Roles of Weight Functions to a Nonlocal Porous Medium Equation with Inner Absorption and Nonlocal Boundary Condition

Abstract

This work is concerned with an initial boundary value problem for a nonlocal porous medium equation with inner absorption and weighted nonlocal boundary condition. We obtain the roles of weight function on whether determining the blowup of nonnegative solutions or not and establish the precise blow-up rate estimates under some suitable condition.

1. Introduction

Our main interest lies in the following nonlocal porous medium equation with inner absorption term:𝑢_𝑡=Δ𝑢^𝑚+𝑢^𝑝_Ω𝑢^𝑞𝑦,𝑡𝑑𝑦−𝑘𝑢^𝑟,(𝑥,𝑡)∈Ω×(0,+∞), (1.1) subjected to weighted linear nonlocal boundary and initial conditions, 𝑢(𝑥,𝑡)=_Ω𝑓𝑥,𝑦𝑢𝑦,𝑡𝑑𝑦,(𝑥,𝑡)∈𝜕Ω×(0,+∞),𝑢(𝑥,0)=𝑢₀(𝑥),𝑥∈Ω, (1.2)(1.3) where 𝑚>1,𝑝≥0,𝑞>0,𝑝+𝑞≥1,𝑟≥1,𝑘>0, and Ω⊂𝑅^𝑁(𝑁≥1) is a bounded domain with smooth boundary. The weight function 𝑓(𝑥,𝑦)≢0 is a nonnegative continuous function defined on 𝜕Ω×Ω, and ∫_Ω𝑓(𝑥,𝑦)𝑑𝑦>0 on 𝜕Ω. The initial value 𝑢₀(𝑥)∈𝐶^2+𝛼(Ω) with 0<𝛼<1 is a nonnegative continuous function satisfying the compatibility condition on 𝜕Ω.

Many natural phenomena have been formulated as nonlocal diffusive equation (1.1), such as the model of non-Newton flux in the mechanics of fluid, the model of population, biological species, and filtration (we refer to [1, 2] and the references therein). For instance, in the diffusion system of some biological species with human-controlled distribution, 𝑢(𝑥,𝑡),Δ𝑢^𝑚,𝑢^𝑝∫_Ω𝑢^𝑞(𝑦,𝑡)𝑑𝑦, and −𝑘 represent the density of the species, the mutation, the human-controlled distribution, and the decrement rate of biological species at location 𝑥 and time 𝑡, respectively. Due to the effect of spatial inhomogeneity, the arising of nonlocal term denotes that the evolution of the species at a point of space depends not only on the density of species in partial region but also on the total region (we refer to [3–5]). However, there are some important phenomena formulated as parabolic equations which are coupled with weighted nonlocal boundary conditions in mathematical models, such as thermoelasticity theory. In this case, the solution 𝑢(𝑥,𝑡) describes entropy per volume of the material (we refer to [6, 7]).

To motivate our work, let us recall some results of global and blow-up solutions to the initial boundary value problems with nonlocal terms or with nonlocal terms in boundary conditions (we refer to [8–16]). For the study of the initial boundary value problems for the parabolic equations with local terms which subject to the weighted nonlocal linear boundary condition (1.2), one can see [8–10]. For example, Friedman [8] studied the linear parabolic 𝑢_𝑡−𝐴𝑢=0,(𝑥,𝑡)∈Ω×(0,𝑇), (1.4) subjected to the nonlocal Dirichlet boundary condition (1.2), where 𝐴 is an elliptic operator, 𝐴=^𝑛_𝑖,𝑗=1𝑎_𝑖,𝑗(𝑥)𝜕²𝜕𝑥_𝑖𝜕𝑥_𝑗+^𝑛_𝑖=1𝑏_𝑖(𝑥)𝜕𝜕𝑥_𝑖+𝑐(𝑥),𝑐(𝑥)≤0. (1.5) He proved that when ∫_Ω𝑓(𝑥,𝑦)𝑑𝑦≤𝜌<1, the solution tends to 0 monotonously and exponentially as 𝑡→∞. With regard to more general discussions on initial boundary value problem for linear parabolic equation with nonlocal Neumann boundary condition, one can see [9] by Pao where the following problem was considered: 𝑢_𝑡−𝐿𝑢=𝑔(𝑥,𝑢),𝑥∈Ω,𝑡>0,𝐵𝑢=_Ω𝑓𝑢𝑢()𝑥,𝑦𝑦,𝑡𝑑𝑦,𝑥∈𝜕Ω,𝑡>0,𝑥,0=𝑢₀(𝑥),𝑥∈Ω, (1.6) where 𝐿𝑢=^𝑛_𝑖,𝑗=1𝑎_𝑖𝑗(𝑥)𝑢_{𝑥𝑖𝑥𝑗}+^𝑛_𝑗=1𝑏_𝑗(𝑥)𝑢_𝑥𝑗,𝐵𝑢=𝛼₀𝜕𝑢𝜕𝑛+𝑢. (1.7) He studied the asymptotic behavior of solutions and found the influence of weight function on the existence of global and blow-up solutions. Wang et al. [10] studied porous medium equation with power form source term 𝑢_𝑡=Δ𝑢^𝑚+𝑢^𝑝,(𝑥,𝑡)∈Ω×(0,+∞), (1.8) subjected to nonlocal boundary condition (1.2). By virtue of the method of upper-lower solutions, they obtained global existence, blow-up properties, and blow-up rate of solutions.

For the study of the initial boundary value problems for the parabolic equations with nonlocal terms which subjected to the weighted nonlocal linear boundary condition (1.2), we refer to [11–16]. Lin and Liu [11] considered the semilinear parabolic equation𝑢_𝑡=Δ𝑢+_Ω𝑔(𝑢)𝑑𝑥,(𝑥,𝑡)∈Ω×(0,+∞), (1.9) with nonlocal boundary condition (1.2). They established local existence, global existence, and blow-up properties of solutions. Moreover, they derived the uniform blow-up estimates for special 𝑔(𝑢) under suitable assumption; Cui and Yang [12] discussed the nonlocal slow diffusion equation 𝑢_𝑡=Δ𝑢^𝑚+𝑎𝑢^𝑝_Ω𝑢^𝑞𝑦,𝑡𝑑𝑦,(𝑥,𝑡)∈Ω×(0,+∞), (1.10) and they built global existence, blow-up properties, and blow-up rate of solutions. For the system of equations, we refer readers to [13] and the references therein.

Recently, Wang et al. [14] studied the following semilinear parabolic equation with nonlocal sources and interior absorption term: 𝑢_𝑡=Δ𝑢+_Ω𝑢^𝑞𝑑𝑥−𝛼𝑢^𝑟,(𝑥,𝑡)∈Ω×(0,+∞), (1.11) with weighted linear nonlocal boundary condition (1.2) and initial condition (1.3), where 𝑞≥1,𝑟≥1, and 𝛼>0. By using comparison principle and the method of upper-lower solutions, they got the following results.(a)If 1≤𝑞<𝑟, then the solution of the problem exists globally.(b)If 𝑞>𝑟≥1, the problem has solutions blowing up in finite time as well as global solutions. That is,(i)if ∫_Ω𝑓(𝑥,𝑦)𝑑𝑦≤1, and 𝑢₀(𝑥)≤(𝛼/|Ω|)^{1/(𝑟−𝑞)}, then the solution exists globally;(ii)if ∫_Ω𝑓(𝑥,𝑦)𝑑𝑦>1, and 𝑢₀(𝑥)>(𝛼/(|Ω|−𝛼))^1/𝑟(|Ω|>𝛼), then the solution blows up in finite time;(iii)for any 𝑓(𝑥,𝑦)≥0, there exists 𝑎₂>0 such that the solution blows up in finite time provided that 𝑢₀(𝑥)>𝑎₂𝜙(𝑥), where 𝜙(𝑥) is the corresponding normalized eigenfunction of −Δ with homogeneous Dirichlet boundary condition, and ∫_Ω𝜙(𝑥)𝑑𝑥=1.(c)If 𝑞=𝑟>1.(i)The solution blows up in finite time for any 𝑓(𝑥,𝑦)≥0 and large enough 𝑢₀.(ii)If ∫_Ω𝑓(𝑥,𝑦)𝑑𝑦<1, the solution exists globally for 𝑢₀(𝑥)≤𝑎₁Φ(𝑥) for some 𝑎₁>0, where Φ(𝑥) solves the following problem: −ΔΦ(𝑥)=𝛿₀Φ(𝑥)=,𝑥∈Ω,_Ω𝑓𝑥,𝑦𝑑𝑦,𝑥∈𝜕Ω, (1.12)here 𝛿₀ is a positive constant such that 0≤Φ(𝑥)≤1.

In addition, for the initial boundary value problem of (1.11) with weighted nonlinear boundary condition and Dirichlet boundary condition, we refer to [15, 16] and references therein, respectively.

The aim of this paper is to obtain the sufficient condition of global and blow-up solutions to problem (1.1)–(1.3) and to extend the results of the semilinear equation (1.11) to the quasilinear ones. The difficulty lies in finding the roles of weighted function in the boundary condition and the competitive relationship of nonlocal source and inner absorption on whether determining the blowup of solutions or not. Our detailed results are as follows.

Theorem 1.1. Suppose that 𝑝+𝑞>𝑟. If ∫_Ω𝑓(𝑥,𝑦)𝑑𝑦≥1 for 𝑥∈𝜕Ω, and the initial data 𝑢₀(𝑥)>(𝑘/(|Ω|−𝑘))^{1/(𝑝+𝑞)}(|Ω|>𝑘), then the solution of problem (1.1)–(1.3) blows up in finite time.

Remark 1.2. There may exist a global solution of problem (1.1)–(1.3) for small enough initial data under the condition of Theorem 1.1. Unfortunately, since the weight function satisfies the condition ∫_Ω𝑓(𝑥,𝑦)𝑑𝑦≥1 on the boundary, we cannot construct a suitable supersolution of problem (1.1)–(1.3).

Theorem 1.3. Suppose that 𝑝+𝑞>𝑟, if ∫_Ω𝑓(𝑥,𝑦)𝑑𝑦≤1 for 𝑥∈𝜕Ω, then the solution of problem (1.1)–(1.3) exists globally for the initial data 𝑢₀(𝑥)<(𝑘/|Ω|)^{1/(𝑝+𝑞−𝑟)}. If 𝑝+𝑞≥max{𝑚,𝑟}, then the solution of problem (1.1)–(1.3) blows up in finite time for large enough initial data and arbitrary 𝑓(𝑥,𝑦)>0.

Theorem 1.4. Suppose that 𝑝+𝑞<𝑟,then the solution of problem (1.1)–(1.3) exists globally for arbitrary 𝑓(𝑥,𝑦)>0.

Theorem 1.5. Suppose that 𝑝+𝑞=𝑟.∫_Ω𝑓(𝑥,𝑦)≤𝜌<1 for 𝑥∈𝜕Ω, where 𝜌 is a positive constant and 𝜌<1.(1)If 𝑚>𝑝+𝑞, then every nonnegative solution of problem (1.1)–(1.3) exists globally.(2)If 𝑚=𝑝+𝑞, then for |Ω|<𝑘+(𝛿/𝑀₁)(1−𝜌^𝑚), the solution of problem (1.1)–(1.3) exists globally, where 𝛿,𝑀₁>0 are as defined in (3.13).(3)If 𝑚<𝑝+𝑞, the solution of problem (1.1)–(1.3) exists globally for sufficient small initial data while it blows up in finite time for large enough initial data.

In order to show blow-up rate estimate of the blow-up solution, we need the following assumptions on the initial data 𝑢₀(𝑥):(C₁) Δ𝑢^𝑚₀+𝑢^𝑝₀∫_Ω𝑢^𝑞₀(𝑦,𝑡)𝑑𝑦−𝑘𝑢^𝑟₀>0, for 𝑥∈Ω;(C₂) there exists a constant 𝛿^′>0, such that Δ𝑢^𝑚₀+𝑢^𝑝_0Ω𝑢^𝑞₀𝑦,𝑡𝑑𝑦−𝑘𝑢^𝑟₀−𝛿^′𝑢₀^𝑝+𝑞≥0, (1.13)where 𝛿^′ will be determined later.

Theorem 1.6. Suppose that 𝑝+𝑞>max{𝑚,𝑟},∫_Ω𝑓(𝑥,𝑦)𝑑𝑦≤1 for 𝑥∈𝜕Ω, and the initial data satisfies the conditions (C₁)-(C₂), then 𝑐(𝑇−𝑡)^{−1/(𝑝+𝑞−1)}≤𝑢(𝑥,𝑡)≤𝐶(𝑇−𝑡)^{−1/(𝑝+𝑞−1)}, (1.14) where 𝑐=[|Ω|(𝑝+𝑞−1)]^{−1/(𝑝+𝑞−1)},𝐶=[𝛿(𝑝+𝑞−1)/𝑚]^{−1/(𝑝+𝑞−1)}.

The rest of our paper is organized as follows. In Section 2, with the definitions of weak upper and lower solutions, we will give the comparison principle of problem (1.1)–(1.3), which is an important tool in our research. The proofs of results of global existence and blow-up of solutions will be given in Section 3. And in Section 4, we will give the blow-up rate estimate of the blow-up solutions.

2. Comparison Principle and Local Existence

In this section, we establish a suitable comparison principle for problem (1.1)–(1.3). Let 𝑄_𝑇=Ω×(0,𝑇),𝑄_𝑇=Ω×(0,𝑇), and 𝑆_𝑇=𝜕Ω×(0,𝑇). Firstly, we start with the precise definitions of upper solution and lower solution of problem (1.1)–(1.3).

Definition 2.1. Suppose that 𝑢(𝑥,𝑡)∈𝐶^2,1(𝑄_𝑇)∩𝐶(𝑄_𝑇) is nonnegative and satisfies 𝑢_𝑡≤Δ𝑢^𝑚+𝑢^𝑝_Ω𝑢^𝑞𝑑𝑦−𝑘𝑢^𝑟,(𝑥,𝑡)∈𝑄_𝑇,𝑢(𝑥,𝑡)≤_Ω𝑓𝑥,𝑦𝑢𝑦,𝑡𝑑𝑦,(𝑥,𝑡)∈𝑆_𝑇,𝑢(𝑥,0)≤𝑢₀(𝑥),𝑥∈Ω, (2.1)(2.2)(2.3) then we call 𝑢(𝑥,𝑡) is the lower solution of problem (1.1)–(1.3), in 𝑄_𝑇.

Similarly, a nonnegative function 𝑢(𝑥,𝑡)∈𝐶^2,1(𝑄_𝑇)∩𝐶(𝑄_𝑇) is an upper solution if it satisfies (2.1)–(2.3) in the reverse order. We say 𝑢(𝑥,𝑡) is a solution of problem (1.1)–(1.3) in 𝑄_𝑇 if it is both an upper solution and a lower solution of problem (1.1)–(1.3) in 𝑄_𝑇 which is called classical solution.

The following comparison principle plays a crucial role in our proofs which can be obtained by establishing suitable test function and Gronwall’s inequality.

Proposition 2.2 (comparison principle). Suppose that 𝑢(𝑥,𝑡) and 𝑢(𝑥,𝑡) are the nonnegative subsolution and supersolution of problem (1.1)–(1.3), respectively. Moreover 𝑢(𝑥,0)≤𝑢(𝑥,0),𝑢(𝑥,0)≥0, and 𝑢(𝑥,0)≥𝛿>0 in Ω, then 𝑢(𝑥,𝑡)≤𝑢(𝑥,𝑡) in 𝑄_𝑇.

Proof. Let 𝜓(𝑥,𝑡)∈𝐶^2,1(𝑄_𝑇) be a nonnegative function with 𝜓=0 on 𝑆_𝑇. Multiplying the inequality in (2.1) by 𝜓(𝑥,𝑡) and integrating it on 𝑄_𝑇, we get _𝑄𝑇𝑢_𝑡𝜓𝑑𝑥𝑑𝑡≤_𝑄𝑇𝑢𝜓_𝑡+𝑢^𝑚Δ𝜓+𝜓𝑢^𝑝_Ω𝑢^𝑞𝑑𝑦−𝑘𝑢^𝑟−𝑑𝑥𝑑𝑡_𝑡0𝜕Ω𝜕𝜓𝜕𝑛_Ω𝑓𝑥,𝑦𝑢𝑑𝑦^𝑚𝑑𝑆𝑑𝑡, (2.4) and similarly, the upper solution satisfies the reversed inequality, _𝑄𝑇𝑢_𝑡𝜓𝑑𝑥𝑑𝑡≤_𝑄𝑇𝑢𝜓_𝑡+𝑢^𝑚Δ𝜓+𝜓𝑢^𝑝_Ω𝑢^𝑞𝑑𝑦−𝑘𝑢^𝑟−𝑑𝑥𝑑𝑡_𝑡0𝜕Ω𝜕𝜓𝜕𝑛_Ω𝑓𝑥,𝑦𝑢𝑑𝑦^𝑚𝑑𝑆𝑑𝑡. (2.5) Let 𝜔(𝑥,𝑡)=𝑢(𝑥,𝑡)−𝑢(𝑥,𝑡), we have _𝑄𝑇𝜔_𝑡𝜓𝑑𝑥𝑑𝑡≤_𝑄𝑇𝜓_𝑡+Φ₁(𝑥,𝑡)Δ𝜓+Φ₂(𝑥,𝑡)_Ω𝑢^𝑞𝑑𝑦−𝑘Φ₃()𝜓+𝑥,𝑡𝜔𝑑𝑥𝑑𝑡_𝑄𝑇𝑢^𝑝𝜓_ΩΦ₄𝜔𝑑𝑦𝑑𝑥𝑑𝑡−^𝑡_0𝜕Ω𝜕𝜓𝜕𝑛𝑚𝜉^𝑚−1_Ω𝑓𝑥,𝑦𝜔𝑑𝑦𝑑𝑆𝑑𝑡, (2.6) where Φ₁(𝑥,𝑡)=¹₀𝑚𝜃𝑢(𝑥,𝑡)+(1−𝜃)𝑢(𝑥,𝑡)^𝑚−1Φ𝑑𝜃,₂(𝑥,𝑡)=¹₀𝑝𝜃𝑢(𝑥,𝑡)+(1−𝜃)𝑢(𝑥,𝑡)^𝑝−1Φ𝑑𝜃,₃(𝑥,𝑡)=¹₀𝑟𝜃𝑢(𝑥,𝑡)+(1−𝜃)𝑢(𝑥,𝑡)^𝑟−1Φ𝑑𝜃,₄(𝑥,𝑡)=¹₀𝑞𝜃𝑢(𝑥,𝑡)+(1−𝜃)𝑢(𝑥,𝑡)^𝑞−1𝑑𝜃, (2.7) and 𝜉 is a function between ∫_Ω𝑓(𝑥,𝑦)𝑢𝑑𝑦 and ∫_Ω𝑓(𝑥,𝑦)𝑢𝑑𝑦. Noticing that 𝑢(𝑥,𝑡) and 𝑢(𝑥,𝑡) are bounded functions, it follows from 𝑚>1,𝑝≥1,𝑞≥1, and 𝑟≥1 that Φ_𝑖(𝑖=1,2,3,4) are bounded nonnegative functions. If 0≤𝑝<1 or 0<𝑞<1, we have Φ₂≤𝛿^𝑝−1 and Φ₄≤𝛿^𝑞−1 by the condition that 𝑢(𝑥,0)≥0 or 𝑢(𝑥,0)≥𝛿>0. Thus, we may choose an appropriate 𝜓(𝑥,𝑡) as in [17, p. 118–123] to obtain _Ω𝜔₊𝑑𝑥≤𝐶_1Ω𝜔(𝑥,0)₊𝑑𝑥+𝐶_2𝑄𝑇𝜔(𝑥,𝑡)𝑑𝑥𝑑𝑡, (2.8) where 𝜔₊=max{𝜔,0} and 𝐶₁,𝐶₂>0. It follows from 𝑢(𝑥,0)≤𝑢(𝑥,0) that _Ω𝜔₊𝑑𝑥≤𝐶_2𝑄𝑇𝜔(𝑥,𝑡)𝑑𝑥𝑑𝑡. (2.9) By Gronwall’s inequality, we know that (𝑢−𝑢)₊≤0, that is, 𝑢(𝑥,𝑡)≤𝑢(𝑥,𝑡) in 𝑄_𝑇.
For 𝑥∈𝜕Ω,𝑦∈Ω,𝑡>0, 𝑢−𝑢≤_Ω𝑓𝑥,𝑦𝑢−𝑢𝑑𝑦≤0. (2.10) This completes the proof.

Next, we state the local existence and uniqueness theorem without proof.

Theorem 2.3 (local existence and uniqueness). Suppose that the nonnegative initial data 𝑢₀(𝑥)∈𝐶^2+𝛼(Ω)∩𝐶(Ω)(0<𝛼<1) satisfies the compatibility condition. Then, there exists a constant 𝑇^∗>0 such that the problem (1.1)–(1.3) admits nonnegative solution 𝑢(𝑥,𝑡)∈𝐶^2,1(𝑄_𝑇)∩𝐶(Ω_𝑇) for each 𝑇<𝑇^∗. Furthermore, either 𝑇^∗=∞ or lim_{𝑡→𝑇∗}sup‖𝑢(𝑥,𝑡)‖_∞=∞. (2.11)

Remark 2.4. The existence of local nonnegative solutions in time to problem (1.1)–(1.3) can be obtained by using the fixed point theorem (see [18]) or the regular theory to get the suitable estimate in a standard limiting process (see [19, 20]). By the previous comparison principle, we can get the uniqueness of solution to the problem (1.1)–(1.3) in the case of 𝑝+𝑞≥1,𝑟≥1.

3. Global Existence and Blowup of Solutions

Comparing problems with the general homogeneous Dirichlet boundary condition, the existence of weight function on the boundary has a great influence on the global and nonglobal existence of solutions.

Proof of Theorem 1.1. Consider the following problem: 𝑣^′(𝑡)=|Ω|𝑣^𝑝+𝑞−𝑘𝑣^𝑟,𝑣(0)=𝑣₀. (3.1) As 𝑝+𝑞>𝑟, we know that 𝑣^𝑝+𝑞+1>𝑣^𝑟, and |Ω|𝑣^𝑝+𝑞−𝑘𝑣^𝑟≥(|Ω|−𝑘)𝑣^𝑝+𝑞−𝑘. Therefore, the solution of (3.1) is an upper solution of the following problem: 𝑣^′(𝑡)=(|Ω|−𝑘)𝑣^𝑝+𝑞−𝑘,𝑣(0)=𝑣₀. (3.2) When |Ω|>𝑘 and 𝑝+𝑞>1, it is known that the solution to the problem (3.2) blows up in finite time if 𝑣₀>(𝑘/(|Ω|−𝑘))^{1/(𝑝+𝑞)}.
It is obvious that the solution of problem (3.1) is a lower solution of problem (1.1)–(1.3) when ∫_Ω𝑓(𝑥,𝑦)𝑑𝑦≥1 and 𝑢₀(𝑥)>𝑣₀. By Proposition 2.2, 𝑢(𝑥,𝑡) is a blow-up solution of problem (1.1)–(1.3).

Proof of Theorem 1.3. (1) The case of 𝑝+𝑞>𝑟. Let 𝑢=(𝑘/|Ω|)^{1/(𝑝+𝑞−𝑟)}. It is easy to show that if ∫_Ω𝑓(𝑥,𝑦)𝑑𝑦<1 and 𝑢₀(𝑥)<(𝑘/|Ω|)^{1/(𝑝+𝑞−𝑟)},𝑢(𝑥,𝑡) is the upper solution of problem (1.1)–(1.3), then we can draw the conclusion.
(2) The case of 𝑝+𝑞≥max{𝑚,𝑟}. We need to establish a self-similar blow-up solution in order to prove the blow-up result. We first suppose that 𝜔∈𝐶¹(Ω),𝜔(𝑥)≥0, and 𝜔(𝑥) is not identically zero, and 𝜔(𝑥)|_𝜕Ω=0. Without loss of generality, we assume that 0∈Ω and 𝜔(0)>0.
Let 𝑢(𝑥,𝑡)=(𝑇−𝑡)^−𝛾𝑉^1/𝑚(𝜉),𝑉(𝜉)=(1−𝜉²/2𝐴)₊, and 𝜉=|𝑥|(𝑇−𝑡)^−𝜇, where 𝐴>1,0<𝑇<1,𝛾 and 𝜇>0. We know that supp𝑢₊(⋅,𝑡)=𝐵0,𝑅(𝑇−𝑡)^𝜇⊂𝐵(0,𝑅𝑇^𝜇)⊂Ω, (3.3) for sufficiently small 𝑇>0 and 𝑅=𝐴(𝐴+2). Calculating the derivative of 𝑢, we obtain 𝑢_𝑡=𝑚𝛾𝑉^1/𝑚(𝜉)+𝜇𝜉𝑉^′(𝜉)𝑉^{(1−𝑚)/𝑚}𝑚(𝑇−𝑡)^𝛾+1,Δ𝑢^𝑚𝑁=−𝐴(𝑇−𝑡)^{𝑚𝛾+2𝜇},𝑢^𝑝_Ω𝑢^𝑞𝑉𝑦,𝑡𝑑𝑦=^𝑝/𝑚(𝑇−𝑡)^{(𝑝+𝑞)𝛾𝐵(0,𝑅(𝑇−𝑡)𝜇)}𝑉^𝑞/𝑚|𝑥|(𝑇−𝑡)^𝜇≥𝑀𝑑𝑥(𝑇−𝑡)^{(𝑝+𝑞)𝛾−𝑁𝜇}, (3.4) where 𝑀=∫_𝐵(0,𝑅)𝑉^𝑞/𝑚(|𝜉|)𝑑𝜉>0.
It is easy to see that 𝑉^′(𝜉)≤0 and 𝑉<1, and 𝑢_𝑡−Δ𝑢^𝑚−𝑢^𝑝_Ω𝑢^𝑞𝑦,𝑡𝑑𝑦+𝑘𝑢^𝑟≤𝑚𝛾𝑉^1/𝑚(𝜉)+𝜇𝜉𝑉^′(𝜉)𝑉^{(1−𝑚)/𝑚}𝑚(𝑇−𝑡)^𝛾+1+𝑁𝐴(𝑇−𝑡)^{𝑚𝛾+2𝜇}−𝑀(𝑇−𝑡)^{(𝑝+𝑞)𝛾−𝑁𝜇}+𝑘(𝑇−𝑡)^−𝑟𝛾𝑉^𝑟/𝑚(𝜉)≤𝛾(𝑇−𝑡)^𝛾+1+𝑁𝐴(𝑇−𝑡)^{𝑚𝛾+2𝜇}−𝑀(𝑇−𝑡)^{(𝑝+𝑞)𝛾−𝑁𝜇}+𝑘(𝑇−𝑡)^𝑟𝛾. (3.5) Since 𝑝+𝑞≥𝑚>1, choosing 𝛾>0 such that 𝑚<(1+𝛾)/𝛾<𝑝+𝑞, and 𝜇 is sufficiently small such that 𝜇<min𝑝+𝑞−1𝛾−1𝑁,𝑝+𝑞−𝑟𝛾𝑁,𝑝+𝑞−𝑚.2+𝑁 (3.6) Then, for small enough 𝑇>0, we have 𝑢_𝑡−Δ𝑢^𝑚−𝑢^𝑝_Ω𝑢^𝑞𝑦,𝑡𝑑𝑦+𝑘𝑢^𝑟≤0. (3.7)
If 𝑥∈𝜕Ω,𝜔(0)>0, and 𝜔 is continuous, it is known that there exist positive 𝜀 and 𝜌 such that 𝜔≥𝜀 for 𝑥∈𝐵(0,𝜌). We can get 𝐵(0,𝑅𝑇^𝜎)⊂𝐵(0,𝜌)⊂Ω if 𝑇 is small enough. Then, 𝑢≤∫_Ω𝑓(𝑥,𝑦)𝑢𝑑𝑦 on 𝜕Ω×(0,𝑇). It follows from (3.3) that 𝑢(𝑥,0)≤𝐾₀𝜔(𝑥) for sufficiently large 𝐾₀. Therefore, one can observe that the solution to (1.1)–(1.3) exists no later than 𝑡=𝑇 provided that 𝑢₀≥𝐾₀𝜔(𝑥). This implies that the solution blows up in finite time for large enough initial data.

Proof of Theorem 1.4. Suppose that 𝜆₁>0 is the first eigenvalue of −Δ with homogeneous Dirichlet boundary condition, and 𝜙(𝑥) is the corresponding eigenfunction. Let 𝑀∫_Ω(1/(𝜙(𝑦)+𝜀))𝑑𝑦≤1 for some 0<𝜀<1, where 𝑀=max_{𝑥∈𝜕Ω,𝑦∈Ω}𝑓(𝑥,𝑦).
Now, we assume that 𝑢^𝑚=𝑣, then (1.1)–(1.3) becomes that𝑣^𝑛_𝑡−Δ𝑣=𝑣^𝑛𝑝_Ω𝑣^𝑞𝑛𝑑𝑦−𝑘𝑣^𝑛𝑟,𝑥∈Ω,𝑡>0,𝑣(𝑥,𝑡)=_Ω𝑓𝑥,𝑦𝑣^𝑛𝑦,𝑡𝑑𝑦^𝑚,𝑥∈𝜕Ω,𝑡>0,𝑣(𝑥,0)=𝑣₀(𝑥)=𝑢^𝑚₀(𝑥),𝑥∈Ω, (∗)(∗∗)(∗∗∗) where 𝑛=1/𝑚.
Set 𝑣(𝑥,𝑡)=𝐶𝑒^{𝛾𝑡/𝑛}/(𝜙(𝑥)+𝜀), where 𝐶 is determined later, then it follows that 𝑣^𝑛_𝑡=𝐶^𝑛𝛾𝑒^𝛾𝑡𝜙(𝑥)+𝜀^𝑛,∇𝑣=−𝐶𝑒^{𝛾𝑡/𝑛}∇𝜙(𝑥)𝜙(𝑥)+𝜀²=−∇𝜙𝜙(𝑥)+𝜀Δ𝑣,−𝑣=𝜙+𝜀𝑣Δ𝜙−∇𝜙⋅∇𝑣+𝑣∇𝜙⋅∇𝜙𝜙+𝜀²=𝜆₁𝜙𝑣−∇𝜙⋅∇𝑣+𝜙+𝜀∇𝜙²𝜙+𝜀²𝜆𝑣=¹𝜙+2𝜙+𝜀∇𝜙²𝜙+𝜀²𝑣. (3.8) And, we can get 𝑣^𝑛_𝑡−Δ𝑣−𝑣^𝑝𝑛_Ω𝑣^𝑞𝑛𝑑𝑦−𝑘𝑣^𝑛𝑟=𝐶^𝑛𝛾𝑒^𝛾𝑡𝜙(𝑥)+𝜀^𝑛−𝜆₁𝜙+2𝜙+𝜀∇𝜙²𝜙+𝜀²𝑣−𝐶𝑒^{𝛾𝑡/𝑛𝑛(𝑝+𝑞)}𝜙+𝜀^𝑛𝑝Ω1𝜙+𝜀^𝑛𝑞𝑑𝑦+𝑘𝐶𝑒^{𝛾𝑡/𝑛𝑛𝑟}𝜙+𝜀^𝑛𝑟=𝐶𝜙+𝜀^𝑛−1𝛾𝑒^{(𝛾−𝛾/𝑛)𝑡}−𝜆₁𝜙+2𝜙+𝜀∇𝜙²𝜙+𝜀²𝑣−𝐶𝑒^{𝛾𝑡/𝑛𝑛(𝑝+𝑞)}𝜙+𝜀^𝑛𝑝Ω1𝜙+𝜀^𝑛𝑞𝑑𝑦+𝑘𝐶^𝑛𝑟𝑒^𝛾𝑟𝑡𝜙+𝜀^𝑛𝑟. (3.9) Choosing 𝐶>max{sup_Ω[((𝜙+𝜀)^{𝑛(𝑟−𝑝)}/𝑘)∫_Ω(1/(𝜙+𝜀)^𝑛𝑞𝑑𝑦)]^{1/𝑛(𝑟−𝑝−𝑞)},sup(𝜙+𝜀)𝑢₀(𝑥)} and the proper 𝜆₁ such that 𝛾𝐶𝜙+𝜀^𝑛−1>𝜆₁𝜙+2𝜙+𝜀∇𝜙²𝜙+𝜀², (3.10) then 𝑣^𝑛_𝑡−Δ𝑣−𝑣^𝑝𝑛_Ω𝑣^𝑞𝑛𝑑𝑦−𝑘𝑣^𝑛𝑟≥0. (3.11)
For 𝑥∈𝜕Ω,𝑡>0, 𝑣(𝑥,𝑡)=𝐶𝑒^{𝛾𝑡/𝑛}𝜀≥_Ω𝐶𝑒^{𝛾𝑡/𝑛}𝑀𝜙𝑦+𝜀𝑑𝑦≥_Ω𝑓𝑥,𝑦𝑣𝑑𝑦, (3.12) so we can obtain that 𝑣(𝑥,𝑡) is the upper solution of problem (∗)–(∗∗∗). From Proposition 2.2, we know that there exists global solution of problem (∗)–(∗∗∗). Since the problem (1.1)–(1.3) has the same solution with problem (∗)–(∗∗∗). We know that there exists a global solution of problem (1.1)–(1.3). This completed the proof.

Proof of Theorem 1.5. Suppose that Φ(𝑥) solves the following problem: Φ(𝑥)−ΔΦ=𝛿,𝑥∈Ω,=0,𝑥∈𝜕Ω, (3.13) where 𝛿 is a positive constant such that 0≤Φ(𝑥)≤1. Then, let 𝑀₁=max_𝑥∈ΩΦ(𝑥)≤1.
Set 𝜔(𝑥,𝑡)=𝐴(𝜌^𝑚/(1−𝜌^𝑚)+Φ(𝑥)/𝑀₁)^1/𝑚, here 𝐴>0 will be determined later and 0<𝜌<1, 𝜔_𝑡−Δ𝜔^𝑚−𝜔^𝑝_Ω𝜔^𝑞𝑦,𝑡𝑑𝑦+𝑘𝜔^𝑟=𝐴^𝑚𝛿𝑀₁−𝐴^𝑝+𝑞𝜌^𝑚1−𝜌^𝑚+Φ(𝑥)𝑀₁^𝑝/𝑚_Ω𝜌^𝑚1−𝜌^𝑚+Φ(𝑥)𝑀₁^𝑞/𝑚𝑑𝑥+𝑘𝐴^𝑝+𝑞𝜌^𝑚1−𝜌^𝑚+Φ(𝑥)𝑀₁^{(𝑝+𝑞)/𝑚}≥𝐴^𝑚𝛿𝑀₁−𝐴^𝑝+𝑞|Ω|𝜌^𝑚1−𝜌^𝑚+Φ(𝑥)𝑀₁^{(𝑝+𝑞)/𝑚}+𝑘𝐴^𝑝+𝑞𝜌^𝑚1−𝜌^𝑚+Φ(𝑥)𝑀₁^{(𝑝+𝑞)/𝑚}=𝐴^𝑚𝛿𝑀₁+𝐴^𝑝+𝑞(|Ω𝜌𝑘−|)^𝑚1−𝜌^𝑚+Φ(𝑥)𝑀₁^{(𝑝+𝑞)/𝑚}. (3.14) It is obvious that the global existence result holds for 𝑘≥|Ω|. For 𝑘<|Ω|, since Φ(𝑥)/𝑀₁≤1, we know that 𝜌^𝑚/(1−𝜌^𝑚)+Φ(𝑥)/𝑀₁≤𝜌^𝑚/(1−𝜌^𝑚)+1=1/(1−𝜌^𝑚), then 𝜔_𝑡−Δ𝜔^𝑚−𝜔^𝑝_Ω𝜔^𝑞𝑦,𝑡𝑑𝑦+𝑘𝜔^𝑟≥𝐴^𝑚𝛿𝑀₁−𝐴^𝑝+𝑞Ω|)1(|−𝑘1−𝜌^{𝑚(𝑝+𝑞)/𝑚}. (3.15)(1)If 𝑚>𝑝+𝑞, choosing 𝐴=max{max|𝑢₀(𝑥)|,(|Ω|−𝑘)(𝑀/𝛿)(1/(1−𝜌^𝑚))^{(𝑝+𝑞)/𝑚}}, we have 𝜔_𝑡−Δ𝜔^𝑚−𝜔^𝑝_Ω𝜔^𝑞𝑦,𝑡𝑑𝑦+𝑘𝜔^𝑟≥0. (3.16)(2)If 𝑚=𝑝+𝑞, selecting |Ω|<𝑘+(𝛿/𝑀₁)(1−𝜌^𝑚), and 𝐴=max𝑢₀(𝑥), we get 𝜔_𝑡−Δ𝜔^𝑚−𝜔^𝑝_Ω𝜔^𝑞𝑦,𝑡𝑑𝑦+𝑘𝜔^𝑟≥0. (3.17)(3)If 𝑚<𝑝+𝑞, we choose max𝑢₀(𝑥)≤𝐴≤(|Ω|−𝑘)(𝑀₁/𝛿)(1/(1−𝜌^𝑚))^{−(𝑝+𝑞)/𝑚} such that 𝜔_𝑡−Δ𝜔^𝑚−𝜔^𝑝_Ω𝜔^𝑞𝑦,𝑡𝑑𝑦+𝑘𝜔^𝑟≥0. (3.18) For 𝑥∈𝜕Ω, 𝜔()𝜌𝑥,𝑡=𝐴^𝑚1−𝜌^𝑚1/𝑚𝜌=𝐴^𝑚1−𝜌^𝑚+1^1/𝑚𝜌𝜌≥𝐴^𝑚1−𝜌^𝑚+Φ(𝑥)𝑀₁^1/𝑚_Ω𝑓=𝑥,𝑦𝑑𝑦_Ω𝜔𝑦,𝑡𝑓𝑥,𝑦𝑑𝑦. (3.19)Through the previous discussion, we know that the global existence results hold.
For the blow-up case of 𝑚<𝑝+𝑞, it holds clearly from the second part of the proof of Theorem 1.3.

4. Blow-Up Rate Estimates

Next, we will get the following precise blow-up rate estimates for slow diffusion case under some suitable conditions.

Let 𝑣=𝑢^𝑚, we just need to consider the problem (∗)–(∗∗∗), and let 𝑝𝑛=𝑝₁,𝑞𝑛=𝑞₁, and 𝑟𝑛=𝑟₁, then (∗) becomes 𝑣^𝑛_𝑡=Δ𝑣+𝑣^𝑝1_Ω𝑣^𝑞1𝑦,𝑡𝑑𝑦−𝑘𝑣^𝑟1,𝑥∈Ω,𝑡>0. (4.1)

Suppose that 𝑣(𝑥,𝑡) is the blow-up solution of problem (∗)–(∗∗∗) in finite time 𝑇, and set 𝑉(𝑡)=max_𝑥∈Ω𝑣(𝑥,𝑡).

Proof of Theorem 1.6. (1) We can easily know that 𝑉(𝑡) is Lip continuous and differential almost everywhere, 𝑉^𝑛_𝑡≤𝑉^𝑝1_Ω𝑉^𝑞1𝑑𝑦−𝑘𝑉^𝑟1≤𝑉^𝑝1+𝑞1|Ω|−𝑘𝑉^𝑟1≤𝑉^𝑝1+𝑞1|Ω|. (4.2) Then, it follows that 𝑉_𝑡≤1𝑛|Ω|𝑉^{𝑝1+𝑞1+1−𝑛}. (4.3) Integrating it over (𝑡,𝑇), we get 𝑉(𝑡)≥1𝑛|Ω|𝑝₁+𝑞₁−𝑛^{−1/(𝑝1+𝑞1−𝑛)}(𝑇−𝑡)^{−1/(𝑝1+𝑞1−𝑛)}, (4.4) then 𝑢(𝑥,𝑡)≥𝑐(𝑇−𝑡)^{−1/(𝑝+𝑞−1)}, (4.5) where 𝑐=[|Ω|(𝑝+𝑞−1)]^{−1/(𝑝+𝑞−1)}.
(2) Next, we set 𝐽=𝑣_𝑡−𝛿𝑣^{𝑝1+𝑞1+1−𝑛}, where 𝛿>0, then 𝐽_𝑡=𝑣_𝑡𝑡−𝛿𝑝₁+𝑞₁+1−𝑛𝑣^{𝑝1+𝑞1−𝑛}𝑣_𝑡=1−𝑛𝑛𝑣^−𝑛𝑣^𝑛_𝑡𝑣_𝑡+1𝑛𝑣^1−𝑛Δ𝑣+𝑣^𝑝1_Ω𝑣^𝑞1𝑑𝑦−𝑘𝑣^𝑟1_𝑡−𝛿𝑝₁+𝑞₁+1−𝑛𝑣^{𝑝1+𝑞1−𝑛}𝑣_𝑡=(1−𝑛)𝑣⁻¹𝑣_𝑡²+1𝑛𝑣^1−𝑛Δ𝑣_𝑡+𝑝₁𝑛𝑣^𝑝1−𝑛𝑣_𝑡Ω𝑣^𝑞1𝑞𝑑𝑦+¹𝑛𝑣^{𝑝1+1−𝑛}_Ω𝑣^𝑞1−1𝑣_𝑡−𝑑𝑦𝑘𝑟₁𝑛𝑣^𝑟1−𝑛𝑣_𝑡−𝛿𝑝₁+𝑞₁+1−𝑛𝑣^{𝑝1+𝑞1−𝑛}𝑣_𝑡,𝐽_𝑡−𝑣^1−𝑛𝑛Δ𝐽−2𝛿(1−𝑛)𝑣^{𝑝1+𝑞1−𝑛}+𝑝₁𝑣^𝑝1−𝑛𝑛_Ω𝑣^𝑞1𝑑𝑦−𝑘𝑟₁𝑛𝑣^𝑟1−𝑛𝑞𝐽−¹𝑣^{𝑝1+1−𝑛}𝑛_Ω𝑣^𝑞1−1=()𝑣𝐽𝑑𝑦1−𝑛⁻¹𝐽²+𝛿²(1−𝑛)𝑣^{2𝑝1+2𝑞1−2𝑛+1}+𝑝₁𝛿𝑛−𝛿𝑛𝑝₁+𝑞₁+1−𝑛𝑣^{2𝑝1+𝑞1−2𝑛+1}_Ω𝑣^𝑞1+𝑑𝑦𝑘𝛿𝑛𝑝₁+𝑞₁+1−𝑛−𝑘𝑟₁𝛿𝑛𝑣^{𝑝1+𝑞1+𝑟1−2𝑛+1}+𝛿𝑛𝑝₁+𝑞₁−𝑛+1𝑝₁+𝑞₁−𝑛|∇𝑣|²+𝑞₁𝛿𝑛𝑣^{𝑝1+1−𝑛}_Ω𝑣^{𝑝1+2𝑞1−𝑛}𝑑𝑦≥𝛿²(1−𝑛)𝑣^{2𝑝1+2𝑞1−2𝑛+1}+𝛿𝑛𝑞₁+1−𝑛𝑣^{2𝑝1+𝑞1−2𝑛+1}_Ω𝑣^𝑞1𝑑𝑦+𝑘𝛿𝑛𝑝₁+𝑞₁+1−𝑛−𝑟₁𝑣^{𝑝1+𝑞1+𝑟1−2𝑛+1}+𝑞₁𝛿𝑛𝑣^{𝑝1+1−𝑛}_Ω𝑣^{𝑝1+2𝑞1−𝑛}𝑑𝑦=𝛿²(1−𝑛)𝑣^{2𝑝1+2𝑞1−2𝑛+1}+𝛿𝑛𝑣^{𝑝1−𝑛+1}𝑞_1Ω𝑣^{𝑝1+2𝑞1−𝑛}𝑑𝑦−𝑞₁+1−𝑛𝑣^{𝑝1+𝑞1−𝑛}_Ω𝑣^𝑞1+𝑑𝑦𝑘𝛿𝑛𝑝₁+𝑞₁+1−𝑛−𝑟₁𝑣^{𝑝1+𝑞1+𝑟1−2𝑛+1}. (4.6) Since 𝑞₁/(𝑝₁+2𝑞₁−𝑛)+(𝑝₁+𝑞₁−𝑛)/(𝑝₁+2𝑞₁−𝑛)=1, by Hölder’s inequality, we know that _Ω𝑣^𝑞1𝑑𝑦≤_Ω𝑣^{𝑝1+2𝑞1−𝑛𝑞1/(𝑝1+2𝑞1−𝑛)}, (4.7) and by Young inequality, we have 𝑣^{𝑝1+𝑞1−𝑛}_Ω𝑣^{𝑝1+2𝑞1−𝑛𝑞1/(𝑝1+2𝑞1−𝑛)}≤𝑝₁+𝑞₁−𝑛𝑝₁+2𝑞₁𝑣−𝑛^{𝑝1+𝑞1−𝑛(𝑝1+2𝑞1−𝑛)/(𝑝1+𝑞1−𝑛)}+𝑞₁𝑝₁+2𝑞₁−𝑛_Ω𝑣^{𝑝1+2𝑞1−𝑛}=𝑝₁+𝑞₁−𝑛𝑝₁+2𝑞₁𝑣−𝑛^{𝑝1+2𝑞1−𝑛}+𝑞₁𝑝₁+2𝑞₁−𝑛_Ω𝑣^{𝑝1+2𝑞1−𝑛}. (4.8)
Since 𝑝₁+𝑞₁>𝑟₁, we get 𝑝₁+2𝑞₁−𝑛>𝑞₁+1−𝑛 and 𝑝₁+𝑞₁−𝑟₁>0, then by (4.7)-(4.8), 𝐽_𝑡−𝑣^1−𝑛𝑛Δ𝐽−2𝛿(1−𝑛)𝑣^{𝑝1+𝑞1−𝑛}+𝑝₁𝑣^𝑝1−𝑛𝑛_Ω𝑣^𝑞1𝑑𝑦−𝑘𝑟₁𝑛𝑣^𝑟1−𝑛𝑞𝐽−¹𝑣^{𝑝1+1−𝑛}𝑛_Ω𝑣^𝑞1−1𝐽𝑑𝑦≥𝛿²(1−𝑛)𝑣^{2𝑝1+2𝑞1−2𝑛+1}+𝑘𝛿𝑛𝑝₁+𝑞₁+1−𝑛−𝑟₁𝑣^{𝑝1+𝑞1+𝑟1−2𝑛+1}−𝛿𝑝₁+𝑞₁−𝑛𝑛𝑣^{2𝑝1+2𝑞1−2𝑛+1}≥𝛿()−𝑝1−𝑛¹+𝑞₁−𝑛𝑛𝛿𝑣^{2𝑝1+2𝑞1−2𝑛+1}. (4.9) Choosing 𝛿≥(𝑝₁+𝑞₁−𝑛)/𝑛(1−𝑛), such that 𝐽_𝑡−𝑣^1−𝑛𝑛Δ𝐽−2𝛿(1−𝑛)𝑣^{𝑝1+𝑞1−𝑛}+𝑝₁𝑣^𝑝1−𝑛𝑛_Ω𝑣^𝑞1𝑑𝑦−𝑘𝑟₁𝑛𝑣^𝑟1−𝑛𝑞𝐽−¹𝑣^{𝑝1+1−𝑛}𝑛_Ω𝑣^𝑞1−1𝐽𝑑𝑦≥0. (4.10)
For (𝑥,𝑡)∈𝑆_𝑇, because of 𝑣_𝑡=𝐽+𝛿𝑣^{𝑝1+𝑞1+1−𝑛}, we then have 𝐽=𝑣_𝑡−𝛿𝑣^{𝑝1+𝑞1+1−𝑛}=𝑚_Ω𝑓𝑥,𝑦𝑣^1/𝑚𝑑𝑦^𝑚−1_Ω𝑓𝑥,𝑦𝑣^1/𝑚_𝑡𝑑𝑦−𝛿_Ω𝑓𝑥,𝑦𝑣^1/𝑚𝑑𝑦^{𝑚(𝑝1+𝑞1+1−𝑛)}=_Ω𝑓𝑥,𝑦𝑣^1/𝑚𝑑𝑦^𝑚−1𝑚_Ω𝑓𝑥,𝑦𝑣^1/𝑚_𝑡𝑑𝑦−𝛿_Ω𝑓𝑥,𝑦𝑣^1/𝑚𝑑𝑦^𝑝+𝑞=_Ω𝑓𝑥,𝑦𝑣^1/𝑚𝑑𝑦^𝑚−1_Ω𝑓𝑥,𝑦𝑣^{(1−𝑚)/𝑚}𝐽𝑑𝑦+𝛿_Ω𝑓𝑥,𝑦𝑣^{(𝑝+𝑞)/𝑚}𝑑𝑦−_Ω𝑓𝑥,𝑦𝑣^1/𝑚𝑑𝑦^𝑝+𝑞. (4.11) Noticing that 0<𝐹(𝑥)=∫_Ω𝑓(𝑥,𝑦)𝑑𝑦≤1 for 𝑥∈𝜕Ω,𝑝+𝑞>max{𝑚,𝑟}, and applying Jensen’s inequality to the last part in the previous inequality, we can get _Ω𝑓𝑥,𝑦𝑣^{(𝑝+𝑞)/𝑚}𝑑𝑦−_Ω𝑓𝑥,𝑦𝑣^1/𝑚𝑑𝑦^𝑝+𝑞≥𝐹(𝑥)_Ω𝑓𝑥,𝑦𝑣^1/𝑚𝑑𝑦𝐹(𝑥)^𝑝+𝑞−_Ω𝑓𝑥,𝑦𝑣^1/𝑚𝑑𝑦^𝑝+𝑞. (4.12) Hence, we can get 𝐽≥_Ω𝑓𝑥,𝑦𝑣^1/𝑚𝑑𝑦^𝑚−1_Ω𝑓𝑥,𝑦𝑣^{(1−𝑚)/𝑚}𝐽𝑑𝑦≥0. (4.13) Since 𝑢₀(𝑥) satisfies the conditions (C₁)-(C₂) and 𝑣=𝑢^𝑚, 𝐽(𝑥,𝑡)=𝑚𝑢^𝑚−1𝑢_𝑡−𝛿𝑢^{𝑚(𝑝1+𝑞1+1−𝑛)}=𝑚𝑢^𝑚−1Δ𝑢^𝑚+𝑢^𝑝_Ω𝑢^𝑞𝑦,𝑡𝑑𝑦−𝑘𝑢^𝑟−𝛿^′𝑢^𝑝+𝑞, (4.14) where 𝛿^′=𝛿/𝑚, then 𝐽(𝑥,0)≥0. Combining (4.10)-(4.13), we can know that 𝐽(𝑥,𝑡)≥0 for (𝑥,𝑡)∈𝑄_𝑇, that is, 𝑣_𝑡≥𝛿𝑣^{𝑝1+𝑞1+1−𝑛}.
Integrating it over (𝑡,𝑇), we have 𝑣≤𝛿𝑝₁+𝑞₁−𝑛^{−1/(𝑝1+𝑞1−𝑛)}(𝑇−𝑡)^{−1/(𝑝1+𝑞1−𝑛)}, (4.15) then it follows that 𝑢(𝑥,𝑡)≤𝐶(𝑇−𝑡)^{−1/(𝑝+𝑞−1)}, (4.16) where 𝐶=[𝛿(𝑝+𝑞−1)/𝑚]^{−1/(𝑝+𝑞−1)}. This completed the proof.

Acknowledgments

This work is supported by the Natural Science Foundation of Shandong Province of China (ZR2012AM018). The authors would like to deeply thank all the reviewers for their insightful and constructive comments.