A Two-Species Cooperative Lotka-Volterra System of Degenerate Parabolic Equations

Abstract

We consider a cooperating two-species Lotka-Volterra model of degenerate parabolic equations. We are interested in the coexistence of the species in a bounded domain. We establish the existence of global generalized solutions of the initial boundary value problem by means of parabolic regularization and also consider the existence of the nontrivial time-periodic solution for this system.

1. Introduction

In this paper, we consider the following two-species cooperative system:𝑣𝑢_𝑡=Δ𝑢^𝑚1+𝑢^𝛼(𝑎−𝑏𝑢+𝑐𝑣),(𝑥,𝑡)∈Ω×ℝ₊,(1.1)_𝑡𝑢=(𝑥Δ,𝑡𝑣)=^𝑚2+0,𝑣^𝛽𝑣(𝑑(𝑥+,𝑡𝑒)𝑢=−0𝑓,𝑣)(,𝑥,(𝑡𝑥),∈𝑡)𝜕∈ΩΩ××ℝℝ₊₊,(1.2)𝑢(𝑥,0)=𝑢₀,(1.3)(𝑥),𝑣(𝑥,0)=𝑣₀(𝑥),𝑥∈Ω,(1.4) where 𝑚₁,𝑚₂>1, 0<𝛼<𝑚₁, 0<𝛽<𝑚₂, 1≤(𝑚₁−𝛼)(𝑚₂−𝛽), 𝑎=𝑎(𝑥,𝑡), 𝑏=𝑏(𝑥,𝑡), 𝑐=𝑐(𝑥,𝑡), 𝑑=𝑑(𝑥,𝑡), 𝑒=𝑒(𝑥,𝑡), 𝑓=𝑓(𝑥,𝑡) are strictly positive smooth functions and periodic in time with period 𝑇>0 and 𝑢₀(𝑥) and 𝑣₀(𝑥) are nonnegative functions and satisfy 𝑢^𝑚1₀,𝑣^𝑚2₀∈𝑊^01,2(Ω).

In dynamics of biological groups, the system (1.1)-(1.2) can be used to describe the interaction of two biological groups. The diffusion terms Δ𝑢^𝑚1 and Δ𝑣^𝑚2 represent the effect of dispersion in the habitat, which models a tendency to avoid crowding and the speed of the diffusion is rather slow. The boundary conditions (1.3) indicate that the habitat is surrounded by a totally hostile environment. The functions 𝑢 and 𝑣 represent the spatial densities of the species at time 𝑡 and 𝑎,𝑑 are their respective net birth rate. The functions 𝑏 and 𝑓 are intra-specific competitions, whereas 𝑐 and 𝑒 are those of interspecific competitions.

As famous models for dynamics of population, two-species cooperative systems like (1.1)-(1.2) have been studied extensively, and there have been many excellent results, for detail one can see [1–6] and references therein. As a special case, men studied the following two-species Lotka-Volterra cooperative system of ODEs:𝑣𝑢^(𝑡)=𝑢(𝑡)(𝑎(𝑡)−𝑏(𝑡)𝑢(𝑡)+𝑐(𝑡)𝑣(𝑡)),_(𝑡)=𝑣(𝑡)(𝑑(𝑡)+𝑒(𝑡)𝑢(𝑡)−𝑓(𝑡)𝑣(𝑡)).(1.5) For this system, Lu and Takeuchi [7] studied the stability of positive periodic solution and Cui [1] discussed the persistence and global stability of it.

When 𝑚₁=𝑚₂=𝛼=𝛽=1, from (1.1)-(1.2) we get the following classical cooperative system:𝑣𝑢_𝑡=Δ𝑢+𝑢(𝑎−𝑏𝑢+𝑐𝑣),_𝑡=Δ𝑣+𝑣(𝑑+𝑒𝑢−𝑓𝑣).(1.6)

For this system, Lin et al. [5] showed the existence and asymptotic behavior of 𝑇- periodic solutions when 𝑎,𝑏,𝑐,𝑒,𝑑,𝑓 are all smooth positive and periodic in time with period 𝑇>0. When 𝑎,𝑏,𝑐,𝑒,𝑑,𝑓 are all positive constants, Pao [6] proved that the Dirichlet boundary value problem of this system admits a unique solution which is uniformly bounded when 𝑐𝑒<𝑏𝑓, while the blowup solutions are possible when the two species are strongly mutualistic (𝑐𝑒>𝑏𝑓). For the homogeneous Neumann boundary value problem of this system, Lou et al. [4] proved that the solution will blow up in finite time under a sufficient condition on the initial data. When 𝑐=𝑒=0 and 𝛼=𝛽=1, from (1.1) we get the single degenerate equation 𝑢_𝑡=Δ𝑢^𝑚+𝑢(𝑎−𝑏𝑢).(1.7) For this equation, Sun et al. [8] established the existence of nontrivial nonnegative periodic solutions by monotonicity method and showed the attraction of nontrivial nonnegative periodic solutions.

In the recent years, much attention has been paid to the study of periodic boundary value problems for parabolic systems; for detail one can see [9–15] and the references therein. Furthermore, many researchers studied the periodic boundary value problem for degenerate parabolic systems, such as [16–19]. Taking into account the impact of periodic factors on the species dynamics, we are also interested in the existence of the nontrivial periodic solutions of the cooperative system (1.1)-(1.2). In this paper, we first show the existence of the global generalized solution of the initial boundary value problem (1.1)–(1.4). Then under the condition that 𝑏_𝑙𝑓_𝑙>𝑐_𝑀𝑒_𝑀,(1.8) where 𝑓_𝑀=sup{𝑓(𝑥,𝑡)∣(𝑥,𝑡)∈Ω×ℝ},𝑓_𝑙=inf{𝑓(𝑥,𝑡)∣(𝑥,𝑡)∈Ω×ℝ}, we show that the generalized solution is uniformly bounded. At last, by the method of monotone iteration, we establish the existence of the nontrivial periodic solutions of the system (1.1)-(1.2), which follows from the existence of a pair of large periodic supersolution and small periodic subsolution. At last, we show the existence and the attractivity of the maximal periodic solution.

Our main efforts center on the discussion of generalized solutions, since the regularity follows from a quite standard approach. Hence we give the following definition of generalized solutions of the problem (1.1)–(1.4).

Definition 1.1. A nonnegative and continuous vector-valued function (𝑢,𝑣) is said to be a generalized solution of the problem (1.1)–(1.4) if, for any 0≤𝜏<𝑇 and any functions 𝜑_𝑖∈𝐶¹(𝑄_𝜏) with 𝜑_𝑖|_{𝜕Ω×[0,𝜏)}=0(𝑖=1,2),∇𝑢^𝑚1,∇𝑣^𝑚2∈𝐿²(𝑄_𝜏), 𝜕𝑢^𝑚1/𝜕𝑡,𝜕𝑣^𝑚2/𝜕𝑡∈𝐿²(𝑄_𝜏) and _𝑄𝜏𝑢𝜕𝜑¹𝜕𝑡−∇𝑢^𝑚1∇𝜑₁+𝑢^𝛼(𝑎−𝑏𝑢+𝑐𝑣)𝜑₁𝑑𝑥𝑑𝑡=_Ω𝑢(𝑥,𝜏)𝜑₁(𝑥,𝜏)𝑑𝑥−_Ω𝑢₀(𝑥)𝜑₁(𝑥,0)𝑑𝑥,_𝑄𝜏𝑣𝜕𝜑²𝜕𝑡−∇𝑣^𝑚2∇𝜑₂+𝑣^𝛽(𝑑+𝑒𝑢−𝑓𝑣)𝜑₂𝑑𝑥𝑑𝑡=_Ω𝑣(𝑥,𝜏)𝜑₂(𝑥,𝜏)𝑑𝑥−_Ω𝑣₀(𝑥)𝜑₂(𝑥,0)𝑑𝑥,(1.9) where 𝑄_𝜏=Ω×(0,𝜏).
Similarly, we can define a weak supersolution (𝑢,𝑣) (subsolution (𝑢,𝑣)) if they satisfy the inequalities obtained by replacing “=” with “≤” (“≥”) in (1.3), (1.4), and (1.9) and with an additional assumption 𝜑_𝑖≥0(𝑖=1,2).

Definition 1.2. A vector-valued function (𝑢,𝑣) is said to be a 𝑇-periodic solution of the problem (1.1)–(1.3) if it is a solution in [0,𝑇] such that 𝑢(⋅,0)=𝑢(⋅,𝑇), 𝑣(⋅,0)=𝑣(⋅,𝑇) in Ω. A vector-valued function (𝑢,𝑣) is said to be a 𝑇-periodic supersolution of the problem (1.1)–(1.3) if it is a supersolution in [0,𝑇] such that 𝑢(⋅,0)≥𝑢(⋅,𝑇),𝑣(⋅,0)≥𝑣(⋅,𝑇) in Ω. A vector-valued function (𝑢,𝑣) is said to be a 𝑇-periodic subsolution of the problem (1.1)–(1.3), if it is a subsolution in [0,𝑇] such that 𝑢(⋅,0)≤𝑢(⋅,𝑇),𝑣(⋅,0)≤𝑣(⋅,𝑇) in Ω.

This paper is organized as follows. In Section 2, we show the existence of generalized solutions to the initial boundary value problem and also establish the comparison principle. Section 3 is devoted to the proof of the existence of the nonnegative nontrivial periodic solutions by using the monotone iteration technique.

2. The Initial Boundary Value Problem

To solve the problem (1.1)–(1.4), we consider the following regularized problem:𝜕𝑢_𝜀𝜕𝑡=div𝑚𝑢^{𝑚1𝜀−1}𝜕𝑣_𝜀+𝜀∇𝑢_𝜀+𝑢^𝛼𝜀𝑎−𝑏𝑢_𝜀+𝑐𝑣_𝜀,(𝑥,𝑡)∈𝑄_𝑇,(2.1)𝜕𝑡=div𝑚𝑣^{𝑚2𝜀−1}𝑢+𝜀∇𝑣_𝜀+𝑣^𝛽𝜀𝑑+𝑒𝑢_𝜀−𝑓𝑣_𝜀,(𝑥,𝑡)∈𝑄_𝑇,(2.2)_𝜀𝑢(𝑥,𝑡)=0,𝑣_𝜀(𝑥,𝑡)=0,(𝑥,𝑡)∈𝜕Ω×(0,𝑇),(2.3)_𝜀(𝑥,0)=𝑢_0𝜀(𝑥),𝑣_𝜀(𝑥,0)=𝑣_0𝜀(𝑥),𝑥∈Ω,(2.4) where 𝑄_𝑇=Ω×(0,𝑇), 0<𝜀<1,𝑢_0𝜀,𝑣_0𝜀∈𝐶^∞0(Ω) are nonnegative bounded smooth functions and satisfy 𝑢_𝑚10𝜀0≤𝑢_0𝜀≤‖‖𝑢₀‖‖_𝐿∞(Ω),0≤𝑣_0𝜀≤‖‖𝑣₀‖‖_𝐿∞(Ω),⟶𝑢^𝑚1₀,𝑣^𝑚2_0𝜀⟶𝑣^𝑚2₀,in𝑊^01,2(Ω)as𝜀⟶0.(2.5) The standard parabolic theory (cf. [20, 21]) shows that (2.1)–(2.4) admits a nonnegative classical solution (𝑢_𝜀,𝑣_𝜀). So, the desired solution of the problem (1.1)–(1.4) will be obtained as a limit point of the solutions (𝑢_𝜀,𝑣_𝜀) of the problem (2.1)–(2.4). In the following, we show some important uniform estimates for (𝑢_𝜀,𝑣_𝜀).

Lemma 2.1. Let (𝑢_𝜀,𝑣_𝜀) be a solution of the problem (2.1)–(2.4). (1) If 1<(𝑚₁−𝛼)(𝑚₂−𝛽), then there exist positive constants 𝑟 and 𝑠 large enough such that 1𝑚₂−𝛽<𝑚¹+𝑟−1𝑚₂+𝑠−1<𝑚¹‖‖𝑢−𝛼,(2.6)_𝜀‖‖_{𝐿𝑟(𝑄𝑇)}≤𝐶,‖‖𝑣_𝜀‖‖_{𝐿𝑠(𝑄𝑇)}≤𝐶,(2.7) where 𝐶 is a positive constant only depending on 𝑚₁,𝑚₂,𝛼,𝛽,𝑟,𝑠, |Ω|, and 𝑇.(2) If 1=(𝑚₁−𝛼)(𝑚₂−𝛽), then (2.7) also holds when |Ω| is small enough.

Proof. Multiplying (2.1) by 𝑢^𝜀𝑟−1(𝑟>1) and integrating over Ω, we have that _Ω𝜕𝑢^𝑟𝜀𝜕𝑡𝑑𝑥=−4𝑟(𝑟−1)𝑚¹𝑚₁+𝑟−1²_Ω|||∇𝑢_{(𝑚1𝜀+𝑟−1)/2}|||²𝑑𝑥+𝑟_Ω𝑢^{𝜀𝛼+𝑟−1}𝑎−𝑏𝑢_𝜀+𝑐𝑣_𝜀𝑑𝑥.(2.8) By Poincaré’s inequality, we have that 𝐾_Ω𝑢^{𝑚1𝜀+𝑟−1}𝑑𝑥≤_Ω|||∇𝑢_{(𝑚1𝜀+𝑟−1)/2}|||²𝑑𝑥,(2.9) where 𝐾 is a constant depending only on |Ω| and 𝑁 and becomes very large when the measure of the domain Ω becomes small. Since 𝛼<𝑚₁, Young's inequality shows that 𝑎𝑢^{𝜀𝛼+𝑟−1}≤𝐾𝑟(𝑟−1)𝑚¹𝑚₁+𝑟−1²𝑐𝑢_{𝜀𝛼+𝑟−1}𝑢^{𝑚1𝜀+𝑟−1}+𝐶𝐾^{−(𝛼+𝑟−1)/(𝑚1−𝛼)},𝑣_𝜀≤𝐾𝑟(𝑟−1)𝑚¹𝑚₁+𝑟−1²𝑢^{𝑚1𝜀+𝑟−1}+𝐶𝐾^{−(𝛼+𝑟−1)/(𝑚1−𝛼)}𝑣^{(𝑚1+𝑟−1)/(𝑚1𝜀−𝛼)}.(2.10) For convenience, here and below, 𝐶 denotes a positive constant which is independent of 𝜀 and may take different values on different occasions. Complying (2.8) with (2.9) and (2.10), we obtain _Ω𝜕𝑢^𝑟𝜀𝜕𝑡𝑑𝑥≤−2𝐾𝑟(𝑟−1)𝑚¹𝑚₁+𝑟−1²_Ω𝑢^{𝑚1𝜀+𝑟−1}𝑑𝑥+𝐶𝐾^{−(𝛼+𝑟−1)/(𝑚1−𝛼)}_Ω+𝐶𝐾_{−(𝛼+𝑟−1)/(𝑚1−𝛼)}𝑣^{(𝑚1+𝑟−1)/(𝑚1𝜀−𝛼)}𝑑𝑥.(2.11) As a similar argument as above, for 𝑣_𝜀 and positive constant 𝑠>1, we have that _Ω𝜕𝑣^𝑠𝜀𝜕𝑡𝑑𝑥≤−2𝐾𝑠(𝑠−1)𝑚²𝑚₂+𝑠−1²_Ω𝑣^{𝑚2𝜀+𝑠−1}𝑑𝑥+𝐶𝐾^{−(𝛽+𝑠−1)/(𝑚2−𝛽)}_Ω+𝐶𝐾_{−(𝛽+𝑠−1)/(𝑚2−𝛽)}𝑢^{(𝑚2+𝑠−1)/(𝑚2𝜀−𝛽)}𝑑𝑥.(2.12) Thus we have that _Ω𝜕𝜕𝑢𝑡^𝑟𝜀+𝜕𝜕𝑣𝑡^𝑠𝜀𝑑𝑥≤−2𝐾𝑟(𝑟−1)𝑚¹𝑚₁+𝑟−1²_Ω𝑢^{𝑚1𝜀+𝑟−1}𝑑𝑥+𝐶𝐾^{−(𝛽+𝑠−1)/(𝑚2−𝛽)}_Ω−2𝐾𝑠(𝑠−1)𝑚₂𝑢^{(𝑚2+𝑠−1)/(𝑚2𝜀−𝛽)}𝑑𝑥𝑚₂+𝑠−1²_Ω𝑣^{𝑚2𝜀+𝑠−1}𝑑𝑥+𝐶𝐾^{−(𝛼+𝑟−1)/(𝑚1−𝛼)}_Ω+𝐶𝐾_{−(𝛼+𝑟−1)/(𝑚1−𝛼)}𝑣^{(𝑚1+𝑟−1)/(𝑚1𝜀−𝛼)}𝑑𝑥+𝐶𝐾^{−(𝛽+𝑠−1)/(𝑚2−𝛽)}.(2.13)
For the case of 1<(𝑚₁−𝛼)(𝑚₂−𝛽), there exist 𝑟,𝑠 large enough such that 1𝑚₁−𝛼<𝑚²+𝑠−1𝑚₁+𝑟−1<𝑚²−𝛽.(2.14) By Young's inequality, we have that _Ω𝑢^{(𝑚2+𝑠−1)/(𝑚2𝜀−𝛽)}𝑑𝑥≤𝑟(𝑟−1)𝑚¹𝐾^{(𝑚2+𝑠−1)/(𝑚2−𝛽)}𝐶𝑚₁+𝑟−1²_Ω_Ω𝑢^{𝑚1𝜀+𝑟−1}𝑑𝑥+𝐶𝐾^−𝛾1,𝑣^{(𝑚1+𝑟−1)/(𝑚1𝜀−𝛼)}𝑑𝑥≤𝑠(𝑠−1)𝑚²𝐾^{(𝑚1+𝑟−1)/(𝑚1−𝛼)}𝐶𝑚₂+𝑠−1^𝑝2_Ω𝑣^{𝑚2𝜀+𝑠−1}𝑑𝑥+𝐶𝐾^−𝛾2,(2.15) where 𝛾₁=𝑚²+𝑠−1²𝛾𝑚₂−𝛽𝑚₂−𝛽𝑚₁+𝑟−1−𝑚₂+𝑠−1₂,=𝑚¹+𝑟−1²𝑚₁−𝛼𝑚₁−𝛼𝑚₂+𝑠−1−𝑚₁+𝑟−1.(2.16) Together with (2.13), we have that _Ω𝜕𝜕𝑢𝑡^𝑟𝜀+𝜕𝜕𝑣𝑡^𝑠𝜀𝑑𝑥≤−𝐾_Ω𝑢^{𝑚1𝜀+𝑟−1}+𝐶𝐾+_{−(𝛼+𝑟−1)/(𝑚1−𝛼)}𝑣^{𝑚2𝜀+𝑠−1}𝑑𝑥+𝐶𝐾^−𝜃1+𝐾^−𝜃2+𝐶𝐾^{−(𝛽+𝑠−1)/(𝑚2−𝛽)},(2.17) where 𝜃₁=𝑚²+𝑠−1+𝑚₁+𝑟−1(𝛽+𝑠−1)𝑚₂−𝛽𝑚₁+𝑟−1−𝑚₂+𝑠−1,𝜃²=𝑚¹+𝑟−1+𝑚₂+𝑠−1(𝛼+𝑟−1)𝑚₁−𝛼𝑚₂+𝑠−1−𝑚₁+𝑟−1.(2.18) Furthermore, by Hölder's and Young's inequalities, from (2.17) we obtain _Ω𝜕𝜕𝑢𝑡^𝑟𝜀+𝜕𝜕𝑣𝑡^𝑠𝜀𝑑𝑥≤−𝐾_Ω𝑢^𝑟𝜀+𝐶𝐾+𝑣^𝑠𝜀𝑑_{−(𝛼+𝑟−1)/(𝑚1−𝛼)}𝑥+𝐶𝐾^−𝜃1+𝐾^−𝜃2+2𝐾||Ω||+𝐶𝐾^{−(𝛽+𝑠−1)/(𝑚2−𝛽)}.(2.19) Then by Gronwall's inequality, we obtain _Ω𝑢^𝑟𝜀+𝑣^𝑠𝜀𝑑𝑥≤𝐶.(2.20)
Now we consider the case of 1=(𝑚₁−𝛼)(𝑚₂−𝛽). It is easy to see that there exist positive constants 𝑟,𝑠 large enough such that 1𝑚₁−𝛼=𝑚²+𝑠−1𝑚₁+𝑟−1=𝑚²−𝛽.(2.21) Due to the continuous dependence of 𝐾 upon |Ω| in (2.9), from (2.13) we have that _Ω𝜕𝜕𝑢𝑡^𝑟𝜀+𝜕𝜕𝑣𝑡^𝑠𝜀𝑑𝑥≤−𝐾_Ω𝑢^{𝑚1𝜀+𝑟−1}+𝑣^{𝑚2(𝑝2𝜀−1)+𝑠−1}𝑑𝑥+𝐶(2.22) when |Ω| is small enough. Then by Young's and Gronwall's inequalities we can also obtain (2.20), and thus we complete the proof of this lemma.

Taking 𝑢^𝑚1_𝜀, 𝑣^𝑚2_𝜀 as the test functions, we can easily obtain the following lemma.

Lemma 2.2. Let (𝑢_𝜀,𝑣_𝜀) be a solution of (2.1)–(2.4); then _𝑄𝑇||∇𝑢^𝑚1_𝜀||²𝑑𝑥𝑑𝑡≤𝐶,_𝑄𝑇||∇𝑣^𝑚2_𝜀||²𝑑𝑥𝑑𝑡≤𝐶,(2.23) where 𝐶 is a positive constant independent of 𝜀.

Lemma 2.3. Let (𝑢_𝜀,𝑣_𝜀) be a solution of (2.1)–(2.4), then ‖‖𝑢_𝜀‖‖_{𝐿∞(𝑄𝑇)}≤𝐶,‖‖𝑣_𝜀‖‖_{𝐿∞(𝑄𝑇)}≤𝐶,(2.24) where 𝐶 is a positive constant independent of 𝜀.

Proof. For a positive constant 𝑘>‖𝑢_0𝜀‖_𝐿^∞_(Ω), multiplying (2.1) by (𝑢_𝜀−𝑘)^𝑚1₊𝜒_{[𝑡1,𝑡2]} and integrating the results over 𝑄_𝑇, we have that 1𝑚₁+1_𝑄𝑇𝜕𝑢_𝜀−𝑘^𝑚1++1𝜒_{[𝑡1,𝑡2]}𝜕𝑡𝑑𝑥𝑑𝑡+_𝑄𝑇≤||∇𝑢_𝜀−𝑘^𝑚1₊𝜒_{[𝑡1,𝑡2]}||²𝑑𝑥𝑑𝑡_𝑄𝑇𝑎𝑢^𝛼+𝑚1_𝜀𝑎+𝑐𝑣_𝜀𝑑𝑥𝑑𝑡,(2.25) where 𝑠₊=max{0,𝑠} and 𝜒_{[𝑡1,𝑡2]} is the characteristic function of [𝑡₁,𝑡₂](0≤𝑡₁<𝑡₂≤𝑇). Let 𝐼_𝑘(𝑡)=_Ω𝑢_𝜀−𝑘^𝑚1++1𝑑𝑥;(2.26) then 𝐼_𝑘(𝑡) is absolutely continuous on [0,𝑇]. Denote by 𝜎 the point where 𝐼_𝑘(𝑡) takes its maximum. Assume that 𝜎>0, for a sufficient small positive constant 𝜖. Taking 𝑡₁=𝜎−𝜖, 𝑡₂=𝜎 in (2.25), we obtain 1𝑚₁+1𝜖^{𝜎𝜎−𝜖}_Ω𝜕𝑢_𝜀−𝑘^𝑚1++1𝜕𝑡𝑑𝑥𝑑𝑡+1𝜖^{𝜎𝜎−𝜖}_Ω≤1||∇𝑢_𝜀−𝑘^𝑚1₊||²𝑑𝑥𝑑𝑡𝜖^{𝜎𝜎−𝜖}_Ω𝑢^𝛼+𝑚1_𝜀𝑎+𝑐𝑣_𝜀𝑑𝑥𝑑𝑡.(2.27) From ^{𝜎𝜎−𝜖}_Ω𝜕𝑢_𝜀−𝑘^𝑚1++1𝜕𝑡𝑑𝑥𝑑𝑡=𝐼^𝑘(𝜎)−𝐼_𝑘(𝜎−𝜖)≥0,(2.28) we have that 1𝜖^{𝜎𝜎−𝜖}_Ω||∇𝑢_𝜀−𝑘^𝑚1₊||²𝑑𝑥𝑑𝑡≤1𝜖^{𝜎𝜎−𝜖}_Ω𝑢^𝛼+𝑚1_𝜀𝑎+𝑐𝑣_𝜀𝑑𝑥𝑑𝑡.(2.29) Letting 𝜖→0⁺, we have that _Ω||∇𝑢_𝜀(𝑥,𝜎)−𝑘^𝑚1₊||²𝑑𝑥≤_Ω𝑢^𝛼+𝑚1_𝜀(𝑥,𝜎)𝑎+𝑐𝑣_𝜀(𝑥,𝜎)𝑑𝑥.(2.30) Denote 𝐴_𝑘(𝑡)={𝑥∶𝑢_𝜀(𝑥,𝑡)>𝑘} and 𝜇_𝑘=sup_{𝑡∈(0,𝑇)}|𝐴_𝑘(𝑡)|; then _{𝐴𝑘(𝜎)}||∇𝑢_𝜀−𝑘^𝑚1₊||²𝑑𝑥≤_{𝐴𝑘(𝜎)}𝑢^𝛼+𝑚1_𝜀𝑎+𝑐𝑣_𝜀𝑑𝑥.(2.31) By Sobolev's theorem, _{𝐴𝑘(𝜎)}𝑢_𝜀−𝑘^𝑚1₊^𝑝𝑑𝑥^1/𝑝≤𝐶_{𝐴𝑘(𝜎)}||∇𝑢_𝜀−𝑘^𝑚1₊||²𝑑𝑥^1/2,(2.32) with 2<𝑝<⎧⎪⎨⎪⎩+2∞𝑁,𝑁≤2,𝑁−2,𝑁>2,(2.33) we obtain _{𝐴𝑘(𝜎)}𝑢_𝜀−𝑘^𝑚1₊^𝑝𝑑𝑥^2/𝑝≤≤𝐶𝐶_{𝐴𝑘(𝜎)𝐴𝑘(𝜎)}||∇𝑢_𝜀−𝑘^𝑚1₊||²𝑑𝑥≤𝐶𝑢^𝛼+𝑚1_{𝜀𝐴𝑘(𝜎)}𝑎+𝑣_𝜀𝑑𝑥𝑢^𝑟𝜀𝑑𝑥^{(𝑚1+𝛼)/𝑟}_{𝐴𝑘(𝜎)}(𝑎+𝑣_𝜀)^{𝑟/(𝑟−𝑚1−𝛼)}𝑑𝑥^{(𝑟−𝑚1−𝛼)/𝑟}≤𝐶_{𝐴𝑘(𝜎)}(𝑎+𝑣_𝜀)^{𝑟/(𝑟−𝑚1−𝛼)}𝑑𝑥^{(𝑟−𝑚1−𝛼)/𝑟}≤𝐶_{𝐴𝑘(𝜎)}𝑎+𝑣_𝜀^𝑠𝑑𝑥^1/𝑠||𝐴_𝑘(𝜎)||^{(𝑠(𝑟−𝑚1−𝛼)−𝑟)/𝑠𝑟}≤𝐶𝜇^{(𝑠(𝑟−𝑚1𝑘−𝛼)−𝑟)/𝑠𝑟},(2.34) where 𝑟>𝑝(𝑚₁+𝛼)/(𝑝−2), 𝑠>𝑝𝑟/(𝑝(𝑟−𝑚₁−𝛼)−2𝑟) and 𝐶 denotes various positive constants independent of 𝜀. By Hölder’s inequality, it yields 𝐼_𝑘(𝜎)≤=_Ω𝑢_𝜀−𝑘^𝑚1++1𝑑𝑥=_{𝐴𝑘(𝜎)}𝑢_𝜀−𝑘^𝑚1++1𝑑𝑥_{𝐴𝑘(𝜎)}𝑢_𝜀−𝑘^𝑚1𝑝+𝑑𝑥^{(𝑚1+1)/𝑚1𝑝}𝜇^{1−(𝑚1+1)/𝑚1𝑝𝑘}≤𝐶𝜇^{1+[𝑠𝑝(𝑟−𝑚1−𝛼)−𝑝𝑟−2𝑠𝑟](𝑚1+1)/2𝑝𝑠𝑟𝑚1}_𝑘.(2.35) Then 𝐼_𝑘(𝑡)≤𝐼_𝑘(𝜎)≤𝐶𝜇^{1+[𝑠𝑝(𝑟−𝑚1−𝛼)−𝑝𝑟−2𝑠𝑟](𝑚1+1)/2𝑝𝑠𝑟𝑚1}_𝑘,𝑡∈[0,𝑇].(2.36) On the other hand, for any ℎ>𝑘 and 𝑡∈[0,𝑇], we have that 𝐼_𝑘(𝑡)≥_{𝐴𝑘(𝑡)}𝑢_𝜀−𝑘^𝑚1++1𝑑𝑥≥(ℎ−𝑘)^𝑚1+1||𝐴_ℎ(𝑡)||.(2.37) Combined with (2.35), it yields (ℎ−𝑘)^𝑚1+1𝜇_ℎ≤𝐶𝜇^{1+[𝑠𝑝(𝑟−𝑚1−𝛼)−𝑝𝑟−2𝑠𝑟](𝑚1+1)/2𝑝𝑠𝑟𝑚1}_𝑘,(2.38) that is, 𝜇_ℎ≤𝐶(ℎ−𝑘)^𝑚1+1𝜇^{1+[𝑠𝑝(𝑟−𝑚1−𝛼)−𝑝𝑟−2𝑠𝑟](𝑚1+1)/2𝑝𝑠𝑟𝑚1}_𝑘.(2.39) It is easy to see that 𝛾=1+𝑠𝑝𝑟−𝑚¹−𝛼−𝑝𝑟−2𝑠𝑟𝑚₁+12𝑝𝑠𝑟𝑚₁>1.(2.40) Then by the De Giorgi iteration lemma [22], we have that 𝜇_𝑙+𝑑=sup||𝐴_𝑙+𝑑(𝑡)||=0,(2.41) where 𝑑=𝐶^1/(𝑚1+1)𝜇^{(𝛾−1)/(𝑚1𝑙+1)}2^{𝛾/(𝛾−1)}. That is, 𝑢_𝜀≤𝑙+𝑑a.e.in𝑄_𝑇.(2.42)
It is the same for the second inequality of (2.24). The proof is completed.

Lemma 2.4. The solution (𝑢_𝜀,𝑣_𝜀) of (2.1)–(2.4) satisfies the following: _𝑄𝑇||||𝜕𝑢^𝑚1_𝜀𝜕𝑡||||²𝑑𝑥𝑑𝑡≤𝐶,_𝑄𝑇||||𝜕𝑣^𝑚2_𝜀𝜕𝑡||||²𝑑𝑥𝑑𝑡≤𝐶,(2.43) where 𝐶 is a positive constant independent of 𝜀.

Proof. Multiplying (2.1) by (𝜕/𝜕𝑡)𝑢^𝑚1_𝜀 and integrating over Ω, by (2.3), (2.4) and Young's inequality we have that 4𝑚₁𝑚₁+1²_𝑄𝑇|||𝜕𝜕𝑡𝑢^{(𝑚1𝜀+1)/2}|||²=𝑑𝑥𝑑𝑡_𝑄𝑇𝜕𝑢_𝜀=1𝜕𝑡𝜕𝜕𝑢^𝑚1𝜀𝑡𝑑𝑥𝑑𝑡2_Ω||∇𝑢^𝑚1_𝜀(𝑥,0)||²𝑑𝑥−12_Ω+||∇𝑢^𝑚1_𝜀(𝑥,𝑇)||²𝑑𝑥_𝑄𝑇𝑚₁𝑢^{𝛼+𝑚1𝜀−1}𝑎−𝑏𝑢_𝜀+𝑐𝑣_𝜀𝜕𝑢^𝜀=1𝜕𝑡𝑑𝑥𝑑𝑡2_Ω||∇𝑢^𝑚1_𝜀(𝑥,0)||²𝑑𝑥−12_Ω+||∇𝑢^𝑚1_𝜀(𝑥,𝑇)||²𝑑𝑥_𝑄𝑇2𝑚₁𝑚₁+1𝑢^{(2𝛼+𝑚1𝜀−1)/2}𝑎−𝑏𝑢_𝜀+𝑐𝑣_𝜀𝜕𝑢^{(𝑚1𝜀+1)/2}≤1𝜕𝑡𝑑𝑥𝑑𝑡2_Ω||∇𝑢^𝑚1_𝜀(𝑥,0)||²𝑑𝑥+2𝑚₁_𝑄𝑇+2𝑚₁𝑢^{2𝛼+𝑚1𝜀−1}𝑎−𝑏𝑢_𝜀+𝑐𝑣_𝜀²𝑑𝑥𝑑𝑡𝑚₁+1²_𝑄𝑇|||𝜕𝜕𝑡𝑢^{(𝑚1𝜀+1)/2}|||²𝑑𝑥𝑑𝑡,(2.44) which together with the bound of 𝑎,𝑏,𝑐,𝑢_𝜀,𝑣_𝜀 shows that _𝑄𝑇||||𝜕𝑢^{(𝑚1𝜀+1)/2}𝜕𝑡||||²𝑑𝑥𝑑𝑡≤𝐶,(2.45) where 𝐶 is a positive constant independent of 𝜀. Noticing the bound of 𝑢_𝜀, we have that _𝑄𝑇||||𝜕𝑢^𝑚1_𝜀𝜕𝑡||||²𝑑𝑥𝑑𝑡=4𝑚²¹𝑚₁+1²_𝑄𝑇𝑢^{𝑚1𝜀−1}|||𝜕𝜕𝑡𝑢^{(𝑚1𝜀+1)/2}|||²𝑑𝑥𝑑𝑡≤𝐶.(2.46) It is the same for the second inequality. The proof is completed.

From the above estimates of 𝑢_𝜀,𝑣_𝜀, we have the following results.

Theorem 2.5. The problem (1.1)–(1.4) admits a generalized solution.

Proof. By Lemmas 2.2, 2.3, and 2.4, we can see that there exist subsequences of {𝑢_𝜀},{𝑣_𝜀} (denoted by themselves for simplicity) and functions 𝑢,𝑣 such that 𝜕𝑢_𝑚1𝜀𝑢_𝜀⟶𝑢,𝑣_𝜀⟶𝑣,a.ein𝑄_𝑇,∇𝜕𝑢𝑡_𝑚1𝜀⟶𝜕𝜕𝑢𝑡^𝑚1,𝜕𝜕𝑣^𝑚2𝜀𝑡⟶𝜕𝜕𝑣𝑡^𝑚2,weaklyin𝐿²𝑄_𝑇,⟶∇𝑢^𝑚1,∇𝑣^𝑚2_𝜀⟶∇𝑣^𝑚2,weaklyin𝐿²𝑄_𝑇,(2.47) as 𝜀→0. Then a rather standard argument as [23] shows that (𝑢,𝑣) is a generalized solution of (1.1)–(1.4) in the sense of Definition 1.1.

In order to prove that the generalized solution of (1.1)–(1.4) is uniformly bounded, we need the following comparison principle.

Lemma 2.6. Let (𝑢,𝑣) be a subsolution of the problem (1.1)–(1.4) with the initial value (𝑢₀,𝑣₀) and (𝑢,𝑣) a supersolution with a positive lower bound of the problem (1.1)–(1.4) with the initial value (𝑢₀,𝑣₀). If 𝑢₀≤𝑣₀, 𝑢₀≤𝑣₀, then 𝑢(𝑥,𝑡)≤𝑢(𝑥,𝑡), 𝑣(𝑥,𝑡)≤𝑣(𝑥,𝑡) on 𝑄_𝑇.

Proof. Without loss of generality, we might assume that ‖𝑢(𝑥,𝑡)‖_𝐿^∞_(𝑄𝑇), ‖𝑢(𝑥,𝑡)‖_𝐿^∞_(𝑄𝑇),‖𝑣(𝑥,𝑡)‖_𝐿^∞_(𝑄𝑇), ‖𝑣(𝑥,𝑡)‖_𝐿^∞_(𝑄𝑇)≤𝑀, where 𝑀 is a positive constant. By the definitions of subsolution and supersolution, we have that ^𝑡0_Ω−𝑢𝜕𝜑𝜕𝑡+∇𝑢^𝑚1≤∇𝜑𝑑𝑥𝑑𝜏+_Ω𝑢(𝑥,𝑡)𝜑(𝑥,𝑡)𝑑𝑥−_Ω𝑢₀(𝑥)𝜑(𝑥,0)𝑑𝑥_𝑡0_Ω𝑢^𝛼𝑎−𝑏𝑢+𝑐𝑣𝜑𝑑𝑥𝑑𝜏,_𝑡0_Ω−𝑢𝜕𝜑𝜕𝑡+∇≥𝑢^𝑚1∇𝜑𝑑𝑥𝑑𝜏+_Ω𝑢(𝑥,𝑡)𝜑(𝑥,𝑡)𝑑𝑥−_Ω𝑣₀(𝑥)𝜑(𝑥,0)𝑑𝑥_𝑡0_Ω𝑢^𝛼𝑎−𝑏𝑢+𝑐𝑣𝜑𝑑𝑥𝑑𝜏.(2.48) Take the test function as 𝜑(𝑥,𝑡)=𝐻_𝜀𝑢^𝑚1(𝑥,𝑡)−𝑢^𝑚1(𝑥,𝑡),(2.49) where 𝐻_𝜀(𝑠) is a monotone increasing smooth approximation of the function 𝐻(𝑠) defined as follows: 𝐻(𝑠)=10,,𝑠ot>he0rw,ise.(2.50) It is easy to see that 𝐻_𝜀′(𝑠)→𝛿(𝑠) as 𝜀→0. Since 𝜕𝑢^𝑚1/𝜕𝑡,𝜕𝑢^𝑚1/𝜕𝑡∈𝐿²(𝑄_𝑇), the test function 𝜑(𝑥,𝑡) is suitable. By the positivity of 𝑎,𝑏,𝑐 we have that _Ω𝑢−𝑢𝐻_𝜀𝑢^𝑚1−𝑢^𝑚1𝑑𝑥−^𝑡0_Ω𝑢−𝑢𝜕𝐻^𝜀𝑢^𝑚1−𝑢^𝑚1+𝜕𝑡𝑑𝑥𝑑𝜏_𝑡0_Ω≤𝐻^𝜀𝑢^𝑚1−𝑢^𝑚1||∇𝑢^𝑚1−𝑢^𝑚1||²𝑑𝑥𝑑𝜏_𝑡0_Ω𝑎𝑢^𝛼−𝑢^𝛼𝐻_𝜀𝑢^𝑚1−𝑢^𝑚1+𝑐𝑢^𝛼𝑣−𝑢^𝛼𝑣𝐻_𝜀𝑢^𝑚1−𝑢^𝑚1𝑑𝑥𝑑𝜏,(2.51) where 𝐶 is a positive constant depending on ‖𝑎(𝑥,𝑡)‖_{𝐶(𝑄𝑡)},‖𝑐(𝑥,𝑡)‖_{𝐶(𝑄𝑡)}. Letting 𝜀→0 and noticing that ^𝑡0_Ω𝐻^𝜀𝑢^𝑚1−𝑢^𝑚1||∇𝑢^𝑚−𝑢^𝑚||²𝑑𝑥𝑑𝜏≥0,(2.52) we arrive at _Ω𝑢(𝑥,𝑡)−𝑢(𝑥,𝑡)₊𝑑𝑥≤𝐶^𝑡0_Ω𝑢^𝛼−𝑢^𝛼₊+𝑣𝑢^𝛼−𝑢^𝛼₊+𝑢^𝛼𝑣−𝑣₊𝑑𝑥𝑑𝜏.(2.53) Let (𝑢,𝑣) be a supsolution with a positive lower bound 𝜎. Noticing that (𝑥_𝛼(𝑥^𝛼−𝑦^𝛼)₊≤𝐶(𝛼)(𝑥−𝑦)₊,for𝛼≥1,−𝑦^𝛼)₊≤𝑥^𝛼−1(𝑥−𝑦)₊≤𝑦^𝛼−1(𝑥−𝑦)₊,for𝛼<1,(2.54) with 𝑥,𝑦>0, we have that ^𝑡0_Ω𝑢^𝛼−𝑢^𝛼₊+𝑣𝑢^𝛼−𝑢^𝛼₊+𝑢^𝛼𝑣−𝑣₊𝑑𝑥𝑑𝜏≤𝐶^𝑡0_Ω𝑢−𝑢₊+𝑣−𝑣₊𝑑𝑥𝑑𝜏,(2.55) where 𝐶 is a positive constant depending upon 𝛼,𝜎,𝑀.
Similarly, we also have that _Ω𝑣(𝑥,𝑡)−𝑣(𝑥,𝑡)₊𝑑𝑥≤𝐶^𝑡0_Ω𝑢−𝑢₊+𝑣−𝑣₊𝑑𝑥𝑑𝜏.(2.56) Combining the above two inequalities, we obtain _Ω𝑢(𝑥,𝑡)−𝑢(𝑥,𝑡)₊+𝑣(𝑥,𝑡)−𝑣(𝑥,𝑡)₊𝑑𝑥≤𝐶^𝑡0_Ω𝑢−𝑢₊+𝑣−𝑣₊𝑑𝑥𝑑𝜏.(2.57) By Gronwall's lemma, we see that 𝑢≤𝑢,𝑣≤𝑣. The proof is completed.

Corollary 2.7. If 𝑏_𝑙𝑓_𝑙>𝑐_𝑀𝑒_𝑀, then the problem (1.1)–(1.4) admits at most one global solution which is uniformly bounded in Ω×[0,∞).

Proof. The uniqueness comes from the comparison principle immediately. In order to prove that the solution is global, we just need to construct a bounded positive supersolution of (1.1)–(1.4).
Let 𝜌₁=(𝑎_𝑀𝑓_𝑙+𝑑_𝑀𝑐_𝑀)/(𝑏_𝑙𝑓_𝑙−𝑐_𝑀𝑒_𝑀) and 𝜌₂=(𝑎_𝑀𝑒_𝑀+𝑑_𝑀𝑏_𝑙)/(𝑏_𝑙𝑓_𝑙−𝑐_𝑀𝑒_𝑀), since 𝑏_𝑙𝑓_𝑙>𝑐_𝑀𝑒_𝑀; then 𝜌₁,𝜌₂>0 and satisfy 𝑎_𝑀−𝑏_𝑙𝜌₁+𝑐_𝑀𝜌₂=0,𝑑_𝑀+𝑒_𝑀𝜌₁−𝑓_𝑙𝜌₂=0.(2.58) Let (𝑢,𝑣)=(𝜂𝜌₁,𝜂𝜌₂), where 𝜂>1 is a constant such that (𝑢₀,𝑣₀)≤(𝜂𝜌₁,𝜂𝜌₂); then we have that 𝑢_𝑡−Δ𝑢^𝑚1=0≥𝑢^𝛼𝑎−𝑏𝑢+𝑐𝑣,𝑣_𝑡−Δ𝑣^𝑚2=0≥𝑣^𝛽𝑑+𝑒𝑢−𝑓𝑣.(2.59) That is, (𝑢,𝑣)=(𝜂𝜌₁,𝜂𝜌₂) is a positive supersolution of (1.1)–(1.4). Since 𝑢,𝑣 are global and uniformly bounded, so are 𝑢 and 𝑣.

3. Periodic Solutions

In order to establish the existence of the nontrivial nonnegative periodic solutions of the problem (1.1)–(1.3), we need the following lemmas. Firstly, we construct a pair of 𝑇-periodic supersolution and 𝑇-periodic subsolution as follows.

Lemma 3.1. In case of 𝑏_𝑙𝑓_𝑙>𝑐_𝑀𝑒_𝑀, there exists a pair of 𝑇-periodic supersolution and 𝑇-periodic subsolution of the problem (1.1)–(1.3).

Proof. We first construct a 𝑇-periodic subsolution of (1.1)–(1.3). Let 𝜆 be the first eigenvalue and 𝜙 be the uniqueness solution of the following elliptic problem: −Δ𝜙=𝜆𝜙,𝑥∈Ω,𝜙=0,𝑥∈𝜕Ω;(3.1) then we have that 𝜆>0,𝜙(𝑥)>0inΩ,||∇𝜙||>0on𝜕Ω,𝑀=max_𝑥∈Ω𝜙(𝑥)<∞.(3.2) Let 𝑢,𝑣=𝜀𝜙^2/𝑚1(𝑥),𝜀𝜙^2/𝑚2(𝑥),(3.3) where 𝜀>0 is a small constant to be determined. We will show that (𝑢,𝑣) is a (time independent, hence 𝑇-periodic) subsolution of (1.1)–(1.3).
Taking the nonnegative function 𝜑₁(𝑥,𝑡)∈𝐶¹(𝑄_𝑇) as the test function, we have that _𝑄𝑇𝑢𝜕𝜑₁𝜕𝑡+Δ𝑢^𝑚1𝜑₁+𝑢^𝛼𝑎−𝑏𝑢+𝑐𝑣𝜑₁+𝑑𝑥𝑑𝑡_Ω=𝑢(𝑥,0)𝜑₁(𝑥,0)−𝑢(𝑥,𝑇)𝜑₁(𝑥,𝑇)𝑑𝑥_𝑄𝑇=𝑢^𝛼𝑎−𝑏𝑢+𝑐𝑣+Δ𝑢^𝑚1𝜑₁𝑑𝑥𝑑𝑡_𝑄𝑇=𝑢^𝛼𝑎−𝑏𝑢+𝑐𝑣𝜑₁𝑑𝑥𝑑𝑡−_𝑄𝑇∇𝑢^𝑚1∇𝜑₁𝑑𝑥𝑑𝑡_𝑄𝑇𝑢^𝛼𝑎−𝑏𝑢+𝑐𝑣𝜑₁𝑑𝑥𝑑𝑡−2𝜀^𝑚1_𝑄𝑇=𝜙∇𝜙⋅∇𝜑₁𝑑𝑥𝑑𝑡_𝑄𝑇𝑢^𝛼𝑎−𝑏𝑢+𝑐𝑣𝜑₁𝑑𝑥𝑑𝑡−2𝜀^𝑚1_𝑄𝑇=∇𝜙∇𝜙𝜑₁−||∇𝜙||²𝜑₁𝑑𝑥𝑑𝑡_𝑄𝑇𝑢^𝛼𝑎−𝑏𝑢+𝑐𝑣𝜑₁𝑑𝑥𝑑𝑡−2𝜀^𝑚1_𝑄𝑇=−𝑑𝑖𝑣(∇𝜙)𝜙𝜑₁−||∇𝜙||²𝜑₁𝑑𝑥𝑑𝑡_𝑄𝑇𝑢^𝛼𝑎−𝑏𝑢+𝑐𝑣𝜑₁𝑑𝑥𝑑𝑡−2𝜀^𝑚1_𝑄𝑇𝜆𝜙²−||∇𝜙||²𝜑¹𝑑𝑥𝑑𝑡.(3.4) Similarly, for any nonnegative test function 𝜑₂(𝑥,𝑡)∈𝐶¹(𝑄_𝑇), we have that _𝑄𝑇𝑣𝜕𝜑₂𝜕𝑡+Δ𝑣^𝑚2𝜑₂+𝑣^𝛽𝑑+𝑒𝑢−𝑓𝑣𝜑₂𝑑𝑥𝑑𝑡+_Ω=𝑣(𝑥,0)𝜑₂(𝑥,0)−𝑣(𝑥,𝑇)𝜑₂(𝑥,𝑇)𝑑𝑥_𝑄𝑇𝑣^𝛽𝑑+𝑒𝑢−𝑓𝑣𝜑₂𝑑𝑥𝑑𝑡−2𝜀^𝑚2_𝑄𝑇𝜆𝜙²−||∇𝜙||²𝜑²𝑑𝑥𝑑𝑡.(3.5) We just need to prove the nonnegativity of the right-hand side of (3.4) and (3.5).
Since 𝜙₁=𝜙₂=0,|∇𝜙₁|,|∇𝜙₂|>0 on 𝜕Ω, then there exists 𝛿>0 such that 𝜆𝜙²−||∇𝜙||²≤0,𝑥∈Ω_𝛿,(3.6) where Ω_𝛿={𝑥∈Ω∣dist(𝑥,𝜕Ω)≤𝛿}. Choosing 𝜀≤min𝑎^𝑙𝑏_𝑀𝑀^2/𝑚1,𝑑^𝑙𝑓_𝑀𝑀^2/𝑚2,(3.7) then we have that 2𝜀^𝑚1^𝑇0_Ω𝛿𝜆𝜙²−||∇𝜙||²𝜑¹𝑑𝑥𝑑𝑡≤0≤^𝑇0_Ω𝛿2𝜀_𝑚2𝑢^𝛼𝑎−𝑏𝑢+𝑐𝑣𝜑₁𝑑𝑥𝑑𝑡,^𝑇0_Ω𝛿𝜆𝜙²−||∇𝜙||²𝜑²𝑑𝑥𝑑𝑡≤0≤^𝑇0_Ω𝛿𝑣^𝛽𝑑+𝑒𝑢−𝑓𝑣𝜑₂𝑑𝑥𝑑𝑡,(3.8) which shows that (𝑢,𝑣) is a positive (time independent, hence 𝑇-periodic) subsolution of (1.1)–(1.3) on Ω_𝛿×(0,𝑇).
Moreover, we can see that, for some 𝜎>0, 𝜙(𝑥)≥𝜎>0,𝑥∈Ω⧵Ω_𝛿.(3.9) Choosing 𝜀≤min𝑎^𝑙2𝑏_𝑀𝑀^2/𝑚1,𝑎^𝑙4𝜆𝑀^{2(𝑚1−𝛼)/𝑚1}^{1/(𝑚1−𝛼)},𝑑^𝑙2𝑓_𝑀𝑀^2/𝑚2,𝑑^𝑙4𝜆𝑀^{2(𝑚2−𝛽)/𝑚2}^{1/(𝑚2−𝛽)},(3.10) then 𝜀𝜀^𝛼𝜙^2𝛼/𝑚1𝑎−𝑏𝜀^𝛼+1𝜙^{2(𝛼+1)/𝑚1}+𝑐𝜀^𝛼𝜙^2𝛼/𝑚1𝜀𝜙^2/𝑚2−2𝜀^𝑚1𝜆𝜙²≥0,_𝛽𝜙^2𝛽/𝑚2𝑑+𝑒𝜀𝜙^2/𝑚1𝜀^𝛽𝜙^2𝛽/𝑚2−𝑓𝜀^𝛽+1𝜙^{2(𝛽+1)/𝑚2}−2𝜀^𝑚2𝜆𝜙²≥0(3.11) on 𝑄_𝑇, that is _𝑄𝑇𝑢^𝛼𝑎−𝑏𝑢+𝑐𝑣𝜑₁𝑑𝑥𝑑𝑡−2𝜀^𝑚1_𝑄𝑇𝜆𝜙²−||∇𝜙||²𝜑¹𝑑𝑥𝑑𝑡≥0,_𝑄𝑇𝑣^𝛽𝑑+𝑒𝑢−𝑓𝑣𝜑₂𝑑𝑥𝑑𝑡−2𝜀^𝑚2_𝑄𝑇𝜆𝜙²−||∇𝜙||²𝜑²𝑑𝑥𝑑𝑡≥0.(3.12) These relations show that (𝑢,𝑣)=(𝜀𝜙^2/𝑚1₁(𝑥),𝜀𝜙^2/𝑚2₂(𝑥)) is a positive (time independent, hence 𝑇-periodic) subsolution of (1.1)–(1.3).
Letting (𝑢,𝑣)=(𝜂𝜌₁,𝜂𝜌₂), where 𝜂,𝜌₁,𝜌₂ are taken as those in Corollary 2.7, it is easy to see that (𝑢,𝑣) is a positive (time independent, hence 𝑇-periodic) subsolution of (1.1)–(1.3).
Obviously, we may assume that 𝑢(𝑥,𝑡)≤𝑢(𝑥,𝑡), 𝑣(𝑥,𝑡)≤𝑣(𝑥,𝑡) by changing 𝜂, 𝜀 appropriately.

Lemma 3.2 (see [24, 25]). Let 𝑢 be the solution of the following Dirichlet boundary value problem 𝜕𝑢𝜕𝑡=Δ𝑢^𝑚𝑢(𝑥,𝑡)+=𝑓0,(𝑥,(𝑡𝑥),,𝑡)(∈𝑥,𝜕𝑡Ω)∈×Ω(0×,𝑇()0,,𝑇),(3.13) where 𝑓∈𝐿^∞(Ω×(0,𝑇)); then there exist positive constants 𝐾 and 𝛼∈(0,1) depending only upon 𝜏∈(0,𝑇) and ‖𝑓‖_𝐿^∞_{(Ω×(0,𝑇))}, such that, for any (𝑥_𝑖,𝑡_𝑖)∈Ω×[𝜏,𝑇](𝑖=1,2), ||𝑢𝑥₁,𝑡₁−𝑢𝑥₂,𝑡₂||≤𝐾||𝑥₁−𝑥₂||^𝛼+||𝑡₁−𝑡₂||^𝛼/2.(3.14)

Lemma 3.3 (see [26]). Define a Poincaré mapping 𝑃𝑃_𝑡∶𝐿^∞(Ω)×𝐿^∞(Ω)⟶𝐿^∞(Ω)×𝐿^∞(Ω),_𝑡𝑢₀(𝑥),𝑣₀(𝑥)∶=(𝑢(𝑥,𝑡),𝑣(𝑥,𝑡))(𝑡>0),(3.15) where (𝑢(𝑥,𝑡),𝑣(𝑥,𝑡)) is the solution of (1.1)–(1.4) with initial value (𝑢₀(𝑥),𝑣₀(𝑥)). According to Lemmas 2.6 and 3.2 and Theorem 2.5, the map 𝑃_𝑡 has the following properties: (i)𝑃_𝑡 is defined for any 𝑡>0 and order preserving;(ii)𝑃_𝑡 is order preserving;(iii)𝑃_𝑡 is compact.

Observe that the operator 𝑃_𝑇 is the classical Poincaré map and thus a fixed point of the Poincaré map gives a 𝑇-periodic solution setting. This will be made by the following iteration procedure.

Theorem 3.4. Assume that 𝑏_𝑙𝑓_𝑙>𝑐_𝑀𝑒_𝑀 and there exists a pair of nontrivial nonnegative 𝑇-periodic subsolution (𝑢(𝑥,𝑡),𝑣(𝑥,𝑡)) and 𝑇-periodic supersolution (𝑢(𝑥,𝑡),𝑣(𝑥,𝑡)) of the problem (1.1)–(1.3) with 𝑢(𝑥,0)≤𝑢(𝑥,0); then the problem (1.1)–(1.3) admits a pair of nontrivial nonnegative periodic solutions (𝑢_∗(𝑥,𝑡),𝑣_∗(𝑥,𝑡)),(𝑢^∗(𝑥,𝑡),𝑣^∗(𝑥,𝑡)) such that 𝑢(𝑥,𝑡)≤𝑢_∗(𝑥,𝑡)≤𝑢^∗(𝑥,𝑡)≤𝑢(𝑥,𝑡),𝑣(𝑥,𝑡)≤𝑣_∗(𝑥,𝑡)≤𝑣^∗(𝑥,𝑡)≤𝑣(𝑥,𝑡),in𝑄_𝑇.(3.16)

Proof. Taking 𝑢(𝑥,𝑡), 𝑢(𝑥,𝑡) as those in Lemma 3.1 and choosing suitable 𝐵(𝑥₀,𝛿),𝐵(𝑥₀,𝛿′),Ω′, 𝑘₁, 𝑘₂, and 𝐾, we can obtain 𝑢(𝑥,0)≤𝑢(𝑥,0). By Lemma 2.6, we have that 𝑃_𝑇(𝑢(⋅,0))≥𝑢(⋅,𝑇). Hence by Definition 1.2 we get 𝑃_𝑇(𝑢(⋅,0))≥𝑢(⋅,0), which implies 𝑃_(𝑘+1)𝑇(𝑢(⋅,0))≥𝑃_𝑘𝑇(𝑢(⋅,0)) for any 𝑘∈ℕ. Similarly we have that 𝑃_𝑇(𝑢(⋅,0))≤𝑢(⋅,𝑇)≤𝑢(⋅,0), and hence 𝑃_(𝑘+1)𝑇(𝑢(⋅,0))≤𝑃_𝑘𝑇(𝑢(⋅,0)) for any 𝑘∈ℕ. By Lemma 2.6, we have that 𝑃_𝑘𝑇(𝑢(⋅,0))≤𝑃_𝑘𝑇(𝑢(⋅,0)) for any 𝑘∈ℕ. Then 𝑢_∗(𝑥,0)=lim_𝑘→∞𝑃_𝑘𝑇𝑢(𝑥,0),𝑢^∗(𝑥,0)=lim_𝑘→∞𝑃_𝑘𝑇𝑢(𝑥,0)(3.17) exist for almost every 𝑥∈Ω. Since the operator 𝑃_𝑇 is compact (see Lemma 3.3), the above limits exist in 𝐿^∞(Ω), too. Moreover, both 𝑢_∗(𝑥,0) and 𝑢^∗(𝑥,0) are fixed points of 𝑃_𝑇. With the similar method as [26], it is easy to show that the even extension of the function 𝑢_∗(𝑥,𝑡), which is the solution of the problem (1.1)–(1.4) with the initial value 𝑢_∗(𝑥,0), is indeed a nontrivial nonnegative periodic solution of the problem (1.1)–(1.3). It is the same for the existence of 𝑢^∗(𝑥,𝑡). Furthermore, by Lemma 2.6, we obtain (3.16) immediately, and thus we complete the proof.

Furthermore, by De Giorgi iteration technique, we can also establish a prior upper bound of all nonnegative periodic solutions of (1.1)–(1.3). Then with a similar method as [18], we have the following remark which shows the existence and attractivity of the maximal periodic solution.

Remark 3.5. If 𝑏_𝑙𝑓_𝑙>𝑐_𝑀𝑒_𝑀, the problem (1.1)–(1.3) admits a maximal periodic solution (𝑈,𝑉). Moreover, if (𝑢,𝑣) is the solution of the initial boundary value problem (1.1)–(1.4) with nonnegative initial value (𝑢₀,𝑣₀), then, for any 𝜀>0, there exists 𝑡 depending on 𝑢₀, 𝑣₀, and 𝜀, such that 0≤𝑢≤𝑈+𝜀,0≤𝑣≤𝑉+𝜀,for𝑥∈Ω,𝑡≥𝑡.(3.18)

Acknowledgments

This work was supported by NSFC (10801061), the Fundamental Research Funds for the Central Universities (Grant no. HIT. NSRIF. 2009049), Natural Sciences Foundation of Heilongjiang Province (Grant no. A200909), and also the 985 project of Harbin Institute of Technology.