Numerical Blow-Up Time for a Semilinear Parabolic Equation with Nonlinear Boundary Conditions

Abstract

We obtain some conditions under which the positive solution for semidiscretizations of the semilinear equation 𝑢_𝑡=𝑢_𝑥𝑥−𝑎(𝑥,𝑡)𝑓(𝑢),0<𝑥<1,𝑡∈(0,𝑇), with boundary conditions 𝑢_𝑥(0,𝑡)=0, 𝑢_𝑥(1,𝑡)=𝑏(𝑡)𝑔(𝑢(1,𝑡)), blows up in a finite time and estimate its semidiscrete blow-up time. We also establish the convergence of the semidiscrete blow-up time and obtain some results about numerical blow-up rate and set. Finally, we get an analogous result taking a discrete form of the above problem and give some computational results to illustrate some points of our analysis.

1. Introduction

In this paper, we consider the following boundary value problem:𝑢_𝑡−𝑢_𝑥𝑥𝑢=−𝑎(𝑥,𝑡)𝑓(𝑢),0<𝑥<1,𝑡∈(0,𝑇),_𝑥(0,𝑡)=0,𝑢_𝑥(1,𝑡)=𝑏(𝑡)𝑔𝑢(1,𝑡),𝑡∈(0,𝑇),𝑢(𝑥,0)=𝑢₀(𝑥)≥0,0≤𝑥≤1, (1.1)where 𝑓∶[0,∞)→[0,∞) is a 𝐶¹ function, 𝑓(0)=0,   𝑔∶[0,∞)→[0,∞) is a 𝐶¹ convex function, 𝑔(0)=0, 𝑎∈𝐶⁰([0,1]×ℝ₊), 𝑎(𝑥,𝑡)≥0 in [0,1]×ℝ₊, 𝑎_𝑡(𝑥,𝑡)≤0 in [0,1]×ℝ₊, 𝑏∈𝐶¹(ℝ₊), 𝑏(𝑡)>0 in ℝ₊, 𝑏^′(𝑡)≥0 in ℝ₊. The initial data 𝑢₀∈𝐶²([0,1]), 𝑢^′₀(0)=0, 𝑢^′₀(1)=𝑏(1)𝑔(𝑢₀(1)).

Here (0,𝑇) is the maximal time interval on which the solution 𝑢 of (1.1) exists. The time 𝑇 may be finite or infinite. Where 𝑇 is infinite, we say that the solution 𝑢 exists globally. When 𝑇 is finite, the solution 𝑢 develops a singularity in a finite time, namelylim_𝑡→𝑇𝑢(⋅,𝑡)_∞=+∞, (1.2)where ‖𝑢(⋅,𝑡)‖_∞=max_0≤𝑥≤1|𝑢(𝑥,𝑡)|.

In this last case, we say that the solution 𝑢 blows up in a finite time and the time 𝑇 is called the blow-up time of the solution 𝑢.

In good number of physical devices, the boundary conditions play a primordial role in the progress of the studied processes. It is the case of the problem described in (1.1) which can be viewed as a heat conduction problem where 𝑢 stands for the temperature, and the heat sources are prescribed on the boundaries. At the boundary 𝑥=0, the heat source has a constant flux whereas at the boundary 𝑥=1, the heat source has a nonlinear radition haw. Intensification of the heat source at the boundary 𝑥=1 is provided by the function 𝑏. The function 𝑔 also gives a dominant strength of the heat source at the boundary 𝑥=1.

The theoretical study of blow-up of solutions for semilinear parabolic equations with nonlinear boundary conditions has been the subject of investigations of many authors (see [1–7], and the references cited therein).

The authors have proved that under some assumptions, the solution of (1.1) blows up in a finite time and the blow-up time is estimated. It is also proved that under some conditions, the blow-up occurs at the point 1. In this paper, we are interested in the numerical study. We give some assumptions under which the solution of a semidiscrete form of (1.1) blows up in a finite time and estimate its semidiscrete blow-up time. We also show that the semidiscrete blow-up time converges to the theoretical one when the mesh size goes to zero. An analogous study has been also done for a discrete scheme. For the semidiscrete scheme, some results about numerical blow-up rate and set have been also given. A similar study has been undertaken in [8, 9] where the authors have considered semilinear heat equations with Dirichlet boundary conditions. In the same way in [10] the numerical extinction has been studied using some discrete and semidiscrete schemes (a solution 𝑢 extincts in a finite time if it reaches the value zero in a finite time). Concerning the numerical study with nonlinear boundary conditions, some particular cases of the above problem have been treated by several authors (see [11–15]). Generally, the authors have considered the problem (1.1) in the case where 𝑎(𝑥,𝑡)=0 and 𝑏(𝑡)=1. For instance in [15], the above problem has been considered in the case where 𝑎(𝑥,𝑡)=0 and 𝑏(𝑡)=1. In [16], the authors have considered the problem (1.1) in the case where 𝑎(𝑥,𝑡)=𝜆>0, 𝑏(𝑡)=1, 𝑓(𝑢)=𝑢^𝑝, 𝑔(𝑢)=𝑢^𝑞. They have shown that the solution of a semidiscrete form of (1.1) blows up in a finite time and they have localized the blow-up set. One may also find in [17–22] similar studies concerning other parabolic problems.

The paper is organized as follows. In the next section, we present a semidiscrete scheme of (1.1). In Section 3, we give some properties concerning our semidiscrete scheme. In Section 4, under some conditions, we prove that the solution of the semidiscrete form of (1.1) blows up in a finite time and estimate its semidiscrete blow-up time. In Section 5, we study the convergence of the semidiscrete blow-up time. In Section 6, we give some results on the numerical blow-up rate and Section 7 is consecrated to the study of the numerical blow-up set. In Section 8, we study a particular discrete form of (1.1). Finally, in the last section, taking some discrete forms of (1.1), we give some numerical experiments.

2. The Semidiscrete problem

Let 𝐼 be a positive integer and define the grid 𝑥_𝑖=𝑖ℎ, 0≤𝑖≤𝐼, where ℎ=1/𝐼. We approximate the solution 𝑢 of (1.1) by the solution 𝑈_ℎ(𝑡)=(𝑈₀(𝑡),𝑈₁(𝑡),…,𝑈_𝐼(𝑡))^𝑇 of the following semidiscrete equations𝑑𝑈_𝑖(𝑡)𝑑𝑡−𝛿²𝑈_𝑖(𝑡)=−𝑎_𝑖(𝑡)𝑓𝑈_𝑖(𝑡),0≤𝑖≤𝐼−1,𝑡∈0,𝑇^ℎ_𝑏,𝑑𝑈_𝐼(𝑡)𝑑𝑡−𝛿²𝑈_𝐼2(𝑡)=ℎ𝑏(𝑡)𝑔𝑈_𝐼(𝑡)−𝑎_𝐼(𝑡)𝑓𝑈_𝐼(𝑡),𝑡∈0,𝑇^ℎ_𝑏,𝑈_𝑖(0)=𝜑_𝑖≥0,0≤𝑖≤𝐼, (2.1)(2.2)(2.3)where 𝜑_𝑖+1≥𝜑_𝑖, 0≤𝑖≤𝐼−1,𝛿²𝑈₀(𝑡)=2𝑈₁(𝑡)−2𝑈₀(𝑡)ℎ²,𝛿²𝑈_𝐼(𝑡)=2𝑈_𝐼−1(𝑡)−2𝑈_𝐼(𝑡)ℎ²,𝛿²𝑈_𝑖𝑈(𝑡)=^𝑖+1(𝑡)−2𝑈_𝑖(𝑡)+𝑈_𝑖−1(𝑡)ℎ². (2.4)Here (0,𝑇^ℎ_𝑏) is the maximal time interval on which ‖𝑈_ℎ(𝑡)‖_∞ is finite where ‖𝑈_ℎ(𝑡)‖_∞=max_{0≤𝑖≤𝐼}𝑈_𝑖(𝑡). When 𝑇^ℎ_𝑏 is finite, we say that the solution 𝑈_ℎ(𝑡) blows up in a finite time and the time 𝑇^ℎ_𝑏 is called the blow-up time of the solution 𝑈_ℎ(𝑡).

3. Properties of the Semidiscrete Scheme

In this section, we give some lemmas which will be used later.

The following lemma is a semidiscrete form of the maximum principle.Lemma 3.1. Let 𝑎_ℎ(𝑡)∈𝐶⁰([0,𝑇),ℝ^𝐼+1) and let 𝑉_ℎ(𝑡)∈𝐶¹([0,𝑇),ℝ^𝐼+1) such that𝑑𝑉_𝑖(𝑡)𝑑𝑡−𝛿²𝑉_𝑖(𝑡)+𝑎_𝑖(𝑡)𝑉_𝑖𝑉(𝑡)≥0,0≤𝑖≤𝐼,𝑡∈(0,𝑇),_𝑖(0)≥0,0≤𝑖≤𝐼. (3.1)Then we have 𝑉_𝑖(𝑡)≥0, 0≤𝑖≤𝐼, 𝑡∈(0,𝑇).

Proof. Let 𝑇₀<𝑇 and define the vector 𝑍_ℎ(𝑡)=𝑒^𝜆𝑡𝑉_ℎ(𝑡) where 𝜆 is large enough that 𝑎_𝑖(𝑡)−𝜆>0 for 𝑡∈[0,𝑇₀], 0≤𝑖≤𝐼. Let 𝑚=min_{0≤𝑖≤𝐼,0≤𝑡≤𝑇0}𝑍_𝑖(𝑡). Since for 𝑖∈{0,…,𝐼}, 𝑍_𝑖(𝑡) is a continuous function, there exists 𝑡₀∈[0,𝑇₀] such that 𝑚=𝑍_𝑖0(𝑡₀) for a certain 𝑖₀∈{0,…,𝐼}. It is not hard to see that𝑑𝑍_𝑖0𝑡₀𝑑𝑡=lim_𝑘→0𝑍_𝑖0𝑡₀−𝑍_𝑖0𝑡₀−𝑘𝑘𝛿≤0,₂𝑍_𝑖0𝑡₀=𝑍_𝑖0+1𝑡₀−2𝑍_𝑖0𝑡₀+𝑍_𝑖0−1𝑡₀ℎ²≥0if1≤𝑖₀𝛿≤𝐼−1,₂𝑍_𝑖0𝑡₀=2𝑍₁𝑡₀−2𝑍₀𝑡₀ℎ²≥0if𝑖₀𝛿=0,₂𝑍_𝑖0𝑡₀=2𝑍_𝐼−1𝑡₀−2𝑍_𝐼𝑡₀ℎ²≥0if𝑖₀=𝐼. (3.2)A straightforward computation reveals that𝑑𝑍_𝑖0𝑡₀𝑑𝑡−𝛿²𝑍_𝑖0𝑡₀+𝑎_𝑖0𝑡₀−𝜆𝑍_𝑖0𝑡₀≥0. (3.3)We observe from (3.2) that (𝑎_𝑖0(𝑡₀)−𝜆)𝑍_𝑖0(𝑡₀)≥0 which implies that 𝑍_𝑖0(𝑡₀)≥0 because 𝑎_𝑖0(𝑡₀)−𝜆>0. We deduce that 𝑉_ℎ(𝑡)≥0 for 𝑡∈[0,𝑇₀] and the proof is complete.

Another form of the maximum principle for semidiscrete equations is the following comparison lemma.Lemma 3.2. Let 𝑉_ℎ(𝑡),𝑈_ℎ(𝑡)∈𝐶¹([0,𝑇),ℝ^𝐼+1) and 𝑓∈𝐶⁰(ℝ×ℝ,ℝ) such that for 𝑡∈(0,𝑇)𝑑𝑉_𝑖(𝑡)𝑑𝑡−𝛿²𝑉_𝑖(𝑡)+𝑓𝑉_𝑖(𝑡),𝑡<𝑑𝑈_𝑖(𝑡)𝑑𝑡−𝛿²𝑈_𝑖(𝑡)+𝑓𝑈_𝑖𝑉(𝑡),𝑡,0≤𝑖≤𝐼,_𝑖(0)<𝑈_𝑖(0),0≤𝑖≤𝐼. (3.4)(3.5)Then we have 𝑉_𝑖(𝑡)<𝑈_𝑖(𝑡), 0≤𝑖≤𝐼, 𝑡∈(0,𝑇).

Proof. Define the vector 𝑍_ℎ(𝑡)=𝑈_ℎ(𝑡)−𝑉_ℎ(𝑡). Let 𝑡₀ be the first 𝑡∈(0,𝑇) such that 𝑍_𝑖(𝑡)>0 for 𝑡∈[0,𝑡₀), 0≤𝑖≤𝐼, but 𝑍_𝑖0(𝑡₀)=0 for a certain 𝑖₀∈{0,…,𝐼}. We observe that𝑑𝑍_𝑖0𝑡₀𝑑𝑡=lim_𝑘→0𝑍_𝑖0𝑡₀−𝑍_𝑖0𝑡₀−𝑘𝑘𝛿≤0,₂𝑍_𝑖0𝑡₀=𝑍_𝑖0+1𝑡₀−2𝑍_𝑖0𝑡₀+𝑍_𝑖0−1𝑡₀ℎ²≥0if1≤𝑖₀𝛿≤𝐼−1,₂𝑍_𝑖0𝑡₀=2𝑍₁𝑡₀−2𝑍₀𝑡₀ℎ²≥0if𝑖₀𝛿=0,₂𝑍_𝑖0𝑡₀=2𝑍_𝐼−1𝑡₀−2𝑍_𝐼𝑡₀ℎ²≥0if𝑖₀=𝐼, (3.6) which implies that𝑑𝑍_𝑖0𝑡₀𝑑𝑡−𝛿²𝑍_𝑖0𝑡₀+𝑓𝑈_𝑖0𝑡₀,𝑡₀−𝑓𝑉_𝑖0𝑡₀,𝑡₀≤0. (3.7)But this inequality contradicts (3.4) and the proof is complete.

4. Semidiscrete Blow-Up Solutions

In this section under some assumptions, we show that the solution 𝑈_ℎ of (2.1)–(2.3) blows up in a finite time and estimate its semidiscrete blow-up time.

Before starting, we need the following two lemmas. The first lemma gives a property of the operator 𝛿² and the second one reveals a property of the semidiscrete solution.Lemma 4.1. Let 𝑈_ℎ∈ℝ^𝐼+1 be such that 𝑈_ℎ≥0. Then we have𝛿²𝑔𝑈_𝑖≥𝑔^′𝑈_𝑖𝛿²𝑈_𝑖for0≤𝑖≤𝐼. (4.1)

Proof. Apply Taylor's expansion to obtain𝑔𝑈₁=𝑔𝑈₀+𝑈₁−𝑈₀𝑔^′𝑈₀+𝑈₁−𝑈₀²2𝑔^′′𝜂₀,𝑔𝑈_𝑖+1=𝑔𝑈_𝑖+𝑈_𝑖+1−𝑈_𝑖𝑔^′𝑈_𝑖+𝑈_𝑖+1−𝑈_𝑖²2𝑔^′′𝜃_𝑖𝑔𝑈,1≤𝑖≤𝐼−1,_𝑖−1=𝑔𝑈_𝑖+𝑈_𝑖−1−𝑈_𝑖𝑔^′𝑈_𝑖+𝑈_𝑖−1−𝑈_𝑖²2𝑔^′′𝜂_𝑖𝑔𝑈,1≤𝑖≤𝐼−1,_𝐼−1=𝑔𝑈_𝐼+𝑈_𝐼−1−𝑈_𝐼𝑔^′𝑈_𝐼+𝑈_𝐼−1−𝑈_𝐼²2𝑔^′′𝜂_𝐼, (4.2)where 𝜃_𝑖 is an intermediate between 𝑈_𝑖 and 𝑈_𝑖+1 and 𝜂_𝑖 the one between 𝑈_𝑖−1 and 𝑈_𝑖. The first and last equalities imply that𝛿²𝑔𝑈₀=𝑔^′𝑈₀𝛿²𝑈₀+𝑈₁−𝑈₀²ℎ²𝑔^′′𝜂₀,𝛿²𝑔𝑈_𝐼=𝑔^′𝑈_𝐼𝛿²𝑈_𝐼+𝑈_𝐼−1−𝑈_𝐼²ℎ²𝑔^′′𝜂_𝐼. (4.3)Combining the second and third equalities, we see that𝛿²𝑔𝑈_𝑖=𝑔^′𝑈_𝑖𝛿²𝑈_𝑖+𝑈_𝑖+1−𝑈_𝑖²2ℎ²𝑔^′′𝜃_𝑖+𝑈_𝑖−1−𝑈_𝑖²2ℎ²𝑔^′′𝜂_𝑖,1≤𝑖≤𝐼−1. (4.4)Use the fact that 𝑔^′′(𝑠)≥0 for 𝑠≥0 and 𝑈_ℎ≥0 to complete the rest of the proof.

Lemma 4.2. Let 𝑈_ℎ be the solution of (2.1)–(2.3). Then we have𝑈_𝑖+1(𝑡)>𝑈_𝑖(𝑡),0≤𝑖≤𝐼−1,𝑡∈0,𝑇^ℎ_𝑏. (4.5)

Proof. Let 𝑡₀ be the first 𝑡>0 such that 𝑈_𝑖+1(𝑡)>𝑈_𝑖(𝑡) for 0≤𝑖≤𝐼−1 but 𝑈_𝑖0+1(𝑡₀)=𝑈_𝑖0(𝑡₀) for a certain 𝑖₀∈{0,…,𝐼−1}. Without loss of generality, we may suppose that 𝑖₀ is the smallest integer which satisfies the equality. Introduce the functions 𝑍_𝑖(𝑡)=𝑈_𝑖+1(𝑡)−𝑈_𝑖(𝑡) for 0≤𝑖≤𝐼−1. We get𝑑𝑍_𝑖0𝑡₀𝑑𝑡=lim_𝑘→0𝑍_𝑖0𝑡₀−𝑍_𝑖0𝑡₀−𝑘𝑘𝛿≤0,₂𝑍_𝑖0𝑡₀=𝑍_𝑖0+1𝑡₀−2𝑍_𝑖0𝑡₀+𝑍_𝑖0−1𝑡₀ℎ²>0if1≤𝑖₀𝛿≤𝐼−2,₂𝑍_𝑖0𝑡₀=𝛿²𝑍₀𝑡₀=𝑍₁𝑡₀−3𝑍₀𝑡₀ℎ²>0if𝑖₀𝛿=0,₂𝑍_𝑖0𝑡₀=𝛿²𝑍_𝐼−1𝑡₀=𝑍_𝐼−2𝑡₀−3𝑍_𝐼−1𝑡₀ℎ²>0if𝑖₀=𝐼−1, (4.6)which implies that𝑑𝑍_𝑖0𝑡₀𝑑𝑡−𝛿²𝑍_𝑖0𝑡₀−𝑎_𝑖0+1𝑡₀𝑓𝑈_𝑖0+1(𝑡₀+𝑎_𝑖0𝑡₀𝑓𝑈_𝑖0𝑡₀<0if0≤𝑖₀≤𝐼−2,𝑑𝑍_𝑖0𝑡₀𝑑𝑡−𝛿²𝑍_𝑖0𝑡₀+2ℎ𝑏𝑡₀𝑔_𝑖0+1𝑡₀−𝑎_𝑖0+1𝑡₀𝑓𝑈_𝑖0+1(𝑡₀+𝑎_𝑖0𝑡₀𝑓𝑈_𝑖0𝑡₀<0if𝑖₀=𝐼−1. (4.7)But this contradicts (2.1)-(2.2) and we have the desired result.

The above lemma says that the semidiscrete solution is increasing in space. This property will be used later to show that the semidiscrete solution attains its minimum at the last node 𝑥_𝐼.

Now, we are in a position to state the main result of this section.Theorem 4.3. Let 𝑈_ℎ be the solution of (2.1)–(2.3). Suppose that there exists a positive integer A such that𝛿²𝜑_𝑖−𝑎_𝑖(0)𝑓𝜑_𝑖𝛿≥0,1≤𝑖≤𝐼−1,₂𝜑_𝐼−𝑎_𝐼(0)𝑓𝜑_𝐼+𝑏(0)𝑔_𝐼𝜑_𝐼≥𝐴𝑔𝜑_𝐼. (4.8)Assume that𝑓(𝑠)𝑔^′(𝑠)−𝑓^′(𝑠)𝑔(𝑠)≥0for𝑠≥0. (4.9)Then the solution 𝑈_ℎ blows up in a finite time 𝑇^ℎ_𝑏 and we have the following estimate𝑇^ℎ_𝑏≤1𝐴^+∞_{‖𝜑ℎ‖∞}𝑑𝑠.𝑔(𝑠) (4.10)

Proof. Since (0,𝑇^ℎ_𝑏) is the maximal time interval on which ‖𝑈_ℎ(𝑡)‖_∞<∞, our aim is to show that 𝑇^ℎ_𝑏 is finite and satisfies the above inequality. Introduce the vector 𝐽_ℎ such that𝐽_𝑖(𝑡)=𝑑𝑈_𝑖(𝑡)𝑑𝑡,0≤𝑖≤𝐼−1,𝐽_𝐼(𝑡)=𝑑𝑈_𝐼(𝑡)𝑈𝑑𝑡−𝐴𝑔_𝐼(𝑡). (4.11)A straightforward calculation gives𝑑𝐽_𝑖𝑑𝑡−𝛿²𝐽_𝑖=𝑑𝑑𝑡𝑑𝑈_𝑖𝑑𝑡−𝛿²𝑈_𝑖,0≤𝑖≤𝐼−1,𝑑𝐽_𝐼𝑑𝑡−𝛿²𝐽_𝐼=𝑑𝑑𝑡𝑑𝑈_𝐼𝑑𝑡−𝛿²𝑈_𝐼−𝐴𝑔^′𝑈_𝐼𝑑𝑈_𝐼𝑑𝑡+𝐴𝛿²𝑔𝑈_𝐼. (4.12)From Lemma 4.1, we have 𝛿²𝑔(𝑈_𝐼)≥𝑔^′(𝑈_𝐼)𝛿²𝑈_𝐼 which implies that𝑑𝐽_𝐼𝑑𝑡−𝛿²𝐽_𝐼≥𝑑𝑑𝑡𝑑𝑈_𝐼𝑑𝑡−𝛿²𝑈_𝐼−𝐴𝑔^′𝑈_𝐼𝑑𝑈_𝐼𝑑𝑡−𝛿²𝑈_𝐼. (4.13)Using (2.1), we get𝑑𝐽_𝑖𝑑𝑡−𝛿²𝐽_𝑖≥−𝑎^′_𝑖(𝑡)𝑓𝑈_𝑖−𝑎_𝑖(𝑡)𝑓^′𝑈_𝑖𝑑𝑈_𝑖𝑑𝑡,0≤𝑖≤𝐼−1,𝑑𝐽_𝐼𝑑𝑡−𝛿²𝐽_𝐼≥−𝑎^′_𝐼(𝑡)𝑓𝑈_𝐼−𝑎_𝐼(𝑡)𝑓^′𝑈_𝐼𝑑𝑈_𝐼+2𝑑𝑡ℎ𝑏^′(𝑡)𝑔𝑈_𝐼+2ℎ𝑏(𝑡)𝑔^′𝑈_𝐼𝑑𝑈_𝐼𝑑𝑡−𝐴𝑔^′𝑈_𝐼−𝑎_𝐼(𝑡)𝑓𝑈_𝐼+2ℎ𝑏(𝑡)𝑔𝑈_𝐼. (4.14)It follows from the fact that 𝑎^′_𝑖(𝑡)≤0, 𝑏^′(𝑡)≥0 and 𝑑𝑈_𝑖/𝑑𝑡=𝐽_𝑖+𝐴𝑔(𝑈_𝑖) that𝑑𝐽_𝐼𝑑𝑡−𝛿²𝐽_𝐼≥−𝑎_𝐼(𝑡)𝑓^′𝑈_𝐼+2ℎ𝑏(𝑡)𝑔^′𝑈_𝐼𝐽_𝐼+𝐴𝑎_𝐼(𝑡)𝑔^′𝑈_𝐼𝑓𝑈_𝐼−𝑓^′𝑈_𝐼𝑔𝑈_𝐼. (4.15)We deduce from (4.9) that𝑑𝐽_𝑖𝑑𝑡−𝛿²𝐽_𝑖≥−𝑎_𝑖(𝑡)𝑓^′𝑈_𝑖𝐽_𝑖,0≤𝑖≤𝐼−1,𝑑𝐽_𝐼𝑑𝑡−𝛿²𝐽_𝐼≥−𝑎_𝐼(𝑡)𝑓^′𝑈_𝐼+2ℎ𝑏(𝑡)𝑔^′𝑈_𝐼𝐽_𝐼. (4.16)From (4.8), we observe that𝐽_𝑖(0)=𝛿²𝜑_𝑖−𝑎_𝑖(0)𝑓𝜑_𝑖𝐽≥0,0≤𝑖≤𝐼−1,_𝐼(0)=𝛿²𝜑_𝐼−𝑎_𝐼(0)𝑓𝜑_𝐼+𝑏(0)𝑔_𝐼𝜑_𝐼−𝐴𝑔𝜑_𝐼≥0. (4.17) We deduce from Lemma 3.1 that 𝐽_𝑖(𝑡)≥0, 0≤𝑖≤𝐼, which implies that 𝑑𝑈_𝐼/𝑑𝑡≥𝑔(𝑈_𝐼), 0≤𝑖≤𝐼. Obviously we have𝑑𝑈_𝐼𝑔𝑈_𝐼≥𝐴𝑑𝑡. (4.18)Integrating this inequality over (𝑡,𝑇^ℎ_𝑏), we arrive at𝑇^ℎ_𝑏1−𝑡≤𝐴_𝑈^+∞_𝐼(𝑡)𝑑𝑠,𝑔(𝑠) (4.19)which implies that𝑇^ℎ_𝑏≤1𝐴^+∞_{‖𝑈ℎ(0)‖∞}𝑑𝑠.𝑔(𝑠) (4.20)Since the quantity on the right hand side of the above inequality is finite, we deduce that the solution 𝑈_ℎ blows up in a finite time. Use the fact that ‖𝑈_ℎ(0)‖_∞=‖𝜑_ℎ‖_∞ to complete the rest of the proof.

Remark 4.4. The inequality (4.19) implies that 𝑇^ℎ_𝑏−𝑡₀≤1𝐴∫^+∞_{‖𝑈ℎ(𝑡0)‖∞}𝑑𝑠𝑔(𝑠)if0<𝑡₀<𝑇^ℎ_𝑏,𝑈_𝑖(𝑡)≤𝐻𝐴𝑇^ℎ_𝑏−𝑡,0≤𝑖≤𝐼, (4.21)where 𝐻(𝑠) is the inverse of 𝐺(𝑠)=∫_𝑠^+∞(𝑑𝑧/𝑔(𝑧)).

Remark 4.5. If 𝑔(𝑠)=𝑠^𝑞, then 𝐺(𝑠)=𝑠^1−𝑞/(𝑞−1) and 𝐻(𝑠)=((𝑞−1)𝑠)^1/(1−𝑞).

5. Convergence of the Semidiscrete Blow-Up Time

In this section, we show the convergence of the semidiscrete blow-up time. Now we will show that for each fixed time interval [0,𝑇] where 𝑢 is defined, the solution 𝑈_ℎ(𝑡) of (2.1)–(2.3) approximates 𝑢, when the mesh parameter ℎ goes to zero.Theorem 5.1. Assume that (1.1) has a solution 𝑢∈𝐶^4,1([0,1]×[0,𝑇]) and the initial condition at (2.3) satisfies𝜑_ℎ−𝑢_ℎ(0)_∞=𝑜(1)asℎ⟶0, (5.1)where 𝑢_ℎ(𝑡)=(𝑢(𝑥₀,𝑡),…,𝑢(𝑥_𝐼,𝑡))^𝑇. Then, for h sufficiently small, the problem (2.1)–(2.3) has a unique solution 𝑈_ℎ∈𝐶¹([0,𝑇],ℝ^𝐼+1) such thatmax_{0≤𝑡≤𝑇}𝑈_ℎ(𝑡)−𝑢_ℎ(𝑡)_∞=𝑂𝜑_ℎ−𝑢_ℎ(0)_∞+ℎ²asℎ⟶0. (5.2)

Proof. Let 𝛼>0 be such that𝑢(⋅,𝑡)_∞≤𝛼for𝑡∈[0,𝑇]. (5.3)The problem (2.1)–(2.3) has for each ℎ, a unique solution 𝑈_ℎ∈𝐶¹([0,𝑇^ℎ_𝑏),ℝ^𝐼+1). Let 𝑡(ℎ)≤min{𝑇,𝑇^ℎ_𝑏} the greatest value of 𝑡>0 such that𝑈_ℎ(𝑡)−𝑢_ℎ(𝑡)_∞<1for𝑡∈0,𝑡(ℎ). (5.4)The relation (5.1) implies that 𝑡(ℎ)>0 for ℎ sufficiently small. By the triangle inequality, we obtain𝑈_ℎ(𝑡)_∞≤𝑢(⋅,𝑡)_∞+𝑈_ℎ(𝑡)−𝑢_ℎ(𝑡)_∞for𝑡∈0,𝑡(ℎ), (5.5)which implies that𝑈_ℎ(𝑡)_∞≤1+𝛼for𝑡∈0,𝑡(ℎ). (5.6)Let 𝑒_ℎ(𝑡)=𝑈_ℎ(𝑡)−𝑢_ℎ(𝑡) be the error of discretization. Using Taylor's expansion, we have for 𝑡∈(0,𝑡(ℎ)),𝑑𝑒_𝑖(𝑡)𝑑𝑡−𝛿²𝑒_𝑖(𝑡)=−𝑎_𝑖(𝑡)𝑓^′𝜉_𝑖(𝑡)𝑒_𝑖(𝑡)+𝑜ℎ²,0≤𝑖≤𝐼−1,𝑑𝑒_𝐼(𝑡)𝑑𝑡−𝛿²𝑒_𝐼(𝑡)=−𝑎_𝐼(𝑡)𝑓^′𝜉_𝐼(𝑡)𝑒_𝐼2(𝑡)+ℎ𝑏(𝑡)𝑔^′𝑈_𝐼(𝑡)𝑒_𝐼(𝑡)+𝑜ℎ², (5.7)where 𝜃_𝐼(𝑡) is an intermediate value between 𝑈_𝐼(𝑡) and 𝑢(𝑥_𝐼,𝑡) and 𝜉_𝑖(𝑡) the one between 𝑈_𝑖(𝑡) and 𝑢(𝑥_𝑖,𝑡). Using (5.3) and (5.6), there exist two positive constants 𝐾 and 𝐿 such that𝑑𝑒_𝑖(𝑡)𝑑𝑡−𝛿²𝑒_𝑖(𝑡)≤𝐿𝑒_𝑖(𝑡)+𝐾ℎ²,0≤𝑖≤𝐼−1,𝑑𝑒_𝐼(𝑡)−𝑑𝑡2𝑒_𝐼−1(𝑡)−2𝑒_𝐼(𝑡)ℎ²≤𝐿𝑒_𝐼(𝑡)ℎ+𝐿𝑒_𝐼(𝑡)+𝐾ℎ². (5.8)Consider the function 𝑧(𝑥,𝑡)=𝑒^{((𝑀+1)𝑡+𝐶𝑥2)}(‖𝜑_ℎ−𝑢_ℎ(0)‖_∞+𝑄ℎ²) where 𝑀, 𝐶, 𝑄 are constants which will be determined later. We get𝑧_𝑡(𝑥,𝑡)−𝑧_𝑥𝑥(𝑥,𝑡)=(𝑀+1−2𝐶−4𝐶²𝑥²𝑧)𝑧(𝑥,𝑡),_𝑥(0,𝑡)=0,𝑧_𝑥(1,𝑡)=2𝐶𝑧(1,𝑡),𝑧(𝑥,0)=𝑒^𝐶𝑥2(‖𝜑_ℎ−𝑢_ℎ(0)‖_∞+𝑄ℎ²). (5.9)By a semidiscretization of the above problem, we may choose 𝑀, 𝐶, 𝑄 large enough that𝑑𝑧𝑥𝑑𝑡^𝑖,𝑡>𝛿²𝑧𝑥_𝑖,𝑡+𝐿𝑧𝑥_𝑖,𝑡+𝐾ℎ²𝑑,0≤𝑖≤𝐼−1,𝑧𝑥𝑑𝑡^𝐼,𝑡>𝛿²𝑧𝑥_𝐼+𝐿,𝑡ℎ𝑧𝑥_𝐼,𝑡+𝐿𝑧𝑥_𝐼,𝑡+𝐾ℎ²,𝑧𝑥_𝑖,0>𝑒_𝑖(0),0≤𝑖≤𝐼. (5.10)It follows from Lemma 3.2 that𝑧𝑥_𝑖,𝑡>𝑒_𝑖(𝑡)for𝑡∈0,𝑡(ℎ),0≤𝑖≤𝐼. (5.11)By the same way, we also prove that𝑧𝑥_𝑖,𝑡>−𝑒_𝑖(𝑡)for𝑡∈0,𝑡(ℎ),0≤𝑖≤𝐼, (5.12)which implies that𝑧𝑥_𝑖,𝑡>𝑒_𝑖(𝑡)for𝑡∈0,𝑡(ℎ),0≤𝑖≤𝐼. (5.13)We deduce that𝑈_ℎ(𝑡)−𝑢_ℎ(𝑡)_∞≤𝑒^{(𝑀𝑡+𝐶)}𝜑_ℎ−𝑢_ℎ(0)_∞+𝑄ℎ²,𝑡∈0,𝑡(ℎ). (5.14)Let us show that 𝑡(ℎ)=𝑇. Suppose that 𝑇>𝑡(ℎ). From (5.4), we obtain1=𝑈_ℎ𝑡(ℎ)−𝑢_ℎ𝑡(ℎ)_∞≤𝑒^{(𝑀𝑇+𝐶)}𝜑_ℎ−𝑢_ℎ(0)_∞+𝑄ℎ². (5.15)Since the term on the right hand side of the above inequality goes to zero as ℎ tends to zero, we deduce that 1≤0, which is impossible. Consequently 𝑡(ℎ)=𝑇, and the proof is complete.

Now, we are in a position to prove the main result of this section.Theorem 5.2. Suppose that the problem (1.1) has a solution u which blows up in a finite time 𝑇_𝑏 such that 𝑢∈𝐶^4,1([0,1]×[0,𝑇_𝑏)) and the initial condition at (2.3) satisfies𝜑_ℎ−𝑢_ℎ(0)_∞=𝑜(1)asℎ⟶0. (5.16)Under the assumptions of Theorem 4.3, the problem (2.1)–(2.3) admits a unique solution 𝑈_ℎ which blows up in a finite time 𝑇^ℎ_𝑏 and we have the following relationlim_ℎ→0𝑇^ℎ_𝑏=𝑇_𝑏. (5.17)

Proof. Let 𝜀>0. There exists a positive constant 𝑁 such that1𝐴_𝑥^+∞𝑑𝑠≤𝜀𝑔(𝑠)2for𝑥∈[𝑁,+∞). (5.18)Since the solution 𝑢 blows up at the time 𝑇_𝑏, then there exists 𝑇₁∈(𝑇_𝑏−𝜀/2,𝑇_𝑏) such that ‖𝑢(⋅,𝑡)‖_∞≥2𝑁 for 𝑡∈[𝑇₁,𝑇_𝑏). Setting 𝑇₂=(𝑇₁+𝑇_𝑏)/2, then we have sup_{𝑡∈[0,𝑇2]}|𝑢(𝑥,𝑡)|<∞. It follows from Theorem 5.1 thatsup_{𝑡∈[0,𝑇2]}𝑈_ℎ(𝑡)−𝑢_ℎ(𝑡)_∞≤𝑁. (5.19)Applying the triangle inequality, we get𝑈_ℎ𝑇_2∞≥𝑢_ℎ𝑇_2∞−𝑈_ℎ𝑇₂−𝑢_ℎ𝑇_2∞, (5.20)which leads to ‖𝑈_ℎ(𝑇₂)‖_∞≥𝑁. From Theorem 4.3, 𝑈_ℎ(𝑡) blows up at the time 𝑇^ℎ_𝑏. We deduce from Remark 4.4 and (5.18) that𝑇_𝑏−𝑇^ℎ_𝑏≤𝑇_𝑏−𝑇₂+𝑇^ℎ_𝑏−𝑇₂≤𝜀2+1𝐴^+∞_{‖𝑈ℎ(𝑇2)‖∞}𝑑𝑠𝑔(𝑠)≤𝜀, (5.21)and the proof is complete.

6. Numerical Blow-Up Rate

In this section, we determine the blow-up rate of the solution 𝑈_ℎ of (2.1)–(2.3) in the case where 𝑏(𝑡)=1. Our result is the following.Theorem 6.1. Let 𝑈_ℎ(𝑡) be the solution of (2.1)–(2.3). Under the assumptions of Theorem 4.3, 𝑈_ℎ(𝑡) blows up in a finite time 𝑇^ℎ_𝑏 and there exist two positive constants 𝐶₁,𝐶₂ such that𝐻𝐶₁𝑇^ℎ_𝑏−𝑡≤𝑈_𝐼(𝑡)≤𝐻𝐶₂𝑇^ℎ_𝑏−𝑡,for𝑡∈0,𝑇^ℎ_𝑏, (6.1)where 𝐻(𝑠) is the inverse of the function 𝐺(𝑠)=∫_𝑠^+∞(𝑑𝜎/𝑔(𝜎)).

Proof. From Theorem 4.3 and Remark 4.4, 𝑈_ℎ(𝑡) blows up in a finite time 𝑇^ℎ_𝑏 and there exists a constant 𝐶₂>0 such that𝑈_𝐼(𝑡)≤𝐻𝐶₂𝑇^ℎ_𝑏−𝑡for𝑡∈0,𝑇^ℎ_𝑏. (6.2)From Lemma 4.2, 𝑈_𝐼−1<𝑈_𝐼. Then using (2.2), we deduce that 𝑑𝑈_𝐼/𝑑𝑡≤(2/ℎ)𝑏(𝑡)𝑔(𝑈_𝐼)−𝑎_𝐼(𝑡)𝑓(𝑈_𝐼), which implies that 𝑑𝑈_𝐼/𝑑𝑡≤(2𝑏(𝑡)/ℎ)𝑔(𝑈_𝐼). Integration this inequality over (𝑡,𝑇^ℎ_𝑏), there exists a positive constant 𝐶₁ such that𝑈_𝐼(𝑡)≥𝐻𝐶₁𝑇^ℎ_𝑏−𝑡for𝑡∈0,𝑇^ℎ_𝑏, (6.3)which leads us to the result.

7. Numerical Blow-Up Set

In this section, we determine the numerical blow-up set of the semidiscrete solution. This is stated in the theorem below.Theorem 7.1. Suppose that there exists a positive constant 𝐶₀ such that 𝑠𝐹^′(𝑠)≤𝐶₀ and𝑑𝑈𝑑𝑡^𝑖−𝛿²𝑈_𝑖≤0,0≤𝑖≤𝐼−1. (7.1)Assume that there exists a positive constant 𝐶 such𝑈_𝑖≤𝐻𝐶(𝑇−𝑡),0≤𝑖≤𝐼. (7.2)Then the numerical blow-up set is 𝐵={1}.

Proof. Let 𝑣(𝑥)=1−𝑥² and define𝑊(𝑥,𝑡)=𝐻𝛿𝑣(𝑥)+𝛿𝐵(𝑇−𝑡)for0≤𝑥≤1,𝑡≥𝑡₀, (7.3)where 𝛿 is small enough. We have𝑊_𝑥(0,𝑡)=0,𝑊(1,𝑡)=𝐻𝛿𝐵(𝑇−𝑡)≥𝑢(1,𝑡), (7.4)and for 𝑡≥𝑡₀, we get𝑊(𝑥,𝑡₀)=𝐻𝛿𝑣(𝑥)+𝛿≥𝐻(2𝛿)=𝐻2𝛿𝐵𝑇−𝑡₀≥𝐻𝐶𝑇−𝑡₀≥𝑢𝑥,𝑡₀. (7.5)A straightforward computation yields𝑊_𝑡(𝑥,𝑡)−𝑊_𝑥𝑥(𝑥,𝑡)=𝛿𝐹𝐻(𝜏)𝐵−2−4𝑥𝐹^′𝐻(𝜏)≥𝛿𝐹𝐻(𝜏)𝐵−2−4𝛿𝐶₀. (7.6)This implies that there exists 𝛼>0 such that𝑊_𝑡(𝑥,𝑡)−𝑊_𝑥𝑥(𝑥,𝑡)≥𝛼𝐹𝐻(𝛿+𝛿𝐵𝑇). (7.7)Using Taylor's expansion, there exists a constant 𝐾>0 such that𝑑𝑊𝑥𝑑𝑡^𝑖,𝑡−𝛿²𝑊𝑥_𝑖,𝑡≥𝛼𝐹𝐻(𝛿+𝛿𝐵𝑇)−𝐾ℎ²,0≤𝑖≤𝐼, (7.8)which implies that𝑑𝑊𝑥_𝑖,𝑡𝑑𝑡−𝛿²𝑊𝑥_𝑖,𝑡≥0. (7.9)The maximum principle implies that𝑈_𝑖(𝑡)≤𝐻𝛿𝑣(𝑥)+𝛿𝐵𝑇−𝑡₀for𝑡≥𝑡₀,0≤𝑖≤𝐼. (7.10)Hence, we get𝑈_𝑖(𝑡)≤𝐻𝛿𝑣(𝑥),0≤𝑖≤𝐼. (7.11)Therefore 𝑈_𝑖(𝑇)<+∞,0≤𝑖≤𝐼−1, and we have the desired result.

8. Full Discretization

In this section, we consider the problem (1.1) in the case where 𝑎(𝑥,𝑡)=1, 𝑏(𝑡)=1, 𝑓(𝑢)=𝑢^𝑝, 𝑔(𝑢)=𝑢^𝑝 with 𝑝= const >1. Thus our problem is equivalent to𝑢_𝑡(𝑥,𝑡)=𝑢_𝑥𝑥(𝑥,𝑡)−𝑢^𝑝𝑢(𝑥,𝑡),0<𝑥<1,𝑡∈(0,𝑇),_𝑥(0,𝑡)=0,𝑢_𝑥(1,𝑡)=𝑢^𝑝(1,𝑡),𝑡∈(0,𝑇),𝑢(𝑥,0)=𝑢₀(𝑥)>0,0≤𝑥≤1, (8.1)where 𝑝>1, 𝑢₀∈𝐶¹([0,1]), 𝑢^′₀(0)=0 and 𝑢^′₀(1)=𝑢^𝑝₀(1).

We start this section by the construction of an adaptive scheme as follows. Let 𝐼 be a positive integer and let ℎ=1/𝐼. Define the grid 𝑥_𝑖=𝑖ℎ, 0≤𝑖≤𝐼 and approximate the solution 𝑢(𝑥,𝑡) of the problem (8.1) by the solution 𝑈_ℎ^(𝑛)=(𝑈₀^(𝑛),𝑈₁^(𝑛),…,𝑈_𝐼^(𝑛))^𝑇 of the following discrete equations𝛿_𝑡𝑈_𝑖^(𝑛)=𝛿²𝑈_𝑖^(𝑛)−𝑈_𝑖^(𝑛)𝑝𝛿,0≤𝑖≤𝐼−1,_𝑡𝑈_𝐼^(𝑛)=𝛿²𝑈_𝐼^(𝑛)−𝑈_𝐼^(𝑛)𝑝+2ℎ𝑈_𝐼^(𝑛)𝑝,𝑈_𝑖⁽⁰⁾=𝜑_𝑖,0≤𝑖≤𝐼, (8.2)(8.3)(8.4)where 𝑛≥0, 𝜑_𝑖+1≥𝜑_𝑖, 0≤𝑖≤𝐼−1,𝛿_𝑡𝑈_𝑖^(𝑛)=𝑈_𝑖^(𝑛+1)−𝑈_𝑖^(𝑛)Δ𝑡_𝑛,𝛿²𝑈_𝑖^(𝑛)=𝑈^(𝑛)_𝑖+1−2𝑈_𝑖^(𝑛)+𝑈^(𝑛)_𝑖−1ℎ²𝛿,1≤𝑖≤𝐼−1,₂𝑈₀^(𝑛)=2𝑈₁^(𝑛)−2𝑈₀^(𝑛)ℎ²,𝛿²𝑈_𝐼^(𝑛)=2𝑈^(𝑛)_𝐼−1−2𝑈_𝐼^(𝑛)ℎ². (8.5) In order to permit the discrete solution to reproduce the property of the continuous one when the time 𝑡 approaches the blow-up time, we need to adapt the size of the time step so that we take Δ𝑡_𝑛=min{(1−𝑝𝜏)ℎ²/3,𝜏/‖𝑈_ℎ^(𝑛)‖_∞^𝑝−1}, 0<𝜏<1/𝑝.

Let us notice that the restriction on the time step ensures the nonnegativity of the discrete solution. The lemma below shows that the discrete solution is increasing in space.Lemma 8.1. Let 𝑈_ℎ^(𝑛) be the solution of (8.2)–(8.4). Then we have𝑈^(𝑛)_𝑖+1≥𝑈_𝑖^(𝑛),0≤𝑖≤𝐼−1. (8.6)

Proof. Let 𝑍_𝑖^(𝑛)=𝑈^(𝑛)_𝑖+1−𝑈_𝑖^(𝑛), 0≤𝑖≤𝐼−1. We observe that𝑍₀^(𝑛+1)−𝑍₀^(𝑛)Δ𝑡_𝑛=𝑍₁^(𝑛)−3𝑍₀^(𝑛)ℎ²−𝑈₁^(𝑛)𝑝−𝑈₀^(𝑛)𝑝,𝑍_𝑖^(𝑛+1)−𝑍_𝑖^(𝑛)Δ𝑡_𝑛=𝑍^(𝑛)_𝑖+1−2𝑍_𝑖^(𝑛)+𝑍^(𝑛)_𝑖−1ℎ²−𝑈^(𝑛)_𝑖+1^𝑝−𝑈_𝑖^(𝑛)𝑝𝑍,1≤𝑖≤𝐼−2,_{(𝑛+1)𝐼−1}−𝑍^(𝑛)_𝐼−1Δ𝑡_𝑛=𝑍^(𝑛)_𝐼−2−3𝑍^(𝑛)_𝐼−1ℎ²−𝑈_𝐼^(𝑛)𝑝−𝑈^(𝑛)_𝐼−1^𝑝+2ℎ𝑈_𝐼^(𝑛)𝑝. (8.7)Using the Taylor's expansion, we find that𝑍₀^(𝑛+1)=Δ𝑡_𝑛ℎ²𝑍₁^(𝑛)+1−3Δ𝑡_𝑛ℎ²𝑍₀^(𝑛)−Δ𝑡_𝑛𝑝𝜉₀^{(𝑛)𝑝−1}𝑍₀^(𝑛),𝑍_𝑖^(𝑛+1)=Δ𝑡_𝑛ℎ²𝑍^(𝑛)_𝑖+1+1−2Δ𝑡_𝑛ℎ²𝑍_𝑖^(𝑛)+Δ𝑡_𝑛ℎ²𝑍^(𝑛)_𝑖−1−Δ𝑡_𝑛𝑝𝜉_𝑖^{(𝑛)𝑝−1}𝑍_𝑖^(𝑛)𝑍,1≤𝑖≤𝐼−2,_{(𝑛+1)𝐼−1}≥Δ𝑡_𝑛ℎ²𝑍^(𝑛)_𝐼−2+1−3Δ𝑡_𝑛ℎ²𝑍^(𝑛)_𝐼−1−Δ𝑡_𝑛𝑝𝜉^(𝑛)_𝐼−1^𝑝−1𝑍^(𝑛)_𝐼−1, (8.8)where 𝜉_𝑖^(𝑛) is an intermediate value between 𝑈_𝑖^(𝑛) and 𝑈^(𝑛)_𝑖+1. If 𝑍_𝑖^(𝑛)≤0, 0≤𝑖≤𝐼−1, we deduce that𝑍₀^(𝑛+1)≥Δ𝑡_𝑛ℎ²𝑍₁^(𝑛)+1−3Δ𝑡_𝑛ℎ²−Δ𝑡_𝑛𝑝𝑈_ℎ^(𝑛)_∞^𝑝−1𝑍₀^(𝑛),𝑍_𝑖^(𝑛+1)≥Δ𝑡_𝑛ℎ²𝑍^(𝑛)_𝑖+1+1−2Δ𝑡_𝑛ℎ²−Δ𝑡_𝑛𝑝𝑈_ℎ^(𝑛)_∞^𝑝−1𝑍_𝑖^(𝑛)+Δ𝑡_𝑛ℎ²𝑍^(𝑛)_𝑖−1𝑍,1≤𝑖≤𝐼−2,_{(𝑛+1)𝐼−1}≥Δ𝑡_𝑛ℎ²𝑍^(𝑛)_𝐼−2+1−3Δ𝑡_𝑛ℎ²−Δ𝑡_𝑛𝑝𝑈_ℎ^(𝑛)_∞^𝑝−1𝑍^(𝑛)_𝐼−1. (8.9)Using the restriction Δ𝑡_𝑛≤𝜏/‖𝑈_ℎ^(𝑛)‖_∞^𝑝−1, we find that𝑍₀^(𝑛+1)≥Δ𝑡_𝑛ℎ²𝑍₁^(𝑛)+1−3Δ𝑡_𝑛ℎ²−𝑝𝜏𝑍₀^(𝑛),𝑍_𝑖^(𝑛+1)≥Δ𝑡_𝑛ℎ²𝑍^(𝑛)_𝑖+1+1−2Δ𝑡_𝑛ℎ²−𝑝𝜏𝑍_𝑖^(𝑛)+Δ𝑡_𝑛ℎ²𝑍^(𝑛)_𝑖−1𝑍,1≤𝑖≤𝐼−2,_{(𝑛+1)𝐼−1}≥Δ𝑡_𝑛ℎ²𝑍^(𝑛)_𝐼−2+1−3Δ𝑡_𝑛ℎ²−𝑝𝜏𝑍^(𝑛)_𝐼−1. (8.10)We observe that 1−3(Δ𝑡_𝑛/ℎ²)−𝑝𝜏 is nonnegative and by induction, we deduce that 𝑍_𝑖^(𝑛)≤0, 0≤𝑖≤𝐼−1. This ends the proof.

The following lemma is a discrete form of the maximum principle.

Lemma 8.2. Let 𝑎_ℎ^(𝑛) be a bounded vector and let 𝑉_ℎ^(𝑛) a sequence such that𝛿_𝑡𝑉_𝑖^(𝑛)−𝛿²𝑉_𝑖^(𝑛)+𝑎_𝑖^(𝑛)𝑉_𝑖^(𝑛)𝑉≥0,0≤𝑖≤𝐼,𝑛≥0,_𝑖(0)≥0,0≤𝑖≤𝐼. (8.11)(8.12)Then 𝑉_𝑖^(𝑛)≥0 for 𝑛≥0, 0≤𝑖≤𝐼 if Δ𝑡_𝑛≤ℎ²/(2+‖𝑎_ℎ^(𝑛)‖_∞ℎ²).

Proof. If 𝑉_ℎ^(𝑛)≥0 then a routine computation yields𝑉₀^(𝑛+1)≥2Δ𝑡_𝑛ℎ²𝑉₁^(𝑛)+1−2Δ𝑡_𝑛ℎ²−Δ𝑡_𝑛𝑎_ℎ^(𝑛)_∞𝑉₀^(𝑛),𝑉_𝑖^(𝑛+1)≥Δ𝑡_𝑛ℎ²𝑉^(𝑛)_𝑖+1+1−2Δ𝑡_𝑛ℎ²−Δ𝑡_𝑛𝑎_ℎ^(𝑛)_∞𝑉_𝑖^(𝑛)+Δ𝑡_𝑛ℎ²𝑉^(𝑛)_𝑖−1𝑉,1≤𝑖≤𝐼−1,_𝐼(𝑛+1)≥2Δ𝑡_𝑛ℎ²𝑉^(𝑛)_𝐼−1+1−2Δ𝑡_𝑛ℎ²−Δ𝑡_𝑛𝑎_ℎ^(𝑛)_∞𝑉_𝐼^(𝑛). (8.13)Since Δ𝑡_𝑛≤ℎ²/(2+‖𝑎_ℎ^(𝑛)‖_∞ℎ²), we see that 1−2(Δ𝑡_𝑛/ℎ²)−Δ𝑡_𝑛‖𝑎_ℎ^(𝑛)‖_∞ is nonnegative. From (8.12), we deduce by induction that 𝑉_ℎ^(𝑛)≥0 which ends the proof.

A direct consequence of the above result is the following comparison lemma. Its proof is straightforward.

Lemma 8.3. Suppose that 𝑎_ℎ^(𝑛) and 𝑏_ℎ^(𝑛) two vectors such that 𝑎_ℎ^(𝑛) is bounded. Let 𝑉_ℎ^(𝑛) and 𝑊_ℎ^(𝑛) two sequences such that𝛿_𝑡𝑉_𝑖^(𝑛)−𝛿²𝑉_𝑖^(𝑛)+𝑎_𝑖^(𝑛)𝑉_𝑖^(𝑛)+𝑏_𝑖^(𝑛)≤𝛿_𝑡𝑊_𝑖^(𝑛)−𝛿²𝑊_𝑖^(𝑛)+𝑎_𝑖^(𝑛)𝑊_𝑖^(𝑛)+𝑏_𝑖^(𝑛)𝑉,0≤𝑖≤𝐼,𝑛≥0,_𝑖(0)≤𝑊_𝑖⁽⁰⁾,0≤𝑖≤𝐼. (8.14)Then 𝑉_𝑖^(𝑛)≤𝑊_𝑖^(𝑛) for 𝑛≥0, 0≤𝑖≤𝐼 if Δ𝑡_𝑛≤ℎ²/(2+‖𝑎_ℎ^(𝑛)‖_∞ℎ²).

Now, let us give a property of the operator 𝛿_𝑡.

Lemma 8.4. Let 𝑈^(𝑛)∈ℝ be such that 𝑈^(𝑛)≥0 for 𝑛≥0. Then we have𝛿_𝑡𝑈^(𝑛)𝑝≥𝑝𝑈^{(𝑛)𝑝−1}𝛿_𝑡𝑈^(𝑛),𝑛≥0. (8.15)

Proof. From Taylor's expansion, we find that𝛿_𝑡𝑈^(𝑛)𝑝=𝑝𝑈^{(𝑛)𝑝−1}𝛿_𝑡𝑈^(𝑛)+𝑝(𝑝−1)2Δ𝑡_𝑛𝛿_𝑡𝑈^(𝑛)2𝜃^{(𝑛)𝑝−2}, (8.16)where 𝜃^(𝑛) is an intermediate value between 𝑈^(𝑛) and 𝑈^(𝑛+1). Use the fact that 𝑈^(𝑛)≥0 for 𝑛≥0 to complete the rest of the proof.

To handle the phenomenon of blow-up for discrete equations, we need the following definition.

Definition 8.5. We say that the solution 𝑈_ℎ^(𝑛) of (8.2)–(8.4) blows up in a finite time if lim_𝑛→+∞𝑈_ℎ^(𝑛)_∞=+∞,𝑇_ℎ^Δ𝑡=lim_𝑛→∞^𝑛−1_𝑖=0Δ𝑡_𝑖<+∞. (8.17)The number 𝑇_ℎ^Δ𝑡 is called the numerical blow-up time of 𝑈_ℎ^(𝑛).

The following theorem reveals that the discrete solution 𝑈_ℎ^(𝑛) of (8.2)–(8.4) blows up in a finite time under some hypotheses.

Theorem 8.6. Let 𝑈_ℎ^(𝑛) be the solution of (8.2)–(8.4). Suppose that there exists a constant 𝐴∈(0,1] such that the initial data at (8.4) satisfies𝛿²𝜑_𝑖−𝜑^𝑝_𝑖𝛿≥0,0≤𝑖≤𝐼−1.₂𝜑_𝐼−𝜑^𝑝_𝐼+2ℎ𝜑^𝑝_𝐼≥𝐴𝜑^𝑝_𝐼. (8.18)Then 𝑈_ℎ^(𝑛) blows up in a finite time 𝑇_ℎ^Δ𝑡 which satisfies the following estimate𝑇_ℎ^Δ𝑡≤𝜏(1+𝜏^′)^𝑝−1(1+𝜏^′)^𝑝−1−1𝜑_ℎ∞^𝑝−1, (8.19)where 𝜏^′=𝐴min{(1−𝑝𝜏)ℎ²‖𝜑_ℎ‖^−𝑝−1_inf/3,𝜏}.

Proof. Introduce the vector 𝐽_ℎ^(𝑛) defined as follows𝐽_𝑖^(𝑛)=𝛿_𝑡𝑈_𝑖^(𝑛)𝐽,0≤𝑖≤𝐼−1,𝑛≥0,_𝐼(𝑛)=𝛿_𝑡𝑈_𝐼^(𝑛)−𝐴𝑈_𝐼^{(𝑛)−𝑝},𝑛≥0. (8.20) A straightforward computation yields𝛿_𝑡𝐽_𝑖^(𝑛)−𝛿²𝐽_𝑖^(𝑛)=𝛿_𝑡𝛿_𝑡𝑈_𝑖^(𝑛)−𝛿²𝑈_𝑖^(𝑛)𝛿,0≤𝑖≤𝐼−1,_𝑡𝐽_𝐼^(𝑛)−𝛿²𝐽_𝐼^(𝑛)=𝛿_𝑡𝛿_𝑡𝑈_𝐼^(𝑛)−𝛿²𝑈_𝐼^(𝑛)−𝐴𝛿_𝑡𝑈_𝐼^(𝑛)𝑝+𝐴𝛿²𝑈_𝐼^(𝑛)𝑝. (8.21)Using (8.2), we arrive at𝛿_𝑡𝐽_𝑖^(𝑛)−𝛿²𝐽_𝑖^(𝑛)=−𝛿_𝑡𝑈_𝑖^(𝑛)𝑝𝛿,0≤𝑖≤𝐼−1,_𝑡𝐽_𝐼^(𝑛)−𝛿²𝐽_𝐼^(𝑛)=2ℎ−1−𝐴𝛿_𝑡𝑈_𝐼^(𝑛)𝑝+𝐴𝛿²𝑈_𝐼^(𝑛)𝑝. (8.22)Due to the mean value theorem, we get𝛿_𝑡𝑈_𝑖^(𝑛)𝑝=𝑝𝜉_𝑖^{(𝑛)𝑝−1}𝛿_𝑡𝑈_𝑖^(𝑛)=𝑝𝜉_𝑖^{(𝑛)𝑝−1}𝐽_𝑖^(𝑛), (8.23)where 𝜉_𝑖^(𝑛) is an intermediate value between 𝑈_𝑖^(𝑛) and 𝑈^(𝑛)_𝑖+1. On the other hand, from Lemmas 2.4 and 2.5, we deduce that𝛿_𝑡𝐽_𝑖^(𝑛)−𝛿²𝐽_𝑖^(𝑛)=−𝑝𝜉_𝑖^{(𝑛)𝑝−1}𝐽_𝑖^(𝑛)𝛿,0≤𝑖≤𝐼−1,_𝑡𝐽_𝐼^(𝑛)−𝛿²𝐽_𝐼^(𝑛)=2ℎ−1−𝐴𝑝𝑈_𝐼^{(𝑛)𝑝−1}𝛿_𝑡𝑈_𝐼^(𝑛)+𝐴𝑝𝛿_𝑡𝑈_𝐼^{(𝑛)𝑝−1}𝛿²𝑈_𝐼^(𝑛). (8.24)It follows from (8.3) that𝛿_𝑡𝐽_𝐼^(𝑛)−𝛿²𝐽_𝐼^(𝑛)=2ℎ−1𝑝𝑈_𝐼^{(𝑛)𝑝−1}𝛿_𝑡𝑈_𝐼^(𝑛)−𝐴𝑝𝛿_𝑡𝑈_𝐼^{(𝑛)𝑝−1}2ℎ−1𝑈_𝐼^(𝑛)𝑝, (8.25)which implies that𝛿_𝑡𝐽_𝑖^(𝑛)−𝛿²𝐽_𝑖^(𝑛)=−𝑝𝜉_𝑖^{(𝑛)𝑝−1}𝐽_𝑖^(𝑛)𝛿,0≤𝑖≤𝐼−1,_𝑡𝐽_𝐼^(𝑛)−𝛿²𝐽_𝐼^(𝑛)=2ℎ−1𝑝𝑈_𝐼^{(𝑛)𝑝−1}𝐽_𝐼^(𝑛). (8.26)From (8.18), we observe that 𝐽_ℎ⁽⁰⁾≥0. It follows from Lemma 8.2 that 𝐽_ℎ^(𝑛)≥0 which implies that𝑈_𝐼^(𝑛+1)≥𝑈_𝐼^(𝑛)1+𝐴Δ𝑡_𝑛𝑈_𝐼^{(𝑛)𝑝−1}. (8.27)From Lemma 8.1, we see that 𝑈_𝐼^(𝑛)=‖𝑈_ℎ^(𝑛)‖_∞ which implies that𝑈_ℎ^(𝑛+1)_∞≥𝑈_ℎ^(𝑛)_∞1+𝐴Δ𝑡_𝑛𝑈_ℎ^(𝑛)_∞^𝑝−1. (8.28)It is not hard to see that𝐴Δ𝑡_𝑛𝑈_ℎ^(𝑛)_∞^𝑝−1=𝐴min(1−𝑝𝜏)ℎ²𝑈_ℎ^(𝑛)_∞^𝑝−13,𝜏. (8.29)From (8.28), we get ‖𝑈_ℎ^(𝑛+1)‖_∞≥‖𝑈_ℎ^(𝑛)‖_∞. By induction, we arrive at ‖𝑈_ℎ^(𝑛+1)‖_∞≥‖𝑈_ℎ⁽⁰⁾‖_∞=‖𝜑_ℎ‖_∞, which implies that ‖𝑈_ℎ^(𝑛)‖_∞^𝑝−1≥‖𝜑_ℎ‖_∞^𝑝−1. Therefore, we find that𝐴Δ𝑡_𝑛𝑈_ℎ^(𝑛)_∞^𝑝−1≥𝐴min(1−𝑝𝜏)ℎ²𝜑_ℎ∞^𝑝−13,𝜏=𝜏^′. (8.30)Consequently, we arrive at𝑈_ℎ^(𝑛+1)_∞≥𝑈_ℎ^(𝑛)_∞(1+𝜏^′) (8.31)and by induction, we get𝑈_ℎ^(𝑛)_∞≥𝑈_ℎ⁽⁰⁾_∞(1+𝜏^′)^𝑛=𝜑_ℎ∞(1+𝜏^′)^𝑛,𝑛≥0. (8.32)Since the term on the right hand side of the above equality tends to infinity as 𝑛 approaches infinity, we conclude that ‖𝑈_ℎ^(𝑛)‖_∞ tends to infinity as 𝑛 approaches infinity. Now, let us estimate the numerical blow-up time. Due to (8.32), the restriction on the time step ensures thatΣ^+∞_𝑛=0Δ𝑡_𝑛≤Σ^+∞_𝑛=0𝜏𝑈_ℎ^(𝑛)_∞^𝑝−1≤𝜏𝜑_ℎ∞^𝑝−1Σ^+∞_𝑛=01(1+𝜏^′)^𝑝−1𝑛. (8.33)Using the fact that the series on the right hand side of the above inequality converges towards 𝜏(1+𝜏^′)^𝑝−1/((1+𝜏^′)^𝑝−1−1), we deduce that Σ^+∞_𝑛=0Δ𝑡_𝑛≤Σ^+∞_𝑛=0(𝜏(1+𝜏^′)^𝑝−1/((1+𝜏^′)^𝑝−1−1)‖𝜑_ℎ‖_∞^𝑝−1) and the proof is complete.

Remark 8.7. Apply Taylor's expansion to obtain (1+𝜏^′)^𝑝−1=1−(𝑝−1)𝜏^′+𝑜(𝜏^′), which implies that 𝜏(1+𝜏^′)^𝑝−1=𝜏−1𝜏^′1≤𝑝−1+𝑜(1)2𝜏𝜏^′.(𝑝−1) (8.34)If we take 𝜏=ℎ², we see that𝜏𝜏^′=𝐴min1−𝑝ℎ²3𝜑_ℎ∞^𝑝−11,1≥𝐴min4𝜑_ℎ∞^𝑝−1,1. (8.35)We deduce that 𝜏/𝜏^′ is bounded from above. We conclude that 𝜏/((1+𝜏^′)^𝑝−1−1) is bounded from above.

Remark 8.8. From (8.31), we get 𝑈_ℎ^(𝑛)_∞≥𝑈_ℎ^(𝑞)_∞(1+𝜏^′)^𝑛−𝑞for𝑛≥𝑞 (8.36)which implies thatΣ^+∞_𝑛=𝑞Δ𝑡_𝑛≤𝜏𝑈_ℎ^(𝑞)_∞^𝑝−1Σ^+∞_𝑛=𝑞1(1+𝜏^′)^{𝑝−1𝑛−𝑞}. (8.37)We deduce that𝑇_ℎ^Δ𝑡−𝑡_𝑞≤𝜏𝑈_ℎ^(𝑞)_∞^𝑝−1(1+𝜏^′)^𝑝−1(1+𝜏^′)^𝑝−1.−1 (8.38)

In the sequel, we take 𝜏=ℎ².

9. Convergence of the Blow-Up Time

In this section, under some conditions, we show that the discrete solution blows up in a finite time and its numerical blow-up time goes to the real one when the mesh size goes to zero. To start, let us prove a result about the convergence of our scheme.Theorem 9.1. Suppose that the problem (1.1) has a solution 𝑢∈𝐶^4,2([0,1]×[0,𝑇]). Assume that the initial data at (8.4) satisfies𝜑_ℎ−𝑢_ℎ(0)_∞=𝑜(1)asℎ⟶0. (9.1)Then the problem (8.2)–(8.4) has a solution 𝑈_ℎ^(𝑛) for h sufficiently small, 0≤𝑛≤𝐽 and we have the following relationmax_{0≤𝑛≤𝐽}𝑈_ℎ^(𝑛)−𝑢_ℎ(𝑡_𝑛)_∞=𝑂𝜑_ℎ−𝑢_ℎ(0)_∞+ℎ²+Δ𝑡_𝑛asℎ⟶0, (9.2)where 𝐽 is such that ∑^𝐽−1_𝑛=0Δ𝑡_𝑛≤𝑇 and 𝑡_𝑛=∑^𝑛−1_𝑗=0Δ𝑡_𝑗.

Proof. For each ℎ, the problem (8.2)–(8.4) has a solution 𝑈_ℎ^(𝑛). Let 𝑁≤𝐽 be the greatest value of 𝑛 such that𝑈_ℎ^(𝑛)−𝑢_ℎ(𝑡_𝑛)_∞<1for𝑛<𝑁. (9.3)We know that 𝑁≥1 because of (9.1). Due to the fact that 𝑢∈𝐶^4,2, there exists a positive constant 𝐾 such that ‖𝑢‖_∞≤𝐾. Applying the triangle inequality, we have𝑈_ℎ^(𝑛)_∞≤𝑢_ℎ𝑡_𝑛∞+𝑈_ℎ^(𝑛)−𝑢_ℎ𝑡_𝑛∞≤1+𝐾for𝑛<𝑁. (9.4)Since 𝑢∈𝐶^4,2, using Taylor's expansion, we find that𝛿_𝑡𝑢𝑥_𝑖,𝑡_𝑛−𝛿²𝑢𝑥_𝑖,𝑡_𝑛=−𝑢^𝑝𝑥_𝑖,𝑡_𝑛+𝑂ℎ²+𝑂Δ𝑡_𝑛𝛿,0≤𝑖≤𝐼−1,_𝑡𝑢𝑥_𝐼,𝑡_𝑛−𝛿²𝑢𝑥_𝐼,𝑡_𝑛=−𝑢^𝑝𝑥_𝐼,𝑡_𝑛+2ℎ𝑢^𝑝𝑥_𝐼,𝑡_𝑛+𝑂ℎ²+𝑂Δ𝑡_𝑛. (9.5)Let 𝑒_ℎ^(𝑛)=𝑈_ℎ^(𝑛)−𝑢_ℎ(𝑡_𝑛) be the error of discretization. From the mean value theorem, we get𝛿_𝑡𝑒_𝑖^(𝑛)−𝛿²𝑒_𝑖^(𝑛)=−𝑝𝜉_𝑖^{(𝑛)𝑝−1}𝑒_𝑖^(𝑛)+𝑂ℎ²+𝑂Δ𝑡_𝑛𝛿,0≤𝑖≤𝐼−1,_𝑡𝑒_𝐼^(𝑛)−𝛿²𝑒_𝐼^(𝑛)2=𝑝ℎ−1𝜉_𝐼^{(𝑛)𝑝−1}𝑒_𝐼^(𝑛)+𝑂ℎ²+𝑂Δ𝑡_𝑛, (9.6) where 𝜉_𝑖^(𝑛) is an intermediate value between 𝑢(𝑥_𝑖,𝑡_𝑛) and 𝑈_𝑖^(𝑛). Hence, there exist positive constants 𝐿 and 𝐾 such that𝛿_𝑡𝑒_𝑖^(𝑛)−𝛿²𝑒_𝑖^(𝑛)≤−𝑝𝜉_𝑖^{(𝑛)𝑝−1}𝑒_𝑖^(𝑛)+𝐿ℎ²+𝐿Δ𝑡_𝑛𝛿,0≤𝑖≤𝐼−1,𝑛<𝑁,_𝑡𝑒_𝐼^(𝑛)−𝛿²𝑒_𝐼^(𝑛)2≤𝑝ℎ−1𝜉_𝐼^{(𝑛)𝑝−1}𝑒_𝐼^(𝑛)+𝐿ℎ²+𝐿Δ𝑡_𝑛,𝑛<𝑁. (9.7)Consider the function 𝑍(𝑥,𝑡)=𝑒^{((𝑀+1)𝑡+𝐶𝑥2)}(‖𝜑_ℎ−𝑢_ℎ(0)‖_∞+𝑄ℎ²+𝑄Δ𝑡_𝑛) where 𝑀, 𝐶, 𝑄 are positive constants which will be determined later. We get𝑍_𝑡(𝑥,𝑡)−𝑍_𝑥𝑥(𝑥,𝑡)=𝑀+1−2𝐶−4𝐶²𝑥²𝑍𝑍(𝑥,𝑡),_𝑥(0,𝑡)=0,𝑍_𝑥(1,𝑡)=2𝐶𝑍(1,𝑡),𝑍(𝑥,0)=𝑒^(𝐶𝑥2)𝜑_ℎ−𝑢_ℎ(0)_∞+𝑄ℎ²+𝑄Δ𝑡_𝑛. (9.8)By a discretization of the above problem, we obtain𝛿_𝑡𝑍𝑥_𝑖,𝑡_𝑛−𝛿²𝑍𝑥_𝑖,𝑡_𝑛=𝑀+1−2𝐶−4𝐶²𝑥²_𝑖𝑍𝑥_𝑖,𝑡_𝑛+ℎ²𝑍12^{𝑥𝑥𝑥𝑥}𝑥_𝑖,𝑡_𝑛−Δ𝑡_𝑛2𝑍_𝑡𝑡𝑥_𝑖,𝑡_𝑛,𝛿_𝑡𝑍𝑥_𝐼,𝑡_𝑛−𝛿²𝑍𝑥_𝐼,𝑡_𝑛=𝑀+1−2𝐶−4𝐶²𝑥²_𝐼𝑍𝑥_𝐼,𝑡_𝑛+4𝐶ℎ𝑍𝑥_𝐼,𝑡_𝑛+ℎ²𝑍12^{𝑥𝑥𝑥𝑥}𝑥_𝐼,𝑡_𝑛−Δ𝑡_𝑛2𝑍_𝑡𝑡𝑥_𝐼,𝑡_𝑛. (9.9)We may choose 𝑀, 𝐶, 𝑄 large enough that𝛿_𝑡𝑍𝑥_𝑖,𝑡_𝑛−𝛿²𝑍𝑥_𝑖,𝑡_𝑛>−𝑝𝜉_𝑖^{(𝑛)𝑝−1}𝑍𝑥_𝑖,𝑡_𝑛+𝐿ℎ²+𝐿Δ𝑡_𝑛𝛿,0≤𝑖≤𝐼−1,_𝑡𝑍𝑥_𝐼,𝑡_𝑛−𝛿²𝑍𝑥_𝐼,𝑡_𝑛2>𝑝ℎ−1𝜉_𝐼^{(𝑛)𝑝−1}𝑍𝑥_𝐼,𝑡_𝑛+𝐿ℎ²+𝐿Δ𝑡_𝑛,𝑍_𝑖⁽⁰⁾>𝑒_𝑖⁽⁰⁾,0≤𝑖≤𝐼. (9.10)It follows from Comparison Lemma 8.3 that𝑍𝑥_𝑖,𝑡_𝑛>𝑒_𝑖^(𝑛),0≤𝑖≤𝐼,𝑛<𝑁. (9.11)By the same way, we also prove that𝑍𝑥_𝑖,𝑡_𝑛>−𝑒_𝑖^(𝑛),0≤𝑖≤𝐼,𝑛<𝑁, (9.12)which implies that𝑈_ℎ^(𝑛)−𝑢_ℎ(𝑡)_∞≤𝑒^{(𝑀𝑡𝑛+𝐶)}𝜑_ℎ−𝑢_ℎ(0)_∞+𝑄ℎ²+𝑄Δ𝑡_𝑛,𝑛<𝑁. (9.13)Let us show that 𝑁=𝐽. Suppose that 𝑁<𝐽. From (9.3), we obtain1≤𝑈_ℎ^(𝑁)−𝑢_ℎ𝑡_𝑁∞≤𝑒^{(𝑀𝑇+𝐶)}𝜑_ℎ−𝑢_ℎ(0)_∞+𝑄ℎ²+𝑄Δ𝑡_𝑛. (9.14)Since the term on the right hand side of the second inequality goes to zero as ℎ goes to zero, we deduce that 1≤0, which is a contradiction and the proof is complete.

Now, we are in a position to state the main theorem of this section.Theorem 9.2. Suppose that the problem (1.1) has a solution 𝑢 which blows up in a finite time 𝑇₀ and 𝑢∈𝐶^4,2([0,1]×[0,𝑇₀)). Assume that the initial data at (2.3) satisfies𝜑_ℎ−𝑢_ℎ(0)_∞=𝑜(1)asℎ⟶0. (9.15)Under the assumption of Theorem 8.6, the problem (8.2)–(8.4) has a solution 𝑈_ℎ^(𝑛) which blows up in a finite time 𝑇_ℎ^Δ𝑡 and the following relation holdslim_ℎ→0𝑇_ℎ^Δ𝑡=𝑇₀. (9.16)

Proof. We know from Remark 8.7 that 𝜏(1+𝜏^′)/((1+𝜏^′)^𝑝−1−1) is bounded. Letting 𝜀>0, there exists a constant 𝑅>0 such that𝜏(1+𝜏^′)^𝑝−1𝑥^𝑝−1(1+𝜏^′)^𝑝−1<𝜀−12for𝑥∈[𝑅,∞). (9.17)Since 𝑢 blows up at the time 𝑇₀, there exists 𝑇₁∈(𝑇₀−𝜀/2,𝑇₀) such that ‖𝑢(⋅,𝑡)‖_∞≥2𝑅 for 𝑡∈[𝑇₁,𝑇₀). Let 𝑇₂=(𝑇₁+𝑇₀)/2 and let 𝑞 be a positive integer such that 𝑡_𝑞=∑^𝑞−1_𝑛=0Δ𝑡_𝑛∈[𝑇₁,𝑇₂] for ℎ small enough. We have 0<‖𝑢_ℎ(𝑡_𝑛)‖_∞<∞ for 𝑛≤𝑞. It follows from Theorem 4.3 that the problem (2.1)–(2.3) has a solution 𝑈_ℎ^(𝑛) which obeys ‖𝑈_ℎ^(𝑛)−𝑢_ℎ(𝑡_𝑛)‖_∞<𝑅 for 𝑛≤𝑞, which implies that𝑈_ℎ^(𝑞)_∞≥𝑢_ℎ𝑡_𝑞∞−𝑈_ℎ^(𝑞)−𝑢_ℎ𝑡_𝑞∞≥𝑅. (9.18)From Theorem 8.6, 𝑈_ℎ^(𝑛) blows up at the time 𝑇_ℎ^Δ𝑡. It follows from Remark 8.8 and (9.17) that |𝑇_ℎ^Δ𝑡−𝑡_𝑞|≤𝜏(1+𝜏^′)^𝑝−1‖𝑈_ℎ^(𝑞)‖_∞^1−𝑝/((1+𝜏^′)^𝑝−1−1)<𝜀/2 because ‖𝑈_ℎ^(𝑞)‖_∞≥𝑅. We deduce that |𝑇₀−𝑇_ℎ^Δ𝑡|≤|𝑇₀−𝑡_𝑞|+|𝑡_𝑞−𝑇_ℎ^Δ𝑡|≤𝜀/2+𝜀/2≤𝜀, which leads us to the result.

10. Numerical Experiments

In this section, we present some numerical approximations to the blow-up time of (1.1) in the case where 𝑎(𝑥,𝑡)=𝜆>0, 𝑓(𝑢)=𝑢^𝑝, 𝑔(𝑢)=𝑢^𝑞, 𝑏(𝑡)=1 with 𝑝= const >1, 𝑞= const >1. We approximate the solution 𝑢 of (1.1) by the solution 𝑈_ℎ^(𝑛) of the following explicit scheme𝛿_𝑡𝑈_𝑖^(𝑛)=𝛿²𝑈_𝑖^(𝑛)−𝜆𝑈_𝑖^{(𝑛)𝑝−1}𝑈_𝑖^(𝑛+1)𝛿,0≤𝑖≤𝐼−1,_𝑡𝑈_𝐼^(𝑛)=𝛿²𝑈_𝐼^(𝑛)+2ℎ𝑈_𝐼^(𝑛)𝑞−𝜆𝑈_𝐼^{(𝑛)𝑝−1}𝑈_𝐼^(𝑛+1),𝑈_𝑖⁽⁰⁾=𝜑_𝑖≥0,0≤𝑖≤𝐼, (10.1)We also approximate the solution 𝑢 of (1.1) by the solution 𝑈_ℎ^(𝑛) of the implicit scheme below𝛿_𝑡𝑈_𝑖^(𝑛)=𝛿²𝑈_𝑖^(𝑛+1)−𝜆𝑈_𝑖^{(𝑛)𝑝−1}𝑈_𝑖^(𝑛+1)𝛿,0≤𝑖≤𝐼−1,_𝑡𝑈_𝐼^(𝑛)=𝛿²𝑈_𝐼^(𝑛+1)+2ℎ𝑈_𝐼^(𝑛)𝑞−𝜆𝑈_𝐼^{(𝑛)𝑝−1}𝑈_𝐼^(𝑛+1),𝑈_𝑖⁽⁰⁾=𝜑_𝑖≥0,0≤𝑖≤𝐼. (10.2)For the time step, we take 𝑛≥0, Δ𝑡_𝑛=min(ℎ²/2,𝜏‖𝑈_ℎ^(𝑛)‖_∞^1−𝑝) for the explicit scheme and Δ𝑡_𝑛=𝜏‖𝑈_ℎ^(𝑛)‖_∞^1−𝑝 for the implicit scheme.

The problem described in (10.1) may be rewritten as follows𝑈₀^(𝑛+1)=2Δ𝑡_𝑛/ℎ²𝑈₁^(𝑛)+1−2Δ𝑡_𝑛/ℎ²𝑈₀^(𝑛)1+𝜆Δ𝑡_𝑛𝑈₀^{(𝑛)𝑝−1},𝑈_𝑖^(𝑛+1)=2Δ𝑡_𝑛/ℎ²𝑈^(𝑛)_𝑖+1+1−2Δ𝑡_𝑛/ℎ²𝑈_𝑖^(𝑛)+2Δ𝑡_𝑛/ℎ²𝑈^(𝑛)_𝑖−11+𝜆Δ𝑡_𝑛𝑈_𝑖^{(𝑛)𝑝−1},𝑈_𝐼^(𝑛+1)=2Δ𝑡_𝑛/ℎ²𝑈^(𝑛)_𝐼−1+1−2Δ𝑡_𝑛/ℎ²𝑈_𝐼^(𝑛)+2Δ𝑡_𝑛/ℎ²𝑈_𝐼^(𝑛)𝑞1+𝜆Δ𝑡_𝑛𝑈_𝐼^{(𝑛)𝑝−1}. (10.3)Let us notice that the restriction on the time step Δ𝑡_𝑛≤ℎ²/2 ensures the nonnegativity of the discrete solution.

The implicit scheme may be rewritten in the following form𝐴^𝑛_ℎ𝑈_ℎ^(𝑛+1)=𝐹^𝑛, (10.4)where𝐴_ℎ^(𝑛)=𝑎₀𝑏₀𝑐0⋯0₁𝑎₁𝑏₁0⋯0⋱⋱⋱⋮⋱⋱⋱𝑏_𝐼−10⋯0𝑐_𝐼𝑎_𝐼,𝑎_𝑖=1+2Δ𝑡_𝑛ℎ²+𝜆Δ𝑡_𝑛𝑈_𝑖^{(𝑛)𝑝−1}𝑏,0≤𝑖≤𝐼,_𝑖=−2Δ𝑡_𝑛ℎ²𝑐,𝑖=0,…,𝐼−1,_𝑖=−2Δ𝑡_𝑛ℎ²,𝑖=1,…,𝐼,(𝐹^𝑛)_𝑖=𝑈_𝑖^(𝑛),𝑖=0,…,𝐼−1,(𝐹^𝑛)_𝐼=𝑈_𝐼^(𝑛)+2ℎΔ𝑡_𝑛𝑈_𝐼^(𝑛)𝑞. (10.5) The matrix 𝐴_ℎ^(𝑛) satisfies the following properties𝐴_ℎ^(𝑛)_𝑖𝑖>0,𝐴_ℎ^(𝑛)_𝑖𝑗𝐴<0,if𝑖≠𝑗,_{ℎ(𝑛)𝑖𝑖}>∑_𝑗≠𝑖𝐴_ℎ^(𝑛)_𝑖𝑗. (10.6) It follows that 𝑈_ℎ^(𝑛) exists for 𝑛≥0. In addition, since 𝑈_ℎ⁽⁰⁾ is nonnegative, 𝑈_ℎ^(𝑛) is also nonnegative for 𝑛≥0. We need the following definition.

Definition 10.1. We say that the discrete solution 𝑈_ℎ^(𝑛) of the explicit scheme or the implicit scheme blows up in a finite time if lim_𝑛→+∞‖𝑈_ℎ^(𝑛)‖_∞=+∞ and the series ∑^+∞_𝑛=0Δ𝑡_𝑛 converges. The quantity ∑^+∞_𝑛=0Δ𝑡_𝑛 is called the numerical blow-up time of the solution 𝑈_ℎ^(𝑛).

In Tables 1, 2, 3, 4, 5, 6, 7, and 8, in rows, we present the numerical blow-up times, values of 𝑛, the CPU times and the orders of the approximations corresponding to meshes of 16, 32, 64, 128, 256. For the numerical blow-up time we take 𝑇^𝑛=∑^𝑛−1_𝑗=0Δ𝑡_𝑗 which is computed at the first time whenΔ𝑡_𝑛=𝑇^𝑛+1−𝑇^𝑛≤10⁻¹⁶. (10.7)The order (𝑠) of the method is computed from𝑇𝑠=log_4ℎ−𝑇_2ℎ/𝑇_2ℎ−𝑇_ℎ.log(2) (10.8)

Table 1 

Numerical blow-up times, numbers of iterations, CPU times (seconds) and orders of the approximations obtained with the explicit Euler method defined in (10.1).

Table 2 

Numerical blow-up times, numbers of iterations, CPU times (seconds) and orders of the approximations obtained with the implicit Euler method defined in (10.2).

Table 3 

Numerical blow-up times, numbers of iterations, CPU times (seconds) and orders of the approximations obtained with the explicit Euler method defined in (10.1).

Table 4 

Numerical blow-up times, numbers of iterations, CPU times (seconds) and orders of the approximations obtained with the implicit Euler method defined in (10.2).

Table 5 

Numerical blow-up times, numbers of iterations, CPU times (seconds) and orders of the approximations obtained with the explicit Euler method defined in (10.1).

Table 6 

Numerical blow-up times, numbers of iterations, CPU times (seconds) and orders of the approximations obtained with the implicit Euler method defined in (10.2).

Table 7 

Numerical blow-up times, numbers of iterations, CPU times (seconds) and orders of the approximations obtained with the explicit Euler method defined in (8.2)–(8.4).

Table 8 

Numerical blow-up times, numbers of iterations, CPU times (seconds) and orders of the approximations obtained with the implicit Euler method defined in (10.2).

Case 1. 𝑝=0, 𝑞=2, 𝜑_𝑖=10+10∗cos(𝜋𝑖ℎ), 𝜆=1.

Case 2. 𝑝=2, 𝑞=4, 𝜑_𝑖=10+10∗cos(𝜋𝑖ℎ), 𝜆=1.

Case 3. 𝑝=2, 𝑞=3, 𝜑_𝑖=10+10∗cos(𝜋𝑖ℎ), 𝜆=1.

Case 4. 𝑝=2, 𝑞=2, 𝜑_𝑖=10+10∗cos(𝜋𝑖ℎ), 𝜆=1.

Remark 10.2. The different cases of our numerical results show that there is a relationship between the flow on the boundary and the absorption in the interior of the domain. Indeed, when there is not an absorption on the interior of the domain, we see that the blow-up time is slightly equal to 0.043 for 𝑞=2 whereas if there is an absorption in the interior of the domain, we observe that the blow-up time is slightly equal to 0.048 for 𝑞=2 and 𝑝=2. We see that there is a diminution of the blow-up time. We also remark that if the power of flow on the boundary increases then the blow-up time diminishes. Thus the flow on the boundary make blow-up occurs whereas the absorption in the interior of domain prevents the blow-up. This phenomenon is well known in a theoretical point of view.

For other illustrations, in what follows, we give some plots to illustrate our analysis. In Figures 1, 2, 3, 4, 5, and 6, we can appreciate that the discrete solution blows up in a finite time at the last node.

Figure 1 

Evolution of the discrete solution, 𝑞=2, 𝑝=2 (explicit scheme).

Figure 2 

Evolution of the discrete solution, 𝑞=2, 𝑝=2 (implicit scheme).

Figure 3 

Evolution of the discrete solution, 𝑞=3, 𝑝=2 (explicit scheme).

Figure 4 

Evolution of the discrete solution, 𝑞=3, 𝑝=2 (implicit scheme).

Figure 5 

Evolution of the discrete solution, 𝑞=4, 𝑝=2 (explicit scheme).

Figure 6 

Evolution of the discrete solution, 𝑞=4, 𝑝=2 (implicit scheme).

Acknowledgments

We want to thank the anonymous referee for the throughout reading of the manuscript and several suggestions that help us to improve the presentation of the paper.