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

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

Jiebao Sun, Dazhi Zhang, and Boying Wu

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: 𝑢 𝑡 = Δ 𝑢 𝑚 1 + 𝑢 𝛼 ( 𝑎 − 𝑏 𝑢 + 𝑐 𝑣 ) , ( 𝑥 , 𝑡 ) ∈ Ω × ℝ + , 𝑣 ( 1 . 1 ) 𝑡 = Δ 𝑣 𝑚 2 + 𝑣 𝛽 ( 𝑑 + 𝑒 𝑢 − 𝑓 𝑣 ) , ( 𝑥 , 𝑡 ) ∈ Ω × ℝ + , ( 1 . 2 ) 𝑢 ( 𝑥 , 𝑡 ) = 0 , 𝑣 ( 𝑥 , 𝑡 ) = 0 , ( 𝑥 , 𝑡 ) ∈ 𝜕 Ω × ℝ + , ( 1 . 3 ) 𝑢 ( 𝑥 , 0 ) = 𝑢 0 ( 𝑥 ) , 𝑣 ( 𝑥 , 0 ) = 𝑣 0 ( 𝑥 ) , 𝑥 ∈ Ω , ( 1 . 4 ) where 𝑚 1 , 𝑚 2 > 1 , 0 < 𝛼 < 𝑚 1 , 0 < 𝛽 < 𝑚 2 , 1 ≤ ( 𝑚 1 − 𝛼 ) ( 𝑚 2 − 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 𝑢 0 ( 𝑥 ) and 𝑣 0 ( 𝑥 ) are nonnegative functions and satisfy 𝑢 𝑚 1 0 , 𝑣 𝑚 2 0 ∈ 𝑊 0 1 , 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 Δ 𝑢 𝑚 1 and Δ 𝑣 𝑚 2 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: 𝑢 𝑣 ( 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 𝑚 1 = 𝑚 2 = 𝛼 = 𝛽 = 1 , from (1.1)-(1.2) we get the following classical cooperative system: 𝑢 𝑡 𝑣 = Δ 𝑢 + 𝑢 ( 𝑎 − 𝑏 𝑢 + 𝑐 𝑣 ) , 𝑡 = Δ 𝑣 + 𝑣 ( 𝑑 + 𝑒 𝑢 − 𝑓 𝑣 ) . ( 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 𝑐 𝑒 < 𝑏 𝑓 , while the blowup solutions are possible when the two species are strongly mutualistic ( 𝑐 𝑒 > 𝑏 𝑓 ) . 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 = Δ 𝑢 𝑚 + 𝑢 ( 𝑎 − 𝑏 𝑢 ) . ( 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 𝑙 𝑓 𝑙 > 𝑐 𝑀 𝑒 𝑀 , ( 1 . 8 ) where 𝑓 𝑀 = s u p { 𝑓 ( 𝑥 , 𝑡 ) ∣ ( 𝑥 , 𝑡 ) ∈ Ω × ℝ } , 𝑓 𝑙 = i n f { 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 ∈ 𝐶 1 ( 𝑄 𝜏 ) with 𝜑 𝑖 | 𝜕 Ω × [ 0 , 𝜏 ) = 0 ( 𝑖 = 1 , 2 ) , ∇ 𝑢 𝑚 1 , ∇ 𝑣 𝑚 2 ∈ 𝐿 2 ( 𝑄 𝜏 ) , 𝜕 𝑢 𝑚 1 / 𝜕 𝑡 , 𝜕 𝑣 𝑚 2 / 𝜕 𝑡 ∈ 𝐿 2 ( 𝑄 𝜏 ) and 𝑄 𝜏 𝑢 𝜕 𝜑 1 𝜕 𝑡 − ∇ 𝑢 𝑚 1 ∇ 𝜑 1 + 𝑢 𝛼 ( 𝑎 − 𝑏 𝑢 + 𝑐 𝑣 ) 𝜑 1 𝑑 𝑥 𝑑 𝑡 = Ω 𝑢 ( 𝑥 , 𝜏 ) 𝜑 1 ( 𝑥 , 𝜏 ) 𝑑 𝑥 − Ω 𝑢 0 ( 𝑥 ) 𝜑 1 ( 𝑥 , 0 ) 𝑑 𝑥 , 𝑄 𝜏 𝑣 𝜕 𝜑 2 𝜕 𝑡 − ∇ 𝑣 𝑚 2 ∇ 𝜑 2 + 𝑣 𝛽 ( 𝑑 + 𝑒 𝑢 − 𝑓 𝑣 ) 𝜑 2 𝑑 𝑥 𝑑 𝑡 = Ω 𝑣 ( 𝑥 , 𝜏 ) 𝜑 2 ( 𝑥 , 𝜏 ) 𝑑 𝑥 − Ω 𝑣 0 ( 𝑥 ) 𝜑 2 ( 𝑥 , 0 ) 𝑑 𝑥 , ( 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 = d i v 𝑚 𝑢 𝑚 1 𝜀 − 1 + 𝜀 ∇ 𝑢 𝜀 + 𝑢 𝛼 𝜀 𝑎 − 𝑏 𝑢 𝜀 + 𝑐 𝑣 𝜀 , ( 𝑥 , 𝑡 ) ∈ 𝑄 𝑇 , ( 2 . 1 ) 𝜕 𝑣 𝜀 𝜕 𝑡 = d i v 𝑚 𝑣 𝑚 2 𝜀 − 1 + 𝜀 ∇ 𝑣 𝜀 + 𝑣 𝛽 𝜀 𝑑 + 𝑒 𝑢 𝜀 − 𝑓 𝑣 𝜀 , ( 𝑥 , 𝑡 ) ∈ 𝑄 𝑇 𝑢 , ( 2 . 2 ) 𝜀 ( 𝑥 , 𝑡 ) = 0 , 𝑣 𝜀 ( 𝑢 𝑥 , 𝑡 ) = 0 , ( 𝑥 , 𝑡 ) ∈ 𝜕 Ω × ( 0 , 𝑇 ) , ( 2 . 3 ) 𝜀 ( 𝑥 , 0 ) = 𝑢 0 𝜀 ( 𝑥 ) , 𝑣 𝜀 ( 𝑥 , 0 ) = 𝑣 0 𝜀 ( 𝑥 ) , 𝑥 ∈ Ω , ( 2 . 4 ) where 𝑄 𝑇 = Ω × ( 0 , 𝑇 ) , 0 < 𝜀 < 1 , 𝑢 0 𝜀 , 𝑣 0 𝜀 ∈ 𝐶 ∞ 0 ( Ω ) are nonnegative bounded smooth functions and satisfy 0 ≤ 𝑢 0 𝜀 ≤ ‖ ‖ 𝑢 0 ‖ ‖ 𝐿 ∞ ( Ω ) , 0 ≤ 𝑣 0 𝜀 ≤ ‖ ‖ 𝑣 0 ‖ ‖ 𝐿 ∞ ( Ω ) , 𝑢 𝑚 1 0 𝜀 ⟶ 𝑢 𝑚 1 0 , 𝑣 𝑚 2 0 𝜀 ⟶ 𝑣 𝑚 2 0 , i n 𝑊 0 1 , 2 ( Ω ) a s 𝜀 ⟶ 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 < ( 𝑚 1 − 𝛼 ) ( 𝑚 2 − 𝛽 ) , then there exist positive constants 𝑟 and 𝑠 large enough such that 1 𝑚 2 < 𝑚 − 𝛽 1 + 𝑟 − 1 𝑚 2 + 𝑠 − 1 < 𝑚 1 ‖ ‖ 𝑢 − 𝛼 , ( 2 . 6 ) 𝜀 ‖ ‖ 𝐿 𝑟 ( 𝑄 𝑇 ) ‖ ‖ 𝑣 ≤ 𝐶 , 𝜀 ‖ ‖ 𝐿 𝑠 ( 𝑄 𝑇 ) ≤ 𝐶 , ( 2 . 7 ) where 𝐶 is a positive constant only depending on 𝑚 1 , 𝑚 2 , 𝛼 , 𝛽 , 𝑟 , 𝑠 , | Ω | , and 𝑇 .(2) If 1 = ( 𝑚 1 − 𝛼 ) ( 𝑚 2 − 𝛽 ) , then (2.7) also holds when | Ω | is small enough.

Proof. Multiplying (2.1) by 𝑢 𝜀 𝑟 − 1 ( 𝑟 > 1 ) and integrating over Ω , we have that Ω 𝜕 𝑢 𝑟 𝜀 𝜕 𝑡 𝑑 𝑥 = − 4 𝑟 ( 𝑟 − 1 ) 𝑚 1 𝑚 1 + 𝑟 − 1 2 Ω | | | ∇ 𝑢 ( 𝑚 1 𝜀 + 𝑟 − 1 ) / 2 | | | 2 𝑑 𝑥 + 𝑟 Ω 𝑢 𝜀 𝛼 + 𝑟 − 1 𝑎 − 𝑏 𝑢 𝜀 + 𝑐 𝑣 𝜀 𝑑 𝑥 . ( 2 . 8 ) By Poincaré’s inequality, we have that 𝐾 Ω 𝑢 𝑚 1 𝜀 + 𝑟 − 1 𝑑 𝑥 ≤ Ω | | | ∇ 𝑢 ( 𝑚 1 𝜀 + 𝑟 − 1 ) / 2 | | | 2 𝑑 𝑥 , ( 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 < 𝑚 1 , Young's inequality shows that 𝑎 𝑢 𝜀 𝛼 + 𝑟 − 1 ≤ 𝐾 𝑟 ( 𝑟 − 1 ) 𝑚 1 𝑚 1 + 𝑟 − 1 2 𝑢 𝑚 1 𝜀 + 𝑟 − 1 + 𝐶 𝐾 − ( 𝛼 + 𝑟 − 1 ) / ( 𝑚 1 − 𝛼 ) , 𝑐 𝑢 𝜀 𝛼 + 𝑟 − 1 𝑣 𝜀 ≤ 𝐾 𝑟 ( 𝑟 − 1 ) 𝑚 1 𝑚 1 + 𝑟 − 1 2 𝑢 𝑚 1 𝜀 + 𝑟 − 1 + 𝐶 𝐾 − ( 𝛼 + 𝑟 − 1 ) / ( 𝑚 1 − 𝛼 ) 𝑣 ( 𝑚 1 + 𝑟 − 1 ) / ( 𝑚 1 𝜀 − 𝛼 ) . ( 2 . 1 0 ) 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 𝑟 𝜀 𝜕 𝑡 𝑑 𝑥 ≤ − 2 𝐾 𝑟 ( 𝑟 − 1 ) 𝑚 1 𝑚 1 + 𝑟 − 1 2 Ω 𝑢 𝑚 1 𝜀 + 𝑟 − 1 𝑑 𝑥 + 𝐶 𝐾 − ( 𝛼 + 𝑟 − 1 ) / ( 𝑚 1 − 𝛼 ) Ω 𝑣 ( 𝑚 1 + 𝑟 − 1 ) / ( 𝑚 1 𝜀 − 𝛼 ) 𝑑 𝑥 + 𝐶 𝐾 − ( 𝛼 + 𝑟 − 1 ) / ( 𝑚 1 − 𝛼 ) . ( 2 . 1 1 ) As a similar argument as above, for 𝑣 𝜀 and positive constant 𝑠 > 1 , we have that Ω 𝜕 𝑣 𝑠 𝜀 𝜕 𝑡 𝑑 𝑥 ≤ − 2 𝐾 𝑠 ( 𝑠 − 1 ) 𝑚 2 𝑚 2 + 𝑠 − 1 2 Ω 𝑣 𝑚 2 𝜀 + 𝑠 − 1 𝑑 𝑥 + 𝐶 𝐾 − ( 𝛽 + 𝑠 − 1 ) / ( 𝑚 2 − 𝛽 ) Ω 𝑢 ( 𝑚 2 + 𝑠 − 1 ) / ( 𝑚 2 𝜀 − 𝛽 ) 𝑑 𝑥 + 𝐶 𝐾 − ( 𝛽 + 𝑠 − 1 ) / ( 𝑚 2 − 𝛽 ) . ( 2 . 1 2 ) Thus we have that Ω 𝜕 𝑢 𝑟 𝜀 + 𝜕 𝑡 𝜕 𝑣 𝑠 𝜀 𝜕 𝑡 𝑑 𝑥 ≤ − 2 𝐾 𝑟 ( 𝑟 − 1 ) 𝑚 1 𝑚 1 + 𝑟 − 1 2 Ω 𝑢 𝑚 1 𝜀 + 𝑟 − 1 𝑑 𝑥 + 𝐶 𝐾 − ( 𝛽 + 𝑠 − 1 ) / ( 𝑚 2 − 𝛽 ) Ω 𝑢 ( 𝑚 2 + 𝑠 − 1 ) / ( 𝑚 2 𝜀 − 𝛽 ) − 𝑑 𝑥 2 𝐾 𝑠 ( 𝑠 − 1 ) 𝑚 2 𝑚 2 + 𝑠 − 1 2 Ω 𝑣 𝑚 2 𝜀 + 𝑠 − 1 𝑑 𝑥 + 𝐶 𝐾 − ( 𝛼 + 𝑟 − 1 ) / ( 𝑚 1 − 𝛼 ) Ω 𝑣 ( 𝑚 1 + 𝑟 − 1 ) / ( 𝑚 1 𝜀 − 𝛼 ) 𝑑 𝑥 + 𝐶 𝐾 − ( 𝛼 + 𝑟 − 1 ) / ( 𝑚 1 − 𝛼 ) + 𝐶 𝐾 − ( 𝛽 + 𝑠 − 1 ) / ( 𝑚 2 − 𝛽 ) . ( 2 . 1 3 )
For the case of 1 < ( 𝑚 1 − 𝛼 ) ( 𝑚 2 − 𝛽 ) , there exist 𝑟 , 𝑠 large enough such that 1 𝑚 1 < 𝑚 − 𝛼 2 + 𝑠 − 1 𝑚 1 + 𝑟 − 1 < 𝑚 2 − 𝛽 . ( 2 . 1 4 ) By Young's inequality, we have that Ω 𝑢 ( 𝑚 2 + 𝑠 − 1 ) / ( 𝑚 2 𝜀 − 𝛽 ) 𝑑 𝑥 ≤ 𝑟 ( 𝑟 − 1 ) 𝑚 1 𝐾 ( 𝑚 2 + 𝑠 − 1 ) / ( 𝑚 2 − 𝛽 ) 𝐶 𝑚 1 + 𝑟 − 1 2 Ω 𝑢 𝑚 1 𝜀 + 𝑟 − 1 𝑑 𝑥 + 𝐶 𝐾 − 𝛾 1 , Ω 𝑣 ( 𝑚 1 + 𝑟 − 1 ) / ( 𝑚 1 𝜀 − 𝛼 ) 𝑑 𝑥 ≤ 𝑠 ( 𝑠 − 1 ) 𝑚 2 𝐾 ( 𝑚 1 + 𝑟 − 1 ) / ( 𝑚 1 − 𝛼 ) 𝐶 𝑚 2 + 𝑠 − 1 𝑝 2 Ω 𝑣 𝑚 2 𝜀 + 𝑠 − 1 𝑑 𝑥 + 𝐶 𝐾 − 𝛾 2 , ( 2 . 1 5 ) where 𝛾 1 = 𝑚 2 + 𝑠 − 1 2 𝑚 2 𝑚 − 𝛽 2 𝑚 − 𝛽 1 − 𝑚 + 𝑟 − 1 2 , 𝛾 + 𝑠 − 1 2 = 𝑚 1 + 𝑟 − 1 2 𝑚 1 𝑚 − 𝛼 1 𝑚 − 𝛼 2 − 𝑚 + 𝑠 − 1 1 . + 𝑟 − 1 ( 2 . 1 6 ) Together with (2.13), we have that Ω 𝜕 𝑢 𝑟 𝜀 + 𝜕 𝑡 𝜕 𝑣 𝑠 𝜀 𝜕 𝑡 𝑑 𝑥 ≤ − 𝐾 Ω 𝑢 𝑚 1 𝜀 + 𝑟 − 1 + 𝑣 𝑚 2 𝜀 + 𝑠 − 1 𝐾 𝑑 𝑥 + 𝐶 − 𝜃 1 + 𝐾 − 𝜃 2 + 𝐶 𝐾 − ( 𝛼 + 𝑟 − 1 ) / ( 𝑚 1 − 𝛼 ) + 𝐶 𝐾 − ( 𝛽 + 𝑠 − 1 ) / ( 𝑚 2 − 𝛽 ) , ( 2 . 1 7 ) where 𝜃 1 = 𝑚 2 + 𝑚 + 𝑠 − 1 1 + 𝑟 − 1 ( 𝛽 + 𝑠 − 1 ) 𝑚 2 𝑚 − 𝛽 1 − 𝑚 + 𝑟 − 1 2 + 𝑠 − 1 , 𝜃 2 = 𝑚 1 + 𝑚 + 𝑟 − 1 2 + 𝑠 − 1 ( 𝛼 + 𝑟 − 1 ) 𝑚 1 𝑚 − 𝛼 2 − 𝑚 + 𝑠 − 1 1 + 𝑟 − 1 . ( 2 . 1 8 ) Furthermore, by Hölder's and Young's inequalities, from (2.17) we obtain Ω 𝜕 𝑢 𝑟 𝜀 + 𝜕 𝑡 𝜕 𝑣 𝑠 𝜀 𝜕 𝑡 𝑑 𝑥 ≤ − 𝐾 Ω 𝑢 𝑟 𝜀 + 𝑣 𝑠 𝜀 𝐾 𝑑 𝑥 + 𝐶 − 𝜃 1 + 𝐾 − 𝜃 2 | | Ω | | + 2 𝐾 + 𝐶 𝐾 − ( 𝛼 + 𝑟 − 1 ) / ( 𝑚 1 − 𝛼 ) + 𝐶 𝐾 − ( 𝛽 + 𝑠 − 1 ) / ( 𝑚 2 − 𝛽 ) . ( 2 . 1 9 ) Then by Gronwall's inequality, we obtain Ω 𝑢 𝑟 𝜀 + 𝑣 𝑠 𝜀 𝑑 𝑥 ≤ 𝐶 . ( 2 . 2 0 )
Now we consider the case of 1 = ( 𝑚 1 − 𝛼 ) ( 𝑚 2 − 𝛽 ) . It is easy to see that there exist positive constants 𝑟 , 𝑠 large enough such that 1 𝑚 1 = 𝑚 − 𝛼 2 + 𝑠 − 1 𝑚 1 + 𝑟 − 1 = 𝑚 2 − 𝛽 . ( 2 . 2 1 ) Due to the continuous dependence of 𝐾 upon | Ω | in (2.9), from (2.13) we have that Ω 𝜕 𝑢 𝑟 𝜀 + 𝜕 𝑡 𝜕 𝑣 𝑠 𝜀 𝜕 𝑡 𝑑 𝑥 ≤ − 𝐾 Ω 𝑢 𝑚 1 𝜀 + 𝑟 − 1 + 𝑣 𝑚 2 ( 𝑝 2 𝜀 − 1 ) + 𝑠 − 1 𝑑 𝑥 + 𝐶 ( 2 . 2 2 ) 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 𝑚 1 𝜀 , 𝑣 𝑚 2 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 𝑄 𝑇 | | ∇ 𝑢 𝑚 1 𝜀 | | 2 𝑑 𝑥 𝑑 𝑡 ≤ 𝐶 , 𝑄 𝑇 | | ∇ 𝑣 𝑚 2 𝜀 | | 2 𝑑 𝑥 𝑑 𝑡 ≤ 𝐶 , ( 2 . 2 3 ) 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 ∞ ( 𝑄 𝑇 ) ‖ ‖ 𝑣 ≤ 𝐶 , 𝜀 ‖ ‖ 𝐿 ∞ ( 𝑄 𝑇 ) ≤ 𝐶 , ( 2 . 2 4 ) where 𝐶 is a positive constant independent of 𝜀 .

Proof. For a positive constant 𝑘 > ‖ 𝑢 0 𝜀 ‖ 𝐿 ∞ ( Ω ) , multiplying (2.1) by ( 𝑢 𝜀 − 𝑘 ) 𝑚 1 + 𝜒 [ 𝑡 1 , 𝑡 2 ] and integrating the results over 𝑄 𝑇 , we have that 1 𝑚 1 + 1 𝑄 𝑇 𝜕 𝑢 𝜀 − 𝑘 𝑚 1 + + 1 𝜒 [ 𝑡 1 , 𝑡 2 ] 𝜕 𝑡 𝑑 𝑥 𝑑 𝑡 + 𝑄 𝑇 | | ∇ 𝑢 𝜀 − 𝑘 𝑚 1 + 𝜒 [ 𝑡 1 , 𝑡 2 ] | | 2 ≤ 𝑑 𝑥 𝑑 𝑡 𝑄 𝑇 𝑎 𝑢 𝛼 + 𝑚 1 𝜀 𝑎 + 𝑐 𝑣 𝜀 𝑑 𝑥 𝑑 𝑡 , ( 2 . 2 5 ) where 𝑠 + = m a x { 0 , 𝑠 } and 𝜒 [ 𝑡 1 , 𝑡 2 ] is the characteristic function of [ 𝑡 1 , 𝑡 2 ] ( 0 ≤ 𝑡 1 < 𝑡 2 ≤ 𝑇 ) . Let 𝐼 𝑘 ( 𝑡 ) = Ω 𝑢 𝜀 − 𝑘 𝑚 1 + + 1 𝑑 𝑥 ; ( 2 . 2 6 ) 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 𝑡 1 = 𝜎 − 𝜖 , 𝑡 2 = 𝜎 in (2.25), we obtain 1 𝑚 1 𝜖 + 1 𝜎 𝜎 − 𝜖 Ω 𝜕 𝑢 𝜀 − 𝑘 𝑚 1 + + 1 1 𝜕 𝑡 𝑑 𝑥 𝑑 𝑡 + 𝜖 𝜎 𝜎 − 𝜖 Ω | | ∇ 𝑢 𝜀 − 𝑘 𝑚 1 + | | 2 ≤ 1 𝑑 𝑥 𝑑 𝑡 𝜖 𝜎 𝜎 − 𝜖 Ω 𝑢 𝛼 + 𝑚 1 𝜀 𝑎 + 𝑐 𝑣 𝜀 𝑑 𝑥 𝑑 𝑡 . ( 2 . 2 7 ) From 𝜎 𝜎 − 𝜖 Ω 𝜕 𝑢 𝜀 − 𝑘 𝑚 1 + + 1 𝜕 𝑡 𝑑 𝑥 𝑑 𝑡 = 𝐼 𝑘 ( 𝜎 ) − 𝐼 𝑘 ( 𝜎 − 𝜖 ) ≥ 0 , ( 2 . 2 8 ) we have that 1 𝜖 𝜎 𝜎 − 𝜖 Ω | | ∇ 𝑢 𝜀 − 𝑘 𝑚 1 + | | 2 1 𝑑 𝑥 𝑑 𝑡 ≤ 𝜖 𝜎 𝜎 − 𝜖 Ω 𝑢 𝛼 + 𝑚 1 𝜀 𝑎 + 𝑐 𝑣 𝜀 𝑑 𝑥 𝑑 𝑡 . ( 2 . 2 9 ) Letting 𝜖 → 0 + , we have that Ω | | ∇ 𝑢 𝜀 ( 𝑥 , 𝜎 ) − 𝑘 𝑚 1 + | | 2 𝑑 𝑥 ≤ Ω 𝑢 𝛼 + 𝑚 1 𝜀 ( 𝑥 , 𝜎 ) 𝑎 + 𝑐 𝑣 𝜀 ( 𝑥 , 𝜎 ) 𝑑 𝑥 . ( 2 . 3 0 ) Denote 𝐴 𝑘 ( 𝑡 ) = { 𝑥 ∶ 𝑢 𝜀 ( 𝑥 , 𝑡 ) > 𝑘 } and 𝜇 𝑘 = s u p 𝑡 ∈ ( 0 , 𝑇 ) | 𝐴 𝑘 ( 𝑡 ) | ; then 𝐴 𝑘 ( 𝜎 ) | | ∇ 𝑢 𝜀 − 𝑘 𝑚 1 + | | 2 𝑑 𝑥 ≤ 𝐴 𝑘 ( 𝜎 ) 𝑢 𝛼 + 𝑚 1 𝜀 𝑎 + 𝑐 𝑣 𝜀 𝑑 𝑥 . ( 2 . 3 1 ) By Sobolev's theorem, 𝐴 𝑘 ( 𝜎 ) 𝑢 𝜀 − 𝑘 𝑚 1 + 𝑝 𝑑 𝑥 1 / 𝑝 ≤ 𝐶 𝐴 𝑘 ( 𝜎 ) | | ∇ 𝑢 𝜀 − 𝑘 𝑚 1 + | | 2 𝑑 𝑥 1 / 2 , ( 2 . 3 2 ) with ⎧ ⎪ ⎨ ⎪ ⎩ 2 < 𝑝 < + ∞ , 𝑁 ≤ 2 , 2 𝑁 𝑁 − 2 , 𝑁 > 2 , ( 2 . 3 3 ) we obtain 𝐴 𝑘 ( 𝜎 ) 𝑢 𝜀 − 𝑘 𝑚 1 + 𝑝 𝑑 𝑥 2 / 𝑝 ≤ 𝐶 𝐴 𝑘 ( 𝜎 ) | | ∇ 𝑢 𝜀 − 𝑘 𝑚 1 + | | 2 𝑑 𝑥 ≤ 𝐶 𝐴 𝑘 ( 𝜎 ) 𝑢 𝛼 + 𝑚 1 𝜀 𝑎 + 𝑣 𝜀 𝑑 𝑥 ≤ 𝐶 𝐴 𝑘 ( 𝜎 ) 𝑢 𝑟 𝜀 𝑑 𝑥 ( 𝑚 1 + 𝛼 ) / 𝑟 𝐴 𝑘 ( 𝜎 ) ( 𝑎 + 𝑣 𝜀 ) 𝑟 / ( 𝑟 − 𝑚 1 − 𝛼 ) 𝑑 𝑥 ( 𝑟 − 𝑚 1 − 𝛼 ) / 𝑟 ≤ 𝐶 𝐴 𝑘 ( 𝜎 ) ( 𝑎 + 𝑣 𝜀 ) 𝑟 / ( 𝑟 − 𝑚 1 − 𝛼 ) 𝑑 𝑥 ( 𝑟 − 𝑚 1 − 𝛼 ) / 𝑟 ≤ 𝐶 𝐴 𝑘 ( 𝜎 ) 𝑎 + 𝑣 𝜀 𝑠 𝑑 𝑥 1 / 𝑠 | | 𝐴 𝑘 | | ( 𝜎 ) ( 𝑠 ( 𝑟 − 𝑚 1 − 𝛼 ) − 𝑟 ) / 𝑠 𝑟 ≤ 𝐶 𝜇 ( 𝑠 ( 𝑟 − 𝑚 1 𝑘 − 𝛼 ) − 𝑟 ) / 𝑠 𝑟 , ( 2 . 3 4 ) where 𝑟 > 𝑝 ( 𝑚 1 + 𝛼 ) / ( 𝑝 − 2 ) , 𝑠 > 𝑝 𝑟 / ( 𝑝 ( 𝑟 − 𝑚 1 − 𝛼 ) − 2 𝑟 ) and 𝐶 denotes various positive constants independent of 𝜀 . By Hölder’s inequality, it yields 𝐼 𝑘 ( 𝜎 ) = Ω 𝑢 𝜀 − 𝑘 𝑚 1 + + 1 𝑑 𝑥 = 𝐴 𝑘 ( 𝜎 ) 𝑢 𝜀 − 𝑘 𝑚 1 + + 1 ≤ 𝑑 𝑥 𝐴 𝑘 ( 𝜎 ) 𝑢 𝜀 − 𝑘 𝑚 1 𝑝 + 𝑑 𝑥 ( 𝑚 1 + 1 ) / 𝑚 1 𝑝 𝜇 1 − ( 𝑚 1 + 1 ) / 𝑚 1 𝑝 𝑘 ≤ 𝐶 𝜇 1 + [ 𝑠 𝑝 ( 𝑟 − 𝑚 1 − 𝛼 ) − 𝑝 𝑟 − 2 𝑠 𝑟 ] ( 𝑚 1 + 1 ) / 2 𝑝 𝑠 𝑟 𝑚 1 𝑘 . ( 2 . 3 5 ) Then 𝐼 𝑘 ( 𝑡 ) ≤ 𝐼 𝑘 ( 𝜎 ) ≤ 𝐶 𝜇 1 + [ 𝑠 𝑝 ( 𝑟 − 𝑚 1 − 𝛼 ) − 𝑝 𝑟 − 2 𝑠 𝑟 ] ( 𝑚 1 + 1 ) / 2 𝑝 𝑠 𝑟 𝑚 1 𝑘 [ ] . , 𝑡 ∈ 0 , 𝑇 ( 2 . 3 6 ) On the other hand, for any ℎ > 𝑘 and 𝑡 ∈ [ 0 , 𝑇 ] , we have that 𝐼 𝑘 ( 𝑡 ) ≥ 𝐴 𝑘 ( 𝑡 ) 𝑢 𝜀 − 𝑘 𝑚 1 + + 1 𝑑 𝑥 ≥ ( ℎ − 𝑘 ) 𝑚 1 + 1 | | 𝐴 ℎ ( | | . 𝑡 ) ( 2 . 3 7 ) Combined with (2.35), it yields ( ℎ − 𝑘 ) 𝑚 1 + 1 𝜇 ℎ ≤ 𝐶 𝜇 1 + [ 𝑠 𝑝 ( 𝑟 − 𝑚 1 − 𝛼 ) − 𝑝 𝑟 − 2 𝑠 𝑟 ] ( 𝑚 1 + 1 ) / 2 𝑝 𝑠 𝑟 𝑚 1 𝑘 , ( 2 . 3 8 ) that is, 𝜇 ℎ ≤ 𝐶 ( ℎ − 𝑘 ) 𝑚 1 + 1 𝜇 1 + [ 𝑠 𝑝 ( 𝑟 − 𝑚 1 − 𝛼 ) − 𝑝 𝑟 − 2 𝑠 𝑟 ] ( 𝑚 1 + 1 ) / 2 𝑝 𝑠 𝑟 𝑚 1 𝑘 . ( 2 . 3 9 ) It is easy to see that 𝛾 = 1 + 𝑠 𝑝 𝑟 − 𝑚 1 𝑚 − 𝛼 − 𝑝 𝑟 − 2 𝑠 𝑟 1 + 1 2 𝑝 𝑠 𝑟 𝑚 1 > 1 . ( 2 . 4 0 ) Then by the De Giorgi iteration lemma [22], we have that 𝜇 𝑙 + 𝑑 | | 𝐴 = s u p 𝑙 + 𝑑 | | ( 𝑡 ) = 0 , ( 2 . 4 1 ) where 𝑑 = 𝐶 1 / ( 𝑚 1 + 1 ) 𝜇 ( 𝛾 − 1 ) / ( 𝑚 1 𝑙 + 1 ) 2 𝛾 / ( 𝛾 − 1 ) . That is, 𝑢 𝜀 ≤ 𝑙 + 𝑑 a . e . i n 𝑄 𝑇 . ( 2 . 4 2 )
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: 𝑄 𝑇 | | | | 𝜕 𝑢 𝑚 1 𝜀 | | | | 𝜕 𝑡 2 𝑑 𝑥 𝑑 𝑡 ≤ 𝐶 , 𝑄 𝑇 | | | | 𝜕 𝑣 𝑚 2 𝜀 | | | | 𝜕 𝑡 2 𝑑 𝑥 𝑑 𝑡 ≤ 𝐶 , ( 2 . 4 3 ) where 𝐶 is a positive constant independent of 𝜀 .

Proof. Multiplying (2.1) by ( 𝜕 / 𝜕 𝑡 ) 𝑢 𝑚 1 𝜀 and integrating over Ω , by (2.3), (2.4) and Young's inequality we have that 4 𝑚 1 𝑚 1 + 1 2 𝑄 𝑇 | | | 𝜕 𝑢 𝜕 𝑡 ( 𝑚 1 𝜀 + 1 ) / 2 | | | 2 = 𝑑 𝑥 𝑑 𝑡 𝑄 𝑇 𝜕 𝑢 𝜀 𝜕 𝑡 𝜕 𝑢 𝑚 1 𝜀 = 1 𝜕 𝑡 𝑑 𝑥 𝑑 𝑡 2 Ω | | ∇ 𝑢 𝑚 1 𝜀 ( | | 𝑥 , 0 ) 2 1 𝑑 𝑥 − 2 Ω | | ∇ 𝑢 𝑚 1 𝜀 ( | | 𝑥 , 𝑇 ) 2 + 𝑑 𝑥 𝑄 𝑇 𝑚 1 𝑢 𝛼 + 𝑚 1 𝜀 − 1 𝑎 − 𝑏 𝑢 𝜀 + 𝑐 𝑣 𝜀 𝜕 𝑢 𝜀 = 1 𝜕 𝑡 𝑑 𝑥 𝑑 𝑡 2 Ω | | ∇ 𝑢 𝑚 1 𝜀 | | ( 𝑥 , 0 ) 2 1 𝑑 𝑥 − 2 Ω | | ∇ 𝑢 𝑚 1 𝜀 | | ( 𝑥 , 𝑇 ) 2 + 𝑑 𝑥 𝑄 𝑇 2 𝑚 1 𝑚 1 𝑢 + 1 ( 2 𝛼 + 𝑚 1 𝜀 − 1 ) / 2 𝑎 − 𝑏 𝑢 𝜀 + 𝑐 𝑣 𝜀 𝜕 𝑢 ( 𝑚 1 𝜀 + 1 ) / 2 ≤ 1 𝜕 𝑡 𝑑 𝑥 𝑑 𝑡 2 Ω | | ∇ 𝑢 𝑚 1 𝜀 | | ( 𝑥 , 0 ) 2 𝑑 𝑥 + 2 𝑚 1 𝑄 𝑇 𝑢 2 𝛼 + 𝑚 1 𝜀 − 1 𝑎 − 𝑏 𝑢 𝜀 + 𝑐 𝑣 𝜀 2 + 𝑑 𝑥 𝑑 𝑡 2 𝑚 1 𝑚 1 + 1 2 𝑄 𝑇 | | | 𝜕 𝑢 𝜕 𝑡 ( 𝑚 1 𝜀 + 1 ) / 2 | | | 2 𝑑 𝑥 𝑑 𝑡 , ( 2 . 4 4 ) which together with the bound of 𝑎 , 𝑏 , 𝑐 , 𝑢 𝜀 , 𝑣 𝜀 shows that 𝑄 𝑇 | | | | 𝜕 𝑢 ( 𝑚 1 𝜀 + 1 ) / 2 | | | | 𝜕 𝑡 2 𝑑 𝑥 𝑑 𝑡 ≤ 𝐶 , ( 2 . 4 5 ) where 𝐶 is a positive constant independent of 𝜀 . Noticing the bound of 𝑢 𝜀 , we have that 𝑄 𝑇 | | | | 𝜕 𝑢 𝑚 1 𝜀 | | | | 𝜕 𝑡 2 𝑑 𝑥 𝑑 𝑡 = 4 𝑚 2 1 𝑚 1 + 1 2 𝑄 𝑇 𝑢 𝑚 1 𝜀 − 1 | | | 𝜕 𝑢 𝜕 𝑡 ( 𝑚 1 𝜀 + 1 ) / 2 | | | 2 𝑑 𝑥 𝑑 𝑡 ≤ 𝐶 . ( 2 . 4 6 ) 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 𝜀 ⟶ 𝑢 , 𝑣 𝜀 ⟶ 𝑣 , a . e i n 𝑄 𝑇 , 𝜕 𝑢 𝑚 1 𝜀 ⟶ 𝜕 𝑡 𝜕 𝑢 𝑚 1 , 𝜕 𝑡 𝜕 𝑣 𝑚 2 𝜀 ⟶ 𝜕 𝑡 𝜕 𝑣 𝑚 2 𝜕 𝑡 , w e a k l y i n 𝐿 2 𝑄 𝑇 , ∇ 𝑢 𝑚 1 𝜀 ⟶ ∇ 𝑢 𝑚 1 , ∇ 𝑣 𝑚 2 𝜀 ⟶ ∇ 𝑣 𝑚 2 , w e a k l y i n 𝐿 2 𝑄 𝑇 , ( 2 . 4 7 ) 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 ( 𝑢 0 , 𝑣 0 ) and ( 𝑢 , 𝑣 ) a supersolution with a positive lower bound of the problem (1.1)–(1.4) with the initial value ( 𝑢 0 , 𝑣 0 ) . If 𝑢 0 ≤ 𝑣 0 , 𝑢 0 ≤ 𝑣 0 , 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 ∞ ( 𝑄 𝑇 ) , ‖ 𝑢 ( 𝑥 , 𝑡 ) ‖ 𝐿 ∞ ( 𝑄 𝑇 ) , ‖ 𝑣 ( 𝑥 , 𝑡 ) ‖ 𝐿 ∞ ( 𝑄 𝑇 ) , ‖ 𝑣 ( 𝑥 , 𝑡 ) ‖ 𝐿 ∞ ( 𝑄 𝑇 ) ≤ 𝑀 , where 𝑀 is a positive constant. By the definitions of subsolution and supersolution, we have that 𝑡 0 Ω − 𝑢 𝜕 𝜑 𝜕 𝑡 + ∇ 𝑢 𝑚 1 ∇ 𝜑 𝑑 𝑥 𝑑 𝜏 + Ω 𝑢 ( 𝑥 , 𝑡 ) 𝜑 ( 𝑥 , 𝑡 ) 𝑑 𝑥 − Ω 𝑢 0 ( ≤ 𝑥 ) 𝜑 ( 𝑥 , 0 ) 𝑑 𝑥 𝑡 0 Ω 𝑢 𝛼 𝑎 − 𝑏 𝑢 + 𝑐 𝑣 𝜑 𝑑 𝑥 𝑑 𝜏 , 𝑡 0 Ω − 𝑢 𝜕 𝜑 𝜕 𝑡 + ∇ 𝑢 𝑚 1 ∇ 𝜑 𝑑 𝑥 𝑑 𝜏 + Ω 𝑢 ( 𝑥 , 𝑡 ) 𝜑 ( 𝑥 , 𝑡 ) 𝑑 𝑥 − Ω 𝑣 0 ≥ ( 𝑥 ) 𝜑 ( 𝑥 , 0 ) 𝑑 𝑥 𝑡 0 Ω 𝑢 𝛼 𝑎 − 𝑏 𝑢 + 𝑐 𝑣 𝜑 𝑑 𝑥 𝑑 𝜏 . ( 2 . 4 8 ) Take the test function as 𝜑 ( 𝑥 , 𝑡 ) = 𝐻 𝜀 𝑢 𝑚 1 ( 𝑥 , 𝑡 ) − 𝑢 𝑚 1 ( 𝑥 , 𝑡 ) , ( 2 . 4 9 ) where 𝐻 𝜀 ( 𝑠 ) is a monotone increasing smooth approximation of the function 𝐻 ( 𝑠 ) defined as follows: 𝐻 ( 𝑠 ) = 1 , 𝑠 > 0 , 0 , o t h e r w i s e . ( 2 . 5 0 ) It is easy to see that 𝐻 𝜀 ′ ( 𝑠 ) → 𝛿 ( 𝑠 ) as 𝜀 → 0 . Since 𝜕 𝑢 𝑚 1 / 𝜕 𝑡 , 𝜕 𝑢 𝑚 1 / 𝜕 𝑡 ∈ 𝐿 2 ( 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 𝑚 1 − 𝑢 𝑚 1 𝑑 𝑥 − 𝑡 0 Ω 𝑢 − 𝑢 𝜕 𝐻 𝜀 𝑢 𝑚 1 − 𝑢 𝑚 1 + 𝜕 𝑡 𝑑 𝑥 𝑑 𝜏 𝑡 0 Ω 𝐻 𝜀 𝑢 𝑚 1 − 𝑢 𝑚 1 | | ∇ 𝑢 𝑚 1 − 𝑢 𝑚 1 | | 2 ≤ 𝑑 𝑥 𝑑 𝜏 𝑡 0 Ω 𝑎 𝑢 𝛼 − 𝑢 𝛼 𝐻 𝜀 𝑢 𝑚 1 − 𝑢 𝑚 1 𝑢 + 𝑐 𝛼 𝑣 − 𝑢 𝛼 𝑣 𝐻 𝜀 𝑢 𝑚 1 − 𝑢 𝑚 1 𝑑 𝑥 𝑑 𝜏 , ( 2 . 5 1 ) where 𝐶 is a positive constant depending on ‖ 𝑎 ( 𝑥 , 𝑡 ) ‖ 𝐶 ( 𝑄 𝑡 ) , ‖ 𝑐 ( 𝑥 , 𝑡 ) ‖ 𝐶 ( 𝑄 𝑡 ) . Letting 𝜀 → 0 and noticing that 𝑡 0 Ω 𝐻 𝜀 𝑢 𝑚 1 − 𝑢 𝑚 1 | | ∇ 𝑢 𝑚 − 𝑢 𝑚 | | 2 𝑑 𝑥 𝑑 𝜏 ≥ 0 , ( 2 . 5 2 ) we arrive at Ω 𝑢 ( 𝑥 , 𝑡 ) − 𝑢 ( 𝑥 , 𝑡 ) + 𝑑 𝑥 ≤ 𝐶 𝑡 0 Ω 𝑢 𝛼 − 𝑢 𝛼 + + 𝑣 𝑢 𝛼 − 𝑢 𝛼 + + 𝑢 𝛼 𝑣 − 𝑣 + 𝑑 𝑥 𝑑 𝜏 . ( 2 . 5 3 ) Let ( 𝑢 , 𝑣 ) be a supsolution with a positive lower bound 𝜎 . Noticing that ( 𝑥 𝛼 − 𝑦 𝛼 ) + ≤ 𝐶 ( 𝛼 ) ( 𝑥 − 𝑦 ) + , f o r 𝛼 ≥ 1 , ( 𝑥 𝛼 − 𝑦 𝛼 ) + ≤ 𝑥 𝛼 − 1 ( 𝑥 − 𝑦 ) + ≤ 𝑦 𝛼 − 1 ( 𝑥 − 𝑦 ) + , f o r 𝛼 < 1 , ( 2 . 5 4 ) with 𝑥 , 𝑦 > 0 , we have that 𝑡 0 Ω 𝑢 𝛼 − 𝑢 𝛼 + + 𝑣 𝑢 𝛼 − 𝑢 𝛼 + + 𝑢 𝛼 𝑣 − 𝑣 + 𝑑 𝑥 𝑑 𝜏 ≤ 𝐶 𝑡 0 Ω 𝑢 − 𝑢 + + 𝑣 − 𝑣 + 𝑑 𝑥 𝑑 𝜏 , ( 2 . 5 5 ) where 𝐶 is a positive constant depending upon 𝛼 , 𝜎 , 𝑀 .
Similarly, we also have that Ω 𝑣 ( 𝑥 , 𝑡 ) − 𝑣 ( 𝑥 , 𝑡 ) + 𝑑 𝑥 ≤ 𝐶 𝑡 0 Ω 𝑢 − 𝑢 + + 𝑣 − 𝑣 + 𝑑 𝑥 𝑑 𝜏 . ( 2 . 5 6 ) Combining the above two inequalities, we obtain Ω 𝑢 ( 𝑥 , 𝑡 ) − 𝑢 ( 𝑥 , 𝑡 ) + + 𝑣 ( 𝑥 , 𝑡 ) − 𝑣 ( 𝑥 , 𝑡 ) + 𝑑 𝑥 ≤ 𝐶 𝑡 0 Ω 𝑢 − 𝑢 + + 𝑣 − 𝑣 + 𝑑 𝑥 𝑑 𝜏 . ( 2 . 5 7 ) By Gronwall's lemma, we see that 𝑢 ≤ 𝑢 , 𝑣 ≤ 𝑣 . The proof is completed.

Corollary 2.7. If 𝑏 𝑙 𝑓 𝑙 > 𝑐 𝑀 𝑒 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 𝜌 1 = ( 𝑎 𝑀 𝑓 𝑙 + 𝑑 𝑀 𝑐 𝑀 ) / ( 𝑏 𝑙 𝑓 𝑙 − 𝑐 𝑀 𝑒 𝑀 ) and 𝜌 2 = ( 𝑎 𝑀 𝑒 𝑀 + 𝑑 𝑀 𝑏 𝑙 ) / ( 𝑏 𝑙 𝑓 𝑙 − 𝑐 𝑀 𝑒 𝑀 ) , since 𝑏 𝑙 𝑓 𝑙 > 𝑐 𝑀 𝑒 𝑀 ; then 𝜌 1 , 𝜌 2 > 0 and satisfy 𝑎 𝑀 − 𝑏 𝑙 𝜌 1 + 𝑐 𝑀 𝜌 2 = 0 , 𝑑 𝑀 + 𝑒 𝑀 𝜌 1 − 𝑓 𝑙 𝜌 2 = 0 . ( 2 . 5 8 ) Let ( 𝑢 , 𝑣 ) = ( 𝜂 𝜌 1 , 𝜂 𝜌 2 ) , where 𝜂 > 1 is a constant such that ( 𝑢 0 , 𝑣 0 ) ≤ ( 𝜂 𝜌 1 , 𝜂 𝜌 2 ) ; then we have that 𝑢 𝑡 − Δ 𝑢 𝑚 1 = 0 ≥ 𝑢 𝛼 𝑎 − 𝑏 𝑢 + 𝑐 𝑣 , 𝑣 𝑡 − Δ 𝑣 𝑚 2 = 0 ≥ 𝑣 𝛽 𝑑 + 𝑒 𝑢 − 𝑓 𝑣 . ( 2 . 5 9 ) That is, ( 𝑢 , 𝑣 ) = ( 𝜂 𝜌 1 , 𝜂 𝜌 2 ) 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 𝑙 𝑓 𝑙 > 𝑐 𝑀 𝑒 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: − Δ 𝜙 = 𝜆 𝜙 , 𝑥 ∈ Ω , 𝜙 = 0 , 𝑥 ∈ 𝜕 Ω ; ( 3 . 1 ) then we have that | | | | 𝜆 > 0 , 𝜙 ( 𝑥 ) > 0 i n Ω , ∇ 𝜙 > 0 o n 𝜕 Ω , 𝑀 = m a x 𝑥 ∈ Ω 𝜙 ( 𝑥 ) < ∞ . ( 3 . 2 ) Let 𝑢 , 𝑣 = 𝜀 𝜙 2 / 𝑚 1 ( 𝑥 ) , 𝜀 𝜙 2 / 𝑚 2 , ( 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 𝜑 1 ( 𝑥 , 𝑡 ) ∈ 𝐶 1 ( 𝑄 𝑇 ) as the test function, we have that 𝑄 𝑇 𝑢 𝜕 𝜑 1 𝜕 𝑡 + Δ 𝑢 𝑚 1 𝜑 1 + 𝑢 𝛼 𝑎 − 𝑏 𝑢 + 𝑐 𝑣 𝜑 1 + 𝑑 𝑥 𝑑 𝑡 Ω 𝑢 ( 𝑥 , 0 ) 𝜑 1 ( 𝑥 , 0 ) − 𝑢 ( 𝑥 , 𝑇 ) 𝜑 1 = ( 𝑥 , 𝑇 ) 𝑑 𝑥 𝑄 𝑇 𝑢 𝛼 𝑎 − 𝑏 𝑢 + 𝑐 𝑣 + Δ 𝑢 𝑚 1 𝜑 1 = 𝑑 𝑥 𝑑 𝑡 𝑄 𝑇 𝑢 𝛼 𝑎 − 𝑏 𝑢 + 𝑐 𝑣 𝜑 1 𝑑 𝑥 𝑑 𝑡 − 𝑄 𝑇 ∇ 𝑢 𝑚 1 ∇ 𝜑 1 = 𝑑 𝑥 𝑑 𝑡 𝑄 𝑇 𝑢 𝛼 𝑎 − 𝑏 𝑢 + 𝑐 𝑣 𝜑 1 𝑑 𝑥 𝑑 𝑡 − 2 𝜀 𝑚 1 𝑄 𝑇 𝜙 ∇ 𝜙 ⋅ ∇ 𝜑 1 = 𝑑 𝑥 𝑑 𝑡 𝑄 𝑇 𝑢 𝛼 𝑎 − 𝑏 𝑢 + 𝑐 𝑣 𝜑 1 𝑑 𝑥 𝑑 𝑡 − 2 𝜀 𝑚 1 𝑄 𝑇 ∇ 𝜙 ∇ 𝜙 𝜑 1 − | | | | ∇ 𝜙 2 𝜑 1 = 𝑑 𝑥 𝑑 𝑡 𝑄 𝑇 𝑢 𝛼 𝑎 − 𝑏 𝑢 + 𝑐 𝑣 𝜑 1 𝑑 𝑥 𝑑 𝑡 − 2 𝜀 𝑚 1 𝑄 𝑇 − 𝑑 𝑖 𝑣 ( ∇ 𝜙 ) 𝜙 𝜑 1 − | | | | ∇ 𝜙 2 𝜑 1 = 𝑑 𝑥 𝑑 𝑡 𝑄 𝑇 𝑢 𝛼 𝑎 − 𝑏 𝑢 + 𝑐 𝑣 𝜑 1 𝑑 𝑥 𝑑 𝑡 − 2 𝜀 𝑚 1 𝑄 𝑇 𝜆 𝜙 2 − | | | | ∇ 𝜙 2 𝜑 1 𝑑 𝑥 𝑑 𝑡 . ( 3 . 4 ) Similarly, for any nonnegative test function 𝜑 2 ( 𝑥 , 𝑡 ) ∈ 𝐶 1 ( 𝑄 𝑇 ) , we have that 𝑄 𝑇 𝑣 𝜕 𝜑 2 𝜕 𝑡 + Δ 𝑣 𝑚 2 𝜑 2 + 𝑣 𝛽 𝑑 + 𝑒 𝑢 − 𝑓 𝑣 𝜑 2 𝑑 𝑥 𝑑 𝑡 + Ω 𝑣 ( 𝑥 , 0 ) 𝜑 2 ( 𝑥 , 0 ) − 𝑣 ( 𝑥 , 𝑇 ) 𝜑 2 = ( 𝑥 , 𝑇 ) 𝑑 𝑥 𝑄 𝑇 𝑣 𝛽 𝑑 + 𝑒 𝑢 − 𝑓 𝑣 𝜑 2 𝑑 𝑥 𝑑 𝑡 − 2 𝜀 𝑚 2 𝑄 𝑇 𝜆 𝜙 2 − | | | | ∇ 𝜙 2 𝜑 2 𝑑 𝑥 𝑑 𝑡 . ( 3 . 5 ) We just need to prove the nonnegativity of the right-hand side of (3.4) and (3.5).
Since 𝜙 1 = 𝜙 2 = 0 , | ∇ 𝜙 1 | , | ∇ 𝜙 2 | > 0 on 𝜕 Ω , then there exists 𝛿 > 0 such that 𝜆 𝜙 2 − | | | | ∇ 𝜙 2 ≤ 0 , 𝑥 ∈ Ω 𝛿 , ( 3 . 6 ) where Ω 𝛿 = { 𝑥 ∈ Ω ∣ d i s t ( 𝑥 , 𝜕 Ω ) ≤ 𝛿 } . Choosing 𝑎 𝜀 ≤ m i n 𝑙 𝑏 𝑀 𝑀 2 / 𝑚 1 , 𝑑 𝑙 𝑓 𝑀 𝑀 2 / 𝑚 2 , ( 3 . 7 ) then we have that 2 𝜀 𝑚 1 𝑇 0 Ω 𝛿 𝜆 𝜙 2 − | | | | ∇ 𝜙 2 𝜑 1 𝑑 𝑥 𝑑 𝑡 ≤ 0 ≤ 𝑇 0 Ω 𝛿 𝑢 𝛼 𝑎 − 𝑏 𝑢 + 𝑐 𝑣 𝜑 1 𝑑 𝑥 𝑑 𝑡 , 2 𝜀 𝑚 2 𝑇 0 Ω 𝛿 𝜆 𝜙 2 − | | | | ∇ 𝜙 2 𝜑 2 𝑑 𝑥 𝑑 𝑡 ≤ 0 ≤ 𝑇 0 Ω 𝛿 𝑣 𝛽 𝑑 + 𝑒 𝑢 − 𝑓 𝑣 𝜑 2 𝑑 𝑥 𝑑 𝑡 , ( 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 𝑎 𝜀 ≤ m i n 𝑙 2 𝑏 𝑀 𝑀 2 / 𝑚 1 , 𝑎 𝑙 4 𝜆 𝑀 2 ( 𝑚 1 − 𝛼 ) / 𝑚 1 1 / ( 𝑚 1 − 𝛼 ) , 𝑑 𝑙 2 𝑓 𝑀 𝑀 2 / 𝑚 2 , 𝑑 𝑙 4 𝜆 𝑀 2 ( 𝑚 2 − 𝛽 ) / 𝑚 2 1 / ( 𝑚 2 − 𝛽 ) , ( 3 . 1 0 ) then 𝜀 𝛼 𝜙 2 𝛼 / 𝑚 1 𝑎 − 𝑏 𝜀 𝛼 + 1 𝜙 2 ( 𝛼 + 1 ) / 𝑚 1 + 𝑐 𝜀 𝛼 𝜙 2 𝛼 / 𝑚 1 𝜀 𝜙 2 / 𝑚 2 − 2 𝜀 𝑚 1 𝜆 𝜙 2 𝜀 ≥ 0 , 𝛽 𝜙 2 𝛽 / 𝑚 2 𝑑 + 𝑒 𝜀 𝜙 2 / 𝑚 1 𝜀 𝛽 𝜙 2 𝛽 / 𝑚 2 − 𝑓 𝜀 𝛽 + 1 𝜙 2 ( 𝛽 + 1 ) / 𝑚 2 − 2 𝜀 𝑚 2 𝜆 𝜙 2 ≥ 0 ( 3 . 1 1 ) on 𝑄 𝑇 , that is 𝑄 𝑇 𝑢 𝛼 𝑎 − 𝑏 𝑢 + 𝑐 𝑣 𝜑 1 𝑑 𝑥 𝑑 𝑡 − 2 𝜀 𝑚 1 𝑄 𝑇 𝜆 𝜙 2 − | | | | ∇ 𝜙 2 𝜑 1 𝑑 𝑥 𝑑 𝑡 ≥ 0 , 𝑄 𝑇 𝑣 𝛽 𝑑 + 𝑒 𝑢 − 𝑓 𝑣 𝜑 2 𝑑 𝑥 𝑑 𝑡 − 2 𝜀 𝑚 2 𝑄 𝑇 𝜆 𝜙 2 − | | | | ∇ 𝜙 2 𝜑 2 𝑑 𝑥 𝑑 𝑡 ≥ 0 . ( 3 . 1 2 ) These relations show that ( 𝑢 , 𝑣 ) = ( 𝜀 𝜙 2 / 𝑚 1 1 ( 𝑥 ) , 𝜀 𝜙 2 / 𝑚 2 2 ( 𝑥 ) ) is a positive (time independent, hence 𝑇 -periodic) subsolution of (1.1)–(1.3).
Letting ( 𝑢 , 𝑣 ) = ( 𝜂 𝜌 1 , 𝜂 𝜌 2 ) , where 𝜂 , 𝜌 1 , 𝜌 2 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 = Δ 𝑢 𝑚 + 𝑓 ( 𝑥 , 𝑡 ) , ( 𝑥 , 𝑡 ) ∈ Ω × ( 0 , 𝑇 ) , 𝑢 ( 𝑥 , 𝑡 ) = 0 , ( 𝑥 , 𝑡 ) ∈ 𝜕 Ω × ( 0 , 𝑇 ) , ( 3 . 1 3 ) 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 𝑥 1 , 𝑡 1 𝑥 − 𝑢 2 , 𝑡 2 | | | | 𝑥 ≤ 𝐾 1 − 𝑥 2 | | 𝛼 + | | 𝑡 1 − 𝑡 2 | | 𝛼 / 2 . ( 3 . 1 4 )

Lemma 3.3 (see [26]). Define a Poincaré mapping 𝑃 𝑡 ∶ 𝐿 ∞ ( Ω ) × 𝐿 ∞ ( Ω ) ⟶ 𝐿 ∞ ( Ω ) × 𝐿 ∞ 𝑃 ( Ω ) , 𝑡 𝑢 0 ( 𝑥 ) , 𝑣 0 ( 𝑥 ) ∶ = ( 𝑢 ( 𝑥 , 𝑡 ) , 𝑣 ( 𝑥 , 𝑡 ) ) ( 𝑡 > 0 ) , ( 3 . 1 5 ) where ( 𝑢 ( 𝑥 , 𝑡 ) , 𝑣 ( 𝑥 , 𝑡 ) ) is the solution of (1.1)–(1.4) with initial value ( 𝑢 0 ( 𝑥 ) , 𝑣 0 ( 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 𝑙 𝑓 𝑙 > 𝑐 𝑀 𝑒 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 ) , i n 𝑄 𝑇 . ( 3 . 1 6 )

Proof. Taking 𝑢 ( 𝑥 , 𝑡 ) , 𝑢 ( 𝑥 , 𝑡 ) as those in Lemma 3.1 and choosing suitable 𝐵 ( 𝑥 0 , 𝛿 ) , 𝐵 ( 𝑥 0 , 𝛿 ′ ) , Ω ′ , 𝑘 1 , 𝑘 2 , 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 𝑘 𝑇 ( 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 𝑘 𝑇 ( 𝑢 ( ⋅ , 0 ) ) for any 𝑘 ∈ ℕ . By Lemma 2.6, we have that 𝑃 𝑘 𝑇 ( 𝑢 ( ⋅ , 0 ) ) ≤ 𝑃 𝑘 𝑇 ( 𝑢 ( ⋅ , 0 ) ) for any 𝑘 ∈ ℕ . Then 𝑢 ∗ ( 𝑥 , 0 ) = l i m 𝑘 → ∞ 𝑃 𝑘 𝑇 𝑢 ( 𝑥 , 0 ) , 𝑢 ∗ ( 𝑥 , 0 ) = l i m 𝑘 → ∞ 𝑃 𝑘 𝑇 𝑢 ( 𝑥 , 0 ) ( 3 . 1 7 ) 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 𝑙 𝑓 𝑙 > 𝑐 𝑀 𝑒 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 ( 𝑢 0 , 𝑣 0 ) , then, for any 𝜀 > 0 , there exists 𝑡 depending on 𝑢 0 , 𝑣 0 , and 𝜀 , such that 0 ≤ 𝑢 ≤ 𝑈 + 𝜀 , 0 ≤ 𝑣 ≤ 𝑉 + 𝜀 , f o r 𝑥 ∈ Ω , 𝑡 ≥ 𝑡 . ( 3 . 1 8 )

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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