1. Introduction

In this paper, we consider the following boundary value problem:where is a function, ,   is a convex function, , , in , in , , in , in . The initial data , , .

Here is the maximal time interval on which the solution of (1.1) exists. The time may be finite or infinite. Where is infinite, we say that the solution exists globally. When is finite, the solution develops a singularity in a finite time, namelywhere .

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

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 stands for the temperature, and the heat sources are prescribed on the boundaries. At the boundary , the heat source has a constant flux whereas at the boundary , the heat source has a nonlinear radition haw. Intensification of the heat source at the boundary is provided by the function . The function also gives a dominant strength of the heat source at the boundary .

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 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 and . For instance in [15], the above problem has been considered in the case where and . In [16], the authors have considered the problem (1.1) in the case where , , , . 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 be a positive integer and define the grid , , where . We approximate the solution of (1.1) by the solution of the following semidiscrete equationswhere , ,Here is the maximal time interval on which is finite where . When is finite, we say that the solution blows up in a finite time and the time is called the blow-up time of the solution .

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 and let such that Then we have , , .

Proof. Let and define the vector where is large enough that for , . Let . Since for , is a continuous function, there exists such that for a certain . It is not hard to see thatA straightforward computation reveals thatWe observe from (3.2) that which implies that because . We deduce that for and the proof is complete.

Another form of the maximum principle for semidiscrete equations is the following comparison lemma.Lemma 3.2. Let , and such that for Then we have , ,

Proof. Define the vector . Let be the first such that for , , but for a certain . We observe that which implies thatBut 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 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 and the second one reveals a property of the semidiscrete solution.Lemma 4.1. Let be such that . Then we have

Proof. Apply Taylor's expansion to obtainwhere is an intermediate between and and the one between and . The first and last equalities imply thatCombining the second and third equalities, we see thatUse the fact that for and to complete the rest of the proof.

Lemma 4.2. Let be the solution of (2.1)–(2.3). Then we have

Proof. Let be the first such that for but for a certain . Without loss of generality, we may suppose that is the smallest integer which satisfies the equality. Introduce the functions for . We getwhich implies thatBut 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 .

Now, we are in a position to state the main result of this section.Theorem 4.3. Let be the solution of (2.1)–(2.3). Suppose that there exists a positive integer A such that Assume that Then the solution blows up in a finite time and we have the following estimate

Proof. Since is the maximal time interval on which , our aim is to show that is finite and satisfies the above inequality. Introduce the vector such thatA straightforward calculation givesFrom Lemma 4.1, we have which implies thatUsing (2.1), we getIt follows from the fact that , and thatWe deduce from (4.9) thatFrom (4.8), we observe that We deduce from Lemma 3.1 that , , which implies that , . Obviously we haveIntegrating this inequality over , we arrive atwhich implies thatSince the quantity on the right hand side of the above inequality is finite, we deduce that the solution blows up in a finite time. Use the fact that to complete the rest of the proof.

Remark 4.4. The inequality (4.19) implies that where is the inverse of .

Remark 4.5. If , then and .

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 where is defined, the solution of (2.1)–(2.3) approximates , when the mesh parameter goes to zero.Theorem 5.1. Assume that (1.1) has a solution and the initial condition at (2.3) satisfies where . Then, for h sufficiently small, the problem (2.1)–(2.3) has a unique solution such that

Proof. Let be such thatThe problem (2.1)–(2.3) has for each , a unique solution . Let the greatest value of such thatThe relation (5.1) implies that for sufficiently small. By the triangle inequality, we obtainwhich implies thatLet be the error of discretization. Using Taylor's expansion, we have for ,where is an intermediate value between and and the one between and