝜏−𝑡)²+𝐶₁𝑒^{−𝐹𝜏𝜏}_𝜏−𝑡𝑒^𝐹𝜉𝑓(𝜉)²𝑑𝜉≤𝑒^−𝐹𝜏𝑒^{𝐹(𝜏−𝑡)}𝑔₀(𝜏−𝑡)²+𝐶₁𝑒^{−𝐹𝜏𝜏}_−∞𝑒^𝐹𝜉𝑓(𝜉)²𝑑𝜉. (31) Notice that 𝑔₀(𝜏−𝑡)∈𝐷(𝜏−𝑡) and 𝐷={𝐷(𝑡)}_𝑡∈ℝ∈𝒟_𝐹. We find that for every 𝜏∈ℝ, there exists 𝑇=𝑇(𝜏,𝐷) such that for all 𝑡≥𝑇, 𝑒^{𝐹(𝜏−𝑡)}𝑔₀(𝜏−𝑡)²≤𝐶₁^𝜏_−∞𝑒^𝐹𝜉𝑓(𝜉)²𝑑𝜉. (32) By (31) and (32), we get that, for all 𝑡≥𝑇, 𝑔𝜏,𝜏−𝑡,𝑔₀(𝜏−𝑡)²+2𝑑𝑒^{−𝐹𝜏𝜏}_𝜏−𝑡𝑒^𝐹𝜉∇𝑔(𝜉)²𝑑𝜉≤2𝐶₁𝑒^{−𝐹𝜏𝜏}_−∞𝑒^𝐹𝜉𝑓(𝜉)²𝑑𝜉, (33) which completes the proof.

The following lemma is useful for deriving uniform estimates of solutions in 𝔼.

Lemma 9. Suppose that (19) holds. Then for every 𝜏∈ℝ and 𝐷={𝐷(𝑡)}_𝑡∈ℝ∈𝒟_𝐹, there exists 𝑇=𝑇(𝜏,𝐷)>2 such that for all 𝑡≥𝑇, ^𝜏_𝜏−2𝑒^𝐹𝜉𝑔𝜉,𝜏−𝑡,𝑔₀(𝜏−𝑡)²𝑑𝜉≤𝑀^𝜏_−∞𝑒^𝐹𝜉𝑓(𝜉)²𝑑𝜉,_𝜏𝜏−2𝑒^𝐹𝜉∇𝑔𝜉,𝜏−𝑡,𝑔₀(𝜏−𝑡)²𝑑𝜉≤𝑀^𝜏_−∞𝑒^𝐹𝜉𝑓(𝜉)²𝑑𝜉, (34) where 𝑔₀(𝜏−𝑡)∈𝐷(𝜏−𝑡) and 𝑀 relies on 𝐺, 𝐹, 𝑁, and 𝑘.

Proof. By (30) we find that 𝑑𝑔(𝑡𝑑𝑡)²+𝐹𝑔(𝑡)²≤𝐶₁𝑓². (35) Let 𝑠∈[𝜏−2,𝜏] and 𝜏>2. Multiplying the above by 𝑒^𝐹𝑡 and integrating over (𝜏−𝑡,𝑠), we get 𝑒^𝐹𝑠𝑔𝑠,𝜏−𝑡,𝑔₀(𝜏−𝑡)²≤𝑒^{𝐹(𝜏−𝑡)}𝑔₀(𝜏−𝑡)²+𝐶₁^𝑠_𝜏−𝑡𝑒^𝐹𝜉𝑓(𝜉)²𝑑𝜉≤𝑒^{𝐹(𝜏−𝑡)}𝑔₀(𝜏−𝑡)²+𝐶₁^𝜏_−∞𝑒^𝐹𝜉𝑓(𝜉)²𝑑𝜉. (36) Therefore, there exists 𝑇=𝑇(𝜏,𝐷)>2 such that for all 𝑡≥𝑇 and 𝑠∈[𝜏−2,𝜏], 𝑒^𝐹𝑠𝑔𝑠,𝜏−𝑡,𝑔₀(𝜏−𝑡)²≤2𝐶₁^𝜏_−∞𝑒^𝐹𝜉𝑓(𝜉)²𝑑𝜉. (37) Integrate the above with respect to 𝑠 on (𝜏−2,𝜏) to obtain ^𝜏_𝜏−2𝑒^𝐹𝑠𝑔𝑠,𝜏−𝑡,𝑔₀(𝜏−𝑡)²𝑑𝑠≤4𝐶₁^𝜏_−∞𝑒^𝐹𝜉𝑓(𝜉)²𝑑𝜉. (38) On the other hand, for 𝑠=𝜏−2, (37) implies that 𝑒^{𝐹(𝜏−2)}𝑔𝜏−2,𝜏−𝑡,𝑔₀(𝜏−𝑡)²≤2𝐶₁^𝜏_−∞𝑒^𝐹𝜉𝑓(𝜉)²𝑑𝜉. (39) Multiplying (30) by 𝑒^𝐹𝜏 and then integrating over (𝜏−2,𝜏), by (38) we get that, for all 𝑡≥𝑇, 𝑒^𝐹𝜏𝑔𝜏,𝜏−𝑡,𝑔₀(𝜏−𝑡)²+2𝑑^𝜏_𝜏−2𝑒^𝐹𝜉∇𝑔(𝜉)²𝑑𝜉≤𝑒^{𝐹(𝜏−2)}𝑔𝜏−2,𝜏−𝑡,𝑔₀(𝜏−𝑡)²+𝐶₁^𝜏_𝜏−𝑡𝑒^𝐹𝜉𝑓(𝜉)²𝑑𝜉≤3𝐶₁^𝜏_−∞𝑒^𝐹𝜉𝑓(𝜉)²𝑑𝜉. (40) which along with (39) completes the proof.

Note that 𝑒^𝜆𝜉≥𝑒^{𝜆𝜏−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∈𝒟_𝐹, there exists 𝑇=𝑇(𝜏,𝐷)>2 such that for all 𝑡≥𝑇, ^𝜏_𝜏−2𝑔𝜉,𝜏−𝑡,𝑔₀(𝜏−𝑡)²𝑑𝑠≤𝑀𝑒^{−𝐹𝜏𝜏}_−∞𝑒^𝐹𝜉𝑓(𝜉)²𝑑𝜉,_𝜏𝜏−2∇𝑔𝜉,𝜏−𝑡,𝑔₀(𝜏−𝑡)²𝑑𝑠≤𝑀𝑒^{−𝐹𝜏𝜏}_−∞𝑒^𝐹𝜉𝑓(𝜉)²𝑑𝜉, (41) where 𝑔₀(𝜏−𝑡)∈𝐷(𝜏−𝑡) and 𝑀 depends on 𝐺, 𝐹, 𝑁, and 𝑘.

Before the derivation of uniform estimates of solutions in 𝔼, 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∈𝒟_𝐹, there exists 𝑇=𝑇(𝜏,𝐷)>0 such that for all 𝑡≥𝑇, 𝑔𝜏,𝜏−𝑡,𝑔₀(𝜏−𝑡)⁶₆≤𝑀𝑒^{−𝐹𝜏𝜏}_−∞𝑒^𝐹𝜉𝑓(𝜉)⁶_𝕍′𝑑𝜉,_𝜏𝜏−2𝑒^𝐹𝜉𝑔𝜉,𝜏−𝑡,𝑔₀(𝜏−𝑡)⁶₆𝑑𝜉≤𝑀^𝜏_−∞𝑒^𝐹𝜉𝑓(𝜉)⁶_𝕍′𝑑𝜉, (42)(43) where 𝑔₀(𝜏−𝑡)∈𝐷(𝜏−𝑡) and 𝑀 depends on 𝐺, 𝐹, 𝑁, and 𝑘.

Proof. Let 𝑉(𝑡,𝑥)=(𝑣(𝑡,𝑥))/𝐺, then (25)–(27) become 𝜕𝑢𝜕𝑡=𝑑₁Δ𝑢−(𝐹+𝑘)𝑢+𝐺𝑢²𝑉−𝐺𝑢³+𝑘𝑊+ℎ₁(𝑡,𝑥),𝜕𝑉𝜕𝑡=𝑑₂Δ𝑉−𝐹𝑉−𝑢²𝑉+𝑢³+ℎ₂(),𝜇𝑡,𝑥𝜕𝑊𝜕𝑡=𝜇𝑑₃Δ𝑊+𝑘𝑢−𝜇𝐹+𝑘𝑊+ℎ₃(𝑡,𝑥). (44) 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 16𝑑‖𝑢‖𝑑𝑡⁶⁶+𝐺‖𝑉‖⁶₆+𝜇‖𝑊‖⁶₆+5𝑑₁𝑢²∇𝑢²+𝑑₂𝐺𝑉²∇𝑉²+𝜇𝑑₃𝐺𝑊²∇𝑊²≤−(𝐹+𝑘)_ℝ𝑛𝑢⁶𝑑𝑠+𝐹_ℝ𝑛𝑉⁵𝑑𝑥−𝐺𝐹_ℝ𝑛𝑉⁶−𝑑𝑥𝜇𝐹+𝑘_ℝ𝑛𝑊⁶𝑑𝑥+𝑘_ℝ𝑛𝑢⁵𝑊𝑑𝑥+𝑘_ℝ𝑛𝑢𝑊⁵𝑑𝑥−𝐺_ℝ𝑛𝑢⁸−𝑢⁷𝑉−𝑢³𝑉⁵+𝑢²𝑉⁶+𝑑𝑥_ℝ𝑛𝑢⁵𝑓₁𝑑𝑥+𝐺_ℝ𝑛𝑉⁵𝑓₂𝑑𝑥+_ℝ𝑛𝑊⁵𝑓₃𝑑𝑥. (45) By Young's inequality, −𝐺_ℝ𝑛𝑢⁸−𝑢⁷𝑉−𝑢³𝑉⁵+𝑢²𝑉⁶𝐺𝑑𝑥≤0,_ℝ𝑛𝑉⁵𝑓₂𝑑𝑥≤5𝐹𝐺6‖𝑉‖⁶₆+𝐺𝑓₂⁶_𝕍′6𝐹⁵_ℝ𝑛,𝑢⁵𝑓₁𝑑𝑥≤5𝐹6‖𝑢‖⁶₆+𝑓₁⁶_𝕍′6𝐹⁵_ℝ𝑛,𝑊⁵ℎ₃𝑑𝑥≤5𝜇𝐹6‖𝑊‖⁶₆+𝑓₃⁶_𝕍′6𝜇𝐹⁵. (46) From (46) then (45) yields 𝑑‖𝑢‖𝑑𝑡⁶⁶+𝐺‖𝑉‖⁶₆+𝜇‖𝑊‖⁶₆+𝐹‖𝑢‖⁶₆+𝐺‖𝑉‖⁶₆+𝜇‖𝑊‖⁶₆≤𝑓₁⁶_𝕍′𝐹⁵+𝐺𝑓₂⁶_𝕍′𝐹⁵+𝑓₃⁶_𝕍′𝜇𝐹⁵. (47) Denote 𝐶₄=max1,𝐺,1/𝜇⁵𝐹⁵min1,1/𝐺⁵,1/𝜇⁵, (48) then from (47) implies that 𝑑𝑔(𝑡𝑑𝑡)⁶₆+𝐹𝑔(𝑡)⁶₆≤𝐶₄𝑓(𝑡)⁶_𝕍′. (49) Multiplying (49) by 𝑒^𝐹𝑡 and then integrating the resulting inequality on (𝜏−𝑡,𝜏) with 𝑡≥0, we find that𝑔𝜏,𝜏−𝑡,𝑔₀(𝜏−𝑡)⁶₆≤𝑒^−𝐹𝜏𝑒^{𝐹(𝜏−𝑡)}𝑔₀(𝜏−𝑡)⁶₆+𝐶₄𝑒^{−𝐹𝜏𝜏}_𝜏−𝑡𝑒^𝐹𝜉𝑓(𝜉)⁶_𝕍′𝑑𝜉≤𝑒^−𝐹𝜏𝑒^{𝐹(𝜏−𝑡)}𝑔₀(𝜏−𝑡)⁶₆+𝐶₄𝑒^{−𝐹𝜏𝜏}_−∞𝑒^𝐹𝜉𝑓(𝜉)⁶_𝕍′𝑑𝜉. (50)
Notice that 𝑔₀(𝜏−𝑡)∈𝐷(𝜏−𝑡) and 𝐷={𝐷(𝑡)}_𝑡∈ℝ∈𝒟_𝐹. We find that, for every 𝜏∈ℝ, there exists 𝑇=𝑇(𝜏,𝐷) such that for all 𝑡≥𝑇, 𝑒^{𝐹(𝜏−𝑡)}𝑔₀(𝜏−𝑡)⁶₆≤𝐶₄^𝜏_−∞𝑒^𝐹𝜉𝑓(𝜉)⁶_𝕍′𝑑𝜉. (51) By (50) and (51), we get that, for all 𝑡≥𝑇, 𝑔𝜏,𝜏−𝑡,𝑔₀(𝜏−𝑡)⁶₆≤2𝐶₄𝑒^{−𝐹𝜏𝜏}_−∞𝑒^𝐹𝜉𝑓(𝜉)⁶_𝕍′𝑑𝜉, (52) 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∈𝒟_𝐹, there exists 𝑇=𝑇(𝜏,𝐷)>0 such that for all 𝑡≥𝑇, 𝑔𝜏,𝜏−𝑡,𝑔₀(𝜏−𝑡)⁴₄≤𝑀𝑒^{−𝐹𝜏𝜏}_−∞𝑒^𝐹𝜉𝑓(𝜉)⁴_𝕌′𝑑𝜉,_𝜏𝜏−2𝑒^𝐹𝜉𝑔𝜉,𝜏−𝑡,𝑔₀(𝜏−𝑡)⁴₄𝑑𝜉≤𝑀^𝜏_−∞𝑒^𝐹𝜉𝑓(𝜉)⁴_𝕌′𝑑𝜉, (53) where 𝑔₀(𝜏−𝑡)∈𝐷(𝜏−𝑡) and 𝑀 depends on 𝐺, 𝐹, 𝑁, and 𝑘.

Proof. The proof is similar to Proposition 11 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∈𝒟_𝐹, there exists 𝑇=𝑇(𝜏,𝐷)>2 such that for all 𝑡≥𝑇, ∇𝑔𝜏,𝜏−𝑡,𝑔₀(𝜏−𝑡)²≤𝑀𝑒^{−𝐹𝜏𝜏}_−∞𝑒^𝐹𝜉𝑓(𝜉)²𝑑𝜉+^𝜏_−∞𝑒^𝐹𝜉𝑓(𝜉)⁶_𝕍′𝑑𝜉_𝑡𝜏−1,𝑔_𝜉𝜉,𝜏−𝑡,𝑔₀(𝜏−𝑡)²𝑑𝜉≤𝑀𝑒^{−𝐹𝜏𝜏}_−∞𝑒^𝐹𝜉𝑓(𝜉)²𝑑𝜉+^𝜏_−∞𝑒^𝐹𝜉𝑓(𝜉)⁶_𝕍′𝑑𝜉, (54) 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 𝑢_𝑡, the second equation with 𝑣_𝑡, and the third equation with 𝑤_𝑡, respectively, in 𝐿²(ℝ^𝑛) and then replacing 𝑡 by 𝜉, we obtain 𝑢_𝜉𝜉,𝜏−𝑡,𝑢₀(𝜏−𝑡)²+𝑑𝑑𝑑𝜉2¹∇𝑢𝜉,𝜏−𝑡,𝑢₀(𝜏−𝑡)²+(𝐹+𝑘)2𝑢𝜉,𝜏−𝑡,𝑢₀(𝜏−𝑡)²≤12𝑢_𝜉𝜉,𝜏−𝑡,𝑢₀(𝜏−𝑡)²+43+2𝐺²𝑢𝜉,𝜏−𝑡,𝑢₀(𝜏−𝑡)⁶₆+23𝑣𝜉,𝜏−𝑡,𝑣₀(𝜏−𝑡)⁶₆+2𝑁²𝑤𝜉,𝜏−𝑡,𝑤₀(𝜏−𝑡)²+2𝑓₁(𝜉)²,𝑣_𝜉𝜉,𝜏−𝑡,𝑣₀(𝜏−𝑡)²+𝑑𝑑𝑑𝜉2²∇𝑣𝜉,𝜏−𝑡,𝑢₀(𝜏−𝑡)²+𝐹2𝑣𝜉,𝜏−𝑡,𝑣₀(𝜏−𝑡)²≤12𝑣_𝜉𝜉,𝜏−𝑡,𝑣₀(𝜏−𝑡)²+31+2𝐺²𝑢𝜉,𝜏−𝑡,𝑢₀(𝜏−𝑡)⁶₆+12𝑣𝜉,𝜏−𝑡,𝑣₀(𝜏−𝑡)⁶₆+32𝑓₂(𝜉)²,𝑤_𝜉𝜉,𝜏−𝑡,𝑤₀(𝜏−𝑡)²+𝑑𝑑𝑑𝜉2³∇𝑤𝜉,𝜏−𝑡,𝑤₀(𝜏−𝑡)²+(𝐹+𝑁)2𝑤𝜉,𝜏−𝑡,𝑤₀(𝜏−𝑡)²≤12𝑤_𝜉𝜉,𝜏−𝑡,𝑤₀(𝜏−𝑡)²+𝑘²𝑢𝜉,𝜏−𝑡,𝑢₀(𝜏−𝑡)²+𝑓₃(𝜉)², (55)(56)(57) 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 𝑢_𝜉𝜉,𝜏−𝑡,𝑢₀²+𝑣_𝜉𝜉,𝜏−𝑡,𝑣₀²+𝑤_𝜉𝜉,𝜏−𝑡,𝑤₀²+𝑑𝑑𝑑𝜉¹∇𝑢𝜉,𝜏−𝑡,𝑢₀²+𝑑₂∇𝑣𝜉,𝜏−𝑡,𝑣₀²+𝑑₃∇𝑤𝜉,𝜏−𝑡,𝑤₀²+(𝐹+𝑘)𝑢𝜉,𝜏−𝑡,𝑢₀²+𝐹𝑣𝜉,𝜏−𝑡,𝑣₀²+(𝐹+𝑁)𝑤𝜉,𝜏−𝑡,𝑤₀²3≤41+2𝐺²𝑢𝜉,𝜏−𝑡,𝑢₀⁶₆+2𝑣𝜉,𝜏−𝑡,𝑣₀⁶₆+4𝑁²𝑤𝜉,𝜏−𝑡,𝑤₀²+2𝑘²𝑢𝜉,𝜏−𝑡,𝑢₀²+4𝑓₁(𝜉)²+3𝑓₂(𝜉)²+2𝑓₃(𝜉)². (58) Let 𝐶₂=4+6𝐺²,𝐶₃=2max2𝑁²,𝑘². (59) Then (58) yields 𝑔_𝜉𝜉,𝜏−𝑡,𝑔₀²+𝑑𝑑𝑑𝜉∇𝑔𝜉,𝜏−𝑡,𝑔₀²+𝐹𝑔𝜉,𝜏−𝑡,𝑔₀²≤𝐶₂𝑔𝜉,𝜏−𝑡,𝑔₀⁶₆+𝐶₃𝑔𝜉,𝜏−𝑡,𝑔₀²+4𝑓(𝜉)². (60) That is, 𝑑𝑑𝑑𝜉∇𝑔𝜉,𝜏−𝑡,𝑔₀²+𝐹𝑔𝜉,𝜏−𝑡,𝑔₀²≤𝐶₂𝑔𝜉,𝜏−𝑡,𝑔₀⁶₆+𝐶₃𝑔𝜉,𝜏−𝑡,𝑔₀²+4𝑓(𝜉)². (61) Let 𝑠≤𝜏 and 𝑡≥2. By integrating (61) over (𝑠,𝜏), we get 𝑑∇𝑔𝜏,𝜏−𝑡,𝑔₀²+𝐹𝑔𝜏,𝜏−𝑡,𝑔₀²≤𝑑∇𝑔𝑠,𝜏−𝑡,𝑔₀²+𝐹𝑔𝑠,𝜏−𝑡,𝑔₀²+𝐶₂^𝜏_𝑠𝑔𝜉,𝜏−𝑡,𝑔₀⁶₆𝑑𝜉+𝐶₃^𝜏_𝑠𝑔𝜉,𝜏−𝑡,𝑔₀²𝑑𝜉+4^𝜏_𝑠𝑓(𝜉)²𝑑𝜉. (62) Now integrating the above with respect to s on (𝜏−1,𝜏), we find 𝑑∇𝑔𝜏,𝜏−𝑡,𝑔₀²+𝐹𝑔𝜏,𝜏−𝑡,𝑔₀²≤𝑑^𝜏_𝜏−1∇𝑔𝑠,𝜏−𝑡,𝑔₀²𝑑𝑠+𝐹^𝜏_𝜏−1𝑔𝑠,𝜏−𝑡,𝑔₀²𝑑𝑠+𝐶₂^𝜏_𝜏−1𝑔𝜉,𝜏−𝑡,𝑔₀⁶₆𝑑𝜉+𝐶₃^𝜏_𝜏−1𝑔𝜉,𝜏−𝑡,𝑔₀²𝑑𝜉+4^𝜏_𝜏−1𝑓(𝜉)²𝑑𝜉, (63) which along with Corollary 10 and Proposition 11 implies that there exists 𝑇=𝑇(𝜏,𝐷)>2 such that for all 𝑡≥𝑇, 𝑑∇𝑔𝜏,𝜏−𝑡,𝑔₀²+𝐹𝑔𝜏,𝜏−𝑡,𝑔₀²≤𝑑+𝐹+𝐶₃+4𝑒^𝐹𝑀𝑒^{−𝐹𝜏𝜏}_−∞𝑒^𝐹𝜉𝑓(𝜉)²𝑑𝜉+𝐶₂𝑒^𝐹𝑀𝑒^{−𝐹𝜏𝜏}_−∞𝑓(𝜉)⁶_𝕍′𝑑𝜉≤𝑀𝑒^{−𝐹𝜏𝜏}_−∞𝑒^𝐹𝜉𝑓(𝜉)²𝑑𝜉+^𝜏_−∞𝑒^𝐹𝜉𝑓(𝜉)⁶_𝕍′𝑑𝜉. (64) 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, 𝑑∇𝑔𝜏−1,𝜏−𝑡,𝑔₀²+𝐹𝑔𝜏−1,𝜏−𝑡,𝑔₀²≤𝑑+𝐹+𝐶₃+4𝑒^𝐹𝑀𝑒^{−𝐹𝜏𝜏}_−∞𝑒^𝐹𝜉𝑓(𝜉)²𝑑𝜉+𝐶₂𝑒^𝐹𝑀𝑒^{−𝐹𝜏𝜏}_−∞𝑓(𝜉)⁶_𝕍′𝑑𝜉≤𝑀𝑒^{−𝐹𝜏𝜏}_−∞𝑒^𝐹𝜉𝑓(𝜉)²𝑑𝜉+^𝜏_−∞𝑒^𝐹𝜉𝑓(𝜉)⁶_𝕍′𝑑𝜉. (65) Now integrating (60) over (𝜏−1,𝜏), we obtain ^𝜏_𝜏−1𝑔_𝜉𝜉,𝜏−𝑡,𝑔₀²𝑑𝜉+𝑑∇𝑔𝜏,𝜏−𝑡,𝑔₀²+𝐹𝑔𝜏,𝜏−𝑡,𝑔₀²≤𝑑∇𝑔𝜏−1,𝜏−𝑡,𝑔₀²+𝐹𝑔𝜏−1,𝜏−𝑡,𝑔₀²+𝐶₂^𝜏_𝜏−1𝑔𝜉,𝜏−𝑡,𝑔₀⁶₆𝑑𝜉+𝐶₃^𝜏_𝜏−1𝑔𝜉,𝜏−𝑡,𝑔₀²𝑑𝜉+4^𝜏_𝜏−1𝑓(𝜉)²𝑑𝜉, (66) which along with (65) shows that, for all 𝑡≥𝑇, ^𝜏_𝜏−1𝑔_𝜉𝜉,𝜏−𝑡,𝑔₀²𝑑𝜉≤𝑀𝑒^{−𝐹𝜏𝜏}_−∞𝑒^𝐹𝜉𝑓(𝜉)²𝑑𝜉+^𝜏_−∞𝑒^𝐹𝜉𝑓(𝜉)⁶_𝕍′𝑑𝜉. (67) Then Lemma 13 follows from (64) and (67) which completes the proof.

Lemma 14. Suppose that (19) and (20) hold, and let 𝑑𝑓∈𝐿𝑑𝑡^2locℝ,𝐿²ℝ^𝑛3. (68) Then for every 𝜏∈ℝ and 𝐷={𝐷(𝑡)}_𝑡∈ℝ∈𝒟_𝐹, there exists 𝑇=𝑇(𝜏,𝐷)>2 such that for all 𝑡≥𝑇, 𝑔_𝜏𝜏,𝜏−𝑡,𝑔₀(𝜏−𝑡)²≤𝑀𝑒^−𝐹𝜏𝑒^{𝑀𝑒−𝐹𝜏∫𝜏−∞𝑒𝐹𝜉‖𝑓(𝜉)‖4𝕌′𝑑𝜉}×^𝜏_−∞𝑒^𝐹𝜉𝑓(𝜉)²+𝑑𝜉_𝜏−∞𝑒^𝐹𝜉𝑓(𝜉)⁶_𝕍′𝑑𝜉+𝑀^𝜏_𝜏−1𝑓_𝜉(𝜉)²𝑑𝜉, (69) 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. Let 𝑢_𝑡=𝑢, 𝑣_𝑡=𝑣̃,  and 𝑊_𝑡=𝑤 and differentiate system (1) with respect to 𝑡 to get that 𝜕𝑢𝜕𝑡=𝑑₁Δ𝑢−(𝐹+𝑘)𝑢+2𝑢𝑣𝑢+𝑢²𝑣̃−3𝐺𝑢²𝑢+𝑘𝑤+𝜕𝑓₁(),𝜕𝑣̃𝜕𝑡𝑡,𝑥𝜕𝑡=𝑑₂Δ𝑣̃−𝐹𝑣̃−2𝑢𝑣𝑢−𝑢²𝑣̃+3𝐺𝑢²𝑢+𝜕𝑓₂(),𝜕𝑤𝜕𝑡𝑡,𝑥𝜕𝑡=𝑑₃Δ𝑤+𝑘𝑢−(𝐹+𝑁)𝑤+𝜕𝑓₃().𝜕𝑡𝑡,𝑥 (70) Taking the inner products (𝜕𝑢/𝜕𝑡,𝑢), (𝜕𝑣/̃𝜕𝑡,𝑣)̃, and (𝜕𝑤/𝜕𝑡,𝑤) in 𝐿²(ℝ^𝑛) and then putting the three equalities together, we have 12𝑑‖𝑢‖𝑑𝑡²+‖𝑣‖̃²+‖𝑤‖²+𝑑₁‖∇𝑢‖²+𝑑₂‖∇𝑣‖̃²+𝑑₃‖∇𝑤‖²+(𝐹+𝑘)‖𝑢‖²+𝐹‖𝑣‖̃²+(𝑁+𝑘)‖𝑤‖²≤32+2𝐺‖𝑢‖²₄+‖𝑣‖²₄‖𝑢‖²₄+‖𝑣‖̃²₄+𝑁‖𝑤‖²₄+𝑘‖𝑢‖²₄+𝑁4‖𝑢‖²₄+𝑘4‖𝑤‖²₄+𝐹2‖𝑢‖²+‖𝑣‖̃²+‖𝑤‖²+1𝑓2𝐹^1𝑡(𝑡)²+𝑓_2𝑡(𝑡)²+𝑓_3𝑡(𝑡)². (71) That is, 𝑑𝑔𝑑𝑡²+2𝑑∇𝑔²+𝐹𝑔²≤(4+3𝐺)𝑔²₄𝑔²₄+12max{𝑁,𝑘}𝑔²+1𝐹𝑓_𝑡(𝑡)². (72) Due to the Hölder inequality and Gagliardo-Nirenberg inequality, 𝑑𝑔𝑑𝑡²+2𝑑∇𝑔²+𝐹𝑔²≤(4+3𝐺)𝑔²₄𝑔²₄+12max{𝑁,𝑘}𝑔²+1𝐹𝑓_𝑡(𝑡)²≤(4+3𝐺)𝜚𝑔²₄𝑔∇𝑔+12max{𝑁,𝑘}𝑔²+1𝐹𝑓_𝑡(𝑡)²≤2𝑑∇𝑔²+(4+3𝐺)²𝜚²𝑔8𝑑⁴⁴𝑔²+𝛿𝑔²+1𝐹𝑓_𝑡(𝑡)², (73) where 𝛿=(1/2)max{𝑁,𝑘}. Then (73) implies that 𝑑𝑔𝑑𝑡²≤(4+3𝐺)²𝜚²𝑔8𝑑⁴⁴+𝛿𝑔²+1𝐹𝑓_𝑡(𝑡)². (74) By the Gronwall lemma, letting 𝑠∈[𝜏−1,𝜏] and 𝑡≥1 and integrating on (𝑠,𝜏), by 𝑔=𝑔_𝑡 we get 𝑔_𝜏𝜏,𝜏−𝑡,𝑔₀(𝜏−𝑡)²≤𝑔_𝜏𝑠,𝜏−𝑡,𝑔₀(𝜏−𝑡)²×𝑒^{((4+3𝐺)2𝜚2/8𝑑)∫𝜏𝑠‖𝑔(𝜉,𝜏−𝑡,𝑔0(𝜏−𝑡))‖44𝑑𝜉+𝛿(𝜏−𝑠)}+1𝐹^𝜏_𝑠𝑓_𝜉(𝜉)²≤𝑔𝑑𝜉_𝜏𝜏,𝜏−𝑡,𝑔₀(𝜏−𝑡)²×𝑒^{((4+3𝐺)2𝜚2/8𝑑)∫𝜏𝑠‖𝑔(𝜉,𝜏−𝑡,𝑔0(𝜏−𝑡))‖44𝑑𝜉+𝛿(𝜏−𝑠)}+1𝐹^𝜏_𝑠𝑓_𝜉(𝜉)²𝑑𝜉. (75) Now integrating (75) with respect to 𝑠 on (𝜏−1,𝜏), we find 𝑔_𝜏𝜏,𝜏−𝑡,𝑔₀(𝜏−𝑡)²≤𝑒^{((4+3𝐺)2𝜚2/8𝑑)𝑒−𝐹𝜏∫𝜏𝜏−1𝑒−𝐹𝜉‖𝑔(𝜉,𝜏−𝑡,𝑔0(𝜏−𝑡))‖44𝑑𝜉+𝛿}×^𝜏_𝜏−1𝑔_𝜏𝑠,𝜏−𝑡,𝑔₀(𝜏−𝑡)²+1𝑑𝑠𝐹^𝜏_𝜏−1𝑓_𝜉(𝜉)²𝑑𝜉≤𝑒^{𝑀𝑒−𝐹𝜏∫𝜏−∞𝑒𝐹𝜉‖𝑓(𝜉)‖4𝕌′𝑑𝜉+𝛿}×^𝜏_𝜏−1𝑔_𝜏𝑠,𝜏−𝑡,𝑔₀(𝜏−𝑡)²+1𝑑𝑠𝐹^𝜏_𝜏−1𝑓_𝜉(𝜉)²𝑑𝜉, (76) which along with Lemma 13 and Proposition 12 shows that there exists 𝑇=(𝜏,𝐷)>2 such that for all 𝑡≥𝑇, 𝑔_𝜏𝜏,𝜏−𝑡,𝑔₀(𝜏−𝑡)²≤𝑀𝑒^−𝐹𝜏𝑒^{𝑀𝑒−𝐹𝜏∫𝜏−∞𝑒𝐹𝜉‖𝑓(𝜉)‖4𝕌′𝑑𝜉}×^𝜏_−∞𝑒^𝐹𝜉𝑓(𝜉)²𝑑𝜉+^𝜏_−∞𝑒^𝐹𝜉𝑓(𝜉)⁶_𝕍′𝑑𝜉+𝑀^𝜏_𝜏−1𝑓_𝜉(𝜉)²𝑑𝜉, (77) which completes the proof.

We now establish uniform estimates on the tails of solutions when 𝑡→∞. 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∈𝒟_𝐹, there exist 𝑇=𝑇(𝜏,𝐷,𝜀)>2 and 𝐾=𝐾(𝜏,𝜀)>0 such that for all 𝑡≥𝑇 and 𝑁≥𝐾, _{|𝑥|≥𝐾}𝑔𝑥,𝜏,𝜏−𝑡,𝑔₀(𝜏−𝑡)²𝑑𝑥≤𝜀, (78) 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 satisfying 0≤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(ℝ^𝑛), respectively, we get 𝐺2𝑑𝑑𝑡_ℝ^𝑛𝜌|𝑥|²𝐾²|𝑢|²−𝑑₁𝐺_ℝ𝑛𝜌|𝑥|²𝐾²()𝑢Δ𝑢+𝐺𝐹+𝑘_ℝ𝑛𝜌|𝑥|²𝐾²|𝑢|²=𝐺_ℝ𝑛𝜌|𝑥|²𝐾²𝑢³𝑣𝑑𝑥−𝐺²_ℝ𝑛𝜌|𝑥|²𝐾²𝑢⁴𝑑𝑥+𝑘𝐺_ℝ𝑛𝜌|𝑥|²𝐾²𝑢𝑊𝑑𝑥+𝐺_ℝ𝑛𝜌|𝑥|²𝐾²𝑓₁1𝑢𝑑𝑥,2𝑑𝑑𝑡_ℝ^𝑛𝜌|𝑥|²𝐾²|𝑣|²−𝑑_2ℝ𝑛𝜌|𝑥|²𝐾²𝑣Δ𝑣𝑑𝑥+𝐹_ℝ𝑛𝜌|𝑥|²𝐾²|𝑣|²𝑑𝑥=−_ℝ𝑛𝜌|𝑥|²𝐾²𝑢²𝑣²𝑑𝑥+𝐺_ℝ𝑛𝜌|𝑥|²𝐾²𝑣𝑢³+𝑑𝑥_ℝ𝑛𝜌|𝑥|²𝐾²𝑓₂𝑣𝑑𝑥,𝜇𝐺2𝑑𝑑𝑡_ℝ^𝑛𝜌|𝑥|²𝐾²|𝑊|²−𝜇𝐺𝑑_3ℝ𝑛𝜌|𝑥|²𝐾²𝑊Δ𝑊+𝐺𝜇𝐹+𝑘_ℝ𝑛𝜌|𝑥|²𝐾²|𝑊|²=𝑘𝐺_ℝ𝑛𝜌|𝑥|²𝐾²𝑊𝑢𝑑𝑥+𝐺_ℝ𝑛𝜌|𝑥|²𝐾²𝑊𝑓₃𝑑𝑥. (79) Add up the three equalities. Then we have 𝑑𝐺𝑑𝑡2_ℝ^𝑛𝜌|𝑥|²𝐾²|𝑢|²+12_ℝ^𝑛𝜌|𝑥|²𝐾²|𝑣|²+𝜇𝐺2_ℝ^𝑛𝜌|𝑥|²𝐾²|𝑊|²−𝑑₁𝐺_ℝ𝑛𝜌|𝑥|²𝐾²𝑢Δ𝑢+𝑑_2ℝ𝑛𝜌|𝑥|²𝐾²𝑣Δ𝑣+𝜇𝐺𝑑_3ℝ𝑛𝜌|𝑥|²𝐾²()𝑊Δ𝑊+𝐺𝐹+𝑘_ℝ𝑛𝜌|𝑥|²𝐾²|𝑢|²+𝐹_ℝ𝑛𝜌|𝑥|²𝐾²|𝑣|²𝑑𝑥+𝐺𝜇𝐹+𝑘_ℝ𝑛𝜌|𝑥|²𝐾²|𝑊|²≤−_ℝ𝑛𝜌|𝑥|²𝐾²𝐺𝑢²−𝑢𝑣²𝑑𝑥+2𝑘𝐺_ℝ𝑛𝜌|𝑥|²𝐾²𝑢𝑊𝑑𝑥+𝐺_ℝ𝑛𝜌|𝑥|²𝐾²𝑓₁+𝑢𝑑𝑥_ℝ𝑛𝜌|𝑥|²𝐾²𝑓₂𝑣𝑑𝑥+𝐺_ℝ𝑛𝜌|𝑥|²𝐾²𝑊𝑓₃𝑑𝑥≤𝐺𝑘_ℝ𝑛𝜌|𝑥|²𝐾²|𝑢|²+𝐺𝑘_ℝ𝑛𝜌|𝑥|²𝐾²|𝑊|²+𝐺𝐹2_ℝ^𝑛𝜌|𝑥|²𝐾²|𝑢|²+𝐹2_ℝ^𝑛𝜌|𝑥|²𝐾²|𝑣|²+𝑑𝑥𝜇𝐺𝐹2_ℝ^𝑛𝜌|𝑥|²𝐾²|𝑊|²+𝐺𝑑𝑥2𝐹_{|𝑥|≥𝐾}𝑓₁(𝑥,𝑡)²𝐺𝑑𝑥+2𝜇𝐹_{|𝑥|≥𝐾}𝑓₃(𝑥,𝑡)²+1𝑑𝑥2𝐹_{|𝑥|≥𝐾}𝑓₂(𝑥,𝑡)²𝑑𝑥. (80) That is, 𝑑𝐺𝑑𝑡_ℝ^𝑛𝜌|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≤2𝑐𝐾𝑑₁𝐺‖𝑢‖²+‖∇𝑢‖²+𝑑₂‖𝑣‖²+‖∇𝑣‖²+𝑑₃𝐺𝜇‖𝑊‖²+‖∇𝑊‖²+𝐺𝐹_{|𝑥|≥𝐾}𝑓₁(𝑥,𝑡)²+𝐺𝑑𝑥𝜇𝐹_{|𝑥|≥𝐾}𝑓₃(𝑥,𝑡)²1𝑑𝑥+𝐹_{|𝑥|≥𝐾}𝑓₂(𝑥,𝑡)²𝑑𝑥. (81) Here, we use the integration by parts. We have _ℝ𝑛𝜌|𝑥|²𝐾²𝑐𝑢Δ𝑢𝑑𝑥≤𝐾‖𝑢‖²+‖∇𝑢‖². (82) Denote 𝑑^o=max{𝑑₁,𝑑₂,𝑑₃}, then we can deduce from (81) that 𝑑𝑑𝑡_ℝ^𝑛𝜌|𝑥|²𝐾²𝑔²𝑑𝑥+𝐹_ℝ𝑛𝜌|𝑥|²𝐾²𝑔²≤2𝑐𝑑^o𝐾𝑔²+∇𝑔²+𝐶_{1|𝑥|≥𝐾}𝑓(𝑥,𝑡)²𝑑𝑥. (83) Multiplying (83) by 𝑒^𝐹𝑡 and then integrating over (𝜏−𝑡,𝜏) with 𝑡≥0, we get _ℝ𝑛𝜌|𝑥|²𝐾²𝑔𝑥,𝜏,𝜏−𝑡,𝑔₀(𝜏−𝑡)²𝑑𝑥≤𝑒^−𝐹𝜏𝑒^{𝐹(𝜏−𝑡)}_ℝ𝑛𝜌|𝑥|²𝐾²𝑔₀(𝑥,𝜏−𝑡)²+𝑑𝑥2𝑐𝑑^o𝐾𝑒^{−𝐹𝜏𝜏}_𝜏−𝑡𝑒^𝐹𝜉𝑔𝜉,𝜏−𝑡,𝑔₀(𝜏−𝑡)²+∇𝑔𝜉,𝜏−𝑡,𝑔₀(𝜏−𝑡)²𝑑𝜉+𝐶₁𝑒^{−𝐹𝜏𝜏}_{𝜏−𝑡|𝑥|≥𝐾}𝑓(𝑥,𝜉)²𝑑𝑥𝑑𝜉≤𝑒^−𝐹𝜏𝑒^{𝐹(𝜏−𝑡)}_ℝ𝑛𝑔₀(𝑥,𝜏−𝑡)²𝑑𝑥+𝐶₁𝑒^{−𝐹𝜏𝜏}_{−∞|𝑥|≥𝐾}𝑓(𝑥,𝜉)²+𝑑𝑥𝑑𝜉2𝑐𝑑^o𝐾𝑒^{−𝐹𝜏𝜏}_𝜏−𝑡𝑒^𝐹𝜉𝑔𝜉,𝜏−𝑡,𝑔₀(𝜏−𝑡)²+∇𝑔𝜉,𝜏−𝑡,𝑔₀(𝜏−𝑡)²𝑑𝜉. (84) Note that for given 𝜀>0, there is 𝑇₁=𝑇₁(𝜏,𝐷,𝜀)>0 such that for all 𝑡≥𝑇₁, 𝑒^−𝐹𝜏𝑒^{𝐹(𝜏−𝑡)}_ℝ𝑛𝑔₀(𝑥,𝜏−𝑡)²𝑑𝑥≤𝜀. (85) By (22), there is 𝐾₁=𝐾₁(𝜏,𝜀)>0 such that for all 𝑁≥𝐾₁, 𝐶₁𝑒^{−𝐹𝜏𝜏}_{−∞|𝑥|≥𝐾}𝑓(𝑥,𝜉)²𝑑𝑥𝑑𝜉≤𝜀. (86) 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, 2𝑐𝑑^o𝐾𝑒^{−𝐹𝜏𝜏}_𝜏−𝑡𝑒^𝐹𝜉𝑔𝜉,𝜏−𝑡,𝑔₀(𝜏−𝑡)²+∇𝑔𝜉,𝜏−𝑡,𝑔₀(𝜏−𝑡)²≤𝑑𝜉2𝑐𝑑^o𝐾𝑀𝑒^{−𝐹𝜏𝜏}_−∞𝑒^𝐹𝜉𝑓(𝜉)²𝑑𝜉. (87) Therefore, there is 𝐾₂=𝐾₂(𝜏,𝜀)>𝐾₁ such that for all 𝐾≥𝐾₂ and 𝑡≥𝑇₂, 2𝑐𝑑^𝑜𝐾𝑒^{−𝐹𝜏𝜏}_𝜏−𝑡𝑒^𝐹𝜉𝑔𝜉,𝜏−𝑡,𝑔₀(𝜏−𝑡)²+∇𝑔𝜉,𝜏−𝑡,𝑔₀(𝜏−𝑡)²𝑑𝜉≤𝜀. (88) Let 𝑇=max{𝑇₁,𝑇₂} and then by (85)–(88), we find that, for all 𝐾≥𝐾₂ and 𝑡≥𝑇, _ℝ𝑛𝜌|𝑥|²𝐾²𝑔𝑥,𝜏,𝜏−𝑡,𝑔₀(𝜏−𝑡)²𝑑𝑥≤3𝜀, (89) and hence for all 𝐾≥𝐾₂ and 𝑡≥𝑇, _{|𝑥|≥√2𝑁}𝜌|𝑥|²𝐾²𝑔𝑥,𝜏,𝜏−𝑡,𝑔₀(𝜏−𝑡)²≤𝑑𝑥_ℝ𝑛𝜌|𝑥|²𝐾²𝑔𝑥,𝜏,𝜏−𝑡,𝑔₀(𝜏−𝑡)²𝑑𝑥≤3𝜀, (90) which completes the proof.

5. Existence of Pullback Attractors

In this section, we prove the existence of a 𝒟_𝐹-pullback global attractor for the nonautonomous three-component reversible Gray-Scott system on ℝ^𝑛. We first establish the 𝒟_𝐹-pullback asymptotic compactness of solutions and prove the existence of a pullback attractor in ℍ. Then we show that this attractor is actually a 𝒟_𝐹-pullback attractor in 𝔼.

Lemma 16. Suppose that (19) and (20) hold. Then 𝜙 is 𝒟_𝐹-pullback asymptotically compact in ℍ. That is, for every 𝜏∈ℝ, 𝐷={𝐷(𝑡)}_𝑡∈ℝ∈𝒟_𝐹, 𝑡_𝑛→∞, and 𝑔_0,𝑛∈𝐷(𝜏−𝑡_𝑛), the sequence 𝜙(𝑡_𝑛,𝜏−𝑡_𝑛,𝑔_0,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 𝒟_𝐹-pullback global attractor {𝒜(𝜏)}_𝜏∈ℝ∈𝒟_𝐹 in ℍ. That is, for every 𝜏∈ℝ,(i)𝒜(𝜏) is compact in ℍ,(ii){𝒜(𝜏)}_𝜏∈ℝ is invariant. That is, 𝜙(𝑡,𝜏,𝒜(𝜏))=𝒜(𝑡+𝜏),∀𝑡≥0, (91)(iii){𝒜(𝜏)}_𝜏∈ℝ attracts every set in 𝒟_𝐹 with respect to the norm of ℍ. That is, for every 𝐵={𝐵(𝜏)}_𝜏∈ℝ∈𝒟_𝐹, lim_𝑡→∞𝑑_𝐻𝜙(𝑡,𝜏−𝑡,𝐵(𝜏−𝑡),𝒜(𝜏))=0, (92)where for any 𝑋,𝑌⊆ℍ, 𝑑_ℍ(𝑋,𝑌)=sup_𝑥∈𝑋inf_𝑦∈𝑌‖𝑥−𝑦‖_ℍ.

Proof. For 𝜏∈ℝ, denote 𝐵(𝜏)=𝑔∶𝑔²≤𝑀𝑒^{−𝐹𝜏𝜏}_−∞𝑒^𝐹𝜉𝑓(𝜉)²𝑑𝜉. (93) Note that 𝐵={𝐵(𝜏)}_𝜏∈ℝ∈𝒟_𝐹 is a 𝒟_𝐹-pullback absorbing for 𝜙 in ℍ by Lemma 8. In addition, 𝜙 is 𝒟_𝐹-pullback asymptotically compact by Lemma 16. Thus, the existence of a 𝒟_𝐹-pullback global attractor for 𝜙 in ℍ follows from Theorem 7.

In what follows, we strengthen Theorem 17 and show that the global attractor {𝒜(𝜏)}_𝜏∈ℝ is actually a 𝒟_𝐹-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 𝑑𝑓∈𝐿𝑑𝑡^2locℝ,𝐿²ℝ^𝑛3. (94) Then 𝜙 is 𝒟_𝐹-pullback asymptotically compact in 𝔼. That is, for every 𝜏∈ℝ, 𝐷={𝐷(𝑡)}_𝑡∈ℝ∈𝒟_𝐹, 𝑡_𝑛→∞, and 𝑔_0,𝑛∈𝐷(𝜏−𝑡_𝑛), the sequence 𝜙(𝑡_𝑛,𝜏−𝑡_𝑛,𝑔_0,𝑛) has a convergent subsequence in 𝔼.

Proof. By Lemma 16, the sequence 𝜙(𝑡_𝑛,𝜏−𝑡_𝑛,𝑔_0,𝑛)=𝑔(𝜏,𝜏−𝑡,𝑔_0,𝑛) has a convergent subsequence in ℍ, and hence there exists 𝑔∈ℍ such that, up to a subsequence, 𝑔𝜏,𝜏−𝑡,𝑔_0,𝑛⟶𝑔inℍ. (95) This shows that 𝜙(𝑡_𝑛,𝜏−𝑡_𝑛,𝑔_0,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 𝑔_𝜏𝜏,𝜏−𝑡_𝑛,𝑔_0,𝑛−𝑔_𝜏𝜏,𝜏−𝑡_𝑚,𝑔_0,𝑚=𝐴𝑔𝜏,𝜏−𝑡_𝑛,𝑔_0,𝑛−𝑔𝜏,𝜏−𝑡_𝑚,𝑔_0,𝑚−𝐻𝑔𝜏,𝜏−𝑡_𝑛,𝑔_0,𝑛−𝑔𝜏,𝜏−𝑡_𝑚,𝑔_0,𝑚. (96) That is, 𝜕𝑢_𝑛−𝑢_𝑚𝜕𝜏=𝑑₁Δ𝑢_𝑛−𝑢_𝑚−(𝐹+𝑘)𝑢_𝑛−𝑢_𝑚+𝑢²_𝑛𝑣_𝑛−𝑢²_𝑚𝑣_𝑚−𝐺𝑢³_𝑛−𝑢³_𝑚+𝑁𝑤_𝑛−𝑤_𝑚,𝜕𝑣_𝑛−𝑣_𝑚𝜕𝜏=𝑑₂Δ𝑣_𝑛−𝑣_𝑚−𝐹𝑣_𝑛−𝑣_𝑚−𝑢²_𝑛𝑣_𝑛−𝑢²_𝑚𝑣_𝑚+𝐺𝑢³_𝑛−𝑢³_𝑚,𝜕𝑤_𝑛−𝑤_𝑚𝜕𝜏=𝑑₃Δ𝑤_𝑛−𝑤_𝑚+𝑘𝑢_𝑛−𝑢_𝑚−(𝐹+𝑁)𝑤_𝑛−𝑤_𝑚, (97) where 𝑢_𝑛, 𝑢_𝑚 denote 𝑢(𝜏,𝜏−𝑡_𝑛,𝑢_0,𝑛), 𝑢(𝜏,𝜏−𝑡_𝑚,𝑢_0,𝑚) and so do as 𝑣_𝑛, 𝑣_𝑚, 𝑤_𝑛, 𝑤_𝑚, and 𝑔_𝑛, 𝑔_𝑚. Taking the inner products (𝜕(𝑢_𝑛−𝑢_𝑚)/𝜕𝜏,𝑢_𝑛−𝑢_𝑚), (𝜕(𝑣_𝑛−𝑣_𝑚)/𝜕𝜏,𝑣_𝑛−𝑣_𝑚) and (𝜕(𝑤_𝑛−𝑤_𝑚)/𝜕𝜏,𝑤_𝑛−𝑤_𝑚), respectively, and summing up the three equalities, we get 𝑑∇𝑢_𝑛−𝑢_𝑚²+∇𝑣_𝑛−𝑣_𝑚²+∇𝑤_𝑛−𝑤_𝑚²+(𝐹+𝑘)𝑢_𝑛−𝑢_𝑚²+𝐹𝑣_𝑛−𝑣_𝑚²+(𝑁+𝑘)𝑤_𝑛−𝑤_𝑚²≤(𝑁+𝑘)𝑤_𝑛−𝑤_𝑚,𝑢_𝑛−𝑢_𝑚+𝑢²_𝑛𝑣_𝑛−𝑢²_𝑚𝑣_𝑚,𝑢_𝑛−𝑢_𝑚−𝑣_𝑛+𝑣_𝑚−𝐺𝑢³_𝑛−𝑢³_𝑚,𝑢_𝑛−𝑢_𝑚−𝑣_𝑛+𝑣_𝑚−𝜕𝑢𝜕𝜏^𝑛−𝑢_𝑚+𝑣_𝑛−𝑣_𝑚+𝑤_𝑛−𝑤_𝑚×𝑢_𝑛−𝑢_𝑚+𝑣_𝑛−𝑣_𝑚+𝑤_𝑛−𝑤_𝑚. (98) Because of the Hölder inequality and Gagliardo-Nirenberg inequality, we have 𝑢²_𝑛𝑣_𝑛−𝑢²_𝑚𝑣_𝑚,𝑢_𝑛−𝑢_𝑚−𝑣_𝑛+𝑣_𝑚−𝐺𝑢³_𝑛−𝑢³_𝑚,𝑢_𝑛−𝑢_𝑚−𝑣_𝑛+𝑣_𝑚≤𝑑2∇𝑔_𝑛−𝑔_𝑚²+𝐶²𝜚²𝑔2𝑑^𝑛2+𝑔_𝑛⁴₄+𝑔_𝑛⁶₆𝑔_𝑛−𝑔_𝑚². (99) From (98) and (99), it yields 𝑑∇𝑢_𝑛−𝑢_𝑚+𝑣_𝑛−𝑣_𝑚+𝑤_𝑛−𝑤_𝑚²+2𝐹𝑢_𝑛−𝑢_𝑚+𝑣_𝑛−𝑣_𝑚+𝑤_𝑛−𝑤_𝑚²𝜕≤−2𝑢𝜕𝜏^𝑛−𝑢_𝑚+𝑣_𝑛−𝑣_𝑚+𝑤_𝑛−𝑤_𝑚×𝑢_𝑛−𝑢_𝑚+𝑣_𝑛−𝑣_𝑚+𝑤_𝑛−𝑤_𝑚+𝐶²𝜚²𝑑𝑔_𝑛²+𝑔_𝑛⁴₄+𝑔_𝑛⁶₆×𝑢+2𝛿_𝑛−𝑢_𝑚+𝑣_𝑛−𝑣_𝑚+𝑤_𝑛−𝑤_𝑚². (100) By Lemma 14 we find that, for every 𝜏∈ℝ, there exists 𝑇=𝑇(𝜏,𝐷) such that for all 𝑡≥𝑇, 𝑔_𝜏𝜏,𝜏−𝑡,𝑔₀(𝜏−𝑡)≤𝑀. (101) Since 𝑡_𝑛→∞, there exists 𝑁=𝑁(𝜏,𝐷) such that 𝑡_𝑛≥𝑇 for all 𝑛≥𝑇. Thus, we obtain that, for all 𝑛≥𝑁, 𝑔_𝜏𝜏,𝜏−𝑡,𝑔_0,𝑛≤𝑀, (102) which along with (100) shows that, for all 𝑛,𝑚≥𝑁, 𝑑∇𝑢_𝑛−𝑢_𝑚+𝑣_𝑛−𝑣_𝑚+𝑤_𝑛−𝑤_𝑚²+2𝐹𝑢_𝑛−𝑢_𝑚+𝑣_𝑛−𝑣_𝑚+𝑤_𝑛−𝑤_𝑚²≤2𝑀𝑢_𝑛−𝑢_𝑚+𝑣_𝑛−𝑣_𝑚+𝑤_𝑛−𝑤_𝑚+𝐶²𝜚²𝑑𝑔_𝑛²+𝑔_𝑛⁴₄+𝑔_𝑛⁶₆×𝑢+2𝛿_𝑛−𝑢_𝑚+𝑣_𝑛−𝑣_𝑚+𝑤_𝑛−𝑤_𝑚². (103) That is, ∇𝑔𝜏,𝜏−𝑡,𝑔_0,𝑛−𝑔𝜏,𝜏−𝑡,𝑔_0,𝑚²≤2𝑀𝑔𝜏,𝜏−𝑡,𝑔_0,𝑛−𝑔𝜏,𝜏−𝑡,𝑔_0,𝑚+𝐶²𝜚²𝑑𝑔_𝑛²+𝑔_𝑛⁴₄+𝑔_𝑛⁶₆×𝑔+2𝛿𝜏,𝜏−𝑡,𝑔_0,𝑛−𝑔𝜏,𝜏−𝑡,𝑔_0,𝑚². (104) By Lemma 8 and Propositions 11 and 12, and combining with the fact that 𝑔(𝜏,𝜏−𝑡,𝑔_0,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 𝑑𝑓∈𝐿𝑑𝑡^2locℝ,𝐿²ℝ^𝑛3. (105) Then problem (9)-(10) has a unique 𝒟_𝐹-pullback global attractor {𝒜(𝜏)}_𝜏∈ℝ∈𝒟_𝐹 in 𝔼. That is, for every 𝜏∈ℝ,(i)𝒜(𝜏) is compact in 𝔼,(ii){𝒜(𝜏)}_𝜏∈ℝ is invariant. That is, 𝜙(𝑡,𝜏,𝒜(𝜏))=𝒜(𝑡+𝜏),∀𝑡≥0, (106)(iii){𝒜(𝜏)}_𝜏∈ℝ attracts every set in 𝒟_𝐹 with respect to the norm of 𝔼. That is, for every 𝐵={𝐵(𝜏)}_𝜏∈ℝ∈𝒟_𝐹, lim_𝑡→∞𝑑_𝔼𝜙(𝑡,𝜏−𝑡,𝐵(𝜏−𝑡),𝒜(𝜏))=0, (107)where, for any 𝑋,𝑌⊆ℍ, 𝑑_ℍ(𝑋,𝑌)=sup_𝑥∈𝑋inf_𝑦∈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]).

Acknowledgments

The author would like to thank the anonymous referees for the very helpful comments and suggestions which largely improve the quality of the paper. Also more thanks are due to Professor Yuncheng You and Professor Shengfan Zhou for their helpful conversation and suggestions in their preparing for the paper. This work has been partially supported by NSFC Grants 11071165 and 11161015, NSF of Guangxi Grants 2013GXNSFBA019008, Guangxi Provincial Department of Research Project Grants 2013YB102, and the Program to Sponsor Teams for Innovation in the Construction of Talent Highlands in Guangxi Institutions of Higher Learning Grants [2011]47.