Asymptotic Behavior of a Competition-Diffusion System with Variable Coefficients and Time Delays

Abstract

A class of time-delay reaction-diffusion systems with variable coefficients which arise from the model of two competing ecological species is discussed. An asymptotic global attractor is established in terms of the variable coefficients, independent of the time delays and the effect of diffusion by the upper-lower solutions and iteration method.

1. Introduction and Main Result

The Lotka-Volterra competition model with diffusion and time delays has been the object of analysis by numerous authors under different approaches. For the case of two species, Ruan and Zhao [1] considered uniform persistence and global extinction; Lu [2] studied global attractivity, and Gourley and Ruan [3] analyzed stability and traveling fronts. The periodic case has also been considered, Feng and Wang [4] studied asymptotic stability and Zhou et al. [5] investigated the Hopf bifurcation. The cases of three and 𝑁-species have also been analyzed in [6–8].

In this paper, we consider the asymptotic behavior of solutions for the competition-diffusion system with time delays of the following two species: 𝜕𝑢₁𝜕𝑡=𝐴Δ𝑢₁(𝑡,𝑥)+𝑢₁(𝑡,𝑥)[𝑎₁(𝑡,𝑥)−𝑏₁(𝑡,𝑥)𝑢₁(𝑡,𝑥)−𝑐₁(𝑡,𝑥)^∞₀𝑢₁(𝑡−𝜏,𝑥)𝑑𝜇₁(𝜏)−𝑑₁𝑢₂(𝑡−𝑟₂,𝑥)],𝑡>0,𝑥∈Ω,𝜕𝑢₂𝜕𝑡=𝐴Δ𝑢₂(𝑡,𝑥)+𝑢₂(𝑡,𝑥)[𝑎₂(𝑡,𝑥)−𝑏₂(𝑡,𝑥)𝑢₂(𝑡,𝑥)−𝑐₂(𝑡,𝑥)^∞₀𝑢₂(𝑡−𝜏,𝑥)𝑑𝜇₂(𝜏)−𝑑₂𝑢₁(𝑡−𝑟₁,𝑥)],𝑡>0,𝑥∈Ω,𝜕𝑢₁=𝜕𝜂𝜕𝑢₂𝑢𝜕𝜂=0,𝑡>0,𝑥∈𝜕Ω,_𝑖(𝑡,𝑥)=𝜙_𝑖(𝑡,𝑥)(𝑖=1,2)𝑡≤0,𝑥∈Ω, (1.1) where Ω⊆ℝ^𝑁 (𝑁≥1) is a bounded domain with smooth boundary 𝜕Ω, 𝜕/𝜕𝜂 denotes the differentiation outward normal on 𝜕Ω, 𝐴≥0, 𝑑_𝑖>0, 𝑟_𝑖>0, 0<𝑎_𝑖≤𝑎_𝑖(𝑡,𝑥)≤𝐴_𝑖, 0<𝑏_𝑖≤𝑏_𝑖(𝑡,𝑥)≤𝐵_𝑖, 0≤𝑐_𝑖≤𝑐_𝑖(𝑡,𝑥)≤𝐶_𝑖, 𝜇_𝑖(⋅) is of bounded variation with 𝜇_𝑖(0)=0, 𝜙_𝑖∈𝐶¹((−∞,0])×Ω is bounded and nonnegative and 0≢𝜙_𝑖(0,⋅)∈𝐶¹(Ω) (𝑖=1,2), and 𝑢₁ and 𝑢₂ are the density functions of two species competing for a shared limited resource.

The functions 𝑎₁(𝑥,𝑡) and 𝑎₂(𝑥,𝑡) denote the intrinsic growth rate of the species, 𝑏₁(𝑥,𝑡) and 𝑏₂(𝑥,𝑡) represent self-limitation rates, and 𝑐₁(𝑥,𝑡) and 𝑐₂(𝑥,𝑡) represent the coefficients of the infinite continuous delay. The constants 𝑑₁, 𝑑₂ represent the competition rates. The distributed time delay should be viewed as the effects of past history.

Let 𝑀_𝑖(𝑡) denote the total variation of 𝜇_𝑖(⋅) on [0,𝑡] and let 𝑀^±_𝑖(𝑡)=(𝑀_𝑖(𝑡)±𝜇_𝑖(𝑡))/2 , for all 𝑡∈ℝ⁺. Then, 𝑀_𝑖(𝑡) and 𝑀^±_𝑖(𝑡) are nonnegative and nondecreasing on ℝ⁺. It easy to see that 𝑀⁺_𝑖(𝑡)+𝑀⁻_𝑖(𝑡)=𝑀_𝑖(𝑡),𝑀⁺_𝑖(𝑡)−𝑀⁻_𝑖(𝑡)=𝜇_𝑖(𝑡). (1.2)Denote 𝑀_0𝑖=lim_𝑡→∞𝑀_𝑖(𝑡), 𝑀^±_0𝑖=lim_𝑡→∞𝑀^±_𝑖(𝑡), and 𝜇_0𝑖=lim_𝑡→∞𝜇_𝑖(𝑡). Then, 𝑀⁺_0𝑖+𝑀⁻_0𝑖=𝑀_0𝑖,𝑀⁺_0𝑖−𝑀⁻_0𝑖=𝜇_0𝑖. (1.3)Our result can be stated as follows.

Theorem 1.1. Assume that 𝑏_𝑖>𝑐_𝑖𝑀⁻_0𝑖,𝑎_𝑖−(𝐶_𝑖𝑀⁺_0𝑖𝐴_𝑖)/(𝑏_𝑖−𝑐_𝑖𝑀⁻_0𝑖)>0, and𝑑₂𝐴₂(𝑏₂−𝑐₂𝑀⁻₀₂)(𝑏₁−𝑐₁𝑀⁻₀₁)[𝑎₂(𝑏₂−𝑐₂𝑀⁻₀₂)−𝐶₂𝑀⁺₀₂𝐴₂]<𝐴₂𝐴₁<(𝑏₂−𝑐₂𝑀⁻₀₂)[𝑎₁(𝑏₁−𝑐₁𝑀⁻₀₁)−𝐶₁𝑀⁺₀₁𝐴₁]𝑑₁𝐴₁(𝑏₁−𝑐₁𝑀⁻₀₁). (1.4) Then, for any 𝜙_𝑖∈𝐶¹((−∞,0])×Ω with 𝜙(0,𝑥)≢0, the solution of (1.1) satisfies0<𝛼₁≤liminf_𝑡→∞min_𝑥∈Ω𝑢₁(𝑡,𝑥)≤limsup_𝑡→∞max_𝑥∈Ω𝑢₁(𝑡,𝑥)≤𝛽₁,0<𝛼₂≤liminf_𝑡→∞min_𝑥∈Ω𝑢₂(𝑡,𝑥)≤limsup_𝑡→∞max_𝑥∈Ω𝑢₂(𝑡,𝑥)≤𝛽₂, (1.5) where 𝛼₁, 𝛼₂, 𝛽₁, 𝛽₂ are constants given by the linear system
𝑎₁−𝐶₁𝑀⁺₀₁𝛽₁−𝑑₁𝛽₂−𝛼₁(𝐵₁−𝑐₁𝑀⁻₀₁𝑎)=0,₂−𝐶₂𝑀⁺₀₂𝛽₂−𝑑₂𝛽₁−𝛼₂(𝐵₂−𝑐₂𝑀⁻₀₂𝐴)=0,₁−𝑐₁𝑀⁺₀₁𝛼₁−𝑑₁𝛼₂−𝛽₁(𝑏₁−𝑐₁𝑀⁻₀₁𝐴)=0,₂−𝑐₂𝑀⁺₀₂𝛼₂−𝑑₂𝛼₁−𝛽₂(𝑏₂−𝑐₂𝑀⁻₀₂)=0. (1.6)

Remark 1.2. If 𝜇_𝑖(𝜏)=∫^𝜏₀𝑓_𝑖(𝑡)𝑑𝑡 with 𝑓_𝑖∈𝐶(ℝ⁺)∩𝐿¹(ℝ⁺), 𝐴=1, and 0<𝑎_𝑖=𝑎_𝑖(𝑡,𝑥)=𝐴_𝑖, 0<𝑏_𝑖=𝑏_𝑖(𝑡,𝑥)=𝐵_𝑖, 𝑐_𝑖=𝑐_𝑖(𝑡,𝑥)=𝐶_𝑖=1, then by Theorem 1.1, we get0<𝛼₁≤liminf_𝑡→∞min_𝑥∈Ω𝑢₁(𝑡,𝑥)≤limsup_𝑡→∞max_𝑥∈Ω𝑢₁(𝑡,𝑥)≤𝛽₁,0<𝛼₂≤liminf_𝑡→∞min_𝑥∈Ω𝑢₂(𝑡,𝑥)≤limsup_𝑡→∞max_𝑥∈Ω𝑢₂(𝑡,𝑥)≤𝛽₂, (1.7)where 𝛼₁, 𝛼₂, 𝛽₁, 𝛽₂ are constants given by the linear system
𝑎₁−𝑀⁺₀₁𝛽₁−𝑑₁𝛽₂−𝛼₁(𝑏₁−𝑀⁻₀₁𝑎)=0,₂−𝑀⁺₀₂𝛽₂−𝑑₂𝛽₁−𝛼₂(𝑏₂−𝑀⁻₀₂𝑎)=0,₁−𝑀⁺₀₁𝛼₁−𝑑₁𝛼₂−𝛽₁(𝑏₁−𝑀⁻₀₁𝑎)=0,₂−𝑀⁺₀₂𝛼₂−𝑑₂𝛼₁−𝛽₂(𝑏₂−𝑀⁻₀₂)=0. (1.8) Solving this system, we have the following result:
𝛼₁=𝛽₁=𝑎₁(𝑏₂+𝑀⁺₀₂−𝑀⁻₀₂)−𝑎₂𝑑₁(𝑏₁+𝑀⁺₀₁−𝑀⁻₀₁)(𝑏₂+𝑀⁺₀₂−𝑀⁻₀₂)−𝑑₁𝑑₂,𝛼₂=𝛽₂=𝑎₂(𝑏₁+𝑀⁺₀₁−𝑀⁻₀₁)−𝑎₁𝑑₂(𝑏₁+𝑀⁺₀₁−𝑀⁻₀₁)(𝑏₂+𝑀⁺₀₂−𝑀⁻₀₂)−𝑑₁𝑑₂, (1.9)which coincides with the result of [9], where the authors considered a system like (1.1) with constant coefficients.

Reaction-diffusion systems with delay have been treated by many authors. There are two ways to approach them. The first one is in the framework of semigroup theory of dynamical systems [1, 10]. The second one is a method of upper and lower solutions, using associated monotone iterations; several authors have studied their dynamic properties [2, 9, 11]. Sometimes the birth and death rates depend on both space and time, so when we consider instantaneous and delayed interference within the species and the diffusive effects of the species, system (1.1) will be the appropriate model.

The way we organize the paper is as follows: we first introduce several results which play an important role in the proof of Theorem 1.1 which we will prove in Section 2. We will provide some numerical simulations in Section 3 in order to illustrate our theory.

The following results are developed in [11]. They considered the Volterra reaction-diffusion equations with variable coefficients: 𝜕𝑢𝜕𝑡=𝐴Δ𝑢(𝑡,𝑥)+𝑢(𝑡,𝑥)[𝑎(𝑡,𝑥)−𝑏(𝑡,𝑥)𝑢(𝑡,𝑥)−𝑐(𝑡,𝑥)^∞₀𝑢(𝑡−𝜏,𝑥)𝑑𝜇(𝜏)],𝑡>0,𝑥∈Ω,𝜕𝑢𝜕𝜂=0,𝑡>0,𝑥∈𝜕Ω,𝑢(𝑡,𝑥)=𝜙(𝑡,𝑥),𝑡≤0,𝑥∈Ω, (1.10) where 0<𝑎₁≤𝑎(𝑡,𝑥)≤𝐴₁, 0<𝑏₁≤𝑏(𝑡,𝑥)≤𝐵₁, 0≤𝑐₁≤𝑐(𝑡,𝑥)≤𝐶₁, and 𝜙∈𝐶¹((−∞,0])×Ω is bounded and nonnegative and 0≢𝜙(0,⋅)∈𝐶¹(Ω). Let 𝑀(𝑡) denote the total variation of 𝜇(⋅) and define 𝑀^±, 𝑀^±₀ of the same form as that of 𝑀^±_𝑖, 𝑀^±_0𝑖, respectively.

Lemma 1.3. Assume that 𝑏₁>𝑐₁𝑀⁻₀ and 𝑎₁−(𝐶₁𝑀⁺₀𝐴₁)/(𝑏₁−𝑐₁𝑀⁻₀)>0, then the solution of (1.10) satisfies 𝛼≤liminf_𝑡→∞min_𝑥∈Ω𝑢(𝑡,𝑥)≤limsup_𝑡→∞max_𝑥∈Ω𝑢(𝑡,𝑥)≤𝛽, (1.11) where 𝛼=(𝑏₁−𝑐₁𝑀⁻₀)𝑎₁−𝐶₁𝑀⁺₀𝐴₁(𝑏₁−𝑐₁𝑀⁻₀)(𝐵₁−𝑐₁𝑀⁻₀)−𝐶²₁(𝑀⁺₀)²,𝛽=(𝐵₁−𝑐₁𝑀⁻₀)𝐴₁−𝐶₁𝑀⁺₀𝑎₁(𝑏₁−𝑐₁𝑀⁻₀)(𝐵₁−𝑐₁𝑀⁻₀)−𝐶²₁(𝑀⁺₀)². (1.12) If 𝑀⁺(𝑡)=0, then 𝜇(𝑡)=−𝑀⁻(𝑡). Thus, 𝜕𝑢𝜕𝑡=𝐴Δ𝑢(𝑡,𝑥)+𝑢(𝑡,𝑥)[𝑎(𝑡,𝑥)−𝑏(𝑡,𝑥)𝑢(𝑡,𝑥)+𝑐(𝑡,𝑥)^∞₀𝑢(𝑡−𝜏,𝑥)𝑑𝑀⁻(𝜏)],𝑡>0,𝑥∈Ω,𝜕𝑢𝜕𝜂=0,𝑡>0,𝑥∈𝜕Ω,𝑢(𝑡,𝑥)=𝜙(𝑡,𝑥),𝑡≤0,𝑥∈Ω. (1.13)

Lemma 1.4. Assume that 𝑏₁>𝑐₁𝑀⁻₀ and 𝑎₁>0, then the solution of (1.13) satisfies0<𝛼≤liminf_𝑡→∞min_𝑥∈Ω𝑢(𝑡,𝑥)≤limsup_𝑡→∞max_𝑥∈Ω𝑢(𝑡,𝑥)≤𝛽, (1.14)where𝑎𝛼=¹𝐵₁−𝑐₁𝑀⁻₀𝐴,𝛽=¹𝑏₁−𝑐₁𝑀⁻₀. (1.15)

Lemma 1.5. If 𝑢(𝑡,𝑥) is the solution of (1.10), then 0<𝑢(𝑡,𝑥).

Now, we introduce the existence-comparison result for the competition-diffusion system (1.1) which is a particular case of Theorem 2.2 in[12].

Definition 1.6. A pair of smooth functions 𝑢=(𝑢₁,𝑢₂) and 𝑢=(𝑢₁,𝑢₂) are called upper-lower solutions of (1.1) if 𝑢_𝑖≥𝑢_𝑖 (𝑖=1,2) in ℝ×Ω and the following differential inequalities hold:𝜕𝑢_𝑖𝑢𝜕𝑡−𝐴Δ_𝑖≥𝑢_𝑖(𝑡,𝑥)[𝑎_𝑖(𝑡,𝑥)−𝑏_𝑖(𝑡,𝑥)𝑢_𝑖(𝑡,𝑥)−𝑐_𝑖(𝑡,𝑥)^∞₀𝑢_𝑖(𝑡−𝜏,𝑥)𝑑𝜇_𝑖(𝜏)−𝑑_𝑖𝑢_𝑗(𝑡−𝑟_𝑗𝜕𝑢,𝑥)],𝑗≠𝑖,𝑡>0,𝑥∈Ω,_𝑖𝑢𝜕𝑡−𝐴Δ_𝑖≤𝑢_𝑖(𝑡,𝑥)[𝑎_𝑖(𝑡,𝑥)−𝑏_𝑖(𝑡,𝑥)𝑢_𝑖(𝑡,𝑥)−𝑐_𝑖(𝑡,𝑥)^∞₀𝑢_𝑖(𝑡−𝜏,𝑥)𝑑𝜇_𝑖(𝜏)−𝑑_𝑖𝑢_𝑗(𝑡−𝑟_𝑗𝜕𝑢,𝑥)],𝑗≠𝑖,𝑡>0,𝑥∈Ω,_𝑖𝜕𝑢𝜕𝜂≤0≤^𝑖𝑢𝜕𝜂,𝑡>0,𝑥∈𝜕Ω,_𝑖(𝑡,𝑥)≤𝜙_𝑖(𝑡,𝑥)≤𝑢_𝑖(𝑡,𝑥),𝑡≤0,𝑥∈Ω(𝑖=1,2). (1.16)

Lemma 1.7. If there exists a pair of upper-lower solutions 𝑢=(𝑢₁,𝑢₂) and 𝑢=(𝑢₁,𝑢₂) of (1.1), then the problem (1.1) has a unique solution 𝑢^∗=(𝑢^∗₁,𝑢^∗₂) and 𝑢_𝑖≤𝑢^∗_𝑖≤𝑢_𝑖, 𝑖=1,2.

2. Proof of the Main Result

The method of proof is via successive improvements of upper-lower solutions of suitable systems.

For given 𝜙=(𝜙₁,𝜙₂) as initial conditions for the system (1.1), let 𝐾₁, 𝐾₂ be constants such that𝐾₁≥max{||𝜙₁𝐴||,¹𝑏₁−𝑐₁𝑀⁻₀₁},𝐾₂≥max{||𝜙₂𝐴||,²𝑏₂−𝑐₂𝑀⁻₀₂}, (2.1)where ||𝜙_𝑖||=sup{|𝜙_𝑖(𝑡,𝑥)|∶(𝑡,𝑥)∈(−∞,0]×Ω}, 𝑖=1,2.

Then, (0,0) and (𝐾₁,𝐾₂) are a pair of lower-upper solutions of (1.1). By Lemma 1.7, there exists a unique global nonnegative solution (𝑢₁,𝑢₂) of (1.1) and it satisfies 0≤𝑢₁(𝑡,𝑥)≤𝐾₁, 0≤𝑢₂(𝑡,𝑥)≤𝐾₂.

Define 𝑢₁⁽¹⁾(𝑥,𝑡) and 𝑢₂⁽¹⁾(𝑥,𝑡) by 𝜕𝑢₁⁽¹⁾𝜕𝑡=𝐴Δ𝑢₁⁽¹⁾+𝑢₁⁽¹⁾[𝑎₁(𝑡,𝑥)−𝑏₁(𝑡,𝑥)𝑢₁⁽¹⁾+𝑐₁(𝑡,𝑥)^∞₀𝑢₁⁽¹⁾(𝑡−𝜏,𝑥)𝑑𝑀⁻₁𝜕(𝜏)]𝑡>0,𝑥∈Ω,𝑢₂⁽¹⁾𝜕𝑡=𝐴Δ𝑢₂⁽¹⁾+𝑢₂⁽¹⁾[𝑎₂(𝑡,𝑥)−𝑏₂(𝑡,𝑥)𝑢₂⁽¹⁾+𝑐₂(𝑡,𝑥)^∞₀𝑢₂⁽¹⁾(𝑡−𝜏,𝑥)𝑑𝑀⁻₂𝜕(𝜏)]𝑡>0,𝑥∈Ω,𝑢₁⁽¹⁾=𝜕𝜕𝜂𝑢₂⁽¹⁾𝜕𝜂=0,𝑡>0,𝑥∈𝜕Ω,𝑢_𝑖⁽¹⁾(𝑡,𝑥)=𝐾_𝑖,𝑡≤0,𝑥∈Ω(𝑖=1,2). (2.2)Then, (0,0) and (𝑢₁⁽¹⁾,𝑢₂⁽¹⁾) are lower and upper solutions, and by Lemma 1.7, 0≤𝑢₁≤𝑢₁⁽¹⁾,0≤𝑢₂≤𝑢_𝑖⁽¹⁾. (2.3) By Lemma 1.4, we can get 0<limsup_𝑡→∞max_𝑥∈Ω𝑢_𝑖⁽¹⁾(𝑡,𝑥)≤𝛽_𝑖⁽⁰⁾, (2.4) where𝛽_𝑖⁽⁰⁾=𝐴_𝑖𝑏_𝑖−𝑐_𝑖𝑀⁻_0𝑖,𝑖=1,2. (2.5) Then from (2.3) and (2.4), we get0≤limsup_𝑡→∞max_𝑥∈Ω𝑢_𝑖(𝑡,𝑥)≤𝛽_𝑖⁽⁰⁾. (2.6) From (2.6) and the definition of 𝑀⁺_0𝑖=lim_𝑡→∞𝑀⁺_𝑖(𝑡), we have that for any sufficiently small 𝜀>0, there exist 𝑡^′₁>0and𝑡₁>𝑡^′₁suchthatmax_𝑥∈Ω𝑢_𝑖⁽¹⁾(𝑡,𝑥)<𝛽_𝑖⁽⁰⁾+𝜀,for𝑡≥𝑡^′₁,𝑢_𝑖⁽¹⁾(𝑡−𝑟_𝑖,𝑥)<𝛽_𝑖⁽⁰⁾+𝜀,for𝑡≥𝑡₁,𝐶_𝑖(𝑀⁺_0𝑖−𝑀⁺_𝑖(𝑡−𝑡^′₁))<𝜀,for𝑡≥𝑡₁. (2.7)Define 𝑢₁⁽¹⁾(𝑡,𝑥) and 𝑢₂⁽¹⁾(𝑡,𝑥) by𝜕𝑢₁⁽¹⁾𝜕𝑡=𝐴Δ𝑢₁⁽¹⁾+𝑢₁⁽¹⁾[𝑎₁(𝑡,𝑥)−𝑏₁(𝑡,𝑥)𝑢₁⁽¹⁾+𝑐₁(𝑡,𝑥)^∞₀𝑢₁⁽¹⁾(𝑡−𝜏,𝑥)𝑑𝑀⁻₁(𝜏)−𝑐₁(𝑡,𝑥)^∞₀𝑢₁⁽¹⁾(𝑡−𝜏,𝑥)𝑑𝑀⁺₁(𝜏)−𝑑₁𝑢₂⁽¹⁾(𝑡−𝑟₂,𝑥)],𝑡>𝑡₁,𝑥∈Ω,𝜕𝑢₂⁽¹⁾𝜕𝑡=𝐴Δ𝑢₂⁽¹⁾+𝑢₂⁽¹⁾[𝑎₂(𝑡,𝑥)−𝑏₂(𝑡,𝑥)𝑢₂⁽¹⁾+𝑐₂(𝑡,𝑥)^∞₀𝑢₂⁽¹⁾(𝑡−𝜏,𝑥)𝑑𝑀⁻₂(𝜏)−𝑐₂(𝑡,𝑥)^∞₀𝑢₂⁽¹⁾(𝑡−𝜏,𝑥)𝑑𝑀⁺₂(𝜏)−𝑑₂𝑢₁⁽¹⁾(𝑡−𝑟₁,𝑥)],𝑡>𝑡₁,𝑥∈Ω,𝜕𝑢₁⁽¹⁾=𝜕𝜂𝜕𝑢₂⁽¹⁾𝜕𝜂=0,𝑡>𝑡₁𝑢,𝑥∈𝜕Ω,_𝑖(1)1(𝑡,𝑥)=2𝑢_𝑖(𝑡,𝑥),(𝑡,𝑥)∈(−∞,𝑡₁]×Ω. (2.8)Then, (𝑢₁⁽¹⁾,𝑢₂⁽¹⁾) and (𝑢₁⁽¹⁾,𝑢₂⁽¹⁾) are a pair of lower-upper solutions of (1.1), and by Lemma 1.7,𝑢₁⁽¹⁾≤𝑢₁≤𝑢₁⁽¹⁾,𝑢₂⁽¹⁾≤𝑢₂≤𝑢₂⁽¹⁾. (2.9) From 𝑢_𝑖⁽¹⁾(𝑡,𝑥)≤𝐾_𝑖, for all 𝑡∈ℝ, and (2.8), for 𝑡≥𝑡₁, we get 𝑐_𝑖(𝑡,𝑥)∫^∞₀𝑢_𝑖⁽¹⁾(𝑡−𝜏,𝑥)𝑑𝑀⁺_𝑖(𝜏)+𝑑_𝑖𝑢_𝑗⁽¹⁾(𝑡−𝑟_𝑗,𝑥)=𝑐_𝑖(𝑡,𝑥)∫^{𝑡−𝑡′1}₀𝑢_𝑖⁽¹⁾(𝑡−𝜏,𝑥)𝑑𝑀⁺_𝑖(𝜏)+𝑐_𝑖(𝑡,𝑥)∫^∞_{𝑡−𝑡′1}𝑢_𝑖⁽¹⁾(𝑡−𝜏,𝑥)𝑑𝑀⁺_𝑖(𝜏)+𝑑_𝑖𝑢_𝑗⁽¹⁾(𝑡−𝑟_𝑗,𝑥)≤𝑐_𝑖(𝑡,𝑥)∫^{𝑡−𝑡′1}₀(𝛽_𝑖⁽⁰⁾+𝜀)𝑑𝑀⁺_𝑖(𝜏)+𝑐_𝑖(𝑡,𝑥)∫^∞_{𝑡−𝑡′1}𝐾_𝑖𝑑𝑀⁺_𝑖(𝜏)+𝑑_𝑖(𝛽_𝑗⁽⁰⁾+𝜀)≤𝐶_𝑖(𝑀⁺_𝑖(𝑡−𝑡₁)−𝑀(0))(𝛽_𝑖⁽⁰⁾+𝜀)+𝐾_𝑖𝐶_𝑖(𝑀₀𝑖⁺−𝑀⁺_𝑖(𝑡−𝑡₁))+𝑑_𝑖(𝛽_𝑗⁽⁰⁾+𝜀)≤𝐶_𝑖𝑀⁺_0𝑖(𝛽_𝑖⁽⁰⁾+𝜀)+𝑑_𝑖(𝛽_𝑗⁽⁰⁾+𝜀)+𝐾_𝑖𝜀. (2.10)It follows from (2.9) that, for 𝑡>𝑡₁, 𝑥∈Ω , 𝜕𝑢₁⁽¹⁾𝜕𝑡≥𝐴Δ𝑢₁⁽¹⁾+𝑢₁⁽¹⁾[𝑎₁(𝑡,𝑥)−𝑏₁(𝑡,𝑥)𝑢₁⁽¹⁾+𝑐₁(𝑡,𝑥)^∞₀𝑢₁⁽¹⁾(𝑡−𝜏,𝑥)𝑑𝑀⁻₁(𝜏)−𝐶₁𝑀⁺₀₁(𝛽₁⁽⁰⁾+𝜀)−𝑑₁(𝛽₂⁽⁰⁾+𝜀)−𝐾₁𝜀],𝜕𝑢₂⁽¹⁾𝜕𝑡≥𝐴Δ𝑢₂⁽¹⁾+𝑢₂⁽¹⁾[𝑎₂(𝑡,𝑥)−𝑏₂(𝑡,𝑥)𝑢₂⁽¹⁾+𝑐₂(𝑡,𝑥)^∞₀𝑢₂⁽¹⁾(𝑡−𝜏,𝑥)𝑑𝑀⁻₂(𝜏)−𝐶₂𝑀⁺₀₂(𝛽₂⁽⁰⁾+𝜀)−𝑑₂(𝛽₁⁽⁰⁾+𝜀)−𝐾₂𝜀]. (2.11)By the comparison principle, we get, for 𝑡>𝑡₁,𝑥∈Ω,𝑢₁⁽¹⁾≥𝑣₁⁽¹⁾,𝑢₂⁽¹⁾≥𝑣₂⁽¹⁾, (2.12)where 𝑣₁⁽¹⁾ and 𝑣₂⁽¹⁾ are the solutions of the following problem, respectively: 𝜕𝑣₁⁽¹⁾𝜕𝑡=𝐴Δ𝑣₁⁽¹⁾+𝑣₁⁽¹⁾[𝑎₁(𝑡,𝑥)−𝑏₁(𝑡,𝑥)𝑣₁⁽¹⁾+𝑐₁(𝑡,𝑥)∫^∞₀𝑣₁⁽¹⁾(𝑡−𝜏,𝑥)𝑑𝑀⁻₁(𝜏)−𝐶₁𝑀⁺₀₁(𝛽₁⁽⁰⁾+𝜀)−𝑑₁(𝛽₂⁽⁰⁾+𝜀)−𝐾₁𝜀],𝜕𝑣₁⁽¹⁾𝜕𝜂=0,𝑡>𝑡₁𝑣,𝑥∈𝜕Ω,₁₍₁₎1(𝑡,𝑥)=2𝑢₁(𝑡,𝑥),(𝑡,𝑥)∈(−∞,𝑡₁]×Ω,𝜕𝑣₂⁽¹⁾𝜕𝑡=𝐴Δ𝑣₂⁽¹⁾+𝑣₂⁽¹⁾[𝑎₂(𝑡,𝑥)−𝑏₂(𝑡,𝑥)𝑣₂⁽¹⁾+𝑐₂(𝑡,𝑥)∫^∞₀𝑣₂⁽¹⁾(𝑡−𝜏,𝑥)𝑑𝑀⁻₂(𝜏)−𝐶₂𝑀⁺₀₂(𝛽₂⁽⁰⁾+𝜀)−𝑑₂(𝛽₁⁽⁰⁾+𝜀)−𝐾₂𝜀],𝜕𝑣₂⁽¹⁾𝜕𝜂=0,𝑡>𝑡₁𝑣,𝑥∈𝜕Ω,₂₍₁₎1(𝑡,𝑥)=2𝑢₂(𝑡,𝑥),(𝑡,𝑥)∈(−∞,𝑡₁]×Ω. (2.13)Using the three initial conditions and 𝜀 sufficiently small, we have𝑎₁−𝐶₁𝑀⁺₀₁(𝛽₁⁽⁰⁾+𝜀)−𝑑₁(𝛽₂⁽⁰⁾+𝜀)−𝐾₁𝑎𝜀>0,₂−𝐶₂𝑀⁺₀₂(𝛽₂⁽⁰⁾+𝜀)−𝑑₂(𝛽₁⁽⁰⁾+𝜀)−𝐾₂𝜀>0. (2.14)By Lemma 1.4, we get 𝑎0<¹−𝐶₁𝑀⁺₀₁𝛽₁⁽⁰⁾−𝑑₁𝛽₂⁽⁰⁾𝐵₁−𝑐₁𝑀⁻₀₁𝐶−𝜀¹𝑀⁺₀₁+𝑑₁+𝐾₁𝐵₁−𝑐₁𝑀⁻₀₁≤liminf_𝑡→∞min_𝑥∈Ω𝑣₁⁽¹⁾𝑎(𝑡,𝑥),0<²−𝐶₂𝑀⁺₀₂𝛽₂⁽⁰⁾−𝑑₂𝛽₁⁽⁰⁾𝐵₂−𝑐₂𝑀⁻₀₂𝐶−𝜀²𝑀⁺₀₂+𝑑₂+𝐾₂𝐵₂−𝑐₂𝑀⁻₀₂≤liminf_𝑡→∞min_𝑥∈Ω𝑣₂⁽¹⁾(𝑡,𝑥). (2.15)Then from (2.12), (2.14), and 𝜀 sufficiently small, we can conclude that0<𝛼₁⁽⁰⁾≤liminf_𝑡→∞min_𝑥∈Ω𝑢₁(𝑡,𝑥)≤limsup_𝑡→∞max_𝑥∈Ω𝑢₁(𝑡,𝑥)≤𝛽₁⁽⁰⁾,0<𝛼₂⁽⁰⁾≤liminf_𝑡→∞min_𝑥∈Ω𝑢₂(𝑡,𝑥)≤limsup_𝑡→∞max_𝑥∈Ω𝑢₂(𝑡,𝑥)≤𝛽₂⁽⁰⁾, (2.16) where𝛼₁⁽⁰⁾=𝑎₁−𝐶₁𝑀⁺₀₁𝛽₁⁽⁰⁾−𝑑₁𝛽₂⁽⁰⁾𝐵₁−𝑐₁𝑀⁻₀₁,𝛼₂⁽⁰⁾=𝑎₂−𝐶₂𝑀⁺₀₂𝛽₂⁽⁰⁾−𝑑₂𝛽₁⁽⁰⁾𝐵₂−𝑐₂𝑀⁻₀₂. (2.17) For any sufficiently small 𝜀>0, there exist 𝑡^′₂>𝑡₁and𝑡₂>𝑡^′₂suchthatmin_𝑥∈Ω𝑢_𝑖⁽¹⁾(𝑡,𝑥)>𝛼_𝑖⁽⁰⁾−𝜀,for𝑡≥𝑡^′₂,𝑢_𝑖⁽¹⁾(𝑡−𝑟_𝑖,𝑥)>𝛼_𝑖⁽⁰⁾+𝜀,for𝑡≥𝑡₂,𝑀⁺_0𝑖−𝜀<𝑀⁺_𝑖(𝑡−𝑡^′₂),for𝑡≥𝑡₂, (2.18)Define 𝑢₁⁽²⁾ and 𝑢₂⁽²⁾ by𝜕𝑢₁⁽²⁾𝜕𝑡=𝐴Δ𝑢₁⁽²⁾+𝑢₁⁽²⁾[𝑎₁(𝑡,𝑥)−𝑏₁(𝑡,𝑥)𝑢₁⁽²⁾+𝑐₁(𝑡,𝑥)∫^∞₀𝑢₁⁽²⁾(𝑡−𝜏,𝑥)𝑑𝑀⁻₁(𝜏)−𝑐₁(𝑡,𝑥)∫^∞₀𝑢₁⁽¹⁾(𝑡−𝜏,𝑥)𝑑𝑀⁺₁(𝜏)−𝑑₁𝑢₂⁽¹⁾(𝑡−𝑟₂,𝑥)],𝑡>𝑡₂𝜕,𝑥∈Ω,𝑢₂⁽²⁾𝜕𝑡=𝐴Δ𝑢₂⁽²⁾+𝑢₂⁽²⁾[𝑎₂(𝑡,𝑥)−𝑏₂(𝑡,𝑥)𝑢₂⁽²⁾+𝑐₂(𝑡,𝑥)∫^∞₀𝑢₂⁽²⁾(𝑡−𝜏,𝑥)𝑑𝑀⁻₂(𝜏)−𝑐₂(𝑡,𝑥)∫^∞₀𝑢₂⁽¹⁾(𝑡−𝜏,𝑥)𝑑𝑀⁺₂(𝜏)−𝑑₂𝑢₁⁽¹⁾(𝑡−𝑟₁,𝑥)],𝑡>𝑡₂𝜕,𝑥∈Ω,𝑢₁⁽²⁾=𝜕𝜕𝜂𝑢₂⁽²⁾𝜕𝜂=0,𝑡>𝑡₂,𝑥∈𝜕Ω,𝑢_𝑖⁽²⁾(𝑡,𝑥)=𝐾_𝑖,(𝑡,𝑥)∈(−∞,𝑡₂]×Ω(𝑖=1,2). (2.19)Then (𝑢₁⁽¹⁾,𝑢₂⁽¹⁾) and (𝑢₁⁽²⁾,𝑢₂⁽²⁾) are a pair of lower-upper solutions of (1.1). By Lemma 1.7,𝑢₁⁽¹⁾≤𝑢₁≤𝑢₁⁽²⁾,𝑢₂⁽¹⁾≤𝑢₂≤𝑢₂⁽²⁾. (2.20) From (2.19) and (2.20), for 𝑡>𝑡₂, 𝑥∈Ω, we get 𝜕𝑢₁⁽²⁾𝜕𝑡≤𝐴Δ𝑢₁⁽²⁾+𝑢₁⁽²⁾[𝑎₁(𝑡,𝑥)−𝑏₁(𝑡,𝑥)𝑢₁⁽²⁾+𝑐₁(𝑡,𝑥)^∞₀𝑢₁⁽²⁾(𝑡−𝜏,𝑥)𝑑𝑀⁻₁(𝜏)−𝑐₁𝑀⁺₀₁(𝛼₁⁽⁰⁾−𝜀)−𝑑₁(𝛼₂⁽⁰⁾−𝜀)+𝜀𝑐₁𝛼₁⁽⁰⁾𝜕],𝑢₂⁽²⁾𝜕𝑡≤𝐴Δ𝑢₂⁽²⁾+𝑢₂⁽²⁾[𝑎₂(𝑡,𝑥)−𝑏₂(𝑡,𝑥)𝑢₂⁽²⁾+𝑐₂(𝑡,𝑥)^∞₀𝑢₂⁽²⁾(𝑡−𝜏,𝑥)𝑑𝑀⁻₂(𝜏)−𝑐₂𝑀⁺₀₂(𝛼₂⁽⁰⁾−𝜀)−𝑑₂(𝛼₁⁽⁰⁾−𝜀)+𝜀𝑐₂𝛼₂⁽⁰⁾]. (2.21)By the comparison principle, for 𝑡>𝑡₂,𝑥∈Ω, we get𝑢₁⁽²⁾≤𝑤₁⁽¹⁾,𝑢₂⁽²⁾≤𝑤₂⁽¹⁾, (2.22) where 𝑤₁⁽¹⁾ and 𝑤₂⁽¹⁾ are the solutions of the following problem, respectively:𝜕𝑤₁⁽¹⁾𝜕𝑡=𝐴Δ𝑤₁⁽¹⁾+𝑤₁⁽¹⁾[𝑎₁(𝑡,𝑥)−𝑏₁(𝑡,𝑥)𝑤₁⁽¹⁾+𝑐₁(𝑡,𝑥)∫^∞₀𝑤₁⁽¹⁾(𝑡−𝜏,𝑥)𝑑𝑀⁻₁(𝜏)−𝑐₁𝑀⁺₀₁(𝛼₁⁽⁰⁾−𝜀)−𝑑₁(𝛼₂⁽⁰⁾−𝜀)+𝜀𝑐₁𝛼₁⁽⁰⁾],𝑡>𝑡₂,𝑥∈Ω,𝜕𝑤₁⁽¹⁾𝜕𝜂=0,𝑡>𝑡₂𝑤,𝑥∈𝜕Ω,₁₍₁₎(𝑡,𝑥)=𝐾₁,(𝑡,𝑥)∈(−∞,𝑡₂]×Ω,𝜕𝑤₂⁽¹⁾𝜕𝑡=𝐴Δ𝑤₂⁽¹⁾+𝑤₂⁽¹⁾[𝑎₂(𝑡,𝑥)−𝑏₂(𝑡,𝑥)𝑤₂⁽¹⁾+𝑐₂(𝑡,𝑥)∫^∞₀𝑤₂⁽¹⁾(𝑡−𝜏,𝑥)𝑑𝑀⁻₂(𝜏)−𝑐₂𝑀⁺₀₂(𝛼₂⁽⁰⁾−𝜀)−𝑑₂(𝛼₁⁽⁰⁾−𝜀)+𝜀𝑐₂𝛼₂⁽⁰⁾],𝑡>𝑡₂,𝑥∈Ω,𝜕𝑤₂⁽¹⁾𝜕𝜂=0,𝑡>𝑡₂𝑤,𝑥∈𝜕Ω,₂₍₁₎(𝑡,𝑥)=𝐾₂,(𝑡,𝑥)∈(−∞,𝑡₂]×Ω. (2.23)From (2.16), (2.18), and sufficiently small 𝜀, we get𝑎₁−𝑐₁𝑀⁺₀₁(𝛼₁⁽⁰⁾−𝜀)−𝑑₁(𝛼₂⁽⁰⁾𝑎−𝜀)>0,₂−𝑐₂𝑀⁺₀₂(𝛼₂⁽⁰⁾−𝜀)−𝑑₂(𝛼₁⁽⁰⁾−𝜀)>0. (2.24)By Lemma 1.4, we get0<limsup_𝑡→∞max_𝑥∈Ω𝑤₁⁽¹⁾𝐴(𝑡,𝑥)≤¹−𝑐₁𝑀⁺₀₁𝛼₁⁽⁰⁾−𝑑₁𝛼₂⁽⁰⁾𝑏₁−𝑐₁𝑀⁻₀₁𝑐−𝜀¹𝑀⁺₀₁+𝑑₁−𝑐₁𝛼₁⁽⁰⁾𝑏₁−𝑐₁𝑀⁻₀₁,0<limsup_𝑡→∞max_𝑥∈Ω𝑤₂⁽¹⁾𝐴(𝑡,𝑥)≤²−𝑐₂𝑀⁺₀₂𝛼₂⁽⁰⁾−𝑑₂𝛼₁⁽⁰⁾𝑏₂−𝑐₂𝑀⁻₀₂𝑐−𝜀²𝑀⁺₀₂+𝑑₂−𝑐₂𝛼₂⁽⁰⁾𝑏₂−𝑐₂𝑀⁻₀₂. (2.25)Then from (2.22), (2.24), and sufficiently small 𝜀,0<𝛼₁⁽⁰⁾≤liminf_𝑡→∞min_𝑥∈Ω𝑢₁(𝑡,𝑥)≤limsup_𝑡→∞max_𝑥∈Ω𝑢₁(𝑡,𝑥)≤𝛽₁⁽¹⁾,0<𝛼₂⁽⁰⁾≤liminf_𝑡→∞min_𝑥∈Ω𝑢₂(𝑡,𝑥)≤limsup_𝑡→∞max_𝑥∈Ω𝑢₂(𝑡,𝑥)≤𝛽₂⁽¹⁾, (2.26)where𝛽₁⁽¹⁾=𝐴₁−𝑐₁𝑀⁺₀₁𝛼₁⁽⁰⁾−𝑑₁𝛼₂⁽⁰⁾𝑏₁−𝑐₁𝑀⁻₀₁,𝛽₂⁽¹⁾=𝐴₂−𝑐₂𝑀⁺₀₂𝛼₂⁽⁰⁾−𝑑₂𝛼₁⁽⁰⁾𝑏₂−𝑐₂𝑀⁻₀₂. (2.27)Then,0<𝛼₁⁽⁰⁾≤𝛽₁⁽¹⁾≤𝛽₁⁽⁰⁾,0<𝛼₂⁽⁰⁾≤𝛽₂⁽¹⁾≤𝛽₂⁽⁰⁾. (2.28)For any sufficiently small 𝜀>0, there exist 𝑡^′₃>𝑡₂and𝑡₃>𝑡^′₃suchthatmax_𝑥∈Ω𝑢_𝑖⁽²⁾(𝑡,𝑥)<𝛽_𝑖⁽¹⁾+𝜀,for𝑡≥𝑡^′₃,𝑢_𝑖⁽¹⁾(𝑡−𝑟_𝑖,𝑥)<𝛽_𝑖⁽¹⁾+𝜀,for𝑡≥𝑡₃,𝐶_𝑖(𝑀⁺_0𝑖−𝑀⁺_𝑖(𝑡−𝑡^′₃))<𝜀,for𝑡≥𝑡₃. (2.29)Define 𝑢₁⁽²⁾(𝑡,𝑥) and 𝑢₂⁽²⁾(𝑡,𝑥) by𝜕𝑢₁⁽²⁾𝜕𝑡=𝐴Δ𝑢₁⁽²⁾+𝑢₁⁽²⁾[𝑎₁(𝑡,𝑥)−𝑏₁(𝑡,𝑥)𝑢₁⁽²⁾+𝑐₁(𝑡,𝑥)^∞₀𝑢₁⁽²⁾(𝑡−𝜏,𝑥)𝑑𝑀⁻₁(𝜏)−𝑐₁(𝑡,𝑥)^∞₀𝑢₁⁽²⁾(𝑡−𝜏,𝑥)𝑑𝑀⁺₁(𝜏)−𝑑₁𝑢₂⁽²⁾(𝑡−𝑟₂,𝑥)],𝑡>𝑡₃,𝑥∈Ω,𝜕𝑢₂⁽²⁾𝜕𝑡=𝐴Δ𝑢₂⁽²⁾+𝑢₂⁽²⁾[𝑎₂(𝑡,𝑥)−𝑏₂(𝑡,𝑥)𝑢₂⁽²⁾+𝑐₂(𝑡,𝑥)^∞₀𝑢₂⁽²⁾(𝑡−𝜏,𝑥)𝑑𝑀⁻₂(𝜏)−𝑐₂(𝑡,𝑥)^∞₀𝑢₂⁽²⁾(𝑡−𝜏,𝑥)𝑑𝑀⁺₂(𝜏)−𝑑₂𝑢₁⁽²⁾(𝑡−𝑟₁,𝑥)],𝑡>𝑡₃,𝑥∈Ω,𝜕𝑢₁⁽²⁾=𝜕𝜂𝜕𝑢₂⁽²⁾𝜕𝜂=0,𝑡>𝑡₃𝑢,𝑥∈𝜕Ω,_𝑖(2)1(𝑡,𝑥)=2𝑢_𝑖(𝑡,𝑥),(𝑡,𝑥)∈(−∞,𝑡₃]×Ω. (2.30)Then, (𝑢₁⁽²⁾,𝑢₂⁽²⁾) and (𝑢₁⁽²⁾,𝑢₂⁽²⁾) are a pair of lower-upper solutions of (1.1). By Lemma 1.7,𝑢₁⁽²⁾≤𝑢₁≤𝑢₁⁽²⁾,𝑢₂⁽²⁾≤𝑢₂≤𝑢₂⁽²⁾. (2.31)From (2.30) and (2.31), for 𝑡>𝑡₃, 𝑥∈Ω, we get𝜕𝑢₁⁽²⁾𝜕𝑡≥𝐴Δ𝑢₁⁽²⁾+𝑢₁⁽²⁾[𝑎₁(𝑡,𝑥)−𝑏₁(𝑡,𝑥)𝑢₁⁽²⁾+𝑐₁(𝑡,𝑥)^∞₀𝑢₁⁽²⁾(𝑡−𝜏,𝑥)𝑑𝑀⁻₁(𝜏)−𝐶₁𝑀⁺₀₁(𝛽₁⁽¹⁾+𝜀)−𝑑₁(𝛽₂⁽¹⁾+𝜀)−𝐾₁𝜀],𝜕𝑢₂⁽²⁾𝜕𝑡≥𝐴Δ𝑢₂⁽²⁾+𝑢₂⁽²⁾[𝑎₂(𝑡,𝑥)−𝑏₂(𝑡,𝑥)𝑢₂⁽²⁾+𝑐₂(𝑡,𝑥)^∞₀𝑢₂⁽²⁾(𝑡−𝜏,𝑥)𝑑𝑀⁻₂(𝜏)−𝐶₂𝑀⁺₀₂(𝛽₂⁽¹⁾+𝜀)−𝑑₂(𝛽₁⁽¹⁾+𝜀)−𝐾₂𝜀]. (2.32) By the comparison principle, for 𝑡>𝑡₃,𝑥∈Ω, we get 𝑢₁⁽²⁾≥𝑣₁⁽²⁾,𝑢₂⁽²⁾≥𝑣₂⁽²⁾, (2.33) where 𝑣₁⁽²⁾ and 𝑣₂⁽²⁾ are the solutions of the following problem, respectively:𝜕𝑣₁⁽²⁾𝜕𝑡=𝐴Δ𝑣₁⁽²⁾+𝑣₁⁽²⁾[𝑎₁(𝑡,𝑥)−𝑏₁(𝑡,𝑥)𝑣₁⁽²⁾+𝑐₁(𝑡,𝑥)^∞₀𝑣₁⁽²⁾(𝑡−𝜏,𝑥)𝑑𝑀⁻₁(𝜏)−𝐶₁𝑀⁺₀₁(𝛽₁⁽¹⁾+𝜀)−𝑑₁(𝛽₂⁽¹⁾+𝜀)−𝐾₁𝜀],𝑡>𝑡₃,𝑥∈Ω,𝜕𝑣₁⁽²⁾𝜕𝜂=0,𝑡>𝑡₃𝑣,𝑥∈𝜕Ω,₁₍₂₎1(𝑡,𝑥)=2𝑢₁(𝑡,𝑥),(𝑡,𝑥)∈(−∞,𝑡₃]×Ω,𝜕𝑣₂⁽²⁾𝜕𝑡=𝐴Δ𝑣₂⁽²⁾+𝑣₂⁽²⁾[𝑎₂(𝑡,𝑥)−𝑏₂(𝑡,𝑥)𝑣₂⁽²⁾+𝑐₂(𝑡,𝑥)^∞₀𝑣₂⁽²⁾(𝑡−𝜏,𝑥)𝑑𝑀⁻₂(𝜏)−𝐶₂𝑀⁺₀₂(𝛽₂⁽¹⁾+𝜀)−𝑑₂(𝛽₁⁽¹⁾+𝜀)−𝐾₂𝜀],𝑡>𝑡₃,𝑥∈Ω,𝜕𝑣₂⁽²⁾𝜕𝜂=0,𝑡>𝑡₃𝑣,𝑥∈𝜕Ω,₂₍₂₎1(𝑡,𝑥)=2𝑢₂(𝑡,𝑥),(𝑡,𝑥)∈(−∞,𝑡₃]×Ω. (2.34)From (2.29), (2.30), and 𝜀 sufficiently small, we get𝑎₁−𝐶₁𝑀⁺₀₁(𝛽₁⁽¹⁾+𝜀)−𝑑₁(𝛽₂⁽¹⁾+𝜀)−𝐾₁𝑎𝜀>0,₂−𝐶₂𝑀⁺₀₂(𝛽₂⁽¹⁾+𝜀)−𝑑₂(𝛽₁⁽¹⁾+𝜀)−𝐾₂𝜀>0. (2.35)By Lemma 1.4 we get𝑎0<¹−𝐶₁𝑀⁺₀₁𝛽₁⁽¹⁾−𝑑₁𝛽₂⁽¹⁾𝐵₁−𝑐₁𝑀⁻₀₁𝐶−𝜀¹𝑀⁺₀₁+𝑑₁+𝐾₁𝐵₁−𝑐₁𝑀⁻₀₁≤liminf_𝑡→∞min_𝑥∈Ω𝑣₁⁽¹⁾𝑎(𝑡,𝑥),0<²−𝐶₂𝑀⁺₀₂𝛽₂⁽¹⁾−𝑑₂𝛽₁⁽¹⁾𝐵₂−𝑐₂𝑀⁻₀₂𝐶−𝜀²𝑀⁺₀₂+𝑑₂+𝐾₂𝐵₂−𝑐₂𝑀⁻₀₂≤liminf_𝑡→∞min_𝑥∈Ω𝑣₂⁽¹⁾(𝑡,𝑥). (2.36)From (2.33), (2.35), and for 𝜀, we conclude that0<𝛼₁⁽¹⁾≤liminf_𝑡→∞min_𝑥∈Ω𝑢₁(𝑡,𝑥)≤limsup_𝑡→∞max_𝑥∈Ω𝑢₁(𝑡,𝑥)≤𝛽₁⁽¹⁾,0<𝛼₂⁽¹⁾≤liminf_𝑡→∞min_𝑥∈Ω𝑢₂(𝑡,𝑥)≤limsup_𝑡→∞max_𝑥∈Ω𝑢₂(𝑡,𝑥)≤𝛽₂⁽¹⁾, (2.37)where𝛼₁⁽¹⁾=𝑎₁−𝐶₁𝑀⁺₀₁𝛽₁⁽¹⁾−𝑑₁𝛽₂⁽¹⁾𝐵₁−𝑐₁𝑀⁻₀₁,𝛼₂⁽¹⁾=𝑎₂−𝐶₂𝑀⁺₀₂𝛽₂⁽¹⁾−𝑑₂𝛽₁⁽¹⁾𝐵₂−𝑐₂𝑀⁻₀₂. (2.38) Then,0<𝛼₁⁽⁰⁾≤𝛼₁⁽¹⁾≤𝛽₁⁽¹⁾≤𝛽₁⁽⁰⁾,0<𝛼₂⁽⁰⁾≤𝛼₂⁽¹⁾≤𝛽₂⁽¹⁾≤𝛽₂⁽⁰⁾. (2.39) Define the sequences 𝛼₁^(𝑘), 𝛼₂^(𝑘), 𝛽₁^(𝑘), and 𝛽₂^(𝑘) as follows: 𝛼₁^(𝑘)=𝑎₁−𝐶₁𝑀⁺₀₁𝛽₁^(𝑘)−𝑑₁𝛽₂^(𝑘)𝐵₁−𝑐₁𝑀⁻₀₁,𝛼₂^(𝑘)=𝑎₂−𝐶₂𝑀⁺₀₂𝛽₂^(𝑘)−𝑑₂𝛽₁^(𝑘)𝐵₂−𝑐₂𝑀⁻₀₂,𝛽₁^(𝑘+1)=𝐴₁−𝑐₁𝑀⁺₀₁𝛼₁^(𝑘)−𝑑₁𝛼₂^(𝑘)𝑏₁−𝑐₁𝑀⁻₀₁,𝛽₂^(𝑘+1)=𝐴₂−𝑐₂𝑀⁺₀₂𝛼₂^(𝑘)−𝑑₂𝛼₁^(𝑘)𝑏₂−𝑐₂𝑀⁻₀₂,𝛽₁⁽⁰⁾=𝐴₁𝑏₁−𝑐₁𝑀⁻₀₁,𝛽₂⁽⁰⁾=𝐴₂𝑏₂−𝑐₂𝑀⁻₀₂. (2.40)

Lemma 2.1. For the above-defined sequences, one has ⊆[𝛼₁^(𝑘),𝛽₁^(𝑘)],[𝛼₂^(𝑘+1),𝛽₂^(𝑘+1)]⊆[𝛼₂^(𝑘),𝛽₂^(𝑘)],𝑘≥0. (2.41)

Proof. For 𝑘=0, it has been shown that [𝛼₁⁽¹⁾,𝛽₁⁽¹⁾]⊆[𝛼₁⁽⁰⁾,𝛽₁⁽⁰⁾] and [𝛼₂⁽¹⁾,𝛽₂⁽¹⁾]⊆[𝛼₂⁽⁰⁾,𝛽₂⁽¹⁾]. Using induction, we can complete the proof.
Lemma 2.1 implies thatlim_𝑘→∞𝛼₁^(𝑘),lim_𝑘→∞𝛼₂^(𝑘),lim_𝑘→∞𝛽₁^(𝑘),lim_𝑘→∞𝛽₂^(𝑘) (2.42)exist, denoted as 𝛼₁, 𝛼₂, 𝛽₁, and 𝛽₂, respectively. From (2.41), we have the following linear system by which we can obtain the numbers 𝛼₁, 𝛼₂, 𝛽₁, and 𝛽₂:𝛼₁=𝑎₁−𝐶₁𝑀⁺₀₁𝛽₁−𝑑₁𝛽₂𝐵₁−𝑐₁𝑀⁻₀₁,𝛼₂=𝑎₂−𝐶₂𝑀⁺₀₂𝛽₂−𝑑₂𝛽₁𝐵₂−𝑐₂𝑀⁻₀₂,𝛽₁=𝐴₁−𝑐₁𝑀⁺₀₁𝛼₁−𝑑₁𝛼₂𝑏₁−𝑐₁𝑀⁻₀₁,𝛽₂=𝐴₂−𝑐₂𝑀⁺₀₂𝛼₂−𝑑₂𝛼₁𝑏₂−𝑐₂𝑀⁻₀₂. (2.43)

Lemma 2.2. For the solutions of (1.1), one has 0<𝛼₁^(𝑘)≤liminf_𝑡→∞min_𝑥∈Ω𝑢₁(𝑡,𝑥)≤limsup_𝑡→∞max_𝑥∈Ω𝑢₁(𝑡,𝑥)≤𝛽₁^(𝑘),𝑘≥0,0<𝛼₂^(𝑘)≤liminf_𝑡→∞min_𝑥∈Ω𝑢₂(𝑡,𝑥)≤limsup_𝑡→∞max_𝑥∈Ω𝑢₂(𝑡,𝑥)≤𝛽₂^(𝑘),𝑘≥0. (2.44)(2.45)

Proof. We have shown that (2.45) and (3.1) are valid for 𝑘=0,1. Using induction and repeating the above process, we can complete the proof.

Combining the above lemmas, we can complete the proof of Theorem 1.1

Remark 2.3. Following the same kind of proof for Theorem 1.1, it can be shown that the same conclusions hold if instead of system (1.1) we work with solutions of 𝜕𝑣₁𝜕𝑡=𝐴Δ𝑣₁(𝑡,𝑥)+𝑣₁(𝑡,𝑥)[𝑎₁(𝑡,𝑥)−𝑏₁(𝑡,𝑥)𝑣₁(𝑡,𝑥)−𝑐₁(𝑡,𝑥)^∞₀𝑣₁(𝑡−𝜏,𝑥)𝑑𝜇₁(𝜏)−𝑑₁𝑣₂(𝑡−𝑟₂,𝑥)],𝑡>0,𝑥∈Ω,𝜕𝑣₂𝜕𝑡=𝐵Δ𝑣₂(𝑡,𝑥)+𝑣₂(𝑡,𝑥)[𝑎₂(𝑡,𝑥)−𝑏₂(𝑡,𝑥)𝑣₂(𝑡,𝑥)−𝑐₂(𝑡,𝑥)^∞₀𝑣₂(𝑡−𝜏,𝑥)𝑑𝜇₂(𝜏)−𝑑₂𝑣₁(𝑡−𝑟₁,𝑥)],𝑡>0,𝑥∈Ω,𝜕𝑣₁=𝜕𝜂𝜕𝑣₂𝑣𝜕𝜂=0,𝑡>0,𝑥∈𝜕Ω,_𝑖(𝑡,𝑥)=𝜙_𝑖(𝑡,𝑥)(𝑖=1,2)𝑡≤0,𝑥∈Ω, (2.46)as expected since they do not include the coefficients of diffusion. We thank one of the referees for his comments regarding this matter.

3. Numerical Simulations

In this section, we present some numerical results that agree with Theorem 1.1 proved above. We used the method of upper and lower solutions as developed by Pao [12, 13] discretizing the systems into finite difference systems. On both examples, the domain used is Ω=(0,12).

Example 3.1. In this example, we work with coefficients that depend on 𝑡 and initial values that depend on (𝑡,𝑥).
Consider the system𝜕𝑢₁=𝜕𝜕𝑡²𝑢₁𝜕𝑥²+𝑢₁1[(sin(𝑡)+2)−(1+𝑥²+10)𝑢₁−(cos(𝑡)+1.01)^∞₀11+𝜏²𝑢₁1(𝑡−𝜏,𝑥)𝑑𝜏−𝑢10²(𝑡−2,𝑥)],𝑡∈[0,𝑇],𝑥∈[0,12],𝜕𝑢₂=𝜕𝜕𝑡²𝑢₂𝜕𝑥²+𝑢₂[(sin²𝑥(𝑡)+1)−(1+𝑥²+7.5)𝑢₂−(sin(𝑡²)+1.01)^∞₀11+𝜏²𝑢₂1(𝑡−𝜏,𝑥)𝑑𝜏−2𝑢₁(𝑡−1.5,𝑥)],𝑡∈[0,𝑇],𝑥∈[0,12],𝜕𝑢_𝑖𝜕𝑥(0,𝑡)=𝜕𝑢_𝑖𝑢𝜕𝑥(12,𝑡)=0(𝑖=1,2),𝑡>0,₁1(𝑡,𝑥)=21sin(2𝑢𝑡)+1.01,𝑡≤0,𝑥∈[0,12],₂1(𝑡,𝑥)=21(sin(21𝑡)+sin(2𝑥))+1.01,𝑡≤0,𝑥∈[0,12]. (3.1)It is easy to see that this system satisfies the conditions of the main theorem (Theorem 1.1), therefore the global attractors are defined by the solutions of the linear system𝛼₁𝜋(11−0)=1−2.012𝛽₁−1𝛽10²,𝛼₂𝜋(1.5−0)=1−2.012𝛽₂−𝛽₁,𝛽₁𝜋(10−0)=3−0.012𝛼₁−1𝛼10²,𝛽₂𝜋(0.5−0)=2−0.012𝛼₂−𝛼₁, (3.2)that is, the global attractors for 𝑢₁ and 𝑢₂ are given by [0.002245,0.299884] and [0.011192,0.285680], respectively. Figure 1 shows the numerical simulation of the solution of this system and on it we can notice the global attractors obtained before.

(a)

(b)

Figure 1 

Numerical simulation of Example 3.1.

Example 3.2. According to the main theorem (Theorem 1.1), the global attractors depend only on the coefficients and not on the initial values. In this example, we only change the initial functions of the above example:𝑢₁𝑢(𝑡,𝑥)=sin(𝑥)+1.01,𝑡≤0,𝑥∈[0,12],₂1(𝑡,𝑥)=200(1+𝑡²)sin(𝜋𝑥12)+0.01,𝑡≤0,𝑥∈[0,12], (3.3)to obtain the same attractors as in Example 3.1. Figure 2 shows the numerical simulation of the solution of this system and we observe that they have the attractors expected.

(a)

(b)

Figure 2 

Numerical simulation of Example 3.2.

Acknowledgments

The research of the first and second authors was supported in part by CONACYT under Grant SEP-2003-C02-44029. The research of the second author was supported in part by SNI under Grant 15284. The research of the third author was supported in part by UADY under Grant PRIORI-FMAT-04-002.