Blow-Up and Global Existence for a Degenerate Parabolic System with Nonlocal Sources

Abstract

This paper investigates the blow-up and global existence of nonnegative solutions for a class of nonlocal degenerate parabolic system. By using the super- and subsolution techniques, the critical exponent of the system is determined. That is, if 𝑃_𝑐=𝑝₁𝑞₁−(𝑚−𝑝₂)(𝑛−𝑞₂)<0, then every nonnegative solution is global, whereas if 𝑃_𝑐>0, there are solutions that blowup and others that are global according to the size of initial values 𝑢₀(𝑥) and 𝑣₀(𝑥). When 𝑃_𝑐=0, we show that if the domain is sufficiently small, every nonnegative solution is global while if the domain large enough that is, if it contains a sufficiently large ball, there is no global solution.

1. Introduction and Description of Results

In this paper, we investigate the blowup and global existence of nonnegative solutions for the following degenerate parabolic system with nonlocal sources:𝑢_𝑡=Δ𝑢^𝑚+𝑣^𝑝1‖𝑢‖^𝑝2_𝛼,()(),𝑣𝑥,𝑡∈Ω×0,𝑇_𝑡=Δ𝑣^𝑛+𝑢^𝑞1‖𝑣‖^𝑞2_𝛽,()(),𝑢()𝑥,𝑡∈Ω×0,𝑇𝑥,0=𝑢₀(𝑥),𝑣(𝑥,0)=𝑣₀(𝑥)𝑢()()()(),,𝑥∈Ω,𝑥,𝑡=0,𝑣𝑥,𝑡=0,𝑥,𝑡∈𝜕Ω×0,𝑇 (1.1) where Ω is a bounded domain in ℝ^𝑁(𝑁≥1) with smooth boundary 𝜕Ω and 𝑢₀(𝑥),𝑣₀(𝑥) are nonnegative bounded functions in Ω, constants 𝑚,𝑛>1,  𝛼,𝛽≥1,  𝑝₁,𝑞₁,𝑝₂,𝑞₂>0, where ‖⋅‖^𝛼_𝛼=∫_Ω|⋅|^𝛼d𝑥.

Equation (1.1) constitutes a simple example of a reaction diffusion system exhibiting a nontrivial coupling on the unknowns 𝑢(𝑥,𝑡),𝑣(𝑥,𝑡), such as heat propagations in a two-component combustible mixture [1], chemical processes [2], and interaction of two biological groups without self-limiting [3]. And they are worth to study because of the applications to heat and mass transport processes. In addition, there exist interesting interactions among the multi-nonlinearities described by the eight exponents 𝑚,𝑛,𝑝₁,𝑝₂,𝑞₁,𝑞₂ and 𝛼,𝛽 in the problem (1.1).

In the past two decades, many physical phenomena were formulated into nonlocal mathematical models (see [4–7] and references therein) and studied by many authors. Degenerate parabolic equations involving a nonlocal source, which arise in a population model that communicates through chemical means, were studied in [8, 9]. At the same time, there are many important results that have appeared on blowup problems for nonlinear parabolic system. We will recall some of those results concerning the first initial-boundary problem. For the other related works on the global existence and blowup of solutions of nonlinear parabolic system, we refer the readers to [10, 11] and references therein.

In [4], Escobedo and Herrero studied the system𝑢_𝑡=Δ𝑢+𝑣^𝑝,𝑣_𝑡=Δ𝑣+𝑢^𝑞,𝑥∈Ω,𝑡>0 (1.2) in a bounded domain Ω⊂ℝ^𝑁 with null Dirichlet boundary conditions. The authors show that if 𝑝𝑞≤1, every solution of (1.2) is global, whereas if 𝑝𝑞>1, there are solutions that blowup and others that are global according to the size of initial values 𝑢₀(𝑥) and 𝑣₀(𝑥).

In [12], Galaktionov et al. considered the system𝑢_𝑡=Δ𝑢^𝜈+1+𝑣^𝑝,𝑣_𝑡=Δ𝑣^𝜇+1+𝑢^𝑞,(𝑥,𝑡)∈Ω×(0,𝑇) (1.3) with homogeneous Dirichlet boundary conditions, and they proved that 𝑝_𝑐=𝑝𝑞−(𝜈+1)(𝜇+1) is the critical exponent. Zheng [13] and Li et al. [14] studied the following systems:𝑢_𝑡=Δ𝑢+𝑢^𝑝1𝑣^𝑞1,𝑣_𝑡=Δ𝑣+𝑢^𝑝2𝑣^𝑞2,(𝑥,𝑡)∈Ω×𝐑₊,𝑢_𝑡=Δ𝑢+_Ω𝑢^𝑚(𝑥,𝑡)𝑣^𝑛(𝑥,𝑡)d𝑥,𝑣_𝑡=Δ𝑣+_Ω𝑢^𝑝(𝑥,𝑡)𝑣^𝑞(𝑥,𝑡)d𝑥,𝑥∈Ω,𝑡>0, (1.4) respectively. They obtained some results on the global solutions, the blowup solutions and the blowup profiles. Lately, Deng et al. in [15] considered the following nonlocal degenerate parabolic system:𝑢_𝑡=Δ𝑢^𝑚+𝑎‖𝑣‖^𝑝_𝛼,𝑣_𝑡=Δ𝑣^𝑛+𝑏‖𝑢‖^𝑞_𝛽,(𝑥,𝑡)∈Ω×(0,𝑇) (1.5) with homogeneous Dirichlet boundary conditions. Several interesting results are established as follows.(i)If 𝑝𝑞<𝑚𝑛, then every nonnegative solution of (1.5) is global.(ii)If 𝑝𝑞=𝑚𝑛, then if the domain is sufficiently small, the nonnegative solution of (1.5) is global, whereas if the domain contains a sufficiently large ball and 𝑢₀(𝑥),𝑣₀(𝑥)>0, the nonnegative solution blows up in finite time.(iii)If 𝑝𝑞>𝑚𝑛, then there are solutions of (1.5) that blowup and others that are global according to the size of initial data 𝑢₀(𝑥) and 𝑣₀(𝑥).

Our present work is motivated by [12–15] mentioned above. The main purpose of this paper is to extend and improve the results in [15]. At the same time, we will prove that 𝑝_𝑐=𝑝₁𝑞₁−(𝑚−𝑝₂)(𝑛−𝑞₂) is also the critical exponent of system (1.1). Our main results are as follows, two theorems concern the global existence and blowup conditions of the solutions.

Theorem 1.1. If one of the following conditions holds, then the nonnegative solution of system (1.1) exists globally.(1)𝑚>𝑝₂,𝑛>𝑞₂ and 𝑝₁𝑞₁<(𝑚−𝑝₂)(𝑛−𝑞₂).(2)𝑚>𝑝₂,𝑛>𝑞₂,𝑝₁𝑞₁=(𝑚−𝑝₂)(𝑛−𝑞₂) and the domain (|Ω|) is sufficiently small.(3)𝑚>𝑝₂,𝑛>𝑞₂,𝑝₁𝑞₁>(𝑚−𝑝₂)(𝑛−𝑞₂) and the initial data 𝑢₀(𝑥),𝑣₀(𝑥) are sufficiently small.(4)𝑚≤𝑝₂ or 𝑛≤𝑞₂ and the initial data 𝑢₀(𝑥),𝑣₀(𝑥) are sufficiently small.

Theorem 1.2. If one of the following conditions holds, then the nonnegative solution of system (1.1) blows up in a finite time.(1)𝑚>𝑝₂,𝑛>𝑞₂, 𝑝₁𝑞₁>(𝑚−𝑝₂)(𝑛−𝑞₂) and the initial data 𝑢₀(𝑥),𝑣₀(𝑥) are sufficiently large.(2)𝑚>𝑝₂,𝑛>𝑞₂, 𝑝₁𝑞₁=(𝑚−𝑝₂)(𝑛−𝑞₂) and the domain contains a sufficiently large ball, moreover, 𝑢₀(𝑥) and 𝑣₀(𝑥) are large enough.(3)𝑚≤𝑝₂ or 𝑛≤𝑞₂ and initial data 𝑢₀(𝑥),𝑣₀(𝑥) are sufficiently large.

This paper is organized as follows. In the next Section, we establish the local existence theorem and give some auxiliary lemmas. In Section 3, which concerns global existence, we prove Theorem 1.1. In Section 4, which deals with the blowup phenomenon, we prove Theorem 1.2.

2. Local Existence and Comparison Principle

Similar to the Propositions 2.1 and 2.2 of [15], we give the maximum principle and the comparison principle for the nonlocal parabolic system. For convenience, we denote 𝑄_𝑇=Ω×(0,𝑇),𝑄_𝑇=Ω×[0,𝑇],𝑆_𝑇=𝜕Ω×(0,𝑇), where 0<𝑇<+∞.

As it is now well known that degenerate equation needs not possess classical solution, we begin by giving a precise definition of a weak solution for system (1.1). To this end, define the class of test functionsΨ≡𝜓(𝑥,𝑡)∈𝐶𝑄_𝑇;𝜓_𝑡,Δ𝜓∈𝐶𝑄_𝑇∩𝐿²𝑄_𝑇;𝜓≥0;𝜓(𝑥,𝑡)_{𝑥∈𝜕Ω}=0. (2.1)

Definition 2.1. A pair of vector function (𝑢(𝑥,𝑡),𝑣(𝑥,𝑡)) defined on 𝑄_𝑇 is called a super-solution of (1.1), if all the following conditions hold:(1)𝑢(𝑥,𝑡),𝑣(𝑥,𝑡)∈𝐿^∞(𝑄_𝑇); (2)if (𝑥,𝑡)∈𝑆_𝑇,𝑢(𝑥,𝑡),𝑣(𝑥,𝑡)≥0, and for all 𝑥∈Ω, 𝑢(𝑥,0)≥𝑢₀(𝑥),𝑣(𝑥,0)≥𝑣₀(𝑥);(3)for every 𝑡∈[0,𝑇] and any 𝜓₁,𝜓₂∈Ψ,  _Ω𝑢(𝑥,𝑡)𝜓₁(𝑥,𝑡)−𝑢₀(𝑥)𝜓₁(𝑥,0)d𝑥≥^𝑡_0Ω𝑢𝜓_1𝑠+𝑢^𝑚Δ𝜓₁+𝑣^𝑝1‖𝑢‖^𝑝2_𝛼𝜓₁d𝑥d𝑠,_Ω𝑣(𝑥,𝑡)𝜓₂(𝑥,𝑡)−𝑣₀(𝑥)𝜓₂(𝑥,0)d𝑥≥^𝑡_0Ω𝑣𝜓_2𝑠+𝑣^𝑛Δ𝜓₂+𝑢^𝑞1‖𝑣‖^𝑞2_𝛽𝜓₂d𝑥d𝑠. (2.2) A subsolution (𝑢(𝑥,𝑡),𝑣(𝑥,𝑡)) can be defined in a similar way.

Next, we state the maximum principle and comparison principle, and the proofs that are quite standard, we omit them here.

Lemma 2.2 (maximum principle). Suppose that 𝜔₁(𝑥,𝑡),𝜔₂(𝑥,𝑡)∈𝐶^2,1(𝑄_𝑇)∩𝐶(𝑄_𝑇) and satisfy 𝑀₁𝜔=𝜔_1𝑡−𝑑₁Δ𝜔₁−^𝑁_𝑗=1𝑎_1𝑗𝜔_1𝑥𝑗−𝑐₁₁𝜔₁−𝑐₁₂𝜔₂−𝑐_13Ω𝑐₁₄𝜔₁d𝑥≥0,(𝑥,𝑡)∈𝑄_𝑇,𝑀₂𝜔=𝜔_2𝑡−𝑑₂Δ𝜔₂−^𝑁_𝑗=1𝑎_2𝑗𝜔_2𝑥𝑗−𝑐₂₁𝜔₂−𝑐₂₂𝜔₁−𝑐_23Ω𝑐₂₄𝜔₂d𝑥≥0,(𝑥,𝑡)∈𝑄_𝑇,𝜔₁(𝑥,𝑡)≥0,𝜔₂(𝑥,𝑡)≥0,(𝑥,𝑡)∈𝑆_𝑇,𝜔₁(𝑥,0)≥0,𝜔₂(𝑥,0)≥0,𝑥∈Ω, (2.3) where 𝑑_𝑖(𝑥,𝑡),𝑐_𝑖𝑗(𝑥,𝑡)(𝑖=1,2;𝑗=1,2,3,4) and 𝑎_𝑖𝑗(𝑥,𝑡),(𝑖=1,2;𝑗=1,2,…,𝑁) are the continuous and the bounded functions on 𝑄_𝑇, respectively, and 𝑑_𝑖(𝑥,𝑡),𝑐_𝑖2(𝑥,𝑡),𝑐_𝑖3(𝑥,𝑡),𝑐_𝑖4(𝑥,𝑡)≥0,𝑖=1,2,(𝑥,𝑡)∈Ω×(0,𝑇]. (2.4) Then 𝜔_𝑖(𝑥,𝑡)≥0 on 𝑄_𝑇.

Lemma 2.3 (comparison principle). Let (𝑢,𝑣) and (𝑢,𝑣) be a nonnegative supersolution and a nonnegative subsolution of system (1.1), respectively. Then (𝑢,𝑣)≤(𝑢,𝑣) on 𝑄_𝑇 if (𝑢(𝑥,0),𝑣(𝑥,0))≤(𝑢(𝑥,0),𝑣(𝑥,0)) and either 𝑢,𝑣≥𝜌>0𝑜𝑟𝑢,𝑣≥𝜌>0 (2.5) hold.

Theorem 2.4 (local existence and continuation). Assume 𝑢₀,𝑣₀≥0,𝑢₀,𝑣₀∈𝐿^∞(Ω), there is a 𝑇^∗=𝑇^∗(𝑢₀,𝑣₀)>0 such that there exists a nonnegative weak solution (𝑢(𝑥,𝑡),𝑣(𝑥,𝑡)) of (1.1) for each 𝑇<𝑇^∗. Furthermore, either 𝑇^∗=+∞ or limsup_{𝑡→𝑇∗}‖𝑢(⋅,𝑡)‖_∞=+∞𝑜𝑟limsup_{𝑡→𝑇∗}‖𝑣(⋅,𝑡)‖_∞=+∞. (2.6)

Proof. Owing to the degeneracy of equations of (1.1), in order to prove the existence of solution, for 𝑘=1,2,…, we first consider the following corresponding regularized system 𝑢_𝑘𝑡=Δ𝑓_𝑘𝑢_𝑘+𝑔^′_𝑘𝑣_𝑘^𝑝1𝑔_𝑘(𝑢_𝑘)^𝑝2_𝛼,(𝑥,𝑡)∈𝑄_𝑇,𝑣_𝑘𝑡=Δ𝑓^′_𝑘𝑣_𝑘+𝑔_𝑘𝑢_𝑘^𝑞1𝑔^′_𝑘(𝑣_𝑘)^𝑞2_𝛽,(𝑥,𝑡)∈𝑄_𝑇,𝑢_𝑘(𝑥,𝑡)=𝑣_𝑘()=1𝑥,𝑡𝑘,(𝑥,𝑡)∈𝑆_𝑇,𝑢_𝑘(𝑥,0)=𝑢_0𝑖(𝑥)+1𝑘,𝑣_𝑘(𝑥,0)=𝑣_0𝑖(𝑥)+1𝑘,𝑥∈Ω, (2.7) where 𝑓_𝑘𝑢_𝑘=𝑢^𝑚_𝑘,𝑢_𝑘≥1𝑘,1𝑘^𝑚,𝑢_𝑘<1𝑘,𝑓^′_𝑘𝑣_𝑘=𝑣^𝑛_𝑘,𝑣_𝑘≥1𝑘,1𝑘^𝑛,𝑣_𝑘<1𝑘,𝑔_𝑘𝑢_𝑘=𝑢_𝑘,𝑢_𝑘≥1𝑘,1𝑘,𝑢_𝑘<1𝑘,𝑔^′_𝑘𝑣_𝑘=𝑣_𝑘,𝑣_𝑘≥1𝑘,1𝑘,𝑣_𝑘<1𝑘, (2.8) and 𝑢_0𝑖(𝑥),𝑣_0𝑖(𝑥) are smooth approximation of 𝑢₀(𝑥),𝑣₀(𝑥) with supp𝑢_0𝑖⊂Ω and supp𝑣_0𝑖⊂Ω, respectively. It is known that the system (2.7) has a unique classical solution (𝑢^𝑖_𝑘,𝑣^𝑖_𝑘)∈𝐶(Ω×[0,𝑇_𝑖(𝑘)))∩𝐶^2,1(Ω×(0,𝑇_𝑖(𝑘))) for 0<𝑇_𝑖(𝑘)≤∞ by the classical theory for parabolic equations, where 𝑇_𝑖(𝑘) is the maximal existence time. By a direct computation and the classical maximum principle, we have 𝑢^𝑖_𝑘,𝑣^𝑖_𝑘≥1/𝑘. Hence (𝑢^𝑖_𝑘,𝑣^𝑖_𝑘) satisfies 𝑢^𝑖_𝑘𝑡=Δ𝑢^𝑖_𝑘^𝑚+𝑣^𝑖_𝑘^𝑝1𝑢^𝑖_𝑘^𝑝2_𝛼,𝑣^𝑖_𝑘𝑡=Δ𝑣^𝑖_𝑘^𝑛+𝑢^𝑖_𝑘^𝑞1𝑣^𝑖_𝑘^𝑞2_𝛽,(𝑥,𝑡)∈𝑄_{𝑇𝑖(𝑘)} (2.9) with the corresponding initial and boundary conditions. At the same time, if 𝑘₁>𝑘₂, according to Lemma 2.2, we get 𝑢^𝑖_𝑘1(𝑥,𝑡),𝑣^𝑖_𝑘1(𝑥,𝑡)≤𝑢^𝑖_𝑘2(𝑥,𝑡),𝑣^𝑖_𝑘2(𝑥,𝑡),(𝑥,𝑡)∈Ω×0,𝑇_𝑖𝑘₂, (2.10) and 𝑇_𝑖(𝑘₁)≥𝑇_𝑖(𝑘₂). On the other hand, passing to the limit 𝑖→∞, it follows that 𝑢_𝑘(𝑥,𝑡)=lim_𝑖→∞𝑢^𝑖_𝑘(𝑥,𝑡),𝑣_𝑘(𝑥,𝑡)=lim_𝑖→∞𝑣^𝑖_𝑘(𝑥,𝑡), (2.11) and (𝑢_𝑘,𝑣_𝑘) is a weak solution of 𝑢_𝑘𝑡=Δ𝑢_𝑘^𝑚+𝑣_𝑘^𝑝1𝑢_𝑘^𝑝2_𝛼,𝑣_𝑘𝑡=Δ𝑣_𝑘^𝑛+𝑢_𝑘^𝑞1𝑣_𝑘^𝑞2_𝛽,(𝑥,𝑡)∈𝑄_𝑇(𝑘) (2.12) with the corresponding initial and boundary conditions on 𝑄_𝑇(𝑘), where 𝑇(𝑘)=lim_𝑖→∞𝑇_𝑖(𝑘) is the maximal existence time. Here a weak solution of (2.12) is defined in a manner similar to that for (1.1), only the equalities for 𝑢 and 𝑣; (2.2) may be replaced with _Ω𝑢_𝑘(𝑥,𝑡)𝜓₁(𝑥,𝑡)−𝑢₀(𝑥)+1𝑘𝜓₁(=𝑥,0)d𝑥_𝑡0Ω𝑢_𝑘𝜓_1𝑠+𝑢^𝑚_𝑘Δ𝜓₁+𝑣^𝑝1_𝑘𝑢_𝑘^𝑝2_𝛼𝜓₁1d𝑥d𝑠+𝑘^𝑡_0𝜕Ω𝜕𝜓₁𝜕𝜈d𝜎d𝑠,_Ω𝑣_𝑘(𝑥,𝑡)𝜓₂(𝑥,𝑡)−𝑣₀(𝑥)+1𝑘𝜓₂()=𝑥,0d𝑥_𝑡0Ω𝑣_𝑘𝜓_2𝑠+𝑣^𝑛_𝑘Δ𝜓₂+𝑢^𝑞1_𝑘𝑣_𝑘^𝑞2_𝛽𝜓₂1d𝑥d𝑠+𝑘^𝑡_0𝜕Ω𝜕𝜓₂𝜕𝜈d𝜎d𝑠, (2.13) respectively. Then, passing to the limit 𝑖→∞, it happens that (𝑢_𝑘1,𝑣_𝑘1)≤(𝑢_𝑘2,𝑣_𝑘2) and 𝑇(𝑘₁)≥𝑇(𝑘₂) if 𝑘₁>𝑘₂.
Therefore, the limit 𝑇^∗=lim_𝑘→∞𝑇(𝑘) exists, and the pointwise limit 𝑢(𝑥,𝑡)=lim_𝑘→∞𝑢_𝑘(𝑥,𝑡),𝑣(𝑥,𝑡)=lim_𝑘→∞𝑣_𝑘(𝑥,𝑡) (2.14) exists for any (𝑥,𝑡)∈Ω×[0,𝑇^∗). Furthermore, as the convergence of the sequence is monotone, passing to the limit 𝑘→∞ in (2.13), we get that (𝑢(𝑥,𝑡),𝑣(𝑥,𝑡)) is a nonnegative weak solution of (1.1). Thus the proof is completed.

Denote𝐴=𝑚−𝑝₂−𝑝₁−𝑞₁𝑛−𝑞₂𝑙,𝐿=¹𝑙₂. (2.15)

We give Lemmas 2.5 and 2.6 that will be used in the following; please see [16] for their proofs.

Lemma 2.5. If 𝑚>𝑝₂,𝑛>𝑞₂ and 𝑝₁𝑞₁<(𝑚−𝑝₂)(𝑛−𝑞₂), then there exist two positive constants 𝑙₁,𝑙₂ such that 𝐴𝐿>(0,0)^𝑇.

Lemma 2.6. If 𝑚≤𝑝₂ or 𝑛≤𝑞₂ or 𝑝₁𝑞₁>(𝑚−𝑝₂)(𝑛−𝑞₂), then there exist two positive constants 𝑙₁,𝑙₂ such that 𝐴𝐿<(0,0)^𝑇.

3. Proof of Theorem 1.1

According to Lemma 2.3, we only need to construct bounded super-solutions for any 𝑇>0. Let 𝜑(𝑥) be the unique position solution of the following linear elliptic problem:−Δ𝜑(𝑥)=1,𝑥∈Ω;𝜑(𝑥)=0,𝑥∈𝜕Ω. (3.1) Denote 𝐶=max_𝑥∈Ω𝜑(𝑥), then 0≤𝜑(𝑥)≤𝐶. We define the functions 𝑢,𝑣 as follows:𝑢(𝑥,𝑡)=𝑘𝜑(𝑥)+1^𝑙1,𝑣(𝑥,𝑡)=𝑘𝜑(𝑥)+1^𝑙2, (3.2) where 𝑙₁,𝑙₂<1 such that 𝑚𝑙₁,𝑛𝑙₂<1, and 𝑘>0 will be fixed later. Clearly, for any 𝑇>0, (𝑢,𝑣) is a bounded function and 𝑢≥𝑘^𝑙1>0,𝑣≥𝑘^𝑙2>0. Then, we have 𝑢_𝑡−Δ𝑢^𝑚=−𝑘^𝑚𝑙1𝑚𝑙₁𝑚𝑙₁−1𝜑+1^{𝑚𝑙1−2}∇𝜑²+𝑘^𝑚𝑙1𝑚𝑙₁𝜑+1^{𝑚𝑙1−1}≥𝑘^𝑚𝑙1𝑚𝑙₁𝜑+1^{𝑚𝑙1−1}≥𝑘^𝑚𝑙1𝑚𝑙₁(𝐶+1)^{𝑚𝑙1−1},𝑣^𝑝1‖𝑢‖^𝑝2_𝛼=𝑘^𝑝1𝑙2𝜑+1^𝑝1𝑙2𝑘𝜑+1^𝑙1𝑝2_𝛼≤𝑘^{𝑝1𝑙2+𝑝2𝑙1}(𝐶+1)^{𝑝1𝑙2+𝑝2𝑙1}|Ω|^𝑝2/𝛼,𝑣_𝑡−Δ𝑣^𝑛≥𝑘^𝑛𝑙2𝑛𝑙₂(𝐶+1)^{𝑛𝑙2−1},𝑢^𝑞1‖𝑣‖^𝑞2_𝛽≤𝑘^{𝑞1𝑙1+𝑞2𝑙2}(𝐶+1)^{𝑞1𝑙1+𝑞2𝑙2}|Ω|^𝑞2/𝛽. (3.3) Denote𝐶₁=|Ω|^𝑝2/𝛼𝑚𝑙₁(𝐶+1)^{𝑝1𝑙2+𝑝2𝑙1−𝑚𝑙1+11/(𝑚𝑙1−𝑝1𝑙2−𝑝2𝑙1)},𝐶₂=|Ω|^𝑞2/𝛽𝑛𝑙₂(𝐶+1)^{𝑞1𝑙1+𝑞2𝑙2−𝑛𝑙2+11/(𝑛𝑙2−𝑞1𝑙1−𝑞2𝑙2)}. (3.4)

(1) If 𝑚>𝑝₂,𝑛>𝑞₂ and 𝑝₁𝑞₁<(𝑚−𝑝₂)(𝑛−𝑞₂), by Lemma 2.5, there exist two positive constants 𝑙₁,𝑙₂<1 such that𝑝₁𝑙₂+𝑝₂𝑙₁<𝑚𝑙₁,𝑞₁𝑙₁+𝑞₂𝑙₂<𝑛𝑙₂,𝑚𝑙₁,𝑛𝑙₂<1. (3.5) Therefore, we can choose 𝑘 sufficiently large that 𝑘>max{𝐶₁,𝐶₂} and𝑘𝜑+1^𝑙1≥𝑢₀(𝑥),𝑘𝜑+1^𝑙2≥𝑣₀(𝑥). (3.6) Now, it follows from (3.3)–(3.6) that (𝑢,𝑣) is a positive super-solution of (1.1).

(2) If 𝑚>𝑝₂,𝑛>𝑞₂ and 𝑝₁𝑞₁=(𝑚−𝑝₂)(𝑛−𝑞₂), then there exist two positive constants 𝑙₁,𝑙₂<1 such that𝑝₁𝑙₂+𝑝₂𝑙₁=𝑚𝑙₁,𝑞₁𝑙₁+𝑞₂𝑙₂=𝑛𝑙₂,𝑚𝑙₁,𝑛𝑙₂<1. (3.7) Without loss of generality, we may assume that Ω⊂⊂𝐵, where 𝐵 is a sufficiently large ball. And we denote 𝜑_𝐵(𝑥) is the unique positive solution of the following linear elliptic problem:−Δ𝜑(𝑥)=1,𝑥∈𝐵;𝜑(𝑥)=0,𝑥∈𝜕𝐵. (3.8) Let 𝐶₀=max_𝑥∈𝐵𝜑_𝐵(𝑥), then 𝐶≤𝐶₀. Therefore, as long as Ω is sufficiently small and such that|Ω|<min𝑚𝑙₁𝐶₀+1^𝛼/𝑝2,𝑛𝑙₂𝐶₀+1^𝛽/𝑞2. (3.9) Furthermore, choose 𝑘 large enough to satisfy (3.6).Then, it follows from (3.3) and (3.6)–(3.9) that (𝑢,𝑣) is a positive super-solution of (1.1).

(3) If 𝑚>𝑝₂,𝑛>𝑞₂ and 𝑝₁𝑞₁>(𝑚−𝑝₂)(𝑛−𝑞₂), by Lemma 2.6, there exist two positive constants 𝑙₁,𝑙₂<1 such that𝑝₁𝑙₂+𝑝₂𝑙₁>𝑚𝑙₁,𝑞₁𝑙₁+𝑞₂𝑙₂>𝑛𝑙₂,𝑚𝑙₁,𝑛𝑙₂<1. (3.10) Hence, we can choose 𝑘 sufficiently small that 𝑘<min{𝐶₁,𝐶₂}, and provided 𝑢₀(𝑥),𝑣₀(𝑥) are also sufficiently small to satisfy (3.6). Then, from (3.3) and (3.6), (3.10), we know that (𝑢,𝑣) is a positive super-solution of (1.1).

(4) Finally, if 𝑚≤𝑝₂ or 𝑛≤𝑞₂, there exist also positive constants 𝑙₁,𝑙₂<1 such that (3.10) and 𝑚𝑙₁,𝑛𝑙₂<1. Similar to the proof of (3), we get that (𝑢,𝑣) is a positive super-solution of (1.1).

Thus the proof of Theorem 1.1 is completed.

4. Proof of Theorem 1.2

Due to the requirement of the comparison principle, we will construct blowup subsolutions in some subdomain of Ω in which 𝑢,𝑣>0. We use an idea from Souplet [17] and apply it to degenerate parabolic equation. By translation, one may assume without loss of generality that 0∈Ω. Let 𝐵=𝐵(0,𝑅)⊂Ω be an open ball with radius 𝑅, and 𝜓(𝑥) is a nontrivial nonnegative continuous function, vanished on 𝜕𝐵 and 𝜓(0)>0. Set()=1𝑢𝑥,𝑡(𝑇−𝑡)^𝑙1𝑉|𝑥|(𝑇−𝑡)^𝜎()=1,𝑣𝑥,𝑡(𝑇−𝑡)^𝑙2𝑉|𝑥|(𝑇−𝑡)^𝜎, (4.1) with𝑉(𝑟)=𝑅³6−𝑅2𝑟²+13𝑟³|𝑥|,𝑟=(𝑇−𝑡)^𝜎,0≤𝑟≤𝑅, (4.2) where 𝑙₁,𝑙₂,𝜎>0 and 0<𝑇<1 are to be determined later. Clearly, 0≤𝑉(𝑟)≤𝑅³/6 and 𝑉(𝑟) is nonincreasing since 𝑉^′(𝑟)=𝑟(𝑟−𝑅)≤0. Note that, for 𝑇 small enough,Supp𝑢(⋅,𝑡)=Supp𝑣(⋅,𝑡)=𝐵0,𝑅(𝑇−𝑡)^𝜎⊂𝐵0,𝑅𝑇^𝜎⊂Ω,0≤𝑡<𝑇. (4.3)

Obviously, (𝑢,𝑣) becomes unbounded as 𝑡→𝑇⁻ at the point 𝑥=0. Calculating directly, we obtain𝑢_𝑡−Δ𝑢^𝑚=𝑙₁𝑉(𝑟)+𝜎𝑟𝑉^′(𝑟)(𝑇−𝑡)^𝑙1+1−𝑚(𝑚−1)𝑉^𝑚−2(𝑟)𝑉^′(𝑟)²(𝑇−𝑡)^{𝑚𝑙1+2𝜎}−𝑚𝑉^𝑚−1(𝑟)(𝑇−𝑡)^{𝑚𝑙1+2𝜎}(()𝑟)≤𝑙−𝑁𝑅+𝑁+1₁𝑅³/6(𝑇−𝑡)^𝑙1+1+𝑚𝑉^𝑚−1(𝑟)(𝑇−𝑡)^{𝑚𝑙1+2𝜎}(𝑁𝑅−(𝑁+1)𝑟),(𝑥,𝑡)∈𝐵×(0,𝑇),𝑣_𝑡−Δ𝑣^𝑛≤𝑙₂𝑅³/6(𝑇−𝑡)^𝑙2+1+𝑛𝑉^𝑛−1(𝑟)(𝑇−𝑡)^{𝑛𝑙2+2𝜎}(𝑁𝑅−(𝑁+1)𝑟),(𝑥,𝑡)∈𝐵×(0,𝑇). (4.4) notice 𝑇<1 is sufficiently small. Therefore, If 0≤𝑟≤𝑟₀=𝑁𝑅/(𝑁+1), we have 𝑉(𝑟)≥𝑅³(3𝑁+1)/(6(𝑁+1)³); then𝑣^𝑝1‖𝑢‖^𝑝2_𝛼=𝑉^𝑝1(𝑟)(𝑇−𝑡)^𝑝1𝑙2_𝐵𝑉^𝛼(𝑟)(𝑇−𝑡)^{𝛼𝑙1𝑝2/𝛼}≥|𝐵|^𝑝2/𝛼(𝑇−𝑡)^{𝑝1𝑙2+𝑝2𝑙1}𝑅³(3𝑁+1)6(𝑁+1)^3𝑝1+𝑝2,𝑢^𝑞1‖𝑣‖^𝑞2_𝛽=𝑉^𝑞1(𝑟)(𝑇−𝑡)^𝑞1𝑙1_𝐵𝑉^𝛽(𝑟)(𝑇−𝑡)^{𝛽𝑙2𝑞2/𝛽}≥|𝐵|^𝑞2/𝛽(𝑇−𝑡)^{𝑞1𝑙1+𝑞2𝑙2}𝑅³(3𝑁+1)6(𝑁+1)^3𝑞1+𝑞2. (4.5) Hence,𝑢_𝑡−Δ𝑢^𝑚−𝑣^𝑝1‖𝑢‖^𝑝2_𝛼≤𝑙₁𝑅³/6(𝑇−𝑡)^𝑙1+1+𝑚𝑉^𝑚−1(𝑟)(𝑇−𝑡)^{𝑚𝑙1+2𝜎}(()𝑟)−|𝐵|𝑁𝑅−𝑁+1^𝑝2/𝛼(𝑇−𝑡)^{𝑝1𝑙2+𝑝2𝑙1}𝑅³(3𝑁+1)6(𝑁+1)^3𝑝1+𝑝2,𝑣_𝑡−Δ𝑣^𝑛−𝑢^𝑞1‖𝑣‖^𝑞2_𝛽≤𝑙₂𝑅³/6(𝑇−𝑡)^𝑙2+1+𝑛𝑉^𝑛−1(𝑟)(𝑇−𝑡)^{𝑛𝑙2+2𝜎}(()𝑟)−|𝐵|𝑁𝑅−𝑁+1^𝑞2/𝛽(𝑇−𝑡)^{𝑞1𝑙1+𝑞2𝑙2}𝑅³(3𝑁+1)6(𝑁+1)^3𝑞1+𝑞2. (4.6) Similarly, if 𝑁𝑅/(𝑁+1)<𝑟≤𝑅, then𝑢_𝑡−Δ𝑢^𝑚−𝑣^𝑝1‖𝑢‖^𝑝2_𝛼≤𝑙₁𝑅³/6(𝑇−𝑡)^𝑙1+1+𝑚𝑉^𝑚−1(𝑟)(𝑇−𝑡)^{𝑚𝑙1+2𝜎}(𝑁𝑅−(𝑁+1)𝑟),𝑣_𝑡−Δ𝑣^𝑛−𝑢^𝑞1‖𝑣‖^𝑞2_𝛽≤𝑙₂𝑅³/6(𝑇−𝑡)^𝑙2+1+𝑛𝑉^𝑛−1(𝑟)(𝑇−𝑡)^{𝑛𝑙2+2𝜎}(𝑁𝑅−(𝑁+1)𝑟). (4.7)(4.8)

(1) If 𝑚>𝑝₂,𝑛>𝑞₂ and 𝑝₁𝑞₁>(𝑚−𝑝₂)(𝑛−𝑞₂), by Lemma 2.6, there exist two positive constants 𝑙₁,𝑙₂ large enough that𝑝₁𝑙₂+𝑝₂𝑙₁>𝑚𝑙₁,𝑞₁𝑙₁+𝑞₂𝑙₂>𝑛𝑙₂,(𝑚−1)𝑙₁>1,(𝑛−1)𝑙₂>1. (4.9)

Then, we can choose 𝜎>0 sufficiently small such that𝑝₁𝑙₂+𝑝₂𝑙₁>𝑚𝑙₁+2𝜎>𝑚𝑙₁>𝑙₁+1,𝑞₁𝑙₁+𝑞₂𝑙₂>𝑛𝑙₂+2𝜎>𝑛𝑙₂>𝑙₂+1. (4.10) Hence, for sufficiently small 𝑇>0, (4.6)–(4.8) imply that𝑢_𝑡−Δ𝑢^𝑚−𝑣^𝑝1‖𝑢‖^𝑝2_𝛼≤0,𝑣_𝑡−Δ𝑣^𝑛−𝑢^𝑞1‖𝑣‖^𝑞2_𝛽≤0,(𝑥,𝑡)∈𝐵×(0,𝑇). (4.11)

Since 𝜓(0)>0 and 𝜓(𝑥) is continuous, there exist two positive constants 𝜌 and 𝜀 such that 𝜓(𝑥)≥𝜀 for all 𝑥∈𝐵(0,𝜌)⊂𝐵(0,𝑅). Choose 𝑇 small enough to insure 𝐵(0,𝑅𝑇^𝜎)⊂𝐵(0,𝜌)⊂Ω, hence 𝑢≤0,𝑣≤0 on 𝜕Ω×(0,𝑇), and from (4.3) it follows that 𝑢(𝑥,0)≤𝐾𝜓(𝑥),𝑣(𝑥,0)≤𝐾𝜓(𝑥) for sufficiently large 𝐾. By comparison principle, we have (𝑢,𝑣)≤(𝑢,𝑣) provided that 𝑢₀(𝑥)≥𝐾𝜓(𝑥) and 𝑣₀(𝑥)≥𝐾𝜓(𝑥). It follows that (𝑢,𝑣) blows up in finite time.

(2) Next, we consider the case 𝑚>𝑝₂,𝑛>𝑞₂ and 𝑝₁𝑞₁=(𝑚−𝑝₂)(𝑛−𝑞₂). Clearly, there exist two positive constants 𝑙₁,𝑙₂ such that𝑚𝑙₁=𝑝₁𝑙₂+𝑝₂𝑙₁,𝑛𝑙₂=𝑞₁𝑙₁+𝑞₂𝑙₂,(𝑚−1)𝑙₁>1,(𝑛−1)𝑙₂>1. (4.12)

Denote by 𝜆_𝐵𝑅>0 and 𝜙_𝑅(𝑟) the first eigenvalue and the corresponding eigenfunction of the following eigenvalue problem:−𝜙^′′(𝑟)−𝑁−1𝑟𝜙^′(𝑟)=𝜆𝜙(𝑟),𝑟∈(0,𝑅);𝜙^′(0)=0,𝜙(𝑅)=0. (4.13) It is well known that 𝜙_𝑅(𝑟) can be normalized as 𝜙_𝑅(𝑟)>0 in 𝐵 and 𝜙_𝑅(0)=max_𝐵𝜙_𝑅(𝑟)=1. By the property (let 𝜏=𝑟/𝑅) of eigenvalues and eigenfunctions we see that 𝜆_𝐵𝑅=𝑅⁻²𝜆_𝐵1 and 𝜙_𝑅(𝑟)=𝜙₁(𝑟/𝑅)=𝜙₁(𝜏), where 𝜆_𝐵1 and 𝜙₁(𝜏) are the first eigenvalue and the corresponding normalized eigenfunction of the eigenvalue problem in the unit ball 𝐵₁(0). Moreover,max_𝐵1𝜙₁(𝜏)=𝜙₁(0)=𝜙_𝑅(0)=max_𝐵𝜙_𝑅(𝑟)=1. (4.14)

Similar to (4.1), we define the functions 𝑢(𝑥,𝑡),𝑣(𝑥,𝑡) in the form()=1𝑢𝑥,𝑡(𝑇−𝑡)^𝑙1𝜙^𝑙1_𝑅𝑥()=1(||),𝑣𝑥,𝑡(𝑇−𝑡)^𝑙2𝜙^𝑙2_𝑅(|𝑥|). (4.15) In the following, we will prove that (𝑢,𝑣) blows up in finite time in the ball 𝐵=𝐵(0,𝑅). Because of so, (𝑢,𝑣) does blow up in the larger domain Ω. Calculating directly, we have𝑢_𝑡−Δ𝑢^𝑚−𝑣^𝑝1‖𝑢‖^𝑝2_𝛼≤𝜙^𝑙1_𝑅(𝑇−𝑡)^𝑙1+1𝑙₁−1(𝑇−𝑡)^{𝑚𝑙1−𝑙1−1}𝑐₁−𝜆_𝐵𝑅𝑚𝑙₁,𝑣_𝑡−Δ𝑣^𝑛−𝑢^𝑞1‖𝑣‖^𝑞2_𝛽≤𝜙^𝑙2_𝑅(𝑇−𝑡)^𝑙2+1𝑙₂−1(𝑇−𝑡)^{𝑛𝑙2−𝑙2−1}𝑐₂−𝜆_𝐵𝑅𝑛𝑙₂, (4.16) where𝑐₁=𝜙^𝑝1𝑙2_𝑅𝜙^𝑙1_𝑅^𝑝2_𝛼≤𝐾₁𝑅^{𝑁𝑝2/𝛼},𝑐₂=𝜙^𝑞1𝑙1_𝑅𝜙^𝑙2_𝑅^𝑞2_𝛽≤𝐾₂𝑅^{𝑁𝑞2/𝛽} (4.17) and 𝐾₁,𝐾₂ are constants independent of 𝑅. Then, in view of 𝜆_𝐵𝑅=𝑅⁻²𝜆_𝐵1, we may assume that 𝑅, that is, the ball 𝐵, is sufficiently large that𝜆_𝐵𝑅𝑐<min¹𝑚𝑙₁,𝑐₂𝑛𝑙₂. (4.18) Hence, for sufficiently small 𝑇>0, (4.16) implies that𝑢_𝑡−Δ𝑢^𝑚−𝑣^𝑝1‖𝑢‖^𝑝2_𝛼≤0,𝑣_𝑡−Δ𝑣^𝑛−𝑢^𝑞1‖𝑣‖^𝑞2_𝛽≤0. (4.19)

Therefore, (𝑢,𝑣) is a positive subsolution of (1.1) in the ball 𝐵, which blows up in finite time provided the initial data is sufficiently large that𝑢(𝑥,0)=𝑇^−𝑙1𝜙^𝑙1_𝑅(|𝑥|)≤𝑢₀(𝑥),𝑣(𝑥,0)=𝑇^−𝑙2𝜙^𝑙2_𝑅(|𝑥|)≤𝑣₀(𝑥) (4.20) in the ball 𝐵.

(3) Finally, if 𝑚≤𝑝₂ or 𝑛≤𝑞₂, there also exist two positive constants 𝑙₁,𝑙₂ to satisfy (4.9). Similar to the proof of case (1), we can get that (𝑢,𝑣) is a subsolution of (1.1), which blows up in finite time.

Thus the proof of Theorem 1.2 is completed.

Acknowledgments

The first author is supported by National Natural Science Foundation of China. The second author would like to express their many thanks to the Editor and Reviewers for their constructive suggestions to improve the previous version of this paper. This work is supported by the NNSF of China (11071100).