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

We obtain some conditions under which the positive solution for semidiscretizations of the semilinear equation ut uxx − a x, t f u , 0 < x < 1, t ∈ 0, T , with boundary conditions ux 0, t 0, ux 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.


Introduction
In this paper, we consider the following boundary value problem: 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.

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 U 0 t , U 1 t , . . ., U I t T of the following semidiscrete equations where

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 .

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.

3.1
Then we have Proof.Let T 0 < T and define the vector Z h t e λt V h t where λ is large enough that a i t −λ > 0 for t ∈ 0, T 0 , 0 ≤ i ≤ I. Let m min 0≤i≤I, 0≤t≤T 0 Z i t .Since for i ∈ {0, . . ., I}, Z i t is a continuous function, there exists t 0 ∈ 0, T 0 such that m Z i 0 t 0 for a certain i 0 ∈ {0, . . ., I}.

3.2
A straightforward computation reveals that We observe from 3.2 that a i 0 t 0 − λ Z i 0 t 0 ≥ 0 which implies that Z i 0 t 0 ≥ 0 because a i 0 t 0 − λ > 0. We deduce that V h t ≥ 0 for t ∈ 0, T 0 and the proof is complete.
Another form of the maximum principle for semidiscrete equations is the following comparison lemma.

3.5
Then we have Proof.Define the vector Z h t U h t − V h t .Let t 0 be the first t ∈ 0, T such that Z i t > 0 for t ∈ 0, t 0 , 0 ≤ i ≤ I, but Z i 0 t 0 0 for a certain i 0 ∈ {0, . . ., I}.We observe that which implies that But this inequality contradicts 3.4 and the proof is complete.

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 δ 2 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 have 4.1 Proof.Apply Taylor's expansion to obtain 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 that

4.3
Combining the second and third equalities, we see that Use the fact that g s ≥ 0 for s ≥ 0 and U h ≥ 0 to complete the rest of the proof.
Proof.Let t 0 be the first t > 0 such that U i 1 t > U i t for 0 ≤ i ≤ I − 1 but U i 0 1 t 0 U i 0 t 0 for a certain i 0 ∈ {0, . . ., I − 1}.Without loss of generality, we may suppose that i 0 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 get which implies that

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 that Assume that f s g s − f s g s ≥ 0 for s ≥ 0. 4.9 Then the solution U h blows up in a finite time T h b and we have the following estimate 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 that A straightforward calculation gives

4.12
From Lemma 4.1, we have Using 2.1 , we get

4.14
It follows from the fact that

4.15
We deduce from 4.9 that

4.16
From 4.8 , we observe that 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 have Integrating this inequality over t, T h b , we arrive at which implies that 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
where

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 satisfies where u h t u x 0 , t , . . ., u x I , t T .Then, for h sufficiently small, the problem The problem 2.1 -2.3 has for each h, a unique solution The relation 5.1 implies that t h > 0 for h sufficiently small.By the triangle inequality, we obtain which implies that

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 , 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 that

5.8
Consider the function z x, t e M 1 t Cx 2 ϕ h −u h 0 ∞ Qh 2 where M, C, Q are constants which will be determined later.We get 5.9 By a semidiscretization of the above problem, we may choose M, C, Q large enough that

5.10
It follows from Lemma 3.2 that z x i , t > e i t for t ∈ 0, t h , 0 ≤ i ≤ I.

5.11
By the same way, we also prove that 12 which implies that z x i , t > e i t for t ∈ 0, t h , 0 ≤ i ≤ I.

5.13
We deduce that Let us show that t h T .Suppose that T > t h .From 5.4 , we obtain 1

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 satisfies 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 relation 5.17 Proof.Let ε > 0. There exists a positive constant N such that Since the solution u blows up at the time T b , then there exists

5.19
Applying the triangle inequality, we get We deduce from Remark 4.4 and 5.18 that and the proof is complete.

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.
which leads us to the result.

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 C 0 such that sF s ≤ C 0 and Assume that there exists a positive constant C such Then the numerical blow-up set is B {1}.
where δ is small enough.We have and for t ≥ t 0 , we get This implies that there exists α > 0 such that W t x, t − W xx x, t ≥ αF H δ δBT .

7.7
Using Taylor's expansion, there exists a constant K > 0 such that The maximum principle implies that Hence, we get Therefore U i T < ∞, 0 ≤ i ≤ I − 1, and we have the desired result.

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 to where p > 1, u 0 ∈ C 1 0, 1 , u 0 0 0 and u 0 1 u p 0 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 where n ≥ 0,

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 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.

8.7
Using the Taylor's expansion, we find that

15
where 8.9 Using the restriction

8.10
We observe that 1 − 3 Δt n /h 2 − pτ is nonnegative and by induction, we deduce that Z n i ≤ 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 n h be a bounded vector and let
8.14 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 have Proof.From Taylor's expansion, we find that 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 n h of 8.2 -8.4 blows up in a finite time if 8.17 The number T Δt h is called the numerical blow-up time of U n h .
The following theorem reveals that the discrete solution U n h of 8.2 -8.4 blows up in a finite time under some hypotheses.Theorem 8.6.Let U n h 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 satisfies

8.18
Then U n h blows up in a finite time T Δt h which satisfies the following estimate 19 Proof.Introduce the vector J n h defined as follows

8.21
Using 8.2 , we arrive at

8.22
Due to the mean value theorem, we get where and U n i 1 .On the other hand, from Lemmas 2.4 and 2.5, we deduce that

18
Journal of Applied Mathematics It follows from 8.3 that

8.26
From 8.18 , we observe that J 0 h ≥ 0. It follows from Lemma 8.2 that J n h ≥ 0 which implies that

8.27
From Lemma 8.1, we see that 8.28 It is not hard to see that

8.29
From 8.28 , we get ∞ .Therefore, we find that

8.30
Consequently, we arrive at and by induction, we get Since the term on the right hand side of the above equality tends to infinity as n approaches infinity, we conclude that U n h ∞ 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 that

8.33
Using the fact that the series on the right hand side of the above inequality converges towards τ and the proof is complete.
Remark 8.8.From 8.31 , we get for n ≥ q 8.36 which implies that We deduce that In the sequel, we take τ h 2 .

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 satisfies Then the problem 8.2 -8.4 has a solution U n h for h sufficiently small, 0 ≤ n ≤ J and we have the following relation where J is such that Proof.For each h, the problem 8.2 -8.4 has a solution U n h .Let N ≤ J be the greatest value of n such that 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 have Since u ∈ C 4,2 , using Taylor's expansion, we find that U n h − u h t n be the error of discretization.From the mean value theorem, we get where ξ n i is an intermediate value between u x i , t n and U n i .Hence, there exist positive constants L and K such that Lh 2 LΔt n , n < N.

9.7
Consider the function Z x, t e M 1 t Cx 2 ϕ h − u h 0 ∞ Qh 2 QΔt n where M, C, Q are positive constants which will be determined later.We get

9.8
By a discretization of the above problem, we obtain 9.9 We may choose M, C, Q large enough that By the same way, we also prove that 12 which implies that Let us show that N J. Suppose that N < J. From 9.3 , we obtain 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.

Journal of Applied Mathematics
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 T 0 and u ∈ C 4,2 0, 1 × 0, T 0 .Assume that the initial data at 2.3 satisfies 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 that Since u blows up at the time T 0 , there exists T 1 T 0 /2 and let q be a positive integer such that t q q−1 n 0 Δt n ∈ T 1 , T 2 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 n h which obeys which leads us to the result.

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 n h of the following explicit scheme

23
We also approximate the solution u of 1.1 by the solution U n h of the implicit scheme below

10.2
For the time step, we take n ≥ 0, for the explicit scheme and Δt n τ U n h 1−p ∞ for the implicit scheme.The problem described in 10.1 may be rewritten as follows

10.3
Let us notice that the restriction on the time step Δt n ≤ h 2 /2 ensures the nonnegativity of the discrete solution.
The implicit scheme may be rewritten in the following form where 10.5 is also nonnegative for n ≥ 0. We need the following definition.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 0 Δt j which is computed at the first time when The order s of the method is computed from Case 1. p 0, 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.

Lemma 8 . 1 .
Let U n h be the solution of 8.2 -8.4 .Then we have

9 . 10 It
follows from Comparison Lemma 8.3 that Z x i , t n > e n i , 0 ≤ i ≤ I, n < N. 9.11

Definition 10 . 1 .∞ and the series ∞ n 0
We say that the discrete solution U n h of the explicit scheme or the implicit scheme blows up in a finite time if lim n → ∞ U n h ∞ Δt n converges.The quantity ∞ n 0 Δt n is called the numerical blow-up time of the solution U n h .

6 Journal of Applied Mathematics Lemma 4.2. Let
U h be the solution of 2.1 -2.3 .Then we have 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 C 1 such that ∞ h 2 , we see that 1 − 2 Δt n /h 2 − Δt n a

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 .