Blowup Analysis for a Nonlocal Diffusion Equation with Reaction and Absorption

Abstract

We investigate a nonlocal reaction diffusion equation with absorption under Neumann boundary. We obtain optimal conditions on the exponents of the reaction and absorption terms for the existence of solutions blowing up in finite time, or for the global existence and boundedness of all solutions. For the blowup solutions, we also study the blowup rate estimates and the localization of blowup set. Moreover, we show some numerical experiments which illustrate our results.

1. Introduction

In this paper, we devote our attention to the singularity analysis of the following nonlocal diffusion equation: 𝑢_𝑡(𝑥,𝑡)=_Ω𝐽(𝑥−𝑦)(𝑢(𝑦,𝑢𝑡)(−𝑥,𝑢0)(𝑥=,𝑡𝑢))₀𝑑𝑦+𝑢^𝑝(𝑥,𝑡)−𝑘𝑢^𝑞(𝑥,𝑡),𝑥∈Ω,𝑡>0,(𝑥),𝑥∈Ω.(1.1) Here Ω is a bounded connected and smooth domain, which contains the origin, and 𝐽∶ℝ^𝑁→ℝ is a nonnegative, bounded, symmetric radially and strictly decreasing function with ∫_ℝ𝑁𝐽(𝑧)𝑑𝑧=1, and 𝑝, 𝑞, 𝑘 are all positive constants. We take the initial datum, 𝑢₀(𝑥), nonnegative and nontrivial.

Equation in (1.1) is called a nonlocal diffusion equation in the sense that the diffusion of the density 𝑢 at a point 𝑥 and time 𝑡 does not only depend on 𝑢(𝑥,𝑡), but on all the values of 𝑢 in a neighborhood of 𝑥 through the convolution term. Maybe the simplest linear version of nonlocal model (1.1) is 𝑢_𝑡(𝑥,𝑡)=_𝑅𝑁𝐽(𝑥−𝑦)(𝑢(𝑦,𝑡)−𝑢(𝑥,𝑡))𝑑𝑦=𝐽∗𝑢(𝑥,𝑡)−𝑢(𝑥,𝑡).(1.2) In recent years, the linear equation (1.2) and its variations have been widely used to model diffusion process, for example, in biology, dislocations dynamics, phase transition model, material science, and network model and so forth (see [1–7] and the references therein). The idea hidden inside those model is simple to understand. As stated in [6], if 𝑢(𝑥,𝑡) is thought of as a density at the point 𝑥 and time 𝑡 and 𝐽(𝑥−𝑦) is thought of as the probability distribution of jumping from location 𝑦 to location 𝑥, then the convolution (𝐽∗𝑢)(𝑥,𝑡)∶=∫_ℝ𝑁𝐽(𝑥−𝑦)𝑢(𝑦,𝑡)𝑑𝑦 is the rate at which individuals are arriving at 𝑥 from all other places and −𝑢(𝑥,𝑡)=−∫_ℝ𝑁𝐽(𝑥−𝑦)𝑢(𝑥,𝑡)𝑑𝑦 is the rate at which they are leaving location 𝑥 to travel to all other sites.

In the past decades, some works have shown that (1.2) shares many properties with the classical heat equation 𝑢_𝑡−Δ𝑢=0(1.3) such as the existence of constant bounded stationary solutions and the maximum principle, but there is no regularizing effect in general for the former [8]. Therefore, as mentioned in [9–11], it is an interesting topic to compare the properties of solutions to the nonlocal diffusion equation with that of the corresponding local diffusion cases.

To motivate our works, we would like to remark that, in recent years, (1.3) and its variations have been extensively studied. In particular, a considerable effort has been devoted to studying the blowup properties of the following classical diffusion equation with reaction (𝛼>0) and/or absorption (𝑘>0) under Dirichlet or Neumann boundary 𝑢_𝑡−Δ𝑢=𝛼𝑢^𝑝−𝑘𝑢^𝑞,𝑥∈Ω,𝑡>0,(1.4) which provides a simple biological or physical model. For instance, by constructing the self-similar weak subsolutions, Bedjaoui and Souplet [12] obtained the following conclusion for (1.4) with 𝛼=1 under Dirichlet or Neumann boundary: if 𝑝<max{𝑞,1}, then all solutions are global. If 𝑝>max{𝑞,1}, there exist solutions of (1.4) which blow up in finite time. In the critical case 𝑝=max{𝑞,1}, the results may depend on the size of the coefficient 𝑘. In the Dirichlet boundary case, Xiang et al. [13] also studied the blowup rate estimates and obtained the following results: if 𝑝>max{𝑞,1} and the solution 𝑢(𝑥,𝑡) of (1.4) blows up at 𝑇, then there exists constants 𝐶>𝑐>0 such that max_[]Ω×0,𝑡𝑢(𝑥,𝜏)≥𝑐(𝑇−𝑡)^1/(𝑝−1),max_[]Ω×0,𝑡𝑢(𝑥,𝜏)≤𝐶(𝑇−𝑡)^1/(𝑝−1)if1<𝑝<1+2𝑁+1.(1.5)

In this paper, we will deal with the blowup properties of the nonlocal diffusion problem (1.1) and compare them with that of problem (1.4). In our model (1.1), the absorption term −𝑘𝑢^𝑞(𝑥,𝑡) represents a rate of consumption due to an internal reaction, and we are integrating in Ω and thus imposing the condition that the diffusion takes place only in Ω, which means that the individual may enter or leave the domain. This is so called Neumann boundary conditions, see [14, 15]. We remark that García-Melián and Quirós [16] recently proved the existence of a critical exponent of Fujita type for the nonlocal diffusion problem 𝑢_𝑡(𝑥,𝑡)==𝐽∗_𝑅𝑁𝑢𝐽(𝑥(𝑥,𝑡)−−𝑦)𝑢((𝑢𝑥(,𝑦𝑡),𝑡+)−𝑢_𝑝𝑢(𝑥,𝑡))𝑑𝑦+𝑢^𝑝(𝑥,𝑡)(𝑥,𝑡).(1.6) As mentioned in [17], nonlocal diffusion systems are more accurate than the classical diffusion systems in modeling the spatial diffusion of the individuals in some biology areas, such as embryological development process. For more study about the nonlocal diffusion operator, we refer to [8, 18–22] and references therein.

It is noteworthy that the method used in [12, 13] for problem (1.4) is invalid in our current setting due to the appearance of the nonlocal diffusion term. For example, instead of constructing self-similar weak subsolutions, we will use technique of integration to prove the finite time blowup. As far as the blowup rate is concerned, the scaling argument in [13] is not applicable.

We now state our main results. Our first result determines the complete classification of the parameters for which the solution blows up in finite time or exists globally.

Theorem 1.1. (i) If 𝑝≤max{𝑞,1}, then all solutions of (1.1) are global. Moreover, if 𝑝<𝑞 or 𝑝=𝑞 and 𝑘≥1, all solutions are uniformly bounded, while if 𝑝=𝑞 and 𝑘<1, there exist unbounded global solutions.
(ii) If 𝑝>max{𝑞,1}, then (1.1) with large initial data have solutions blowing up in finite time, while the solutions of (1.1) with small initial data exist globally.

Once we have obtained the values of the parameters for which blowup occurs, the next step is to concern the blowup rate, that is the speed at which solutions are blowing up. Different from the result of problem (1.4), we could have a unified upper and low estimate here.

Theorem 1.2. Let 𝑝>max{𝑞,1} and 𝑢(𝑥,𝑡) be a solution of (1.1) blowing up at time 𝑇. Then lim_𝑡→𝑇(𝑇−𝑡)^1/(𝑝−1)max_𝑥∈Ω𝑢(𝑥,𝑡)=(𝑝−1)^{−1/(𝑝−1)}.(1.7)

Remark 1.3. From this result we find that the nonlocal diffusion term plays no role when determining the blowup rate and the blowup rate is just same as that of the ODE 𝑢_𝑡=𝑢^𝑝. And this phenomena is the same as that of local diffusion case.

Next we consider the spacial location of the blowup set. As usual, the blowup set of solution 𝑢(𝑥,𝑡) is defined as follows: 𝐵(𝑢)=𝑥∈Ω;thereexist𝑥_𝑛,𝑡_𝑛⟶(𝑥,𝑇)suchthat𝑢𝑥_𝑛,𝑡_𝑛⟶∞,(1.8) where 𝑇 is the maximal existence time of 𝑢. For a general domain Ω we can localize the blowup set near any pint in Ω just by taking an initial condition being very large near that point and not so large in the rest of the domain. This is the following result.

Theorem 1.4. Let 𝑝>max{𝑞,2}. For any 𝑥₀∈Ω and 𝜀>0, there exists an initial data 𝑢₀ such that the corresponding solution 𝑢(𝑥,𝑡) of (1.1) blows up at finite time 𝑇 and its blowup set 𝐵(𝑢) is contained in 𝐵_𝜖(𝑥₀)={𝑥∈Ω;‖𝑥−𝑥₀‖<𝜖}.

Next we consider the radially symmetric case. In this case, single point blowup occurs.

Theorem 1.5. Let 𝑝>max{𝑞,2} and Ω=𝐵_𝑅={|𝑥|<𝑅}. If the initial data 𝑢₀∈𝐶¹(𝐵_𝑅) is a radial nonnegative function with a unique maximum at the origin, that is, 𝑢₀=𝑢₀(𝑟)≥0, 𝑢^0(𝑟)<0 for 0<𝑟≤𝑅, 𝑢′(0)=0 and 𝑢^0(0)<0, then the blowup set 𝐵(𝑢) of the solution 𝑢 of (1.1) consists only of the original point 𝑥=0.

The remainder of this paper is organized as follows. In Section 2, we give the existence and uniqueness of the solutions as well as the comparison principle. In Section 3, we prove the blowup and global existence condition. And then we prove the blowup rate and blowup set results in Sections 4 and 5, respectively. And in the last section we will give some numerical experiments to demonstrate our results.

2. Existence, Uniqueness, and Comparison Principle

We begin our study of problem (1.1) with a result of existence and uniqueness of continuous solutions and comparison principle.

Firstly, existence and uniqueness of solutions is a consequence of Banach's fixed point theorem. We look for 𝑢∈𝐶¹((0,𝑇);𝐶(Ω))∩𝐶([0,𝑇);𝐶(Ω)) satisfying (1.1). Fix 𝑡₀>0, and consider the Banach space 𝑋_𝑡0=𝑢∈𝐶¹((0,𝑇);𝐶(Ω))∩𝐶([0,𝑇);𝐶(Ω)) with the norm ‖𝜔‖_𝑋𝑡0=max_{0≤𝑡≤𝑡0}‖𝜔(⋅,𝑡)‖_𝐶(Ω)+max_{0<𝑡≤𝑡0}‖‖𝜔_𝑡(⋅,𝑡)‖‖_𝐶(Ω).(2.1)

We define the following operator 𝑇∶𝑋_𝑡0→𝑋_𝑡0𝑇_𝜔0(𝜔)(𝑥,𝑡)=𝜔₀(𝑥)+^𝑡0_Ω+𝐽(𝑥−𝑦)(𝜔(𝑦,𝑠)−𝜔(𝑥,𝑠))𝑑𝑦𝑑𝑠_𝑡0|𝜔|^𝑝−1𝜔(𝑥,𝑠)−𝑘|𝜔|^𝑞−1𝜔(𝑥,𝑠)𝑑𝑠.(2.2)

Similar to [10, 15] we could prove the solution to (1.1) is a fixed point of operator 𝑇 in a convenient ball of 𝑋_𝑡0. Thus, we could obtain the following result.

Theorem 2.1. For every 𝑢₀∈𝐶(Ω) there exists a unique solution 𝑢 of (1.1) such that 𝑢∈𝐶¹((0,𝑇);𝐶(Ω))∩𝐶([0,𝑇);𝐶(Ω)) and 𝑇 (finite or infinite) is the maximal existence time of solution.

Next, we will study the comparison principle. As usual we first give the definition of supersolution and subsolution.

Definition 2.2. A function 𝑢∈𝐶¹((0,𝑇);𝐶(Ω))∩𝐶([0,𝑇);𝐶(Ω)) is a supersolution of (1.1) if it satisfies 𝑢_𝑡(𝑥,𝑡)≥_Ω𝐽(𝑥−𝑦)𝑢(𝑦,𝑡)−𝑢(𝑥,𝑡)𝑑𝑦+𝑢^𝑝(𝑥,𝑡)−𝑘𝑢^𝑞(𝑥,𝑡),𝑢(𝑥,0)≥𝑢₀(𝑥).(2.3) Subsolutions are defined similarly by reversing the inequalities.

To obtain the comparison principle for problem (1.1), we first give a maximum principle.

Lemma 2.3. Suppose that 𝑤(𝑥,𝑡)∈𝐶¹((0,𝑇);𝐶(Ω))∩𝐶([0,𝑇);𝐶(Ω)) is nontrivial and satisfies𝑤_𝑡(𝑥,𝑡)≥_Ω𝐽(𝑥−𝑦)(𝑤(𝑦,𝑡)−𝑤(𝑥,𝑡))𝑑𝑦+𝑐₁𝑤(𝑥,𝑡),𝑥∈Ω,𝑡>0(2.4) with 𝑤(𝑥,0)≥0 for 𝑥∈Ω, where 𝑐₁ is a bounded function, then 𝑤(𝑥,𝑡)>0 for 𝑥∈Ω and 𝑡>0.

Proof. We first show 𝑤(𝑥,𝑡)≥0 for 𝑥∈Ω,𝑡>0. Assume that 𝑤(𝑥,𝑡) is negative somewhere. Let 𝜃(𝑥,𝑡)=𝑒^−𝜆𝑡𝑤(𝑥,𝑡)(𝜆>0,𝜆≥2sup|𝑐₁|). If we take (𝑥₀,𝑡₀) a point where 𝜃 attains its negative minimum, there holds 𝑡₀>0 and 𝜃_𝑡𝑥₀,𝑡₀=≥−𝑒𝜆𝑒^−𝜆𝑡0_−𝜆𝑡0𝑤𝑥₀,𝑡₀+𝑒^−𝜆𝑡0𝑤_𝑡𝑥₀,𝑡₀_Ω𝐽𝑥₀−𝑦𝑤𝑦,𝑡₀−𝑤𝑥₀,𝑡₀𝑑𝑦+−𝜆+𝑐₁𝑤𝑥₀,𝑡₀>0,(2.5) which is a contradiction. Thus 𝜃(𝑥,𝑡)≥0 for 𝑥∈Ω,𝑡>0. And so does 𝑤(𝑥,𝑡).
Now, we suppose 𝜃(𝑥₁,𝑡₁)=0 for some (𝑥₁,𝑡₁); that is, 𝜃 attains its minimum at (𝑥₁,𝑡₁) from the first step. Notice that the hypotheses on 𝐽 imply 𝐽(0)>0, so that 𝜃(𝑥₁,𝑡₁)=0 implies that 𝜃(𝑥,𝑡₁)=0 for 𝑥 in a neighborhood of 𝑥₁. Thus a standard connectedness argument yields 𝜃≡0. This is a contradiction. So we obtain our conclusion.

Lemma 2.4. If 𝑝≥1, 𝑞≥1 and 𝑢,  𝑢_{_}are super and subsolutions to (1.1), respectively then 𝑢(𝑥,𝑡)≥𝑢_{_}(𝑥,𝑡) for every (𝑥,𝑡)∈Ω×[0,𝑇).

Proof. Letting  𝑤(𝑥,𝑡)=𝑢−𝑢_{_}, it is easy to verify that 𝑤(𝑥,𝑡) satisfies (2.4) when 𝑝≥1,𝑞≥1. We could obtain our conclusion from Lemma 2.3.

Remark 2.5. When 𝑝<1 or 𝑞<1, the conclusion is also validity if 𝑢 and  𝑢_{_}  are bounded away from 0.

3. Blowup and Global Existence

In this section, we will analyze the blowup condition and give the proof of Theorem 1.1.

Proof of Theorem 1.1. (i) We only need to look for a global supersolution of (1.1). Indeed, it is easy to construct spacial homogeneous global supersolution of (1.1). To see this, we set 𝑢=𝐶𝑒^𝛼𝑡, where 𝐶 and 𝛼 are positive constants to be determined.
For any given initial data 𝑢₀, we note that 𝑢(𝑡₀)≥‖𝑢₀‖_∞ for 𝑡₀ sufficiently large and 𝑢 is bounded away from 0. Thus by the comparison principle and Remark 2.5, to make 𝑢 a supersolution of (1.1), we only need to show the existence of 𝐶 and 𝛼 satisfying𝐶^𝑝𝑒^𝑝𝛼𝑡≤𝑘𝐶^𝑞𝑒^𝑞𝛼𝑡+𝛼𝐶𝑒^𝛼𝑡.(3.1) If 𝑝≤1<𝑞, for any given 𝛼, we can take 𝐶=𝑘^{1/(𝑝−𝑞)} such that (3.1) holds.  If 𝑞≤1 and thus 𝑝≤1, we can choose 𝐶 and 𝛼 satisfying 𝐶^𝑝−1=𝛼, which make (3.1) validity.
Next, we show all global solutions are uniformly bounded when 𝑝<𝑞 or 𝑝=𝑞 and 𝑘≥1. In fact, (1.1) has constant supersolution 𝑢=𝐴 whenever 𝑝<𝑞 or 𝑝=𝑞 and 𝑘≥1. To see this, we choose 𝐴 large enough such that 𝑘𝐴^𝑞≥𝐴^𝑝,𝐴≥‖‖𝑢₀‖‖_∞,(3.2) which imply that 𝑢 is a supersolution of (1.1).
At last we show there exist global unbounded solutions when 𝑝=𝑞 and 𝑘<1. Indeed, let 𝑓(𝑡)=(1−𝑘)(1−𝑝)𝑡+𝑓^1−𝑝𝑒(0)^1/(1−𝑝),𝑝=𝑞<1,_{((1−𝑘)/2)𝑡},𝑝=𝑞=1.(3.3) It is easy to see that if 𝑓(0)≤max_Ω𝑢₀(𝑥), 𝑓(𝑡) is a subsolution of (1.1). It is obvious that when 𝑝<1, 𝑓(𝑡) is unbounded.
(ii) We first show that if the initial data 𝑢₀(𝑥) is large enough, solutions of (1.1) blow up in finite time.
In the case of 𝑝>𝑞>1. Integrating (1.1)1 in Ω and applying Fubini's theorem, we get 𝑑𝑑𝑡_Ω𝑢(𝑥,𝑡)𝑑𝑥=_Ω𝑢^𝑝(𝑥,𝑡)𝑑𝑥−𝑘_Ω𝑢^𝑞(𝑥,𝑡)𝑑𝑥.(3.4) Using Hölder's inequality, we could get 𝑑𝑑𝑡_Ω𝑢(𝑥,𝑡)𝑑𝑥≥_Ω𝑢^𝑝(𝑥,𝑡)𝑑𝑥−𝑘||Ω||^{(𝑝−𝑞)/𝑝}_Ω𝑢^𝑝(𝑥,𝑡)𝑑𝑥^𝑞/𝑝=_Ω𝑢^𝑝(𝑥,𝑡)𝑑𝑥^𝑞/𝑝_Ω𝑢^𝑝(𝑥,𝑡)𝑑𝑥^{(𝑝−𝑞)/𝑝}−𝑘||Ω||^{(𝑝−𝑞)/𝑝},(3.5) where |Ω| is assumed to be the measure of Ω. Given positive constant 𝑚>𝑘^{1/(𝑝−𝑞)} and 𝑢₀≥𝑚, we have by the comparison principle that the solution 𝑢(𝑥,𝑡) of problem (1.1) satisfies 𝑢(𝑥,𝑡)≥𝑚. Thus 𝑑𝑑𝑡_Ω𝑢(𝑥,𝑡)𝑑𝑥≥_Ω𝑢^𝑝(𝑥,𝑡)𝑑𝑥^𝑞/𝑝𝑚^𝑝−𝑞||Ω||^{(𝑝−𝑞)/𝑝}−𝑘||Ω||^{(𝑝−𝑞)/𝑝}.(3.6) Then we use Jensen's inequality to obtain 𝑑𝑑𝑡_Ω𝑢(𝑥,𝑡)𝑑𝑥>𝐶_Ω𝑢(𝑥,𝑡)𝑑𝑥^𝑞,(3.7) where 𝐶 is a positive constant independent of the solution 𝑢. From this inequality, we could easily obtain that 𝑢(𝑥,𝑡) blow up in finite time.
In the case of 𝑝>1≥𝑞, it follows from 𝑢^𝑞≤𝑢+1 and Jensen's inequality that _Ω𝑢^𝑝(𝑥,𝑡)𝑑𝑥−𝑘_Ω𝑢^𝑞(𝑥,𝑡)𝑑𝑥≥≥||Ω_Ω||𝑢^𝑝(𝑥,𝑡)𝑑𝑥−𝑘_Ω𝑢(𝑥,𝑡)𝑑𝑥−𝑘_1−𝑝||Ω||_Ω𝑢(𝑥,𝑡)𝑑𝑥^𝑝−𝑘_Ω𝑢(𝑥,𝑡)𝑑𝑥−𝑘||Ω||.(3.8) Substituting this inequality into (3.4), we obtain 𝑑𝑑𝑡_Ω𝑢(𝑥,𝑡)𝑑𝑥≥||Ω||^1−𝑝_Ω𝑢(𝑥,𝑡)𝑑𝑥^𝑝−𝑘_Ω𝑢(𝑥,𝑡)𝑑𝑥−𝑘||Ω||.(3.9)
Therefore, if we take the initial data 𝑢₀ large enough such that |Ω|^1−𝑝(∫_Ω𝑢₀(𝑥)𝑑𝑥)^𝑝−𝑘∫_Ω𝑢₀(𝑥)𝑑𝑥−𝑘|Ω|>0, then ∫_Ω𝑢(𝑥,𝑡)𝑑𝑥 blows up in finite time. So does 𝑢(𝑥,𝑡).
Next we show when the initial data 𝑢₀(𝑥) is small, solutions of (1.1) exist globally. Consider constant 𝐵. Let 0<𝐵≤𝑘^{1/(𝑝−𝑞)}. Then 𝐵_𝑡≥𝐵^𝑝−𝑘𝐵^𝑞. Henceforth, if 𝑢₀(𝑥)≤𝐵, 𝐵 is a supersolution of (1.1). From the comparison principle, we know solutions of (1.1) are global in this case.

4. Blowup Rate Estimate

In this section, we study the blowup rate and prove Theorem 1.2.

Proof of Theorem 1.2. Let 𝑈(𝑡)=𝑢(𝑥(𝑡),𝑡)=max_𝑥∈Ω𝑢(𝑥,𝑡). It is easy to see that 𝑈(𝑡) is Lipschitz continuous and thus it is differential almost everywhere [23]. From the first equality of (1.1) we have 𝑈′(𝑡)≤_Ω𝐽(𝑥−𝑦)(𝑢(𝑦,𝑡)−𝑢(𝑥(𝑡),𝑡))𝑑𝑦+𝑢^𝑝(𝑥(𝑡),𝑡)−𝑘𝑢^𝑞(𝑥(𝑡),𝑡)≤𝑢^𝑝(𝑥(𝑡),𝑡)(4.1) at any point of differentiability of 𝑈(𝑡). Here we used ∇𝑢(𝑥(𝑡),𝑡)=0. Noticing that 𝑝>1 and integrating (4.1) from 𝑡 to 𝑇, we obtain max_𝑥∈Ω𝑢(𝑥,𝑡)≥(𝑝−1)^{−1/(𝑝−1)}(𝑇−𝑡)^{−1/(𝑝−1)}.(4.2)
Next we will establish the upper estimate. For any (𝑥,𝑡)∈Ω×[0,𝑇), we have 𝑢_𝑡(𝑥,𝑡)≥−𝑢(𝑥,𝑡)+𝑢^𝑝(𝑥,𝑡)−𝑘𝑢^𝑞(𝑥,𝑡)=𝑢^𝑝(𝑥,𝑡)1−𝑢^{−(𝑝−1)}(𝑥,𝑡)−𝑘𝑢^{−(𝑝−𝑞)}(𝑥,𝑡).(4.3) In particular, 𝑈^(𝑡)≥𝑈^𝑝(𝑡)1−𝑈(𝑡)^{−(𝑝−1)}−𝑘𝑈(𝑡)^{−(𝑝−𝑞)}.(4.4)
From the lower estimate (4.2) we get 𝑈^(𝑡)≥𝑈^𝑝(𝑡)1−(𝑝−1)(𝑇−𝑡)−𝑘(𝑝−1)^{(𝑝−𝑞)/(𝑝−1)}(𝑇−𝑡)^{(𝑝−𝑞)/(𝑝−1)}.(4.5)
Integrating in (𝑡,𝑇), we get max_𝑥∈Ω𝑢≤(𝑥,𝑡()𝑝−1)(𝑇−𝑡)−(𝑝−1)₂2(𝑇−𝑡)²−𝑘(𝑝−1)^{(3𝑝−𝑞−2)/(𝑝−1)}2𝑝−𝑞−1(𝑇−𝑡)^{(2𝑝−𝑞−1)/(𝑝−1)}^{−1/(𝑝−1)},(4.6) combining with (4.2), the conclusion of Theorem 1.2 is proved if one takes the limit as 𝑡→𝑇.

5. Blowup Set

Next we will concern the blowup set for the solution to problem (1.1). We will first localize the blowup set near any point in Ω just by taking an initial condition being very large near that point and not so large in the rest of the domain.

Proof of Theorem 1.4. Given 𝑥₀∈Ω and 𝜀>0, we could construct an initial condition 𝑢₀ such that 𝐵(𝑢)⊂𝐵_𝜀𝑥₀=𝑥∈Ω∶‖‖𝑥−𝑥₀‖‖<𝜀.(5.1)
In fact, we will consider 𝑢₀ concentrated near 𝑥₀ and small away from 𝑥₀.
Let 𝜑 be a nonnegative smooth function such that supp(𝜑)⊂𝐵_𝜀/2(𝑥₀) and 𝜑(𝑥)>0 for 𝑥∈𝐵_𝜀/2(𝑥₀).
Next, let 𝑢₀(𝑥)=𝑀𝜑(𝑥)+𝛿.(5.2) We want to choose 𝑀 large and 𝛿 small such that (5.1) holds.
First we can assume that 𝑇 is as small as we need by taking 𝑀 large enough. Indeed, we have 𝑇≤𝐶(Ω,𝑝,𝜑)𝑀^𝑞−1or𝑇≤𝐶(Ω,𝑝,𝜑)𝑀^𝑝−1(5.3) from the proof of Theorem 1.1.
Now, from the proof of blowup rate, we havemax_𝑥∈Ω𝑢≤(𝑥,𝑡()𝑝−1)(𝑇−𝑡)−(𝑝−1)₂2(𝑇−𝑡)²−𝑘(𝑝−1)^{(3𝑝−𝑞−2)/(𝑝−1)}2𝑝−𝑞−1(𝑇−𝑡)^{(2𝑝−𝑞−1)/(𝑝−1)}^{−1/(𝑝−1)}≤𝐶(𝑇−𝑡)^{−1/(𝑝−1)}.(5.4) Henceforth, for any 𝑥∈Ω, 𝑢_𝑡𝑥,𝑡=≤_Ω𝐽(𝑥,𝑦)𝑢(𝑦,𝑡)−𝑢𝑥,𝑡𝑑𝑦+𝑢^𝑝𝑥,𝑡−𝑘𝑢^𝑞𝑥,𝑡_Ω𝐽𝑥,𝑦𝑢(𝑦,𝑡)𝑑𝑦+𝑢^𝑝𝑥,𝑡≤𝐶(𝑇−𝑡)^{−1/(𝑝−1)}+𝑢^𝑝𝑥,𝑡.(5.5) Therefore, if 𝑥∈Ω⧵𝐵_𝜖(𝑥₀), then 𝑢(𝑥,𝑡) is a subsolution to 𝑤_𝑡=𝐶(𝑇𝑤−(𝑡0))=𝛿,^{−1/(𝑝−1)}+𝑤^𝑝(𝑡),(5.6) which shows 𝑢𝑥,𝑡≤𝑤(𝑡).(5.7)
Next, we only need to prove that a solution 𝑤 to (5.6) remains bounded up to 𝑡=𝑇, provided that 𝛿 and 𝑇 are small enough.
Let 𝑧(𝑠)=(𝑇−𝑡)^1/(𝑝−1)𝑤(𝑡),𝑠=−ln(𝑇−𝑡).(5.8) Then 𝑧(𝑠) satisfies 𝑧^(𝑠)=𝐶𝑒^−𝑠−1𝑝−1𝑧(𝑠)+𝑧^𝑝(𝑠),𝑧(−ln𝑇)=𝛿𝑇^1/(𝑝−1),(5.9) which show that for 𝑇 and 𝛿 small (𝑇 is small if 𝑀 is large), we have 𝐶𝑇−1𝑝−1𝛿𝑇^1/(𝑝−1)+𝛿^𝑝𝑇^{𝑝/(𝑝−1)}<0.(5.10) So 𝑧′(𝑠)<0 for all 𝑠>−ln𝑇. From this and Lemma 4.2 of [24], we know 𝑧(𝑠)⟶0,𝑠⟶∞.(5.11) Combining the equation verified by 𝑧 we obtain that, for given positive constant 𝛾(<1/𝑝(𝑝−1)), there exists 𝑠₀>0 such that 𝑧^(𝑠)≤𝐶𝑒^−𝑠−1𝑝−1−𝛾𝑧(𝑠)(5.12) for 𝑠>𝑠₀.
Let 𝑣(𝑠) be a solution of 𝑣^(𝑠)=𝐶𝑒^−𝑠−1𝑝−1−𝛾𝑣(𝑠)(5.13) with 𝑣(𝑠₀)≥𝑧(𝑠₀). Integrating this equation we get 𝑣(𝑠)=𝐶₁𝑒^−𝑠+𝐶₂𝑒^{−(1/(𝑝−1)−𝛾)𝑠}.(5.14) By a comparison argument we could get that for every 𝑠>𝑠₀, 𝑧(𝑠)≤𝑣(𝑠)=𝐶₁𝑒^−𝑠+𝐶₂𝑒^{−(1/(𝑝−1)−𝛾)𝑠}.(5.15) Now we go back to 𝑧′(𝑠)=𝐶𝑒^−𝑠−(1/(𝑝−1))𝑧(𝑠)+𝑧^𝑝(𝑠). We have 𝑧^(𝑠)+(1/(𝑝−1))𝑧(𝑠)=𝐶𝑒^−𝑠+𝑧^𝑝(𝑠),(5.16) then 𝑒^{(1/(𝑝−1))𝑠}𝑧(𝑠)^=𝑒^{(1/(𝑝−1))𝑠}(𝐶𝑒^−𝑠+𝑧^𝑝).(5.17) Integrating form 𝑠₀ to 𝑠, one could get 𝑧(𝑠)=𝑒^{−(1/(𝑝−1))𝑠}𝐶¹=𝑒_{−(1/(𝑝−1))𝑠}+^𝑠𝑠₀𝑒^{(1/(𝑝−1))𝜎}(𝐶𝑒^−𝜎+𝑧^𝑝(𝑠))𝑑𝜎𝐶¹+^𝑠𝑠₀𝑒^{−((𝑝−2)/(𝑝−1))𝜎}(𝐶+𝑒^𝜎𝑧^𝑝(𝑠))𝑑𝜎.(5.18) Using (5.15) and 𝛾<1/𝑝(𝑝−1), we have 𝑒^𝑠𝑧^𝑝≤𝐶^𝑝1𝑒^{−(𝑝−1)𝑠}+𝐶^𝑝2𝑒^{−(𝑝/(𝑝−1)−𝑝𝛾−1)𝑠}⟶0(5.19) as 𝑠→+∞.
And thus from (5.18), we get 𝑧(𝑠)≤𝑒^{−(1/(𝑝−1))𝑠}𝐶¹+𝐶₃^𝑠𝑠₀≤𝐶₁𝑒^{−((𝑝−2)/(𝑝−1))𝜎}𝑑𝜎𝑒^{−(1/(𝑝−1))𝑠}+𝐶₄𝑒^−𝑠.(5.20)
As 𝑝>2, we have 𝑧(𝑠)≤𝐶𝑒^{−(1/(𝑝−1))}𝑠.(5.21) This implies that 𝑤(𝑡)≤𝐶, for 0≤𝑡<𝑇. From the boundedness of 𝑤 and (5.7) we get 𝑢(𝑥,𝑡)≤𝑤(𝑡)≤𝐶 for every 𝑥∈Ω⧵𝐵_𝜖(𝑥₀), as we wished.

Next, we will consider the radial symmetric case, that is, the proof of Theorem 1.5. For the convenience of writing, we only deal with the one dimensional case, Ω=(−𝑙,𝑙). The radial case is analogous; we leave the details to the reader.

Proof of Theorem 1.5. Under the hypothesis on the initial condition imposed in Theorem 1.5 we have that the solution is symmetric and 𝑢_𝑥<0 in (0,𝑙]×(0,𝑇) from the stand parabolic theorem and Lemma 4.1 of [10]. Therefore the solution has a unique maximum at the origin for every 𝑡∈(0,𝑇).
Let us perform the following change of variables 𝑧(𝑥,𝑠)=(𝑇−𝑡)^1/(𝑝−1)𝑢(𝑥,𝑡),𝑠=−ln(𝑇−𝑡).(5.22) Our remainder proof consist of two steps.
Step 1. We first prove the only blowup point that verifies the blowup estimate (1.7) is 𝑥=0. And this shows that for 𝑥≠0, 𝑧(𝑥,𝑠) does not converge to 𝐶_𝑝=(𝑝−1)^{−1/(𝑝−1)} as 𝑠→+∞.
We conclude by contradiction. Assume that (𝑇−𝑡)^1/(𝑝−1)𝑢(𝑥₀,𝑡)→𝐶_𝑝 for a 𝑥₀>0.
Let 𝑣(𝑡)=𝑢(0,𝑡)−𝑢(𝑥₀,𝑡). Then 𝑣^(𝑡)=+𝑝^𝑙−𝑙𝜉𝐽(_𝑝−1−𝑦)(𝑢(𝑦,𝑡)−𝑢(0,𝑡))𝑑𝑦−^𝑙−𝑙𝐽𝑥₀−𝑦𝑢(𝑦,𝑡)−𝑢𝑥₀,𝑡𝑑𝑦(𝑡)𝑣(𝑡)−𝑘𝑞𝜂^𝑞−1(𝑡)𝑣(𝑡),(5.23) where 𝜉(𝑡) and 𝜂(𝑡) are between 𝑢(0,𝑡) and 𝑢(𝑥₀,𝑡). Hence 𝑣^(𝑡)≥+^𝑙−𝑙𝐽(−𝑦)−𝐽𝑥₀−𝑦𝑢(𝑦,𝑡)𝑑𝑦_𝑙−𝑙=𝐽𝑦−𝑥₀−𝐽(𝑦)𝑢(0,𝑡)𝑑𝑦−𝑣(𝑡)+𝑝𝜉^𝑝−1(𝑡)𝑣(𝑡)−𝑘𝑞𝜂^𝑞−1(𝑡)𝑣(𝑡)_𝑙−𝑙≥−𝐶₁𝐽𝑦−𝑥₀−𝐽(𝑦)(𝑢(0,𝑡)−𝑢(𝑦,𝑡))𝑑𝑦−𝑣(𝑡)+𝑝𝜉^𝑝−1(𝑡)𝑣(𝑡)−𝑘𝑞𝜂^𝑞−1(𝑡)𝑣(𝑡)+𝑝𝜉^𝑝−1(𝑡)−𝑘𝑞𝜂^𝑞−1(𝑡)𝑣(𝑡),(5.24) for some positive constant.
Integrating the above inequality, we obtain ln(𝑣)(𝑡)−ln(𝑣)𝑡₀≥^𝑡𝑡₀−𝐶₁+𝑝𝜉^𝑝−1(𝑠)−𝑘𝑞𝜂^𝑞−1(𝑠)𝑑𝑠.(5.25) Remember that (𝑇−𝑡)^1/(𝑝−1)𝑢(𝑥₀,𝑡)→𝐶_𝑝, (𝑇−𝑡)^1/(𝑝−1)𝑢(0,𝑡)→𝐶_𝑝, we have lim_𝑡→𝑇𝜉(𝑡)(𝑇−𝑡)^1/(𝑝−1)=lim_𝑡→𝑇𝜂(𝑡)(𝑇−𝑡)^1/(𝑝−1)=𝐶_𝑝.(5.26) And this implies that ^𝑡𝑡₀−𝐶₁+𝑝𝜉^𝑝−1(𝑠)−𝑘𝑞𝜂^𝑞−1(𝑠)𝑑𝑠≥𝑝^𝑡𝑡₀𝐶^𝑝𝑝−1−𝛿₁−𝑘𝑞_𝑡𝑡0𝑇−𝑠𝑑𝑠𝐶^𝑝𝑞−1+𝛿₂(𝑇−𝑠)^{−(𝑞−1)/(𝑝−1)}𝑑𝑠−𝐶₂.(5.27)𝑝>𝑞 implies that (𝑇−𝑠)^{−(𝑞−1)/(𝑝−1)}≤𝛿₃(𝑇−𝑠)⁻¹ as 𝑠→𝑇 for given 𝛿₃>0. Hence ^𝑡𝑡₀−𝐶₁+𝑝𝜉^𝑝−1(𝑠)−𝑘𝑞𝜂^𝑞−1(𝑠)𝑑𝑠≥𝑝^𝑡𝑡₀𝐶^𝑝𝑝−1−𝛿𝑇−𝑠𝑑𝑠−𝐶²=−𝑝𝐶^𝑝𝑝−1−𝛿ln(𝑇−𝑡)−𝐶₂(5.28) for some 𝛿>0.
Hence 𝑣(𝑡)≥𝐶(𝑇−𝑡)^{−𝑝(𝐶𝑝𝑝−1−𝛿)}=𝐶(𝑇−𝑡)^{𝑝𝛿−𝑝/(𝑝−1)}.(5.29) Using this fact, we have 0=lim_𝑡→𝑇(𝑇−𝑡)^1/(𝑝−1)𝑣(𝑡)≥𝐶lim_𝑡→𝑇(𝑇−𝑡)^{1/(𝑝−1)−𝑝/(𝑝−1)+𝑝𝛿}=+∞.(5.30) This contradiction proves our claim.
Step 2. We will show the only possible blowup point is 𝑥=0.
Remembering the transform (5.22), 𝑧(𝑥,𝑠) satisfies 𝑧_𝑠=𝑒^−𝑠^𝑙−𝑙𝐽(𝑥−𝑦)(𝑧(𝑦,𝑠)−𝑧(𝑥,𝑠))𝑑𝑦−1𝑝−1𝑧+𝑧^𝑝−𝑘𝑒^{((𝑞−𝑝)/(𝑝−1))𝑠}𝑧^𝑞.(5.31) Note that the blowup rate of 𝑢 implies that 𝑧(𝑥,𝑠)≤𝐶 for every (𝑥,𝑠)∈[−𝑙,𝑙]×(−ln𝑇,∞). Therefore, 𝑧_𝑠(𝑥,𝑠)≤𝐶𝑒^−𝑠−1𝑝−1𝑧(𝑥,𝑠)+𝑧^𝑝(𝑥,𝑠).(5.32) From this we know that if there exists 𝑠₀ such that 𝑧^𝑝(𝑥,𝑠₀)−(1/(𝑝−1))𝑧(𝑥,𝑠₀)<−𝐶𝑒^−𝑠0, then 𝑧(𝑥,𝑠)→0 as 𝑠→∞ (see Lemma 4.2 in [24]).
Moreover, if there exists 𝑠₀ such that 𝑧^𝑝(𝑥,𝑠₀)−(1/(𝑝−1))𝑧(𝑥,𝑠₀)>𝐶𝑒^−𝑠0 then 𝑧(𝑥,𝑠) blows up in finite time 𝑠. This follows from Lemma 4.3 of [24] using that 𝑧_𝑠(𝑥,𝑠)≥−𝐶𝑒^−𝑠−1𝑝−1𝑧(𝑥,𝑠)+𝑧^𝑝(𝑥,𝑠).(5.33)
Thus if 𝑧(𝑥,𝑠) does not converge to zero and does not blow up in finite time, then 𝑧(𝑥,𝑠) satisfies 𝐶𝑒^−𝑠≥𝑧^𝑝(𝑥,𝑠)−1𝑝−1𝑧(𝑥,𝑠)≥−𝐶𝑒^−𝑠.(5.34) Henceforth, 𝑧^𝑝(𝑥,𝑠)−1𝑝−1𝑧(𝑥,𝑠)⟶0(𝑠⟶+∞).(5.35) As 𝑧(𝑥,𝑠) is continuous, bounded and does not go to zero, we conclude that 𝑧(𝑥,𝑠)→𝐶_𝑝.
Now we could conclude that 𝑧(𝑥,𝑠) verifies 𝑧(𝑥,𝑠)→0(𝑠→+∞), or 𝑧(𝑥,𝑠)→𝐶_𝑝(𝑠→+∞), or 𝑧(𝑥,𝑠) blows up in finite time.
From Step 1 we know for 𝑥≠0, 𝑧(𝑥,𝑠) is bounded and does not converge to 𝐶_𝑝, so 𝑧(𝑥,𝑠)→0 as 𝑠→+∞. Combined with inequality (5.32), we could get 𝑧_𝑠(𝑥,𝑠)≤𝐶𝑒^−𝑠−1𝑝−1−𝜃𝑧(𝑥,𝑠)(5.36) for any 𝜃>0.
By a comparison argument as in the proof of Theorem 1.4, it follows that 𝑧(𝑥,𝑠)≤𝐶₁𝑒^−𝑠+𝐶₂𝑒^{−(1/(𝑝−1)−𝜃)𝑠}.(5.37) Going back to the equation verified by 𝑧(𝑥,𝑡) we obtain 𝑒^{(1/(𝑝−1))𝑠}𝑧(𝑥,𝑠)_𝑠=𝑒^{(1/(𝑝−1))𝑠}𝑒+^−𝑠𝑧_𝑝^𝑙−𝑙𝐽(𝑥−𝑦)(𝑧(𝑦,𝑠)−𝑧(𝑥,𝑠))𝑑𝑦(𝑥,𝑠)−𝑘𝑒^{((𝑞−𝑝)/(𝑝−1))𝑠}𝑧(𝑥,𝑠).(5.38) Integrating we get 𝑧(𝑥,𝑠)=𝑒^{−(1/(𝑝−1))𝑠}𝐶¹+^𝑠𝑠₀𝑒^{−((𝑝−2)/(𝑝−1))𝜎}^𝑙−𝑙+𝑒𝐽_𝜎(𝑥−𝑦)(𝑧(𝑦,𝑠)−𝑧(𝑥,𝑠))𝑑𝑦𝑧^𝑝(𝑥,𝑠)−𝑘𝑒^{((𝑞−1)/(𝑝−1))𝜎}𝑧(𝑥,𝑠)𝑑𝜎.(5.39) On the other hand, (5.37) implies that 𝑒^𝑠𝑧^𝑝(𝑥,𝑠)→0 as 𝑠→∞. Henceforth, 𝑧(𝑥,𝑠)≤𝑒^{−(1/(𝑝−1))𝑠}𝐶¹+𝐶₂^𝑠𝑠₀𝑒^{−((𝑝−2)/(𝑝−1))𝜎}𝑑𝜎.(5.40) Using that 𝑝>2, one could have 𝑧(𝑥,𝑠)≤𝐶₃𝑒^{−(1/(𝑝−1))𝑠}.(5.41) Remembering the transform (5.22), we have 𝑢(𝑥,𝑡)=𝑒^{(1/(𝑝−1))𝑠}𝑧(𝑥,𝑠)≤𝑐₃.(5.42) And so our proof is complete.

6. Numerical Experiments

At the end of this paper, we will use several numerical examples to demonstrate our results about the location of blowup points. For this purpose, we discretize the problem in the spacial variable to obtain an ODE system. Taking Ω=[−4,4] and −4=𝑥_−𝑁<⋯<𝑥_𝑁=4,𝑁=100, we consider the following system: 𝑢^𝑖(𝑡)=^𝑁_𝑗=_−𝑁𝐽𝑥_𝑖−𝑥_𝑗𝑢𝑢_𝑗(𝑡)−𝑢_𝑖(𝑡)+𝑢_𝑖^𝑝(𝑡)−𝑘𝑢_𝑖^𝑞(𝑡),_𝑖(0)=𝑢₀𝑥_𝑖.(6.1) Next we choose 𝑝=3, 𝑞=1, 𝑘=1 and 𝐽(𝑧)=⎧⎪⎨⎪⎩01,,||𝑧𝑧||>≤1110,10.(6.2)

In Figure 1 we choose a nonsymmetric initial condition very large near the point 𝑥₀=1,  𝑢₀(𝑥)=1/4+100(1−|𝑥−1|)₊. We observe that the blowup set is localized in a neighborhood of 𝑥₀=1.

(a)

(b)

Figure 1 

Evolution in time, nonsymmetric datum.

Next we choose a symmetric initial condition with a unique maximum at the origin, 𝑢₀(𝑥)=16−𝑥²⁰. We observe that the solution blows up only at the origin, Figure 2.

(a)

(b)

Figure 2 

Evolution in time, symmetric datum.

Acknowledgments

Y. Wang is supported by the Key Scientific Research Foundation of Xihua University (no.  Z0912611), the Scientific Research Found of Sichuan Provincial Education Department (no. 09ZB081) and the Research Fund of Key Disciplinary of Application Mathematics of Xihua University (no. XZD0910-09-1). Z. Xiang is supported by NNSF of China (11101068), the Sichuan Youth Science & Technology Foundation (2011JQ0003), the Fundamental Research Funds for the Central Universities (ZYGX2009X019) and the SRF for ROCS, SEM. J. Hu is supported by the Scientific Research Found of Sichuan Provincial Education Department (no. 11ZB009).