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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 714248, 18 pages
http://dx.doi.org/10.1155/2011/714248
Research Article

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

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

Received 13 December 2010; Revised 13 February 2011; Accepted 24 February 2011

Academic Editor: Elena Braverman

Copyright © 2011 Jiebao Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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:𝑢𝑡=Δ𝑢𝑚1+𝑢𝛼(𝑎𝑏𝑢+𝑐𝑣),(𝑥,𝑡)Ω×+,𝑣(1.1)𝑡=Δ𝑣𝑚2+𝑣𝛽(𝑑+𝑒𝑢𝑓𝑣),(𝑥,𝑡)Ω×+,(1.2)𝑢(𝑥,𝑡)=0,𝑣(𝑥,𝑡)=0,(𝑥,𝑡)𝜕Ω×+,(1.3)𝑢(𝑥,0)=𝑢0(𝑥),𝑣(𝑥,0)=𝑣0(𝑥),𝑥Ω,(1.4) where 𝑚1,𝑚2>1, 0<𝛼<𝑚1, 0<𝛽<𝑚2, 1(𝑚1𝛼)(𝑚2𝛽), 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 𝑢𝑚10,𝑣𝑚20𝑊01,2(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 Δ𝑢𝑚1 and Δ𝑣𝑚2 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 [16] and references therein. As a special case, men studied the following two-species Lotka-Volterra cooperative system of ODEs:𝑢𝑣(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:𝑢𝑡𝑣=Δ𝑢+𝑢(𝑎𝑏𝑢+𝑐𝑣),𝑡=Δ𝑣+𝑣(𝑑+𝑒𝑢𝑓𝑣).(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 𝑢𝑡=Δ𝑢𝑚+𝑢(𝑎𝑏𝑢).(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 [915] and the references therein. Furthermore, many researchers studied the periodic boundary value problem for degenerate parabolic systems, such as [1619]. 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 𝑏𝑙𝑓𝑙>𝑐𝑀𝑒𝑀,(1.8) where 𝑓𝑀=sup{𝑓(𝑥,𝑡)(𝑥,𝑡)Ω×},𝑓𝑙=inf{𝑓(𝑥,𝑡)(𝑥,𝑡)Ω×}, 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 0𝜏<𝑇 and any functions 𝜑𝑖𝐶1(𝑄𝜏) with 𝜑𝑖|𝜕Ω×[0,𝜏)=0(𝑖=1,2),𝑢𝑚1,𝑣𝑚2𝐿2(𝑄𝜏), 𝜕𝑢𝑚1/𝜕𝑡,𝜕𝑣𝑚2/𝜕𝑡𝐿2(𝑄𝜏) and 𝑄𝜏𝑢𝜕𝜑1𝜕𝑡𝑢𝑚1𝜑1+𝑢𝛼(𝑎𝑏𝑢+𝑐𝑣)𝜑1𝑑𝑥𝑑𝑡=Ω𝑢(𝑥,𝜏)𝜑1(𝑥,𝜏)𝑑𝑥Ω𝑢0(𝑥)𝜑1(𝑥,0)𝑑𝑥,𝑄𝜏𝑣𝜕𝜑2𝜕𝑡𝑣𝑚2𝜑2+𝑣𝛽(𝑑+𝑒𝑢𝑓𝑣)𝜑2𝑑𝑥𝑑𝑡=Ω𝑣(𝑥,𝜏)𝜑2(𝑥,𝜏)𝑑𝑥Ω𝑣0(𝑥)𝜑2(𝑥,0)𝑑𝑥,(1.9) where 𝑄𝜏=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 𝜑𝑖0(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:𝜕𝑢𝜀𝜕𝑡=div𝑚𝑢𝑚1𝜀1+𝜀𝑢𝜀+𝑢𝛼𝜀𝑎𝑏𝑢𝜀+𝑐𝑣𝜀,(𝑥,𝑡)𝑄𝑇,(2.1)𝜕𝑣𝜀𝜕𝑡=div𝑚𝑣𝑚2𝜀1+𝜀𝑣𝜀+𝑣𝛽𝜀𝑑+𝑒𝑢𝜀𝑓𝑣𝜀,(𝑥,𝑡)𝑄𝑇𝑢,(2.2)𝜀(𝑥,𝑡)=0,𝑣𝜀(𝑢𝑥,𝑡)=0,(𝑥,𝑡)𝜕Ω×(0,𝑇),(2.3)𝜀(𝑥,0)=𝑢0𝜀(𝑥),𝑣𝜀(𝑥,0)=𝑣0𝜀(𝑥),𝑥Ω,(2.4) where 𝑄𝑇=Ω×(0,𝑇), 0<𝜀<1,𝑢0𝜀,𝑣0𝜀𝐶0(Ω) are nonnegative bounded smooth functions and satisfy 0𝑢0𝜀𝑢0𝐿(Ω),0𝑣0𝜀𝑣0𝐿(Ω),𝑢𝑚10𝜀𝑢𝑚10,𝑣𝑚20𝜀𝑣𝑚20,in𝑊01,2(Ω)as𝜀0.(2.5) The standard parabolic theory (cf. [20, 21]) shows that (2.1)–(2.4) admits a nonnegative classical solution (𝑢𝜀,𝑣𝜀). So, the desired solution of the problem (1.1)–(1.4) will be obtained as a limit point of the solutions (𝑢𝜀,𝑣𝜀) of the problem (2.1)–(2.4). In the following, we show some important uniform estimates for (𝑢𝜀,𝑣𝜀).

Lemma 2.1. Let (𝑢𝜀,𝑣𝜀) be a solution of the problem (2.1)–(2.4). (1) If 1<(𝑚1𝛼)(𝑚2𝛽), then there exist positive constants 𝑟 and 𝑠 large enough such that 1𝑚2<𝑚𝛽1+𝑟1𝑚2+𝑠1<𝑚1𝑢𝛼,(2.6)𝜀𝐿𝑟(𝑄𝑇)𝑣𝐶,𝜀𝐿𝑠(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=(𝑚1𝛼)(𝑚2𝛽), then (2.7) also holds when |Ω| is small enough.

Proof. Multiplying (2.1) by 𝑢𝜀𝑟1(𝑟>1) and integrating over Ω, we have that Ω𝜕𝑢𝑟𝜀𝜕𝑡𝑑𝑥=4𝑟(𝑟1)𝑚1𝑚1+𝑟12Ω|||𝑢(𝑚1𝜀+𝑟1)/2|||2𝑑𝑥+𝑟Ω𝑢𝜀𝛼+𝑟1𝑎𝑏𝑢𝜀+𝑐𝑣𝜀𝑑𝑥.(2.8) By Poincaré’s inequality, we have that 𝐾Ω𝑢𝑚1𝜀+𝑟1𝑑𝑥Ω|||𝑢(𝑚1𝜀+𝑟1)/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 𝑎𝑢𝜀𝛼+𝑟1𝐾𝑟(𝑟1)𝑚1𝑚1+𝑟12𝑢𝑚1𝜀+𝑟1+𝐶𝐾(𝛼+𝑟1)/(𝑚1𝛼),𝑐𝑢𝜀𝛼+𝑟1𝑣𝜀𝐾𝑟(𝑟1)𝑚1𝑚1+𝑟12𝑢𝑚1𝜀+𝑟1+𝐶𝐾(𝛼+𝑟1)/(𝑚1𝛼)𝑣(𝑚1+𝑟1)/(𝑚1𝜀𝛼).(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𝜕𝑢𝑟𝜀𝜕𝑡𝑑𝑥2𝐾𝑟(𝑟1)𝑚1𝑚1+𝑟12Ω𝑢𝑚1𝜀+𝑟1𝑑𝑥+𝐶𝐾(𝛼+𝑟1)/(𝑚1𝛼)Ω𝑣(𝑚1+𝑟1)/(𝑚1𝜀𝛼)𝑑𝑥+𝐶𝐾(𝛼+𝑟1)/(𝑚1𝛼).(2.11) As a similar argument as above, for 𝑣𝜀 and positive constant 𝑠>1, we have that Ω𝜕𝑣𝑠𝜀𝜕𝑡𝑑𝑥2𝐾𝑠(𝑠1)𝑚2𝑚2+𝑠12Ω𝑣𝑚2𝜀+𝑠1𝑑𝑥+𝐶𝐾(𝛽+𝑠1)/(𝑚2𝛽)Ω𝑢(𝑚2+𝑠1)/(𝑚2𝜀𝛽)𝑑𝑥+𝐶𝐾(𝛽+𝑠1)/(𝑚2𝛽).(2.12) Thus we have that Ω𝜕𝑢𝑟𝜀+𝜕𝑡𝜕𝑣𝑠𝜀𝜕𝑡𝑑𝑥2𝐾𝑟(𝑟1)𝑚1𝑚1+𝑟12Ω𝑢𝑚1𝜀+𝑟1𝑑𝑥+𝐶𝐾(𝛽+𝑠1)/(𝑚2𝛽)Ω𝑢(𝑚2+𝑠1)/(𝑚2𝜀𝛽)𝑑𝑥2𝐾𝑠(𝑠1)𝑚2𝑚2+𝑠12Ω𝑣𝑚2𝜀+𝑠1𝑑𝑥+𝐶𝐾(𝛼+𝑟1)/(𝑚1𝛼)Ω𝑣(𝑚1+𝑟1)/(𝑚1𝜀𝛼)𝑑𝑥+𝐶𝐾(𝛼+𝑟1)/(𝑚1𝛼)+𝐶𝐾(𝛽+𝑠1)/(𝑚2𝛽).(2.13)
For the case of 1<(𝑚1𝛼)(𝑚2𝛽), there exist 𝑟,𝑠 large enough such that 1𝑚1<𝑚𝛼2+𝑠1𝑚1+𝑟1<𝑚2𝛽.(2.14) By Young's inequality, we have that Ω𝑢(𝑚2+𝑠1)/(𝑚2𝜀𝛽)𝑑𝑥𝑟(𝑟1)𝑚1𝐾(𝑚2+𝑠1)/(𝑚2𝛽)𝐶𝑚1+𝑟12Ω𝑢𝑚1𝜀+𝑟1𝑑𝑥+𝐶𝐾𝛾1,Ω𝑣(𝑚1+𝑟1)/(𝑚1𝜀𝛼)𝑑𝑥𝑠(𝑠1)𝑚2𝐾(𝑚1+𝑟1)/(𝑚1𝛼)𝐶𝑚2+𝑠1𝑝2Ω𝑣𝑚2𝜀+𝑠1𝑑𝑥+𝐶𝐾𝛾2,(2.15) where 𝛾1=𝑚2+𝑠12𝑚2𝑚𝛽2𝑚𝛽1𝑚+𝑟12,𝛾+𝑠12=𝑚1+𝑟12𝑚1𝑚𝛼1𝑚𝛼2𝑚+𝑠11.+𝑟1(2.16) Together with (2.13), we have that Ω𝜕𝑢𝑟𝜀+𝜕𝑡𝜕𝑣𝑠𝜀𝜕𝑡𝑑𝑥𝐾Ω𝑢𝑚1𝜀+𝑟1+𝑣𝑚2𝜀+𝑠1𝐾𝑑𝑥+𝐶𝜃1+𝐾𝜃2+𝐶𝐾(𝛼+𝑟1)/(𝑚1𝛼)+𝐶𝐾(𝛽+𝑠1)/(𝑚2𝛽),(2.17) where 𝜃1=𝑚2+𝑚+𝑠11+𝑟1(𝛽+𝑠1)𝑚2𝑚𝛽1𝑚+𝑟12+𝑠1,𝜃2=𝑚1+𝑚+𝑟12+𝑠1(𝛼+𝑟1)𝑚1𝑚𝛼2𝑚+𝑠11+𝑟1.(2.18) Furthermore, by Hölder's and Young's inequalities, from (2.17) we obtain Ω𝜕𝑢𝑟𝜀+𝜕𝑡𝜕𝑣𝑠𝜀𝜕𝑡𝑑𝑥𝐾Ω𝑢𝑟𝜀+𝑣𝑠𝜀𝐾𝑑𝑥+𝐶𝜃1+𝐾𝜃2||Ω||+2𝐾+𝐶𝐾(𝛼+𝑟1)/(𝑚1𝛼)+𝐶𝐾(𝛽+𝑠1)/(𝑚2𝛽).(2.19) Then by Gronwall's inequality, we obtain Ω𝑢𝑟𝜀+𝑣𝑠𝜀𝑑𝑥𝐶.(2.20)
Now we consider the case of 1=(𝑚1𝛼)(𝑚2𝛽). It is easy to see that there exist positive constants 𝑟,𝑠 large enough such that 1𝑚1=𝑚𝛼2+𝑠1𝑚1+𝑟1=𝑚2𝛽.(2.21) Due to the continuous dependence of 𝐾 upon |Ω| in (2.9), from (2.13) we have that Ω𝜕𝑢𝑟𝜀+𝜕𝑡𝜕𝑣𝑠𝜀𝜕𝑡𝑑𝑥𝐾Ω𝑢𝑚1𝜀+𝑟1+𝑣𝑚2(𝑝2𝜀1)+𝑠1𝑑𝑥+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 𝑢𝑚1𝜀, 𝑣𝑚2𝜀 as the test functions, we can easily obtain the following lemma.

Lemma 2.2. Let (𝑢𝜀,𝑣𝜀) be a solution of (2.1)–(2.4); then 𝑄𝑇||𝑢𝑚1𝜀||2𝑑𝑥𝑑𝑡𝐶,𝑄𝑇||𝑣𝑚2𝜀||2𝑑𝑥𝑑𝑡𝐶,(2.23) where 𝐶 is a positive constant independent of 𝜀.

Lemma 2.3. Let (𝑢𝜀,𝑣𝜀) be a solution of (2.1)–(2.4), then 𝑢𝜀𝐿(𝑄𝑇)𝑣𝐶,𝜀𝐿(𝑄𝑇)𝐶,(2.24) where 𝐶 is a positive constant independent of 𝜀.

Proof. For a positive constant 𝑘>𝑢0𝜀𝐿(Ω), multiplying (2.1) by (𝑢𝜀𝑘)𝑚1+𝜒[𝑡1,𝑡2] and integrating the results over 𝑄𝑇, we have that 1𝑚1+1𝑄𝑇𝜕𝑢𝜀𝑘𝑚1++1𝜒[𝑡1,𝑡2]𝜕𝑡𝑑𝑥𝑑𝑡+𝑄𝑇||𝑢𝜀𝑘𝑚1+𝜒[𝑡1,𝑡2]||2𝑑𝑥𝑑𝑡𝑄𝑇𝑎𝑢𝛼+𝑚1𝜀𝑎+𝑐𝑣𝜀𝑑𝑥𝑑𝑡,(2.25) where 𝑠+=max{0,𝑠} and 𝜒[𝑡1,𝑡2] is the characteristic function of [𝑡1,𝑡2](0𝑡1<𝑡2𝑇). Let 𝐼𝑘(𝑡)=Ω𝑢𝜀𝑘𝑚1++1𝑑𝑥;(2.26) then 𝐼𝑘(𝑡) is absolutely continuous on [0,𝑇]. Denote by 𝜎 the point where 𝐼𝑘(𝑡) takes its maximum. Assume that 𝜎>0, for a sufficient small positive constant 𝜖. Taking 𝑡1=𝜎𝜖, 𝑡2=𝜎 in (2.25), we obtain 1𝑚1𝜖+1𝜎𝜎𝜖Ω𝜕𝑢𝜀𝑘𝑚1++11𝜕𝑡𝑑𝑥𝑑𝑡+𝜖𝜎𝜎𝜖Ω||𝑢𝜀𝑘𝑚1+||21𝑑𝑥𝑑𝑡𝜖𝜎𝜎𝜖Ω𝑢𝛼+𝑚1𝜀𝑎+𝑐𝑣𝜀𝑑𝑥𝑑𝑡.(2.27) From 𝜎𝜎𝜖Ω𝜕𝑢𝜀𝑘𝑚1++1𝜕𝑡𝑑𝑥𝑑𝑡=𝐼𝑘(𝜎)𝐼𝑘(𝜎𝜖)0,(2.28) we have that 1𝜖𝜎𝜎𝜖Ω||𝑢𝜀𝑘𝑚1+||21𝑑𝑥𝑑𝑡𝜖𝜎𝜎𝜖Ω𝑢𝛼+𝑚1𝜀𝑎+𝑐𝑣𝜀𝑑𝑥𝑑𝑡.(2.29) Letting 𝜖0+, we have that Ω||𝑢𝜀(𝑥,𝜎)𝑘𝑚1+||2𝑑𝑥Ω𝑢𝛼+𝑚1𝜀(𝑥,𝜎)𝑎+𝑐𝑣𝜀(𝑥,𝜎)𝑑𝑥.(2.30) Denote 𝐴𝑘(𝑡)={𝑥𝑢𝜀(𝑥,𝑡)>𝑘} and 𝜇𝑘=sup𝑡(0,𝑇)|𝐴𝑘(𝑡)|; then 𝐴𝑘(𝜎)||𝑢𝜀𝑘𝑚1+||2𝑑𝑥𝐴𝑘(𝜎)𝑢𝛼+𝑚1𝜀𝑎+𝑐𝑣𝜀𝑑𝑥.(2.31) By Sobolev's theorem, 𝐴𝑘(𝜎)𝑢𝜀𝑘𝑚1+𝑝𝑑𝑥1/𝑝𝐶𝐴𝑘(𝜎)||𝑢𝜀𝑘𝑚1+||2𝑑𝑥1/2,(2.32) with 2<𝑝<+,𝑁2,2𝑁𝑁2,𝑁>2,(2.33) we obtain 𝐴𝑘(𝜎)𝑢𝜀𝑘𝑚1+𝑝𝑑𝑥2/𝑝𝐶𝐴𝑘(𝜎)||𝑢𝜀𝑘𝑚1+||2𝑑𝑥𝐶𝐴𝑘(𝜎)𝑢𝛼+𝑚1𝜀𝑎+𝑣𝜀𝑑𝑥𝐶𝐴𝑘(𝜎)𝑢𝑟𝜀𝑑𝑥(𝑚1+𝛼)/𝑟𝐴𝑘(𝜎)(𝑎+𝑣𝜀)𝑟/(𝑟𝑚1𝛼)𝑑𝑥(𝑟𝑚1𝛼)/𝑟𝐶𝐴𝑘(𝜎)(𝑎+𝑣𝜀)𝑟/(𝑟𝑚1𝛼)𝑑𝑥(𝑟𝑚1𝛼)/𝑟𝐶𝐴𝑘(𝜎)𝑎+𝑣𝜀𝑠𝑑𝑥1/𝑠||𝐴𝑘||(𝜎)(𝑠(𝑟𝑚1𝛼)𝑟)/𝑠𝑟𝐶𝜇(𝑠(𝑟𝑚1𝑘𝛼)𝑟)/𝑠𝑟,(2.34) where 𝑟>𝑝(𝑚1+𝛼)/(𝑝2), 𝑠>𝑝𝑟/(𝑝(𝑟𝑚1𝛼)2𝑟) and 𝐶 denotes various positive constants independent of 𝜀. By Hölder’s inequality, it yields 𝐼𝑘(𝜎)=Ω𝑢𝜀𝑘𝑚1++1𝑑𝑥=𝐴𝑘(𝜎)𝑢𝜀𝑘𝑚1++1𝑑𝑥𝐴𝑘(𝜎)𝑢𝜀𝑘𝑚1𝑝+𝑑𝑥(𝑚1+1)/𝑚1𝑝𝜇1(𝑚1+1)/𝑚1𝑝𝑘𝐶𝜇1+[𝑠𝑝(𝑟𝑚1𝛼)𝑝𝑟2𝑠𝑟](𝑚1+1)/2𝑝𝑠𝑟𝑚1𝑘.(2.35) Then 𝐼𝑘(𝑡)𝐼𝑘(𝜎)𝐶𝜇1+[𝑠𝑝(𝑟𝑚1𝛼)𝑝𝑟2𝑠𝑟](𝑚1+1)/2𝑝𝑠𝑟𝑚1𝑘[].,𝑡0,𝑇(2.36) On the other hand, for any >𝑘 and 𝑡[0,𝑇], we have that 𝐼𝑘(𝑡)𝐴𝑘(𝑡)𝑢𝜀𝑘𝑚1++1𝑑𝑥(𝑘)𝑚1+1||𝐴(||.𝑡)(2.37) Combined with (2.35), it yields (𝑘)𝑚1+1𝜇𝐶𝜇1+[𝑠𝑝(𝑟𝑚1𝛼)𝑝𝑟2𝑠𝑟](𝑚1+1)/2𝑝𝑠𝑟𝑚1𝑘,(2.38) that is, 𝜇𝐶(𝑘)𝑚1+1𝜇1+[𝑠𝑝(𝑟𝑚1𝛼)𝑝𝑟2𝑠𝑟](𝑚1+1)/2𝑝𝑠𝑟𝑚1𝑘.(2.39) It is easy to see that 𝛾=1+𝑠𝑝𝑟𝑚1𝑚𝛼𝑝𝑟2𝑠𝑟1+12𝑝𝑠𝑟𝑚1>1.(2.40) Then by the De Giorgi iteration lemma [22], we have that 𝜇𝑙+𝑑||𝐴=sup𝑙+𝑑||(𝑡)=0,(2.41) where 𝑑=𝐶1/(𝑚1+1)𝜇(𝛾1)/(𝑚1𝑙+1)2𝛾/(𝛾1). That is, 𝑢𝜀𝑙+𝑑a.e.in𝑄𝑇.(2.42)
It is the same for the second inequality of (2.24). The proof is completed.

Lemma 2.4. The solution (𝑢𝜀,𝑣𝜀) of (2.1)–(2.4) satisfies the following: 𝑄𝑇||||𝜕𝑢𝑚1𝜀||||𝜕𝑡2𝑑𝑥𝑑𝑡𝐶,𝑄𝑇||||𝜕𝑣𝑚2𝜀||||𝜕𝑡2𝑑𝑥𝑑𝑡𝐶,(2.43) where 𝐶 is a positive constant independent of 𝜀.

Proof. Multiplying (2.1) by (𝜕/𝜕𝑡)𝑢𝑚1𝜀 and integrating over Ω, by (2.3), (2.4) and Young's inequality we have that 4𝑚1𝑚1+12𝑄𝑇|||𝜕𝑢𝜕𝑡(𝑚1𝜀+1)/2|||2=𝑑𝑥𝑑𝑡𝑄𝑇𝜕𝑢𝜀𝜕𝑡𝜕𝑢𝑚1𝜀=1𝜕𝑡𝑑𝑥𝑑𝑡2Ω||𝑢𝑚1𝜀(||𝑥,0)21𝑑𝑥2Ω||𝑢𝑚1𝜀(||𝑥,𝑇)2+𝑑𝑥𝑄𝑇𝑚1𝑢𝛼+𝑚1𝜀1𝑎𝑏𝑢𝜀+𝑐𝑣𝜀𝜕𝑢𝜀=1𝜕𝑡𝑑𝑥𝑑𝑡2Ω||𝑢𝑚1𝜀||(𝑥,0)21𝑑𝑥2Ω||𝑢𝑚1𝜀||(𝑥,𝑇)2+𝑑𝑥𝑄𝑇2𝑚1𝑚1𝑢+1(2𝛼+𝑚1𝜀1)/2𝑎𝑏𝑢𝜀+𝑐𝑣𝜀𝜕𝑢(𝑚1𝜀+1)/21𝜕𝑡𝑑𝑥𝑑𝑡2Ω||𝑢𝑚1𝜀||(𝑥,0)2𝑑𝑥+2𝑚1𝑄𝑇𝑢2𝛼+𝑚1𝜀1𝑎𝑏𝑢𝜀+𝑐𝑣𝜀2+𝑑𝑥𝑑𝑡2𝑚1𝑚1+12𝑄𝑇|||𝜕𝑢𝜕𝑡(𝑚1𝜀+1)/2|||2𝑑𝑥𝑑𝑡,(2.44) which together with the bound of 𝑎,𝑏,𝑐,𝑢𝜀,𝑣𝜀 shows that 𝑄𝑇||||𝜕𝑢(𝑚1𝜀+1)/2||||𝜕𝑡2𝑑𝑥𝑑𝑡𝐶,(2.45) where 𝐶 is a positive constant independent of 𝜀. Noticing the bound of 𝑢𝜀, we have that 𝑄𝑇||||𝜕𝑢𝑚1𝜀||||𝜕𝑡2𝑑𝑥𝑑𝑡=4𝑚21𝑚1+12𝑄𝑇𝑢𝑚1𝜀1|||𝜕𝑢𝜕𝑡(𝑚1𝜀+1)/2|||2𝑑𝑥𝑑𝑡𝐶.(2.46) It is the same for the second inequality. The proof is completed.

From the above estimates of 𝑢𝜀,𝑣𝜀, 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 {𝑢𝜀},{𝑣𝜀} (denoted by themselves for simplicity) and functions 𝑢,𝑣 such that 𝑢𝜀𝑢,𝑣𝜀𝑣,a.ein𝑄𝑇,𝜕𝑢𝑚1𝜀𝜕𝑡𝜕𝑢𝑚1,𝜕𝑡𝜕𝑣𝑚2𝜀𝜕𝑡𝜕𝑣𝑚2𝜕𝑡,weaklyin𝐿2𝑄𝑇,𝑢𝑚1𝜀𝑢𝑚1,𝑣𝑚2𝜀𝑣𝑚2,weaklyin𝐿2𝑄𝑇,(2.47) as 𝜀0. 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 𝑢0𝑣0, 𝑢0𝑣0, then 𝑢(𝑥,𝑡)𝑢(𝑥,𝑡), 𝑣(𝑥,𝑡)𝑣(𝑥,𝑡) on 𝑄𝑇.

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 𝑡0Ω𝑢𝜕𝜑𝜕𝑡+𝑢𝑚1𝜑𝑑𝑥𝑑𝜏+Ω𝑢(𝑥,𝑡)𝜑(𝑥,𝑡)𝑑𝑥Ω𝑢0(𝑥)𝜑(𝑥,0)𝑑𝑥𝑡0Ω𝑢𝛼𝑎𝑏𝑢+𝑐𝑣𝜑𝑑𝑥𝑑𝜏,𝑡0Ω𝑢𝜕𝜑𝜕𝑡+𝑢𝑚1𝜑𝑑𝑥𝑑𝜏+Ω𝑢(𝑥,𝑡)𝜑(𝑥,𝑡)𝑑𝑥Ω𝑣0(𝑥)𝜑(𝑥,0)𝑑𝑥𝑡0Ω𝑢𝛼𝑎𝑏𝑢+𝑐𝑣𝜑𝑑𝑥𝑑𝜏.(2.48) Take the test function as 𝜑(𝑥,𝑡)=𝐻𝜀𝑢𝑚1(𝑥,𝑡)𝑢𝑚1(𝑥,𝑡),(2.49) where 𝐻𝜀(𝑠) is a monotone increasing smooth approximation of the function 𝐻(𝑠) defined as follows: 𝐻(𝑠)=1,𝑠>0,0,otherwise.(2.50) It is easy to see that 𝐻𝜀(𝑠)𝛿(𝑠) as 𝜀0. Since 𝜕𝑢𝑚1/𝜕𝑡,𝜕𝑢𝑚1/𝜕𝑡𝐿2(𝑄𝑇), the test function 𝜑(𝑥,𝑡) is suitable. By the positivity of 𝑎,𝑏,𝑐 we have that Ω𝑢𝑢𝐻𝜀𝑢𝑚1𝑢𝑚1𝑑𝑥𝑡0Ω𝑢𝑢𝜕𝐻𝜀𝑢𝑚1𝑢𝑚1+𝜕𝑡𝑑𝑥𝑑𝜏𝑡0Ω𝐻𝜀𝑢𝑚1𝑢𝑚1||𝑢𝑚1𝑢𝑚1||2𝑑𝑥𝑑𝜏𝑡0Ω𝑎𝑢𝛼𝑢𝛼𝐻𝜀𝑢𝑚1𝑢𝑚1𝑢+𝑐𝛼𝑣𝑢𝛼𝑣𝐻𝜀𝑢𝑚1𝑢𝑚1𝑑𝑥𝑑𝜏,(2.51) where 𝐶 is a positive constant depending on 𝑎(𝑥,𝑡)𝐶(𝑄𝑡),𝑐(𝑥,𝑡)𝐶(𝑄𝑡). Letting 𝜀0 and noticing that 𝑡0Ω𝐻𝜀𝑢𝑚1𝑢𝑚1||𝑢𝑚𝑢𝑚||2𝑑𝑥𝑑𝜏0,(2.52) we arrive at Ω𝑢(𝑥,𝑡)𝑢(𝑥,𝑡)+𝑑𝑥𝐶𝑡0Ω𝑢𝛼𝑢𝛼++𝑣𝑢𝛼𝑢𝛼++𝑢𝛼𝑣𝑣+𝑑𝑥𝑑𝜏.(2.53) Let (𝑢,𝑣) be a supsolution with a positive lower bound 𝜎. Noticing that (𝑥𝛼𝑦𝛼)+𝐶(𝛼)(𝑥𝑦)+,for𝛼1,(𝑥𝛼𝑦𝛼)+𝑥𝛼1(𝑥𝑦)+𝑦𝛼1(𝑥𝑦)+,for𝛼<1,(2.54) with 𝑥,𝑦>0, we have that 𝑡0Ω𝑢𝛼𝑢𝛼++𝑣𝑢𝛼𝑢𝛼++𝑢𝛼𝑣𝑣+𝑑𝑥𝑑𝜏𝐶𝑡0Ω𝑢𝑢++𝑣𝑣+𝑑𝑥𝑑𝜏,(2.55) where 𝐶 is a positive constant depending upon 𝛼,𝜎,𝑀.
Similarly, we also have that Ω𝑣(𝑥,𝑡)𝑣(𝑥,𝑡)+𝑑𝑥𝐶𝑡0Ω𝑢𝑢++𝑣𝑣+𝑑𝑥𝑑𝜏.(2.56) Combining the above two inequalities, we obtain Ω𝑢(𝑥,𝑡)𝑢(𝑥,𝑡)++𝑣(𝑥,𝑡)𝑣(𝑥,𝑡)+𝑑𝑥𝐶𝑡0Ω𝑢𝑢++𝑣𝑣+𝑑𝑥𝑑𝜏.(2.57) By Gronwall's lemma, we see that 𝑢𝑢,𝑣𝑣. The proof is completed.

Corollary 2.7. If 𝑏𝑙𝑓𝑙>𝑐𝑀𝑒𝑀, 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=(𝑎𝑀𝑓𝑙+𝑑𝑀𝑐𝑀)/(𝑏𝑙𝑓𝑙𝑐𝑀𝑒𝑀) and 𝜌2=(𝑎𝑀𝑒𝑀+𝑑𝑀𝑏𝑙)/(𝑏𝑙𝑓𝑙𝑐𝑀𝑒𝑀), since 𝑏𝑙𝑓𝑙>𝑐𝑀𝑒𝑀; then 𝜌1,𝜌2>0 and satisfy 𝑎𝑀𝑏𝑙𝜌1+𝑐𝑀𝜌2=0,𝑑𝑀+𝑒𝑀𝜌1𝑓𝑙𝜌2=0.(2.58) Let (𝑢,𝑣)=(𝜂𝜌1,𝜂𝜌2), where 𝜂>1 is a constant such that (𝑢0,𝑣0)(𝜂𝜌1,𝜂𝜌2); then we have that 𝑢𝑡Δ𝑢𝑚1=0𝑢𝛼𝑎𝑏𝑢+𝑐𝑣,𝑣𝑡Δ𝑣𝑚2=0𝑣𝛽𝑑+𝑒𝑢𝑓𝑣.(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 𝑏𝑙𝑓𝑙>𝑐𝑀𝑒𝑀, 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,𝑥Ω,𝜙=0,𝑥𝜕Ω;(3.1) then we have that ||||𝜆>0,𝜙(𝑥)>0inΩ,𝜙>0on𝜕Ω,𝑀=max𝑥Ω𝜙(𝑥)<.(3.2) Let 𝑢,𝑣=𝜀𝜙2/𝑚1(𝑥),𝜀𝜙2/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(𝑄𝑇) as the test function, we have that 𝑄𝑇𝑢𝜕𝜑1𝜕𝑡+Δ𝑢𝑚1𝜑1+𝑢𝛼𝑎𝑏𝑢+𝑐𝑣𝜑1+𝑑𝑥𝑑𝑡Ω𝑢(𝑥,0)𝜑1(𝑥,0)𝑢(𝑥,𝑇)𝜑1=(𝑥,𝑇)𝑑𝑥𝑄𝑇𝑢𝛼𝑎𝑏𝑢+𝑐𝑣+Δ𝑢𝑚1𝜑1=𝑑𝑥𝑑𝑡𝑄𝑇𝑢𝛼𝑎𝑏𝑢+𝑐𝑣𝜑1𝑑𝑥𝑑𝑡𝑄𝑇𝑢𝑚1𝜑1=𝑑𝑥𝑑𝑡𝑄𝑇𝑢𝛼𝑎𝑏𝑢+𝑐𝑣𝜑1𝑑𝑥𝑑𝑡2𝜀𝑚1𝑄𝑇𝜙𝜙𝜑1=𝑑𝑥𝑑𝑡𝑄𝑇𝑢𝛼𝑎𝑏𝑢+𝑐𝑣𝜑1𝑑𝑥𝑑𝑡2𝜀𝑚1𝑄𝑇𝜙𝜙𝜑1||||𝜙2𝜑1=𝑑𝑥𝑑𝑡𝑄𝑇𝑢𝛼𝑎𝑏𝑢+𝑐𝑣𝜑1𝑑𝑥𝑑𝑡2𝜀𝑚1𝑄𝑇𝑑𝑖𝑣(𝜙)𝜙𝜑1||||𝜙2𝜑1=𝑑𝑥𝑑𝑡𝑄𝑇𝑢𝛼𝑎𝑏𝑢+𝑐𝑣𝜑1𝑑𝑥𝑑𝑡2𝜀𝑚1𝑄𝑇𝜆𝜙2||||𝜙2𝜑1𝑑𝑥𝑑𝑡.(3.4) Similarly, for any nonnegative test function 𝜑2(𝑥,𝑡)𝐶1(𝑄𝑇), we have that 𝑄𝑇𝑣𝜕𝜑2𝜕𝑡+Δ𝑣𝑚2𝜑2+𝑣𝛽𝑑+𝑒𝑢𝑓𝑣𝜑2𝑑𝑥𝑑𝑡+Ω𝑣(𝑥,0)𝜑2(𝑥,0)𝑣(𝑥,𝑇)𝜑2=(𝑥,𝑇)𝑑𝑥𝑄𝑇𝑣𝛽𝑑+𝑒𝑢𝑓𝑣𝜑2𝑑𝑥𝑑𝑡2𝜀𝑚2𝑄𝑇𝜆𝜙2||||𝜙2𝜑2𝑑𝑥𝑑𝑡.(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||||𝜙20,𝑥Ω𝛿,(3.6) where Ω𝛿={𝑥Ωdist(𝑥,𝜕Ω)𝛿}. Choosing 𝑎𝜀min𝑙𝑏𝑀𝑀2/𝑚1,𝑑𝑙𝑓𝑀𝑀2/𝑚2,(3.7) then we have that 2𝜀𝑚1𝑇0Ω𝛿𝜆𝜙2||||𝜙2𝜑1𝑑𝑥𝑑𝑡0𝑇0Ω𝛿𝑢𝛼𝑎𝑏𝑢+𝑐𝑣𝜑1𝑑𝑥𝑑𝑡,2𝜀𝑚2𝑇0Ω𝛿𝜆𝜙2||||𝜙2𝜑2𝑑𝑥𝑑𝑡0𝑇0Ω𝛿𝑣𝛽𝑑+𝑒𝑢𝑓𝑣𝜑2𝑑𝑥𝑑𝑡,(3.8) which shows that (𝑢,𝑣) is a positive (time independent, hence 𝑇-periodic) subsolution of (1.1)–(1.3) on Ω𝛿×(0,𝑇).
Moreover, we can see that, for some 𝜎>0, 𝜙(𝑥)𝜎>0,𝑥ΩΩ𝛿.(3.9) Choosing 𝑎𝜀min𝑙2𝑏𝑀𝑀2/𝑚1,𝑎𝑙4𝜆𝑀2(𝑚1𝛼)/𝑚11/(𝑚1𝛼),𝑑𝑙2𝑓𝑀𝑀2/𝑚2,𝑑𝑙4𝜆𝑀2(𝑚2𝛽)/𝑚21/(𝑚2𝛽),(3.10) then 𝜀𝛼𝜙2𝛼/𝑚1𝑎𝑏𝜀𝛼+1𝜙2(𝛼+1)/𝑚1+𝑐𝜀𝛼𝜙2𝛼/𝑚1𝜀𝜙2/𝑚22𝜀𝑚1𝜆𝜙2𝜀0,𝛽𝜙2𝛽/𝑚2𝑑+𝑒𝜀𝜙2/𝑚1𝜀𝛽𝜙2𝛽/𝑚2𝑓𝜀𝛽+1𝜙2(𝛽+1)/𝑚22𝜀𝑚2𝜆𝜙20(3.11) on 𝑄𝑇, that is 𝑄𝑇𝑢𝛼𝑎𝑏𝑢+𝑐𝑣𝜑1𝑑𝑥𝑑𝑡2𝜀𝑚1𝑄𝑇𝜆𝜙2||||𝜙2𝜑1𝑑𝑥𝑑𝑡0,𝑄𝑇𝑣𝛽𝑑+𝑒𝑢𝑓𝑣𝜑2𝑑𝑥𝑑𝑡2𝜀𝑚2𝑄𝑇𝜆𝜙2||||𝜙2𝜑2𝑑𝑥𝑑𝑡0.(3.12) These relations show that (𝑢,𝑣)=(𝜀𝜙2/𝑚11(𝑥),𝜀𝜙2/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=Δ𝑢𝑚+𝑓(𝑥,𝑡),(𝑥,𝑡)Ω×(0,𝑇),𝑢(𝑥,𝑡)=0,(𝑥,𝑡)𝜕Ω×(0,𝑇),(3.13) where 𝑓𝐿(Ω×(0,𝑇)); then there exist positive constants 𝐾 and 𝛼(0,1) depending only upon 𝜏(0,𝑇) and 𝑓𝐿(Ω×(0,𝑇)), such that, for any (𝑥𝑖,𝑡𝑖)Ω×[𝜏,𝑇](𝑖=1,2), ||𝑢𝑥1,𝑡1𝑥𝑢2,𝑡2||||𝑥𝐾1𝑥2||𝛼+||𝑡1𝑡2||𝛼/2.(3.14)

Lemma 3.3 (see [26]). Define a Poincaré mapping 𝑃𝑡𝐿(Ω)×𝐿(Ω)𝐿(Ω)×𝐿𝑃(Ω),𝑡𝑢0(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 𝑃𝑡 has the following properties: (i)𝑃𝑡 is defined for any 𝑡>0 and order preserving;(ii)𝑃𝑡 is order preserving;(iii)𝑃𝑡 is compact.

Observe that the operator 𝑃𝑇 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 𝑏𝑙𝑓𝑙>𝑐𝑀𝑒𝑀 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),in𝑄𝑇.(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 𝑃𝑇(𝑢(,0))𝑢(,𝑇). Hence by Definition 1.2 we get 𝑃𝑇(𝑢(,0))𝑢(,0), which implies 𝑃(𝑘+1)𝑇(𝑢(,0))𝑃𝑘𝑇(𝑢(,0)) for any 𝑘. Similarly we have that 𝑃𝑇(𝑢(,0))𝑢(,𝑇)𝑢(,0), and hence 𝑃(𝑘+1)𝑇(𝑢(,0))𝑃𝑘𝑇(𝑢(,0)) for any 𝑘. By Lemma 2.6, we have that 𝑃𝑘𝑇(𝑢(,0))𝑃𝑘𝑇(𝑢(,0)) for any 𝑘. Then 𝑢(𝑥,0)=lim𝑘𝑃𝑘𝑇𝑢(𝑥,0),𝑢(𝑥,0)=lim𝑘𝑃𝑘𝑇𝑢(𝑥,0)(3.17) exist for almost every 𝑥Ω. Since the operator 𝑃𝑇 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 𝑃𝑇. 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 𝑏𝑙𝑓𝑙>𝑐𝑀𝑒𝑀, 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 0𝑢𝑈+𝜀,0𝑣𝑉+𝜀,for𝑥Ω,𝑡𝑡.(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.

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