Blowup for Nonlocal Nonlinear Diffusion Equations with Dirichlet Condition and a Source

Abstract

This paper is concerned with a nonlocal nonlinear diffusion equation with Dirichlet boundary condition and a source 𝑢_𝑡(𝑥,𝑡)=∫^+∞_−∞𝐽((𝑥−𝑦)/𝑢(𝑦,𝑡))𝑑𝑦−𝑢(𝑥,𝑡)+𝑢^𝑝(𝑥,𝑡), 𝑥∈(−𝐿,𝐿), 𝑡>0, 𝑢(𝑥,𝑡)=0, 𝑥∉(−𝐿,𝐿), 𝑡≥0, and 𝑢(𝑥,0)=𝑢₀(𝑥)≥0, 𝑥∈(−𝐿,𝐿), which is analogous to the local porous medium equation. First, we prove the existence and uniqueness of the solution as well as the validity of a comparison principle. Next, we discuss the blowup phenomena of the solution to this problem. Finally, we discuss the blowup rates and sets of the solution.

1. Introduction

Since the long-range effects are taken into account, nonlocal diffusion equations of the form 𝜕𝑢()()=𝜕𝑡𝑥,𝑡=𝐽∗𝑢−𝑢𝑥,𝑡_ℝ𝑁𝐽𝑥−𝑦𝑢𝑦,𝑡−𝑢(𝑥,𝑡)𝑑𝑦 (1) have been widely used to model the dispersal of a species (see [1–7] and references therein). In fact, as stated in [7], if 𝑢(𝑥,𝑡) is thought of as the density of a species at the point 𝑥 at time 𝑡 and 𝐽(𝑥−𝑦) is thought of as the probability distribution of jumping from location 𝑦 to location 𝑥, then ∫_ℝ𝑁𝐽(𝑥−𝑦)𝑢(𝑦,𝑡)𝑑𝑦 is the rate at which individuals are arriving to position 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. It is known that (1) shares many properties with the classical heat equation 𝑢_𝑡=Δ𝑢, such that bounded stationary solutions are constant, a maximum principle holds for both of them, and perturbations propagate with infinite speed (see [7]). However, there is no regularizing effect in general (see [8]).

Another classical equation that has been used to model diffusion is the well-known porous medium equation 𝑢_𝑡=Δ𝑢^𝑚 with 𝑚>1. This equation also shares several properties with the heat equation, but there is a fundamental difference; in this case we have finite speed of propagation. Properties of solutions to the porous medium equation, particularly the blowup phenomena of the solution, have been largely studied over the past years. See, for example, [9–12] and references therein.

In [13, 14], a nonlocal model for diffusion that is analogous to the local porous medium equation is studied. In this model the probability distribution of jumping from location 𝑦 to location 𝑥 is given by 𝐽((𝑥−𝑦)/𝑢(𝑦,𝑡))(1/𝑢(𝑦,𝑡)) when 𝑢(𝑦,𝑡)>0 and 0 otherwise. In this case the rate at which individuals are arriving to position 𝑥 from all other places is ∫_ℝ𝐽((𝑥−𝑦)/𝑢(𝑦,𝑡))𝑑𝑦, and the rate at which they are leaving location 𝑥 to travel to all other sites is −𝑢(𝑥,𝑡)=−∫_ℝ𝐽((𝑥−𝑦)/𝑢(𝑦,𝑡))𝑑𝑦. As before this consideration, in the absence of external sources, leads immediately to the fact that the density 𝑢(𝑥,𝑡) has to satisfy 𝑢_𝑡(𝑥,𝑡)=_ℝ𝐽𝑥−𝑦𝑢()𝑦,𝑡𝑑𝑦−𝑢𝑥,𝑡. (2)

In [15], Bogoya and Elorreaga studied the following nonlocal equation: 𝑢_𝑡(𝑥,𝑡)=_ℝ𝑁𝐽𝑥−𝑦𝑢^𝛼𝑢𝑦,𝑡^1−𝑁𝛼𝑦,𝑡−𝐽𝑥−𝑦𝑢^𝛼()𝑢𝑥,𝑡^1−𝑁𝛼()(𝑢(𝑢()𝑢()𝑥,𝑡𝑑𝑦+𝑓𝑥,𝑡)),𝑥∈Ω,𝑥,𝑡=0,𝑥∉Ω,𝑥,0=𝑢₀(𝑥),𝑥∈Ω. (3) They proved the existence and uniqueness of the solution as well as the validity of a comparison principle and also discussed the blowup phenomena of the solution for some sources.

In the present paper, we are concerned with the following nonlocal “Dirichlet” boundary value problem with a source: 𝑢_𝑡(𝑥,𝑡)=^+∞_−∞𝐽𝑥−𝑦𝑢()𝑦,𝑡𝑑𝑦−𝑢𝑥,𝑡+𝑢^𝑝()()𝑢()()𝑢()𝑥,𝑡,𝑥∈−𝐿,𝐿,𝑡>0,𝑥,𝑡=0,𝑥∉−𝐿,𝐿,𝑡≥0,𝑥,0=𝑢₀(𝑥)=𝑐+𝑤₀(𝑥),𝑥∈(−𝐿,𝐿). (4) Here 𝑝≥1 and 𝑐≥0. Let 𝐽∶ℝ→ℝ be a nonnegative, smooth function, with ∫_ℝ𝐽(𝑧)𝑑𝑧=1, supported in [−1,1], symmetric, and strictly decreasing in [0,1]. We assume that 𝑤₀∈𝐿¹(ℝ) is a nonnegative function.

In this model, it is assumed that no individual can survive outside of the domain (−𝐿,𝐿). Therefore, the density must be zero there. However, individuals are allowed to jump outside the domain (where they die instantaneously). This is what we call Dirichlet boundary conditions.

For the convenience of the statement of our results, denoting Ω=(−𝐿,𝐿), some related definitions are introduced in the following.

Definition 1. A nonnegative function 𝑢∈𝐶([0,𝑇);𝐿¹) is a supersolution of the problem (4) if it satisfies 𝜕𝜕𝑡𝑢(𝑥,𝑡)≥^+∞_−∞𝐽𝑥−𝑦𝑢−𝑦,𝑡𝑑𝑦𝑢(𝑥,𝑡)+𝑢^𝑝(𝑥,𝑡),𝑥∈(−𝐿,𝐿),𝑡>0,𝑢(𝑥,𝑡)=0,𝑥∉(−𝐿,𝐿),𝑡≥0,𝑢(𝑥,0)≥𝑐+𝑤₀(𝑥),𝑥∈(−𝐿,𝐿). (5) The subsolution is defined similarly by reversing the inequalities. Furthermore, if 𝑢 is a supersolution as well as subsolution, then we call it a solution of the problem (4).

Definition 2. A solution 𝑢(𝑥,𝑡) of the problem (4) is called a global solution if supremum norm ‖𝑢(⋅,𝑡)‖_𝐿¹_(Ω) is finite for all 𝑡≥0.

Definition 3. If there is a time 𝑇_∞<∞ such that a solution 𝑢(𝑥,𝑡) of the problem (4) is bounded for all 𝑇<𝑇_∞ with lim_{𝑡→𝑇∞}‖𝑢(⋅,𝑡)‖_𝐿¹_(Ω)=∞, then 𝑢 blows up in finite time 𝑇_∞.

Now our main results can be stated as follows.

Theorem 4. If 𝑝≥1, 𝑤₀∈𝐿¹(ℝ), and 𝑐≥0, then there exists a unique solution 𝑢∈𝐶([0,𝑇);𝐿¹(ℝ)) to the problem (4).

Theorem 5 (blowup). Let 𝑤₀∈𝐶(Ω) be nonnegative and nontrivial. If 𝑝>1, then the solution to the problem (4) blows up in finite time. If 𝑝=1, the solution to the problem (4) is global. Moreover, if 𝑝>1, one has the following estimate for the blowup time: 1𝑇≤|Ω|𝑝−1∫_Ω𝑤₀(𝑥)𝑑𝑥+𝑐|Ω|^𝑝−1. (6)

The rest of the paper is organized as follows. In Section 2, we prove the existence and uniqueness of the solutions for the problem (4) and show a comparison principle for the solution. In Section 3, we deal with the blowup phenomenon for the problem (4) by the method of supersolutions and subsolutions. That is, the estimate of the blowup time, the blowup rates, and sets of the solution of the problem (4) are discussed.

2. Existence and Uniqueness

This section is devoted to the proof of the existence and uniqueness of the solution to the problem (4) via Banach’s fixed point theorem. Simultaneously, the comparison principle for the solution of the problem (4) is also proved. To this end, it is convenient to give some preliminaries before giving its proof.

Fix 𝑡₀>0 and consider the Banach space 𝑌_𝑡0∶=𝐶([0,𝑡₀];𝐿¹(ℝ)) with the norm |‖𝑤‖|_𝑌𝑡0=max_{0≤𝑡≤𝑡0}‖𝑤(⋅,𝑡)‖_𝐿¹_(ℝ). (7) We assume that 0≤𝑤₀(𝑥)≤𝑀 a.e. in Ω and 𝐴=2𝑀+1. Let 𝑋_𝑡0=𝑤∈𝐶0,𝑡₀;𝐿¹(ℝ)∣0≤𝑤≤𝐴, (8) which is a closed subset of 𝑌_𝑡0. We will obtain the solution of the problem (4) in the form 𝑢(𝑥,𝑡)=𝑤(𝑥,𝑡)+𝑐, where 𝑤 is a fixed point of the operator 𝐹_𝑤0∶𝑋_𝑡0→𝑋_𝑡0 defined by 𝐹_𝑤0(𝑤)(𝑥,𝑡)=^𝑡₀𝑒^{−(𝑡−𝑠)}_ℝ𝐽𝑥−𝑦𝑤+(𝑤())𝑦,𝑠+𝑐𝑑𝑦𝑥,𝑠+𝑐^𝑝𝑑𝑠+𝑒^−𝑡𝑤₀−𝑐1−𝑒^−𝑡. (9) The following lemma is the main ingredient of the proof of Theorem 4.

Lemma 6. Let 𝑤₀ and 𝑧₀ be nonnegative functions such that 𝑤₀,𝑧₀∈𝐿¹(ℝ) and 𝑤,𝑧∈𝑋_𝑡0, and then 𝐹_𝑤0(𝑤)(𝑥,𝑡)−𝐹_𝑧0(𝑧≤())(𝑥,𝑡)1+𝑝𝐴+𝑐^𝑝−11−𝑒^−𝑡0+𝑤|‖𝑤−𝑧‖|₀−𝑧_0𝐿1(Ω). (10) Therefore, if 𝑡₀ is small enough, 𝐹_𝑤0 is a strict contraction in 𝑋_𝑡0.

Proof. From the definition of 𝐹_𝑤0, we have _ℝ𝐹_𝑤0(𝑤)(𝑥,𝑡)−𝐹_𝑧0(𝑧=)(𝑥,𝑡)𝑑𝑥_ℝ𝑡0𝑒^{−(𝑡−𝑠)}_ℝ𝐽𝑥−𝑦𝑤𝑦,𝑠+𝑐−𝐽𝑥−𝑦𝑧+(𝑤())𝑦,𝑠+𝑐𝑑𝑦𝑥,𝑠+𝑐^𝑝−(𝑧(𝑥,𝑠)+𝑐)^𝑝𝑑𝑠+𝑒^−𝑡𝑤₀−𝑧₀≤𝑑𝑥_𝑡0𝑒^{−(𝑡−𝑠)}_ℝℝ𝐽𝑥−𝑦𝑤𝑦,𝑠+𝑐−𝐽𝑥−𝑦𝑧+𝑦,𝑠+𝑐𝑑𝑦𝑑𝑥𝑑𝑠_𝑡0𝑒^{−(𝑡−𝑠)}_ℝ(𝑤(𝑥,𝑠)+𝑐)^𝑝−(𝑧(𝑥,𝑠)+𝑐)^𝑝𝑑𝑥𝑑𝑠+𝑒^−𝑡_ℝ𝑤₀−𝑧₀𝑑𝑥. (11)
Now set 𝐴⁺(𝑠)={𝑦∣𝑤(𝑦,𝑠)≥𝑧(𝑦,𝑠)} and 𝐴⁻(𝑠)={𝑦∣𝑤(𝑦,𝑠)<𝑧(𝑦,𝑠)}. We have _ℝℝ𝐽𝑥−𝑦𝑤𝑦,𝑠+𝑐−𝐽𝑥−𝑦𝑧≤𝑦,𝑠+𝑐𝑑𝑦𝑑𝑥_{ℝ𝐴+(𝑠)}𝐽𝑥−𝑦𝑤𝑦,𝑠+𝑐−𝐽𝑥−𝑦𝑧+𝑦,𝑠+𝑐𝑑𝑦𝑑𝑥_{ℝ𝐴−(𝑠)}𝐽𝑥−𝑦𝑧𝑦,𝑠+𝑐−𝐽𝑥−𝑦𝑤𝑦,𝑠+𝑐𝑑𝑦𝑑𝑥. (12) Since the integrands are nonnegative, applying Fubini’s theorem, we can get _{ℝ𝐴+(𝑠)}𝐽𝑥−𝑦𝑤𝑦,𝑠+𝑐−𝐽𝑥−𝑦𝑧=𝑦,𝑠+𝑐𝑑𝑦𝑑𝑥_𝐴+(𝑠)𝑤𝑦,𝑠−𝑧𝑦,𝑠𝑑𝑦,_{ℝ𝐴−(𝑠)}𝐽𝑥−𝑦𝑧𝑦,𝑠+𝑐−𝐽𝑥−𝑦𝑤=𝑦,𝑠+𝑐𝑑𝑦𝑑𝑥_{𝐴−(𝑠)}𝑧𝑦,𝑠−𝑤𝑦,𝑠𝑑𝑦. (13) Therefore, we obtain _ℝℝ𝐽𝑥−𝑦𝑤𝑦,𝑠+𝑐−𝐽𝑥−𝑦𝑧≤𝑦,𝑠+𝑐𝑑𝑦𝑑𝑥_ℝ𝑤𝑦,𝑠−𝑧𝑦,𝑠𝑑𝑦. (14)
Since 𝑝≥1 and 𝑤,𝑧∈𝑋_𝑡0, by the differential mid-value theorem, we deduce that (𝑤_ℝ(𝑥,𝑠)+𝑐)^𝑝−(𝑧(𝑥,𝑠)+𝑐)^𝑝()𝑑𝑥≤𝑝𝐴+𝑐^𝑝−1_ℝ|𝑤(𝑥,𝑠)−𝑧(𝑥,𝑠)|𝑑𝑥. (15)
Furthermore, from the estimate (14) and (15), we get the desired estimate (10).
Next, we check that 𝐹_𝑤0 maps 𝑋_𝑡0 into 𝑋_𝑡0. Since 0≤𝑤≤𝐴, for any 𝑤∈𝑋_𝑡0, we have 𝐽𝑥−𝑦𝐴+𝑐≥𝐽𝑥−𝑦𝑤𝑦,𝑠+𝑐≥𝐽𝑥−𝑦𝑐. (16) Hence, taking 𝑡₀≤ln((𝐴+(𝐴+𝑐)^𝑝)/(𝑀+(𝐴+𝑐)^𝑝)), we get that 𝐹_𝑤0(𝑤)(𝑥,𝑡)≥^𝑡₀𝑒^{−(𝑡−𝑠)}_ℝ𝐽𝑥−𝑦𝑐+𝑑𝑦𝑑𝑠_𝑡0𝑒^{−(𝑡−𝑠)}(𝑤(𝑥,𝑠)+𝑐)^𝑝𝑑𝑠+𝑒^−𝑡𝑤₀−𝑐1−𝑒^−𝑡=𝑐𝑒^−𝑡𝑒^𝑡−𝑒⁰+𝑒^−𝑡𝑤₀(𝑡)−𝑐1−𝑒^−𝑡+^𝑡₀𝑒^{−(𝑡−𝑠)}(𝑤(𝑥,𝑠)+𝑐)^𝑝𝑑𝑠=𝑒^−𝑡𝑤₀(𝑡)+^𝑡₀𝑒^{−(𝑡−𝑠)}(𝑤(𝑥,𝑠)+𝑐)^𝑝𝐹𝑑𝑠≥0,_𝑤0(𝑤)(𝑥,𝑡)≤^𝑡₀𝑒^{−(𝑡−𝑠)}_ℝ𝐽𝑥−𝑦+𝐴+𝑐𝑑𝑦𝑑𝑠_𝑡0𝑒^{−(𝑡−𝑠)}(𝑤(𝑥,𝑠)+𝑐)^𝑝𝑑𝑠+𝑒^−𝑡𝑤₀−𝑐1−𝑒^−𝑡≤𝑒^−𝑡(𝐴+𝑐)𝑒^𝑡−1+𝑒^−𝑡(𝐴+𝑐)^𝑝𝑒^𝑡−1+𝑒^−𝑡𝑤₀−𝑐1−𝑒^−𝑡=𝐴+(𝐴+𝑐)^𝑝1−𝑒^−𝑡+𝑒^−𝑡𝑤₀≤𝐴+(𝐴+𝑐)^𝑝1−𝑒^−𝑡0+𝑤₀≤𝐴. (17) Thus, we conclude that 𝐹_𝑤0(𝑤)(𝑥,𝑡)∈𝑋_𝑡0.
Finally, choosing 𝑡₀≤ln((2𝑝(𝐴+𝑐)^𝑝−1+2)/(2𝑝(𝐴+𝑐)^𝑝−1+1)), we have 𝐹_𝑤0(𝑤)(𝑥,𝑡)−𝐹_𝑤0(𝑧≤())(𝑥,𝑡)1+𝑝𝐴+𝑐^𝑝−11−𝑒^−𝑡0≤1|‖𝑤−𝑧‖|2|‖𝑤−𝑧‖|. (18)
Therefore, if 𝑡₀ is small enough, 𝐹_𝑤0 is a strict contraction in 𝑋_𝑡0. The proof is completed.

2.1. The Proof of Theorem 4

Proof. From Lemma 6, 𝐹_𝑤0 is a strict contraction in 𝑋_𝑡0 for 𝑡₀ small enough. By the Banach fixed point theorem, there exists only one fixed point of 𝐹_𝑤0 in 𝑋_𝑡0. This proves the existence and uniqueness of the solution of (4) in the time interval [0,𝑡₀]. To continue, we may take 𝑢(𝑥,𝑡₀) as initial data and obtain a unique solution of (4) in the time interval [0,𝑡₁]. If |‖𝑤‖|_𝑌𝑡1<∞, arguing as before with 𝑢(⋅,𝑡₁) as the initial datum, it is possible to extend the solution up to some interval [0,𝑡₂) for certain 𝑡₂>𝑡₁. Hence, we can conclude that if the maximal time of the existence of the solution, 𝑇, is finite, then the solution blows up in 𝐿¹(Ω)norm; that is, limsup_𝑡↗𝑇‖𝑢(𝑥,𝑡)‖_𝐿¹_(Ω)=+∞. (19) Otherwise, the solution of the problem (4) is global.

Remark 7. From the proof of Theorem 4, the solution of (4) 𝑢(𝑥,𝑡) is nonnegative.

Remark 8. If 𝑤₀∈𝐶(Ω), then the solution of (4) 𝑢(⋅,𝑡)∈𝐶(Ω) for all 𝑡≥0.

To complete the proof of Theorem 5, we introduce the comparison principle for the problem (4) which is a very useful tool in studying diffusion problems.

Lemma 9. Let 𝑢 and 𝑢 be continuous supersolution and subsolution of the problem (4), respectively, and then 𝑢(𝑥,𝑡)≤𝑢(𝑥,𝑡) for all (𝑥,𝑡)∈Ω×[0,𝑇).

Proof. By an approximation procedure we restrict ourselves to consider strict inequalities for the supersolution. Indeed, we can take 𝑢(𝑥,𝑡)+𝛿𝑡+𝛿 (𝛿>0) as a strict supersolution and take limit as 𝛿→0 at the end.
Hence, we consider strict inequalities for the supersolution and subsolution. Let 𝑢(𝑥,0)−𝑢(𝑥,0)>0 for all 𝑥∈Ω. Arguing by contradiction, we assume that there exist a first time 𝑡₀>0 and some point 𝑥₀∈Ω such that 𝑢(𝑥₀,𝑡₀)=𝑢(𝑥₀,𝑡₀) and 𝑢(𝑥,𝑡)≤𝑢(𝑥,𝑡) for all (𝑥,𝑡)∈Ω×[0,𝑡₀]. Then, it holds that 0≤𝜕𝑢𝑥𝜕𝑡⁰,𝑡₀−𝜕𝑢𝑥𝜕𝑡⁰,𝑡₀<_Ω𝐽𝑥₀−𝑦𝑢𝑦,𝑡₀𝑥−𝐽⁰−𝑦𝑢𝑦,𝑡₀𝑑𝑦≤0, (20) and we reach the desired contradiction.

3. Blowup Time, Blowup Rates, and Sets

Once the existence and uniqueness of the solutions to the problem (4) are proved, we begin to analyze the blowup phenomenon for the problem (4).

3.1. The Proof of Theorem 5

Proof. For 𝑝>1, integrating in 𝑥∈ℝ and 𝑠∈(0,𝑡) in (4), we get ^+∞_−∞𝑢(𝑥,𝑡)𝑑𝑥=^+∞_−∞𝑢₀(𝑥)+𝑑𝑥_{+∞−∞𝑡0+∞−∞}𝐽𝑥−𝑦𝑢+𝑦,𝑠𝑑𝑦𝑑𝑠𝑑𝑥_{+∞−∞𝑡0}𝑢^𝑝(𝑥,𝑠)−𝑢(𝑥,𝑠)𝑑𝑠𝑑𝑥. (21) Applying Fubini’s theorem, we can obtain _Ω𝑢(𝑥,𝑡)𝑑𝑥=_Ω𝑢₀(𝑥)+𝑑𝑥_{𝑡0+∞−∞+∞−∞}𝐽𝑥−𝑦𝑢+𝑦,𝑠𝑑𝑥𝑑𝑦𝑑𝑠_{𝑡0+∞−∞}𝑢^𝑝(𝑥,𝑠)−𝑢(𝑥,𝑠)𝑑𝑠𝑑𝑥. (22) Let 𝜏=(𝑥−𝑦)/𝑢(𝑦,𝑠); then 𝑥=𝜏𝑢(𝑦,𝑠)+𝑦, and we have _Ω𝑢()=𝑥,𝑡𝑑𝑥_Ω𝑢₀(𝑥)𝑑𝑥+^𝑡₀^+∞_−∞𝑢𝑦,𝑠𝑑𝑦𝑑𝑠+^𝑡₀^+∞_−∞𝑢^𝑝(𝑥,𝑠)−𝑢(𝑥,𝑠)𝑑𝑠𝑑𝑥. (23) That is, _Ω𝑢(𝑥,𝑡)𝑑𝑥=_Ω𝑢₀(𝑥)𝑑𝑥+^𝑡_0Ω𝑢^𝑝(𝑥,𝑠)𝑑𝑠𝑑𝑥. (24) Using Hölder inequality, we have 𝑑𝑑𝑡_Ω𝑢(𝑥,𝑡)𝑑𝑥=_Ω𝑢^𝑝()≥|Ω|𝑥,𝑡𝑑𝑥_1−𝑝Ω𝑢(𝑥,𝑡)𝑑𝑥^𝑝. (25) Since 𝑝>1, we have that ∫_Ω𝑢(𝑥,𝑡)𝑑𝑥 cannot be global; thus 𝑢 cannot be global either. Note that, by Theorem 4, in this case, we have that 𝑢(𝑥,𝑡) blows up in finite time in 𝐿^∞(Ω) norm. Moreover, let ∫_Ω𝑢(𝑥,𝑡)𝑑𝑥=𝑧(𝑡), and we obtain 𝑑𝑧≥|Ω|𝑑𝑡^1−𝑝𝑧^𝑝. (26) Integrating the above inequality, we have 𝑧^1−𝑝(𝑡)−𝑧^1−𝑝(0)≥|Ω|1−𝑝^1−𝑝𝑡. (27) Hence, it holds that |Ω|𝑡≤^𝑝−1𝑧^1−𝑝(0)−𝑧^1−𝑝(𝑡)≤|Ω|𝑝−1_𝑝−1𝑧^1−𝑝(0)=1𝑝−1|Ω|𝑝−1𝑧(0)^𝑝−1, (28) where 𝑧(0)=∫_Ω𝑢(𝑥,0)𝑑𝑥=∫_Ω𝑤₀(𝑥)𝑑𝑥+𝑐|Ω|. Therefore, we can obtain the following estimate for the blowup time: 1𝑇≤|Ω|𝑝−1∫_Ω𝑤₀(𝑥)𝑑𝑥+𝑐|Ω|^𝑝−1. (29) For 𝑝=1, let us consider the ODE problem 𝑧^′(𝑡)(𝑡),𝑧(0)=𝑧=max_𝑥∈Ω𝑤₀(𝑥)+𝑐,1. (30) Then, it follows that 𝑧(𝑡)=𝑧(0)𝑒^𝑡 and 𝑧(𝑡)≥1 for 𝑡>0. We observe that ∫^∞_−∞𝐽((𝑥−𝑦)/𝑧(𝑡))𝑑𝑦=𝑧(𝑡). Therefore, 𝑧(𝑡) is a global supersolution of the problem (4). Thus, 𝑢 is global by comparison.

Concerning the blowup rate, that is, the speed at which solutions are blowing up, we find the following result.

Theorem 10 (blowup rates). Let 𝑝>1 and 𝑢 be a continuous solution to the problem (4) which blows up at time 𝑇. Then lim_{𝑡→𝑇−}(𝑇−𝑡)^1/(𝑝−1)max_𝑥∈Ω𝑢()=1𝑥,𝑡𝑝−1^1/(𝑝−1). (31)

Proof. For every 𝑡<𝑇, let 𝑥₀(𝑡)∈[−𝐿,𝐿] be such that max_𝑥∈Ω𝑢(⋅,𝑡)=𝑢(𝑥₀(𝑡),𝑡). Since 𝑢(𝑦,𝑡)≤𝑢(𝑥₀,𝑡) for any 𝑦∈ℝ, we have 𝐽𝑥₀−𝑦𝑢𝑥𝑦,𝑡≤𝐽⁰−𝑦𝑢𝑥₀,𝑡. (32) Changing the variable 𝜏=(𝑦−𝑥₀)/𝑢(𝑥₀,𝑡), then 𝑦=𝑥₀+𝑢𝜏,𝑑𝑦=𝑢(𝑥₀,𝑡)𝑑𝜏. Thus, we have 𝑢_𝑡𝑥₀,𝑡=^+∞_−∞𝐽𝑥₀−𝑦𝑢𝑦,𝑡𝑑𝑦+𝑢^𝑝𝑥₀,𝑡−𝑢𝑥₀,𝑡=^+∞_−∞𝐽(𝜏)𝑢𝑥₀,𝑡𝑑𝜏+𝑢^𝑝𝑥₀,𝑡−𝑢𝑥₀𝑥,𝑡=𝑢₀,𝑡+𝑢^𝑝𝑥₀,𝑡−𝑢𝑥₀,𝑡=𝑢^𝑝𝑥₀,𝑡. (33) Integrating the above inequality from (𝑡,𝑇) and taking into account 𝑝>1, we obtain max_𝑥∈Ω𝑢(𝑥,𝑡)≥(𝑇−𝑡)^{−1/(𝑝−1)}1𝑝−1^1/(𝑝−1). (34) To get the upper estimate, for any (𝑥,𝑡)∈Ω×[0,𝑇), we observe that 𝑢_𝑡(𝑥,𝑡)=^+∞_−∞𝐽𝑥−𝑦𝑢𝑦,𝑡𝑑𝑦+𝑢^𝑝()()()𝑥,𝑡−𝑢𝑥,𝑡≥−𝑢𝑥,𝑡+𝑢^𝑝(𝑥,𝑡)=𝑢^𝑝(𝑥,𝑡)1−𝑢^{−(𝑝−1)}(𝑥,𝑡). (35) In particular max_𝑥∈ℝ𝑢_𝑡(𝑥,𝑡)≥𝑢_𝑡(𝑥,𝑡)≥max_𝑥∈ℝ𝑢^𝑝(𝑥,𝑡)1−max_𝑥∈ℝ𝑢^{−(𝑝−1)}(𝑥,𝑡). (36) Taking into account (34) in this expression, we get max_𝑥∈Ω𝑢_𝑡(𝑥,𝑡)≥max_𝑥∈Ω𝑢^𝑝(𝑥,𝑡)1−𝑝−1(𝑇−𝑡). (37)Integrating (37) in (𝑡,𝑇), we obtain max_𝑥∈Ω𝑢()≤()−1𝑥,𝑡𝑝−1𝑇−𝑡2𝑝−1²(𝑇−𝑡)^{2−1/(𝑝−1)}. (38) Taking the limit as 𝑡→𝑇, we will get the results.

The blowup set, that is, the set of points at which the solutions blow up, is defined as follows: 𝐵(𝑢)={()}.𝑥∈Ω;thereexistsafinitetime𝑇with𝑢𝑥,𝑡⟶∞as𝑡↗𝑇 (39) Finally, we give the result concerning the blowup sets for the solution to the problem (4).

Theorem 11 (blowup sets). Let us consider the problem (4) with 𝑝>2. Given 𝑥₀∈Ω and 𝜀>0, there exists an initial condition, 𝑢₀, such that the blowup set 𝐵(𝑢)⊂𝐵_𝜀(𝑥₀)={𝑥∈Ω;|𝑥−𝑥₀|<𝜀}.

Proof. Given 𝑥₀∈Ω and 𝜀>0 we want to construct an initial condition 𝑢₀ such that 𝐵(𝑢)⊂𝐵_𝜀𝑥₀=𝑥∈Ω;𝑥−𝑥₀<𝜀. (40) To this end, we will consider 𝑢₀ concentrated near 𝑥₀ and small enough away from 𝑥₀.
Let 𝜑 be a nonnegative smooth function such that supp𝜑⊂𝐵_𝜀/2𝑥₀,𝜑(𝑥)>0for𝑥∈𝐵_𝜀/2𝑥₀. (41) Now, let 𝑢₀(𝑥)=𝑀𝜑(𝑥)+𝛿. (42) We want to choose 𝑀 large and 𝛿 small in such a way that (40) holds.
First, note that, thanks to the estimate (6), 1𝑇≤|Ω|𝑝−1∫_Ω𝑀𝜑(𝑥)+𝛿𝑑𝑥^𝑝−1≤1𝑝−1𝑀^𝑝−1|Ω|∫_Ω𝜑(𝑥)𝑑𝑥^𝑝−1, (43) and taking 𝑀 large enough we can assume that 𝑇 is as small as we need. Now, using the upper bound for the blowup rate, max_𝑥∈Ω𝑢()≤()−1𝑥,𝑡𝑝−1𝑇−𝑡2𝑝−1²(𝑇−𝑡)^{2−1/(𝑝−1)}≤𝐶(𝑇−𝑡)^{−1/(𝑝−1)}, (44) we can obtain 𝑢_𝑡(𝑥,𝑡)=_ℝ𝐽𝑥−𝑦𝑢(𝑦,𝑡𝑑𝑦−𝑢𝑥,𝑡)+𝑢^𝑝()≤𝑥,𝑡_ℝ𝐽𝑥−𝑦𝑢𝑦,𝑡𝑑𝑦+𝑢^𝑝()()𝑥,𝑡≤𝐶𝑇−𝑡^{−1/(𝑝−1)}+𝑢^𝑝(𝑥,𝑡), (45) for any 𝑥∈Ω, where 𝑇 is small enough. Therefore, 𝑢(𝑥,𝑡) is a subsolution to 𝑤_𝑡(𝑡)=𝐶(𝑇−𝑡)^{−1/(𝑝−1)}+𝑤^𝑝(𝑥,𝑡). (46) And hence, if 𝑢(𝑥,0)≤𝑤(0), we have 𝑢(𝑥,𝑡)≤𝑤(𝑡). (47)
Now we just have to prove that a solution 𝑤 to (46) beginning with 𝑤(0)=𝛿 remains bounded up to 𝑡=𝑇, provided that 𝛿 and 𝑇 are small enough. To see this we use 𝑧(𝑠)=(𝑇−𝑡)^1/(𝑝−1)𝑤(𝑡),𝑠=−ln(𝑇−𝑡). (48)
So we have 𝑑𝑧=−1𝑑𝑠()𝑝−1𝑇−𝑡^{(1/(𝑝−1))−1}𝑤+(𝑇−𝑡)^1/(𝑝−1)𝐶(𝑇−𝑡)^{−(1/(𝑝−1))}+𝑤^𝑝𝑒^−𝑠1=−()𝑝−1𝑇−𝑡^1/(𝑝−1)𝑤𝑒^−𝑠𝑇−𝑡+𝑐𝑒^−𝑠+(𝑇−𝑡)^{−(1/(𝑝−1))}𝑒^−𝑠𝑤^𝑝=𝐶𝑒^−𝑠+𝑧^𝑝(𝑠)−1𝑧(𝑠)()𝑝−1,𝑧−ln𝑇=𝑇^{(1/(𝑝−1))}𝛿. (49)
Note that for 𝑇 and 𝛿 small it holds that 𝑧^′(−ln𝑇)<0. Indeed, we need 1𝐶𝑇−𝑝−1𝛿𝑇^1/(𝑝−1)+𝛿^𝑝𝑇^{𝑝/(𝑝−1)}<0. (50) For 𝑝>2, we have (2−𝑝)/(𝑝−1)<0, 1/(𝑝−1)>0. If we choose the initial of the solution large enough to make sure 𝑀 large enough, then 𝑇 is small enough. So, for any fixed small 𝛿>0, it holds that 𝐶<(1/(𝑝−1))𝛿𝑇^2−𝑝𝑝−1−𝛿^𝑝𝑇^{𝑎/(𝑝−1)}. So the (50) holds. From this, it is easy to prove that 𝑧^′(𝑠)<0 for all 𝑠>−ln𝑇. Therefore, 𝑧(𝑠)→0 as 𝑠→∞. Going back to the equation verified by 𝑧, for any 𝜀>0, we have 𝑒^{(1/(𝑝−1))𝑠}𝑧(𝑠)_𝑠≤𝐶𝑒^−𝑠−1𝑧(𝑠).𝑝−1−𝜀 (51) Integrating the above inequality from (𝑠₀,𝑠) and using that 𝑝>2, we have 𝑧(𝑠)≤𝐶𝑒^{(1/(𝑝−1))𝑠}. (52) In terms of 𝑤(𝑡) this bound implies that 𝑤(𝑡)≤𝐶, for 0≤𝑡<𝑇. From the boundedness of 𝑤 and (47), we get 𝑢(𝑥,𝑡)≤𝑤(𝑡)≤𝐶 for every 𝑥∈Ω−𝐵_𝜀(𝑥₀), as we desired.