Blowup Properties for a Semilinear Reaction-Diffusion System with Nonlinear Nonlocal Boundary Conditions

Abstract

We investigate the blowup properties of the positive solutions for a semilinear reaction-diffusion system with nonlinear nonlocal boundary condition. We obtain some sufficient conditions for global existence and blowup by utilizing the method of subsolution and supersolution.

1. Introduction

In this paper,we deal with the following semilinear reaction-diffusion system with nonlinear nonlocal boundary conditions and nontrivial nonnegative continuous initial data:𝑢_𝑡=Δ𝑢+𝑣^𝑝,𝑣_𝑡=Δ𝑣+𝑢^𝑞,𝑥∈Ω,𝑡>0,𝑢(𝑥,𝑡)=_Ω𝜑(𝑥,𝑦)𝑢^𝑚(𝑦,𝑡)𝑑𝑦,𝑥∈𝜕Ω,𝑡>0,𝑣(𝑥,𝑡)=_Ω𝜓(𝑥,𝑦)𝑣^𝑛(𝑦,𝑡)𝑑𝑦,𝑥∈𝜕Ω,𝑡>0,𝑢(𝑥,0)=𝑢₀(𝑥),𝑣(𝑥,0)=𝑣₀(𝑥),𝑥∈Ω,(1.1) where Ω is a bounded domain in ℝ^𝑁 for 𝑁≥1 with a smooth boundary 𝜕Ω,𝑝,𝑞,𝑚,𝑛>0, the weight functions 𝜑(𝑥,𝑦) and 𝜓(𝑥,𝑦) are nonnegative continuous defined in 𝜕Ω×Ω, and ∫_Ω𝜑(𝑥,𝑦)𝑑𝑦, ∫_Ω𝜓(𝑥,𝑦)𝑑𝑦>0 on 𝜕Ω. Moreover, for 𝑥∈𝜕Ω, the initial data 𝑢₀(𝑥), 𝑣₀(𝑥) satisfy the compatibility conditions 𝑢₀(𝑥)=∫_Ω𝜑(𝑥,𝑦)𝑢^𝑚₀(𝑦)𝑑𝑦 and 𝑣₀(𝑥)=∫_Ω𝜓(𝑥,𝑦)𝑣^𝑛₀(𝑦)𝑑𝑦, respectively.

System (1.1) has been formulated from physical models arising in various fields of applied sciences. For example, it can be interpreted as a heat conduction problem with nonlocal nonlinear sources on the boundary of the material body (see [1, 2]). In this case, 𝑢(𝑥,𝑡) and 𝑣(𝑥,𝑡) represent the temperatures of the interacting components in the evolution processes.

The local (in time) existence of classical solutions of system (1.1) can be derived easily by standard parabolic theory. We say that the solution (𝑢(𝑥,𝑡),𝑣(𝑥,𝑡)) of problem (1.1) blows up in finite time if there exists a positive constant 𝑇<∞ such that lim_{𝑡→𝑇−}(|𝑢(⋅,𝑡)|_𝐿^∞_(Ω)+|𝑣(⋅,𝑡)|_𝐿^∞_(Ω))=+∞.(1.2) In this case, 𝑇 is called the blowup time. We say that the solution (𝑢(𝑥,𝑡),𝑣(𝑥,𝑡)) exists globally if sup_{𝑡∈(0,+∞)}(|𝑢(⋅,𝑡)|_𝐿^∞_(Ω)+|𝑣(⋅,𝑡)|_𝐿^∞_(Ω))<+∞.(1.3)

In the last few years, a lot of efforts have been devoted to the study of properties of solutions to the semilinear parabolic equation 𝑢_𝑡=Δ𝑢+𝑢^𝑝 with homogeneous Dirichlet boundary condition (see, e.g., the classical works in [3, 4]) and to the heat equation 𝑢_𝑡=Δ𝑢 with Neumann boundary condition 𝜕𝑢/𝜕𝜈=𝑢^𝑝 (see, e.g., [5]).

Blowup properties for the problem of systems𝑢_𝑡=Δ𝑢+𝑣^𝑝,𝑣_𝑡=Δ𝑣+𝑢^𝑞,𝑥∈Ω,𝑡>0,𝑢(𝑥,𝑡)=𝑣(𝑥,𝑡)=0,𝑥∈𝜕Ω,𝑡>0,𝑢(𝑥,0)=𝑢₀(𝑥),𝑣(𝑥,0)=𝑣₀(𝑥),𝑥∈𝑢Ω,_𝑡=Δ𝑢+𝑣^𝑝,𝑣_𝑡=Δ𝑣+𝑢^𝑞,𝑥∈Ω,𝑡>0,𝜕𝑢=𝜕𝜈𝜕𝑣𝜕𝜈=0,𝑥∈𝜕Ω,𝑡>0,𝑢(𝑥,0)=𝑢₀(𝑥),𝑣(𝑥,0)=𝑣₀(𝑥),𝑥∈𝑢Ω,_𝑡=Δ𝑢,𝑣_𝑡=Δ𝑣,𝑥∈Ω,𝑡>0,𝜕𝑢𝜕𝜈=𝑣^𝑝,𝜕𝑣𝜕𝜈=𝑢^𝑞,𝑥∈𝜕Ω,𝑡>0,𝑢(𝑥,0)=𝑢₀(𝑥),𝑣(𝑥,0)=𝑣₀(𝑥),𝑥∈Ω(1.4) have been studied very extensively over past years by many researchers. Here 𝑝,𝑞>0,𝜈 denotes the unit outer normal vector on 𝜕Ω. They were concerned with the existence, uniqueness, and regularity of solutions. Furthermore, they investigated the global and nonglobal existence, the blowup set, and the blowup rate for the above systems (see, e.g., [1, 2, 6–9] and the references cited therein). For blowup results for other parabolic systems, we refer the readers to [10–13] and the references cited therein.

Moreover, in recent years, many authors (see studies such as those in [14, 15] and the references cited therein) considered semilinear reaction-diffusion systems with nonlocal Dirichlet boundary conditions of the form𝑢_𝑡=Δ𝑢+𝑓(𝑢,𝑣),𝑣_𝑡=Δ𝑣+𝑔(𝑢,𝑣),𝑥∈Ω,𝑡>0,𝑢=_Ω𝜑(𝑥,𝑦)𝑢(𝑦,𝑡)𝑑𝑦,𝑣=_Ω𝜓(𝑥,𝑦)𝑣(𝑦,𝑡)𝑑𝑦,𝑥∈𝜕Ω,𝑡>0,𝑢(𝑥,0)=𝑢₀(𝑥),𝑣(𝑥,0)=𝑣₀(𝑥),𝑥∈Ω.(1.5) They studied how the weight functions 𝜑(𝑥,𝑦) and 𝜓(𝑥,𝑦) in the nonlocal boundary conditions affect the blowup properties of the solutions of (1.5).

However, reaction-diffusion problems coupled with nonlocal nonlinear boundary conditions, to our knowledge, have not been well studied. Recently, Gladkov and Kim [16] considered the following problem for a single semilinear heat equation:𝑢_𝑡=Δ𝑢+𝑐(𝑥,𝑡)𝑢^𝑝,𝑥∈Ω,𝑡>0,𝑢(𝑥,𝑡)=_Ω𝑓(𝑥,𝑦,𝑡)𝑢^𝑙(𝑦,𝑡)𝑑𝑦,𝑥∈𝜕Ω,𝑡>0,𝑢(𝑥,0)=𝑢₀(𝑥),𝑥∈Ω,(1.6) where 𝑝, 𝑙>0. They obtained some criteria for the existence of the global solution as well as for blowup of the solution in finite time.

The main purpose of this paper is to understand how the reaction terms, the weight functions and the nonlinear terms in the boundary conditions affect the blowup properties for problem (1.1). We will show that the weight functions 𝜑(𝑥,𝑦),𝜓(𝑥,𝑦) and the nonlinear terms 𝑢^𝑚(𝑦,𝑡), 𝑣^𝑛(𝑦,𝑡) in the boundary conditions of (1.1) play substantial roles in determining blowup or not of solutions.

Before starting the main results, we introduce some useful symbols. Throughout this paper, we let 𝜆 be the first eigenvalue of the eigenvalue problem−Δ𝜙(𝑥)=𝜆𝜙,𝑥∈Ω;𝜙(𝑥)=0,𝑥∈𝜕Ω,(1.7) and 𝜙(𝑥) the corresponding eigenfunction with ∫_Ω𝜙(𝑥)𝑑𝑥=1, 𝜙(𝑥)>0 in Ω. In addition, for convenience, we denote that 𝐿=sup_Ω𝜙(𝑥) and 𝑀₁=sup_𝜕Ω×Ω𝜑(𝑥,𝑦),𝑀₂=inf_𝜕Ω×Ω𝜑(𝑥,𝑦),𝐾₁=sup_𝜕Ω×Ω𝜓(𝑥,𝑦),𝐾₂=inf_𝜕Ω×Ω𝜓(𝑥,𝑦).(1.8)

The main results of this paper are stated as follows.

Theorem 1.1. Assume that 0<𝑝𝑞≤1 and 𝑚,𝑛≤1. Then the solution of problem (1.1) exists globally for any positive initial data.

Theorem 1.2. Assume that 𝑝𝑞>1 or 𝑚,𝑛>1. Then for any 𝜑(𝑥,𝑦), 𝜓(𝑥,𝑦)>0, the solution of problem (1.1) blows up in finite time for sufficiently large initial data.

Theorem 1.3. Assume that 𝑝>1, 𝑞>1, 𝑚>1, and 𝑛>1. Then for any nonnegative continuous 𝜑(𝑥,𝑦) and 𝜓(𝑥,𝑦), the solution of problem (1.1) exists globally for sufficiently small initial data.

Remark 1.4. When 𝑝=𝑞,𝑚=𝑛,𝜑(𝑥,𝑦)=𝜓(𝑥,𝑦), and 𝑢₀(𝑥)=𝑣₀(𝑥), system (1.1) is then reduced to a single equation 𝑢_𝑡=Δ𝑢+𝑢^𝑝 with nonlocal nonlinear boundary condition 𝑢(𝑥,𝑡)=∫_Ω𝜑(𝑥,𝑦)𝑢^𝑚(𝑦,𝑡)𝑑𝑦. In this case, our above results are still true and consistent with those in [16].

The rest of this paper is organized as follows. In Section 2, we establish the comparison principle for problem (1.1). In Sections 3 and 4, we will give the proofs of Theorems 1.1 and 1.2, respectively. Finally, Theorem 1.3 will be proved in Section 5.

2. Preliminaries

In this section, we will give a suitable comparison principle for problem (1.1). Let Ω_𝑇=Ω×(0,𝑇), 𝑆_𝑇=𝜕Ω×(0,𝑇), and Ω_𝑇=Ω×[0,𝑇). We begin with the precise definitions of a subsolution and supersolution of problem (1.1).

Definition 2.1. A pair of functions (𝑢,𝑣)∈𝐶^2,1(Ω_𝑇)∩𝐶(Ω_𝑇)×𝐶^2,1(Ω_𝑇)∩𝐶(Ω_𝑇) is called a subsolution of problem (1.1) in Ω_𝑇 if 𝑢_𝑡≤Δ𝑢+𝑣^𝑝,𝑣_𝑡≤Δ𝑣+𝑢^𝑞,(𝑥,𝑡)∈Ω_𝑇,𝑢(𝑥,𝑡)≤_Ω𝜑(𝑥,𝑦)𝑢^𝑚(𝑦,𝑡)𝑑𝑦,(𝑥,𝑡)∈𝑆_𝑇,𝑣(𝑥,𝑡)≤_Ω𝜓(𝑥,𝑦)𝑣^𝑛(𝑦,𝑡)𝑑𝑦,(𝑥,𝑡)∈𝑆_𝑇,𝑢(𝑥,0)≤𝑢₀(𝑥),𝑣(𝑥,0)≤𝑣₀(𝑥),𝑥∈Ω.(2.1)

Similarly, a pair of functions (𝑢,𝑣)∈𝐶^2,1(Ω_𝑇)∩𝐶(Ω_𝑇)×𝐶^2,1(Ω_𝑇)∩𝐶(Ω_𝑇) is a supersolution of system (1.1) if the reversed inequalities hold in (2.1). We say that (𝑢,𝑣) is a solution of system (1.1) in Ω_𝑇 if it is both a subsolution and a supersolution of problem (1.1) in Ω_𝑇.

Let 𝑔_𝑖(𝑥,𝑡),ℎ_𝑖(𝑥,𝑡)∈𝐶^2,1(Ω_𝑇)∩𝐶(Ω_𝑇), 𝜒_𝑖(𝑥,𝑦)≥0 on 𝜕Ω×Ω, 𝑖=1,2. We first give some hypotheses as follows, which will be used in the sequel.(H1) For 𝑥∈𝜕Ω, 𝑦∈Ω, 𝑡>0, 𝜒₁(𝑥,𝑦)𝑔₁^𝑚−1(𝑦,𝑡), 𝜒₁(𝑥,𝑦)ℎ₁^𝑚−1(𝑦,𝑡), 𝜒₂(𝑥,𝑦)𝑔₂^𝑛−1(𝑦,𝑡), and 𝜒₂(𝑥,𝑦)ℎ₂^𝑛−1(𝑦,𝑡) are nonnegative. Further, ∫_Ω𝑚𝜒₁(𝑥,𝑦)𝑔₁^𝑚−1(𝑦,𝑡)𝑑𝑦≤1, ∫_Ω𝑚𝜒₁(𝑥,𝑦)ℎ₁^𝑚−1(𝑦,𝑡)𝑑𝑦≤1, ∫_Ω𝑛𝜒₂(𝑥,𝑦)𝑔₂^𝑛−1(𝑦,𝑡)𝑑𝑦≤1, and ∫_Ω𝑛𝜒₂ (𝑥,𝑦)ℎ₂^𝑛−1(𝑦,𝑡)𝑑𝑦≤1.(H2) For 𝑥∈𝜕Ω, 𝑦∈Ω, 𝑡>0, there exists 𝑀>0 such that 0≤𝑚𝜒₁(𝑥,𝑦)𝑔₁^𝑚−1(𝑦,𝑡)≤𝑀, 0≤𝑚𝜒₁(𝑥,𝑦)ℎ₁^𝑚−1(𝑦,𝑡)≤𝑀, 0≤𝑛𝜒₂(𝑥,𝑦)𝑔₂^𝑛−1(𝑦,𝑡)≤𝑀, and 0≤𝑛𝜒₂(𝑥,𝑦) ℎ₂^𝑛−1(𝑦,𝑡)≤𝑀.

Lemma 2.2. Let (H1) hold, and 𝑐_𝑖𝑗+𝑑_𝑖𝑗(𝑖,𝑗=1,2) be bounded in Ω_𝑇 and let𝑐_𝑖𝑗+𝑑_𝑖𝑗≥0(𝑖≠𝑗,𝑖,𝑗=1,2). Further, assume that 𝑤_𝑖(𝑥,𝑡)≥𝑠_𝑖(𝑥,𝑡)(𝑖=1,2). If 𝜒_𝑖(𝑥,𝑦)≥0 on 𝜕Ω×Ω; and 𝑔_𝑖,ℎ_𝑖∈𝐶^2,1(Ω_𝑇)∩𝐶(Ω_𝑇)(𝑖=1,2) satisfy 𝑔_1𝑡−(^𝑛_𝑘,𝑙=1𝑎⁽¹⁾_𝑘,𝑙𝜕²𝑔₁𝜕𝑥_𝑘𝜕𝑥_𝑙+^𝑛_𝑘=1𝑏_𝑘⁽¹⁾𝜕𝑔₁𝜕𝑥_𝑘)≥²_𝑖=1𝑐_1𝑖𝑔_𝑖−²_𝑖=1𝑑_1𝑖ℎ_𝑖+𝑤₁(𝑥,𝑡),(𝑥,𝑡)∈Ω_𝑇,𝑔_2𝑡−(^𝑛_𝑘,𝑙=1𝑎⁽²⁾_𝑘,𝑙𝜕²𝑔₂𝜕𝑥_𝑘𝜕𝑥_𝑙+^𝑛_𝑘=1𝑏_𝑘⁽²⁾𝜕𝑔₂𝜕𝑥_𝑘)≥²_𝑖=1𝑐_2𝑖𝑔_𝑖−²_𝑖=1𝑑_2𝑖ℎ_𝑖+𝑤₂(𝑥,𝑡),(𝑥,𝑡)∈Ω_𝑇,ℎ_1𝑡−(^𝑛_𝑘,𝑙=1𝑎⁽¹⁾_𝑘,𝑙𝜕²ℎ₁𝜕𝑥_𝑘𝜕𝑥_𝑙+^𝑛_𝑘=1𝑏_𝑘⁽¹⁾𝜕ℎ₁𝜕𝑥_𝑘)≤²_𝑖=1𝑐_1𝑖ℎ_𝑖−²_𝑖=1𝑑_1𝑖𝑔_𝑖+𝑠₁(𝑥,𝑡),(𝑥,𝑡)∈Ω_𝑇,ℎ_2𝑡−(^𝑛_𝑘,𝑙=1𝑎⁽²⁾_𝑘,𝑙𝜕²ℎ₂𝜕𝑥_𝑘𝜕𝑥_𝑙+^𝑛_𝑘=1𝑏_𝑘⁽²⁾𝜕ℎ₂𝜕𝑥_𝑘)≤²_𝑖=1𝑐_2𝑖ℎ_𝑖−²_𝑖=1𝑑_2𝑖𝑔_𝑖+𝑠₂(𝑥,𝑡),(𝑥,𝑡)∈Ω_𝑇,𝑔₁(𝑥,𝑡)≥_Ω𝜒₁(𝑥,𝑦)𝑔^𝑚₁(𝑦,𝑡)𝑑𝑦,𝑔₂(𝑥,𝑡)≥_Ω𝜒₂(𝑥,𝑦)𝑔^𝑛₂(𝑦,𝑡)𝑑𝑦,(𝑥,𝑡)∈𝑆_𝑇,ℎ₁(𝑥,𝑡)≤_Ω𝜒₁(𝑥,𝑦)ℎ^𝑚₁(𝑦,𝑡)𝑑𝑦,ℎ₂(𝑥,𝑡)≤_Ω𝜒₂(𝑥,𝑦)ℎ^𝑛₂(𝑦,𝑡)𝑑𝑦,(𝑥,𝑡)∈𝑆_𝑇,𝑔₁(𝑥,0)≥ℎ₁(𝑥,0),𝑔₂(𝑥,0)≥ℎ₂(𝑥,0),𝑥∈Ω.(2.2) Then (𝑔₁(𝑥,𝑡),𝑔₂(𝑥,𝑡))≥(ℎ₁(𝑥,𝑡),ℎ₂(𝑥,𝑡)) in Ω_𝑇.

Proof. For any given 𝜀>0, define 𝑔_𝑖=𝑔_𝑖+𝜀𝑒^𝛾𝑡,ℎ_𝑖=ℎ_𝑖−𝜀𝑒^𝛾𝑡,𝑖=1,2,(2.3) where 𝛾>max{𝑐₁₁+𝑐₁₂+𝑑₁₁+𝑑₁₂,𝑐₂₁+𝑐₂₂+𝑑₂₁+𝑑₂₂}.(2.4) Then, a direct computation yields 𝑔_1𝑡−(^𝑛_𝑙,𝑘=1𝑎⁽¹⁾_𝑙,𝑘𝜕²𝑔₁𝜕𝑥_𝑘𝜕𝑥_𝑙+^𝑛_𝑘=1𝑏_𝑘⁽¹⁾𝜕𝑔₁𝜕𝑥_𝑘)>²_𝑖=1𝑐_1𝑖𝑔_𝑖−²_𝑖=1𝑑_1𝑖ℎ_𝑖+𝑤₁(𝑥,𝑡),(𝑥,𝑡)∈Ω_𝑇,𝑔_2𝑡−(^𝑛_𝑙,𝑘=1𝑎⁽²⁾_𝑙,𝑘𝜕²𝑔₂𝜕𝑥_𝑘𝜕𝑥_𝑙+^𝑛_𝑘=1𝑏_𝑘⁽²⁾𝜕𝑔₂𝜕𝑥_𝑘)>²_𝑖=1𝑐_2𝑖𝑔_𝑖−²_𝑖=1𝑑_2𝑖ℎ_𝑖+𝑤₂(𝑥,𝑡),(𝑥,𝑡)∈Ω_𝑇,ℎ_1𝑡−(^𝑛_𝑙,𝑘=1𝑎⁽¹⁾_𝑙,𝑘𝜕²ℎ₁𝜕𝑥_𝑘𝜕𝑥_𝑙+^𝑛_𝑘=1𝑏_𝑘⁽¹⁾𝜕ℎ₁𝜕𝑥_𝑘)<²_𝑖=1𝑐_1𝑖ℎ_𝑖−²_𝑖=1𝑑_1𝑖𝑔_𝑖+𝑠₁(𝑥,𝑡),(𝑥,𝑡)∈Ω_𝑇,ℎ_2𝑡−(^𝑛_𝑙,𝑘=1𝑎⁽²⁾_𝑙,𝑘𝜕²ℎ₂𝜕𝑥_𝑘𝜕𝑥_𝑙+^𝑛_𝑘=1𝑏_𝑘⁽²⁾𝜕ℎ₂𝜕𝑥_𝑘)<²_𝑖=1𝑐_2𝑖ℎ_𝑖−²_𝑖=1𝑑_2𝑖𝑔_𝑖+𝑠₂(𝑥,𝑡),(𝑥,𝑡)∈Ω_𝑇.(2.5) On the other hand, for (𝑥,𝑡)∈𝑆_𝑇, we have 𝑔₁(𝑥,𝑡)≥𝜀𝑒^𝛾𝑡+_Ω𝜒₁(𝑥,𝑦)𝑔^𝑚₁=(𝑦,𝑡)𝑑𝑦_Ω𝜒₁(𝑥,𝑦)𝑔^𝑚₁(𝑦,𝑡)𝑑𝑦+𝜀𝑒^𝛾𝑡−_Ω𝜒₁(𝑥,𝑦)(𝑔^𝑚₁(𝑦,𝑡)−𝑔^𝑚₁(𝑦,𝑡))𝑑𝑦=_Ω𝜒₁(𝑥,𝑦)𝑔^𝑚₁(𝑦,𝑡)𝑑𝑦+𝜀𝑒^𝛾𝑡−𝜀𝑒^𝛾𝑡_Ω𝑚𝜒₁(𝑥,𝑦)𝜃₁^𝑚−1(𝑦,𝑡)𝑑𝑦,(2.6) where 𝜃₁ is an intermediate value between 𝑔₁ and 𝑔₁. From (H1), it follows that 𝑔₁(𝑥,𝑡)>_Ω𝜒₁(𝑥,𝑦)𝑔^𝑚₁(𝑦,𝑡)𝑑𝑦for(𝑥,𝑡)∈𝑆_𝑇.(2.7) Likewise, for any (𝑥,𝑡)∈𝑆_𝑇, we have 𝑔₂(𝑥,𝑡)>_Ω𝜒₂(𝑥,𝑦)𝑔^𝑛₂ℎ(𝑦,𝑡)𝑑𝑦,₁(𝑥,𝑡)<_Ω𝜒₁(𝑥,𝑦)ℎ^𝑚₁(𝑦,𝑡)𝑑𝑦,ℎ₂(𝑥,𝑡)<_Ω𝜒₂(𝑥,𝑦)ℎ^𝑛₂(𝑦,𝑡)𝑑𝑦.(2.8) In addition, it is obvious that 𝑔₁(𝑥,0)−𝜀≥ℎ₁(𝑥,0)+𝜀,𝑔₂(𝑥,0)−𝜀≥ℎ₂(𝑥,0)+𝜀,(2.9) and hence, we know that 𝑔₁(𝑥,0)>ℎ₁(𝑥,0),𝑔₂(𝑥,0)>ℎ₂(𝑥,0).(2.10)
Put ℎ(𝑥,𝑡)=𝑔₁(𝑥,𝑡)−𝑔₂(𝑥,𝑡).(2.11) Next, our task is to show that (𝜂₁(𝑥,𝑡),𝜂₂(𝑥,𝑡))>(0,0).(2.12) Actually, if (2.12) is true; then we can immediately get 𝑔_𝑖(𝑥,𝑡)+𝜀𝑒^𝛾𝑡≥ℎ_𝑖(𝑥,𝑡)−𝜀𝑒^𝛾𝑡(𝑖=1,2),∀(𝑥,𝑡)∈Ω_𝑇,(2.13) which means that (𝑔₁(𝑥,𝑡),𝑔₂(𝑥,𝑡))≥(ℎ₁(𝑥,𝑡),ℎ₂(𝑥,𝑡)) in Ω_𝑇 as desired.
In order to prove (2.12), we set 𝜂₁=𝜂₁𝑒^−𝜎𝑡,𝜂₂=𝜂₂𝑒^−𝜎𝑡(2.14) with 𝜎>𝑚𝑎𝑥{sup_Ω𝑇(𝑐₁₁+𝑑₁₁),sup_Ω𝑇(𝑐₂₂+𝑑₂₂)}. Then from (2.5)–(2.10), we have 𝜂_1𝑡−(^𝑛_𝑙,𝑘=1𝑎⁽¹⁾_𝑙,𝑘𝜕²𝜂₁𝜕𝑥_𝑘𝜕𝑥_𝑙+^𝑛_𝑘=1𝑏_𝑘⁽¹⁾𝜕𝜂₁𝜕𝑥_𝑘)>(𝑐₁₁+𝑑₁₁−𝜎)𝜂₁+(𝑐₁₂+𝑑₁₂)𝜂₂,(𝑥,𝑡)∈Ω_𝑇,𝜂_2𝑡−(^𝑛_𝑙,𝑘=1𝑎⁽¹⁾_𝑙,𝑘𝜕²𝜂₂𝜕𝑥_𝑘𝜕𝑥_𝑙+^𝑛_𝑘=1𝑏_𝑘⁽¹⁾𝜕𝜂₂𝜕𝑥_𝑘)>(𝑐₂₁+𝑑₂₁)𝜂₁+(𝑐₂₂+𝑑₂₂−𝜎)𝜂₂,(𝑥,𝑡)∈Ω_𝑇,𝜂₁(𝑥,𝑡)>_Ω𝑚𝜒₁(𝑥,𝑦)𝜃₂^𝑚−1𝜂₁(𝑦,𝑡)𝑑𝑦,(𝑥,𝑡)∈𝑆_𝑇,𝜂₂(𝑥,𝑡)>_Ω𝑛𝜒₂(𝑥,𝑦)𝜃₃^𝑛−1𝜂₂(𝑦,𝑡)𝑑𝑦,(𝑥,𝑡)∈𝑆_𝑇,𝜂₁(𝑥,0)>0,𝜂₂(𝑥,0)>0,𝑥∈Ω,(2.15) where 𝜃₂ is an intermediate value between 𝑔₁ and ℎ₁, 𝜃₃ is an intermediate value between 𝑔₂ and ℎ₂.
Since 𝜂₁(𝑥,0),𝜂₂(𝑥,0)>0, there exists 𝛿>0 such that 𝜂₁,𝜂₂>0 for (𝑥,𝑡)∈Ω×(0,𝛿). Suppose a contradiction that𝑡=sup{𝑡∈(0,𝑇)∶𝜂₁,𝜂₂>0onΩ×[0,𝑡]}<𝑇.(2.16) Then 𝜂₁,𝜂₂≥0 on Ω_𝑡, and at least one of 𝜂₁, 𝜂₂ vanishes at (𝑥,𝑡) for some 𝑥∈Ω. Without loss of generality, suppose that 𝜂₁(𝑥,𝑡)=0=inf_Ω𝑡𝜂₁. If (𝑥,𝑡)∈Ω_𝑡, by virtue of the first inequality of (2.15), we find that 𝜂_1𝑡−(^𝑛_𝑘,𝑙=1𝑎⁽¹⁾_𝑙,𝑘𝜕²𝜂₁𝜕𝑥_𝑘𝜕𝑥_𝑙+^𝑛_𝑘=1𝑏_𝑘⁽¹⁾𝜕𝜂₁𝜕𝑥_𝑘)>(𝑎₁₁+𝑏₁₁−𝜎)𝜂₁,(𝑥,𝑡)∈Ω_𝑡.(2.17) This leads us to conclude that 𝜂₁≡0 in Ω_𝑡 by the strong maximum principle, a contradiction. If (𝑥,𝑡)∈𝑆_𝑡, this also results in a contradiction, that is 0=𝜂₁(𝑥,𝑡)>_Ω𝑚𝜑(𝑥,𝑦)𝜃₂^𝑚−1𝜂₁(𝑦,𝑡)𝑑𝑦≥0.(2.18) This proves that 𝜂₁,𝜂₂>0, and in turn (𝑔₁(𝑥,𝑡),𝑔₂(𝑥,𝑡))≥(ℎ₁(𝑥,𝑡),ℎ₂(𝑥,𝑡)) in Ω_𝑇. The proof of Lemma 2.2 is complete.

Lemma 2.3. Letthe hypotheses of Lemma2.2, with (H1) replaced by (H2), be satisfied. Then (𝑔₁(𝑥,𝑡),𝑔₂(𝑥,𝑡))≥(ℎ₁(𝑥,𝑡),ℎ₂(𝑥,𝑡))inΩ_𝑇.(2.19)

Proof. Choose a positive function 𝑓 satisfying 𝑓|_{𝑥∈𝜕Ω}=1 and ∫_Ω𝑓(𝑦)𝑑𝑦<1/𝑀. Set 𝑔_𝑖(𝑥,𝑡)=𝑓(𝑥)𝜌_𝑖(𝑥,𝑡),ℎ_𝑖(𝑥,𝑡)=𝑓(𝑥)𝜁_𝑖(𝑥,𝑡),𝑖=1,2.(2.20) Then from (2.2), we have 𝜌_1𝑡−ℒ₁𝜌₁≥²_𝑖=1𝑐_1𝑖𝑓𝜌_𝑖−²_𝑖=1𝑑_1𝑖𝑓𝜁_𝑖+𝑤₁𝑓,(𝑥,𝑡)∈Ω_𝑇,𝜌_2𝑡−ℒ₂𝜌₂≥²_𝑖=1𝑐_2𝑖𝑓𝜌_𝑖−²_𝑖=1𝑑_2𝑖𝑓𝜁_𝑖+𝑤₂𝑓,(𝑥,𝑡)∈Ω_𝑇,𝜁_1𝑡−ℒ₁𝜁₁≤²_𝑖=1𝑐_1𝑖𝑓𝜁_𝑖−²_𝑖=1𝑑_1𝑖𝑓𝜌_𝑖+𝑠₁𝑓,(𝑥,𝑡)∈Ω_𝑇,𝜁_2𝑡−ℒ₂𝜁₂≤²_𝑖=1𝑐_2𝑖𝑓𝜁_𝑖−²_𝑖=1𝑑_2𝑖𝑓𝜌_𝑖+𝑠₂𝑓,(𝑥,𝑡)∈Ω_𝑇,𝜌₁(𝑥,𝑡)≥_Ω𝜒₁(𝑥,𝑦)𝑓^𝑚(𝑦)𝜌^𝑚₁(𝑦,𝑡)𝑑𝑦,(𝑥,𝑡)∈𝑆_𝑇,𝜌₂(𝑥,𝑡)≥_Ω𝜒₂(𝑥,𝑦)𝑓^𝑛(𝑦)𝜌^𝑛₂(𝑦,𝑡)𝑑𝑦,(𝑥,𝑡)∈𝑆_𝑇,𝜁₁(𝑥,𝑡)≤_Ω𝜒₁(𝑥,𝑦)𝑓^𝑚(𝑦)𝜁^𝑚₁(𝑦,𝑡)𝑑𝑦,(𝑥,𝑡)∈𝑆_𝑇,𝜁₂(𝑥,𝑡)≤_Ω𝜒₂(𝑥,𝑦)𝑓^𝑛(𝑦)𝜁^𝑛₂(𝑦,𝑡)𝑑𝑦,(𝑥,𝑡)∈𝑆_𝑇,𝜌₁(𝑥,0)≥𝜁₁(𝑥,0),𝜌₂(𝑥,0)≥𝜁₂(𝑥,0),𝑥∈Ω,(2.21) where ℒ_𝑠=^𝑛_𝑙,𝑘=1𝑎^(𝑠)_𝑙,𝑘𝜕²𝜕𝑥_𝑘𝜕𝑥_𝑙+^𝑛_𝑘=1(^𝑛_𝑙=12𝑎^(𝑠)_𝑘,𝑙𝜕𝑓𝜕𝑥_𝑙+𝑏_𝑘^(𝑠)1𝑓)𝑓𝜕𝜕𝑥_𝑘,𝑠=1,2,(2.22) is a uniformly elliptic operator. By (H2), it is easy to see that _Ω𝑚𝜒₁(𝑥,𝑦)𝜌₁^𝑚−1(𝑦,𝑡)𝑓^𝑚(𝑦)𝑑𝑦≤𝑀_Ω𝑓(𝑦)𝑑𝑦≤1,_Ω𝑛𝜒₂(𝑥,𝑦)𝜌₂^𝑛−1(𝑦,𝑡)𝑓^𝑛(𝑦)𝑑𝑦≤𝑀_Ω𝑓(𝑦)𝑑𝑦≤1.(2.23) Similarly, we have _Ω𝑚𝜒₁(𝑥,𝑦)𝜁₁^𝑚−1(𝑦,𝑡)𝑓^𝑚(𝑦)𝑑𝑦≤1,_Ω𝑛𝜒₂(𝑥,𝑦)𝜁₂^𝑛−1(𝑦,𝑡)𝑓^𝑛(𝑦)𝑑𝑦≤1.(2.24) Therefore, in view of Lemma 2.2, we have 𝜌_𝑖(𝑥,𝑡)≥𝜁_𝑖(𝑥,𝑡),𝑖=1,2,(2.25) which implies that 𝑔_𝑖(𝑥,𝑡)≥ℎ_𝑖(𝑥,𝑡),𝑖=1,2.(2.26) The proof of Lemma 2.3 is complete.

On the basis of the above lemmas, we obtain the following comparison principle for problem (1.1).

Proposition 2.4 (Comparison principle). Let (𝑢,𝑣) and (𝑢,𝑣) be a nonnegative supersolution and a nonnegative subsolution of problem (1.1) in Ω_𝑇, respectively. Suppose that (𝑢,𝑣)>(0,0) and (𝑢,𝑣)>(0,0) in Ω_𝑇 if min{𝑝,𝑞,𝑚,𝑛}<1. If (𝑢(𝑥,0),𝑣(𝑥,0))≥(𝑢(𝑥,0),𝑣(𝑥,0)) for 𝑥∈Ω, then (𝑢,𝑣)≥(𝑢,𝑣) in Ω_𝑇.

Proof. It is easy to check that 𝑢, 𝑣, 𝑢, 𝑣 and 𝜑, 𝜓 satisfy hypotheses (H2).

Next, we state the local existence theorem, and its proof is standard; hence we omit it.

Theorem 2.5 (Local existence). For any nonnegative nontrivial 𝑢₀(𝑥),𝑣₀(𝑥)∈𝐶(Ω), there exists a constant 𝑇^∗=𝑇^∗(𝑢₀,𝑣₀)>0 such that problem (1.1) admits nonnegative solution (𝑢(𝑥,𝑡),𝑣(𝑥,𝑡))∈𝐶^2,1(Ω_𝑇)∩𝐶(Ω_𝑇)×𝐶^2,1(Ω_𝑇)∩𝐶(Ω_𝑇) for each 𝑇<𝑇^∗. Furthermore, either 𝑇^∗=∞ or lim_{𝑡→𝑇∗}sup(‖𝑢(𝑥,𝑡)‖_∞+‖v(𝑥,𝑡)‖_∞)=∞.(2.27)

Remark 2.6. From maximum principle, we know that the solution of system (1.1) is positive when 𝑢₀(𝑥) and 𝑣₀(𝑥) are positive. Indeed, since 𝑢_𝑡−Δ𝑢−𝑣^𝑝≥0 and 𝑣_𝑡−Δ𝑣−𝑢^𝑞≥0, the minimum of (𝑢,𝑣) in Ω_𝑇 should be obtained at a parabolic boundary point by maximum principle. Furthermore, ∫_Ω𝜑(𝑥,𝑦)𝑑𝑦, ∫_Ω𝜓(𝑥,𝑦)𝑑𝑦>0 on 𝜕Ω imply that 𝜑(𝑥,𝑡)≢0 and 𝜓(𝑥,𝑡)≢0, then we have (𝑢,𝑣)>(0,0) for (𝑥,𝑡)∈𝜕Ω×(0,𝑇]. Thus (𝑢,𝑣)>(0,0) provided that 𝑢₀(𝑥) and 𝑣₀(𝑥) are positive. In the rest of this paper, we assume that (𝑢₀(𝑥),𝑣₀(𝑥))>(0,0).

Remark 2.7. If 𝑝𝑞≥1,  𝑚≥1, and 𝑛≥1, we could obtain the uniqueness of the solution easily by comparison principle.

3. Proof of Theorem 1.1

In this section, by constructing special supersolution, we will give the sufficient condition for the existence of global solution of problem (1.1) under the hypotheses 0<𝑝𝑞≤1 and 𝑚,𝑛<1.

Proof of Theorem 1.1. Since 0<𝑝𝑞≤1, there exist 𝛼, 𝛽∈(0,1) such that 1𝑝≥𝛽𝛼,1𝑞≥𝛼𝛽.(3.1) Define 𝑘=1/𝛼+1/𝛽. Let Φ(𝑥,𝑦)≥max{𝜑(𝑥,𝑦),𝜓(𝑥,𝑦)} be a continuous function defined for (𝑥,𝑦)∈𝜕Ω×Ω, and 𝑎(𝑥)=(_ΩΦ(𝑥,𝑦)𝑑𝑦)^{(1−𝛼)/𝛼},𝑏(𝑥)=(_ΩΦ(𝑥,𝑦)𝑑𝑦)^{(1−𝛽)/𝛽},𝑥∈𝜕Ω.(3.2) Suppose that 𝜔 is the solution of the following problem: 𝜔_𝑡=Δ𝜔+𝑘𝜔,𝑥∈Ω,𝑡>0,𝜔(𝑥,𝑡)=(𝑎(𝑥)+𝑏(𝑥)+1)_Ω1(Φ(𝑥,𝑦)+|Ω|)𝜔(𝑦,𝑡)𝑑𝑦,𝑥∈𝜕Ω,𝑡>0,𝜔(𝑥,0)=𝑢₀^1/𝛼(𝑥)+𝑣₀^1/𝛽(𝑥)+1,𝑥∈Ω,(3.3) where |Ω| denotes the measure of Ω. By Theorem 2.5 in [16], we know that 𝜔 is a global solution of (3.3). Moreover, 𝜔(𝑥,𝑡)>1 in (𝑥,𝑡)∈Ω×[0,+∞) by the maximum principle.
Set (𝑢,𝑣)=(𝜔^𝛼,𝜔^𝛽). A simple computation shows that𝑢_𝑡=𝛼𝜔^𝛼−1𝜔_𝑡=𝛼𝜔^𝛼−1𝛼Δ𝜔+(1+𝛽)𝜔^𝛼,Δ𝑢=𝛼𝜔^𝛼−1Δ𝜔+𝛼(𝛼−1)𝜔^𝛼−2|∇𝜔|²,(3.4) and thus 𝑢_𝑡−Δ𝛼𝑢=(1+𝛽)𝜔^𝛼−𝛼(𝛼−1)𝜔^𝛼−2|∇𝜔|²𝛼≥(1+𝛽)𝜔^𝛼≥(𝜔^𝛽)^𝛼/𝛽≥𝑣^𝑝.(3.5) here, we used 𝑝≤𝛼/𝛽 and 𝜔>1.
When (𝑥,𝑡)∈𝜕Ω×(0,+∞), according to Hölder's inequality, we have that 𝑢(𝑥,𝑡)≥𝑎^𝛼(𝑥){_ΩΦ(𝑥,𝑦)𝜔(𝑦,𝑡)𝑑𝑦}^𝛼={_ΩΦ(𝑥,𝑦)𝑑𝑦}^1−𝛼{_ΩΦ(𝑥,𝑦)𝜔(𝑦,𝑡)𝑑𝑦}^𝛼≥{_Ω𝜑(𝑥,𝑦)𝑑𝑦}^1−𝛼{_Ω𝜑(𝑥,𝑦)𝜔(𝑦,𝑡)𝑑𝑦}^𝛼≥_Ω𝜑(𝑥,𝑦)𝜔^𝛼(𝑦,𝑡)𝑑𝑦=_Ω𝜑(𝑥,𝑦)𝑢(𝑥,𝑡)𝑑𝑦≥_Ω𝜑(𝑥,𝑦)𝑢^𝑚(𝑥,𝑡)𝑑𝑦.(3.6) Likewise, we also have for 𝑣 that 𝑣_𝑡−Δ𝑣≥𝑢^𝑞,𝑥∈Ω,𝑡>0,𝑣(𝑥,𝑡)≥_Ω𝜓(𝑥,𝑦)𝑣^𝑛(𝑥,𝑡)𝑑𝑦,𝑥∈𝜕Ω,𝑡>0.(3.7) On the other hand, since 𝛼<1, we have 𝑢(𝑥,0)=𝜔^𝛼(𝑥,0)=(𝑢₀^1/𝛼(𝑥)+𝑣₀^1/𝛽(𝑥)+1)^𝛼≥𝑢₀(𝑥),(3.8) and similarly, 𝑣(𝑥,0)=𝜔^𝛽(𝑥,0)=(𝑢₀^1/𝛼(𝑥)+𝑣₀^1/𝛽(𝑥)+1)^𝛽≥𝑣₀(𝑥).(3.9) Therefore, (𝑢,𝑣) is a global supersolution of (1.1); by Proposition 2.4, the solution of (1.1) exists globally. The proof of Theorem 1.1 is complete.

4. Proof of Theorem 1.2

In this section, we will establish that the solution of system (1.1) blows up in finite time for the case 𝑝𝑞>1 or 𝑚,𝑛>1. We employ a variant of Kaplan's method (see [17] for more details) to obtain our blowup conclusion.

Proof of Theorem 1.2. Let 𝐽₁(𝑡)=_Ω𝑢(𝑥,𝑡)𝜙(𝑥)𝑑𝑥,𝐽₂(𝑡)=_Ω𝑣(𝑥,𝑡)𝜙(𝑥)𝑑𝑥,(4.1) where 𝜙(𝑥) is defined in (1.7). Taking the derivative of 𝐽₁(𝑡) with respect to t, we could obtain 𝐽^′₁(𝑡)=_Ω𝑢_𝑡(𝑥,𝑡)𝜙(𝑥)𝑑𝑥=_Ω(Δ𝑢+𝑣^𝑝)𝜙𝑑𝑥=−_Ω𝐷𝑢⋅𝐷𝜑𝑑𝑥+_𝜕Ω𝜕𝑢𝜕𝜈𝜙𝑑𝑆+_Ω𝑣^𝑝=𝜙(𝑥)𝑑𝑥_Ω𝑢Δ𝜙𝑑𝑥−_𝜕Ω𝜕𝜙𝜕𝜈𝑢𝑑𝑆+_Ω𝑣^𝑝𝜙𝑑𝑥=−_Ω𝜆𝑢𝜙𝑑𝑥−_𝜕Ω𝜕𝜙(𝜕𝜈_Ω𝜑(𝑥,𝑦)𝑢^𝑚(𝑦,𝑡)𝑑𝑦)𝑑𝑆+_Ω𝑣^𝑝𝜙𝑑𝑥.(4.2) Applying the equality ∫_𝜕Ω(𝜕𝜙/𝜕𝜈)𝑑𝑆=−𝜆 to (4.2), we find that 𝐽^′₁(𝑡)≥−𝜆_Ω𝑢𝜙𝑑𝑥+𝜆𝑀₂𝐿_Ω𝜙𝑢^𝑚𝑑𝑥+_Ω𝑣^𝑝𝜙𝑑𝑥.(4.3) Symmetrically, we deduce that 𝐽^′₂(𝑡)≥−𝜆_Ω𝑣𝜙𝑑𝑥+𝜆𝐾₂𝐿_Ω𝜙𝑣^𝑛𝑑𝑥+_Ω𝑢^𝑞𝜙𝑑𝑥.(4.4)

Case 1. For the case 𝑝𝑞>1; we first prove the assertion under the stronger assumption 𝑝,𝑞>1. Without loss of generality, we assume that 𝑞=max{𝑝,𝑞}>1. Then using Jensen's inequality to (4.3) and (4.4), we see that 𝐽^′₁(𝑡)≥−𝜆𝐽₁+𝐽₂^𝑝,𝐽^′₂(𝑡)≥−𝜆𝐽₂+𝐽₁^𝑞.(4.5) Therefore 𝐽=𝐽₁+𝐽₂ satisfies 𝐽^′=𝐽^′₁+𝐽^′₂≥−𝜆𝐽+𝐽₁^𝑞+𝐽₂^𝑝≥−𝜆𝐽+𝐽₁^𝑝+𝐽₂^𝑝−𝐽₁.(4.6) In view of the inequality 𝐽₁^𝑝+𝐽₂^𝑝≥2^1−𝑝𝐽^𝑝, we discover that 𝐽^′≥−(𝜆+1)𝐽+2^1−𝑝𝐽^𝑝.(4.7) It follows that 𝐽 blows up in finite time whenever 𝐽(0)=_Ω(𝑢₀(𝑥)+𝑣₀(𝑥))𝜙(𝑥)𝑑𝑥(4.8) is sufficiently large. Furthermore, the solution of system (1.1) blows up in finite time.
If 𝑝<1 or 𝑞<1, in order to obtain our conclusion, we consider system (1.1) with zero Dirichlet boundary condition; then in light of Theorem 2 in [1], we obtain our result immediately.

Case 2. Consider now the case that 𝑚,𝑛>1. Since 𝑚,𝑛>1, Jensen's inequality can be applied to (4.3) and (4.4) like Step 1 to get 𝐽^′₁(𝑡)≥−𝜆𝐽₁+𝜆𝑀₁𝐿𝐽₁^𝑚,𝐽^′₂(𝑡)≥−𝜆𝐽₂+𝜆𝑀₂𝐿𝐽₂^𝑛.(4.9) Then the left arguments are the same as those for Case 1, we omit the details. The proof of Theorem 1.2 is complete.

5. Proof of Theorem 1.3

In this section, we will use an idea from Gladkov and Kim [16] to prove Theorem 1.3.

Proof of Theorem 1.3. Let Ω₁ be a bounded domain in ℝ^𝑁 satisfying the property that Ω⋐Ω₁ and let 𝜆₁ be the first eigenvalue of −Δ on Ω₁ with null Dirichlet boundary condition which satisfies the inequality 0<𝜆₁<𝜆.
Since 𝜑(𝑥,𝑦) and 𝜓(𝑥,𝑦) are nonnegative continuous defined in 𝜕Ω×Ω; then there exist some constants 0<𝐴,𝐵<+∞ such that_Ω𝜑(𝑥,𝑦)𝑑𝑦≤𝐴,_Ω𝜓(𝑥,𝑦)𝑑𝑦≤𝐵.(5.1) Denote 𝜙 an eigenfunction corresponding the eigenvalue 𝜆₁; then it is obviously that sup_Ω1𝜙inf_Ω𝜙≤sup_Ω1𝜙inf_Ω1𝜙<𝛿,(5.2) where 𝛿>1 is some constant. Choosing any 𝜀 which satisfies the inequality 0<𝜀≤min{(𝐴𝛿^𝑚)^{−1/(𝑚−1)},(𝐵𝛿^𝑛)^{−1/(𝑛−1)}},(5.3) and taking sup_Ω1𝜙=𝛿𝜀,(5.4) then, from (5.2), it follows easily that inf_𝜕Ω𝜙>𝜀.(5.5)Case 1. For 𝑞≥𝑝>1, set 𝑓(𝑡)=𝑒^−𝜆1𝑡[1+max{(𝛿𝜀)^𝑝−1,(𝛿𝜀)^𝑞−1}𝑒^{−(𝑝−1)𝜆1𝑡}𝜆₁]^{−1/(𝑝−1)}.(5.6) It is easy to check that 𝑓(𝑡) satisfy the following ordinary differential equation: 𝑓^′(𝑡)+𝜆₁𝑓−max{(𝛿𝜀)^𝑝−1,(𝛿𝜀)^𝑞−1}𝑓^𝑝=0.(5.7) Observe next that 𝑓(𝑡)<1, and so 𝑓^𝑝≥𝑓^𝑞 under the condition 𝑞≥𝑝>1.
Let𝑢(𝑥,𝑡)=𝑣(𝑥,𝑡)=𝜙(𝑥)𝑓(𝑡).(5.8) A series of computations yields 𝑢_𝑡−Δ𝑢−𝑣^𝑞=𝜙(𝑓^′+𝜆₁𝑓−𝜙^𝑞−1𝑓^𝑞)≥𝜙(𝑓^′+𝜆₁𝑓−max{(𝛿𝜀)^𝑝−1,(𝛿𝜀)^𝑞−1}𝑓^𝑝)≥0.(5.9) And similarly, we have 𝑣_𝑡−Δ𝑣−𝑢^𝑝=𝜙(𝑓^′+𝜆₁𝑓−𝜙^𝑝−1𝑓^𝑝)≥𝜙(𝑓^′+𝜆₁𝑓−max{(𝛿𝜀)^𝑝−1,(𝛿𝜀)^𝑞−1}𝑓^𝑝)≥0.(5.10) On the other hand, since ∫_Ω𝜑(𝑥,𝑦)𝑑𝑦≤𝐴, we have on the boundary that 𝑢(𝑥,𝑡)>𝜀𝑓(𝑡)≥𝐴(𝛿𝜀)^𝑚𝑓(𝑡)≥_Ω𝜑(𝑥,𝑦)(𝜙(𝑦)𝑓(𝑡))^𝑚𝑑𝑦.(5.11) Likewise, we have that 𝑣(𝑥,𝑡)>_Ω𝜓(𝑥,𝑦)(𝜙(𝑦)𝑓(𝑡))^𝑛𝑑𝑦.(5.12) Thus, by exploiting (5.9)–(5.12) and comparison principle, the solution of (1.1) exists globally provided that max{𝑢₀(𝑥),𝑣₀(𝑥)}≤[1+max{(𝛿𝜀)^𝑝−1,(𝛿𝜀)^𝑞−1}𝜆₁]^{−1/(𝑝−1)}𝜙(𝑥).(5.13)

Case 2. For 𝑝>𝑞>1, set 𝑓(𝑡)=𝑒^−𝜆1𝑡[1+max{(𝛿𝜀)^𝑝−1,(𝛿𝜀)^𝑞−1}𝑒^{−(𝑞−1)𝜆1𝑡}𝜆₁]^{−1/(𝑞−1)}.(5.14) We can immediately verify that 𝑓(𝑡) satisfy the following ordinary differential equation: 𝑓^′(𝑡)+𝜆₁𝑓−max{(𝛿𝜀)^𝑝−1,(𝛿𝜀)^𝑞−1}𝑓^𝑞=0.(5.15) In addition, it is obvious that 𝑓(𝑡)<1. Then we have that 𝑓^𝑞≥𝑓^𝑝 under the condition 𝑝>𝑞>1.
Let 𝑢(𝑥,𝑡)=𝑣(𝑥,𝑡)=𝜙(𝑥)𝑓(𝑡).(5.16) Similar to the arguments for the case 𝑞≥𝑝>1, we can prove that (𝑢(𝑥,𝑡),𝑣(𝑥,𝑡)) is a global supersolution of problem (1.1) provided that max{𝑢₀(𝑥),𝑣₀(𝑥)}≤[1+max{(𝛿𝜀)^𝑝−1,(𝛿𝜀)^𝑞−1}𝜆₁]^{−1/(𝑞−1)}𝜙(𝑥).(5.17)

The proof of Theorem 1.3 is complete.

Acknowledgments

The authors are very grateful to the anonymous referees for their careful reading and useful suggestions, which greatly improved the presentation of the paper. This work is funded by Innovative Talent Training Project, the Third Stage of “211 Project”, Chongqing University, project no. S-09110; C. Mu is supported in part by NSF of China (10771226) and in part by Natural Science Foundation Project of CQ CSTC (2007BB0124).