Journal of Applied Mathematics Volume 2008, Article ID 753518,29pages doi:10.1155/2008/753518

*Research Article*

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

**Louis A. Assal ´e,**^{1}**Th ´eodore K. Boni,**^{1}**and Diabate Nabongo**^{2}

*1**Institut National Polytechnique Houphou¨et-Boigny de Yamoussoukro, BP 1093,*
*Yamoussoukro, Cote D’Ivoire*

*2**D´epartement de Math´ematiques et Informatiques, Universit´e d’Abobo-Adjam´e, UFR-SFA,*
*16 BP 372 Abidjan 16, Cote D’Ivoire*

Correspondence should be addressed to Diabate Nabongo,nabongo diabate@yahoo.fr Received 29 April 2008; Revised 15 December 2008; Accepted 29 December 2008 Recommended by Jacek Rokicki

We obtain some conditions under which the positive solution for semidiscretizations of the
semilinear equation*u**t* *u**xx*−*ax, tfu,* 0 *< x <* 1, t ∈ 0, T, with boundary conditions
*u**x*0, t 0,*u**x*1, t *btgu1, 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.

Copyrightq2008 Louis A. Assal´e et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**1. Introduction**

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

*u** _{t}*−

*u*

*−ax, tfu, 0*

_{xx}*< x <*1, t∈0, T,

*u*

*0, t 0,*

_{x}*u*

*1, t*

_{x}*btg*

*u1, t*

*,* *t*∈0, T,
*ux,*0 *u*0x≥0, 0≤*x*≤1,

1.1

where*f* : 0,∞ → 0,∞is a*C*^{1} function,*f0 * 0, *g* : 0,∞ → 0,∞is a*C*^{1} convex
function,*g0 * 0,*a* ∈ *C*^{0}0,1×R,*ax, t* ≥ 0 in0,1×R,*a** _{t}*x, t ≤ 0 in0,1×R,

*b*∈

*C*

^{1}R,

*bt>*0 inR,

*b*

^{}t≥0 inR. The initial data

*u*0 ∈

*C*

^{2}0,1,

*u*

^{}

_{0}0 0,

*u*

^{}

_{0}1

*b1gu*01.

Here0, Tis the maximal time interval on which the solution*u*of1.1exists. 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, namely

*t→*lim*T**u·, t*

∞ ∞, 1.2

whereu·, t∞max_{0≤x≤1}|ux, 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 functiong*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 authorssee 1–7, and the references cited therein.

The authors have proved that under some assumptions, the solution of1.1blows 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 of1.1 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 in8,9where the authors have considered semilinear heat equations
with Dirichlet boundary conditions. In the same way in10 the numerical extinction has
been studied using some discrete and semidiscrete schemesa 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 authorssee11–15. Generally, the authors have considered the problem1.1in
the case where*ax, t * 0 and *bt * 1. For instance in15, the above problem has been
considered in the case where*ax, t * 0 and*bt * 1. In16, the authors have considered
the problem1.1in the case where*ax, t λ >*0,*bt *1,*fu u** ^{p}*,

*gu u*

*. 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 in17–22similar studies concerning other parabolic problems.*

^{q}The paper is organized as follows. In the next section, we present a semidiscrete scheme of1.1. InSection 3, we give some properties concerning our semidiscrete scheme. In Section 4, under some conditions, we prove that the solution of the semidiscrete form of1.1 blows up in a finite time and estimate its semidiscrete blow-up time. InSection 5, we study the convergence of the semidiscrete blow-up time. InSection 6, we give some results on the numerical blow-up rate andSection 7is consecrated to the study of the numerical blow-up

set. InSection 8, we study a particular discrete form of1.1. Finally, in the last section, taking some discrete forms of1.1, 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*of1.1by the solution*U** _{h}*t U0t, U1t, . . . , U

*I*t

*of the following semidiscrete equations*

^{T}*dU** _{i}*t

*dt* −*δ*^{2}*U** _{i}*t −a

*i*tf

*U*

*t*

_{i}*,* 0≤*i*≤*I*−1, t∈
0, T_{b}^{h}

*,* 2.1

*dU** _{I}*t

*dt* −*δ*^{2}*U**I*t 2
*hbtg*

*U**I*t

−*a**I*tf
*U**I*t

*,* *t*∈
0, T_{b}^{h}

*,* 2.2

*U** _{i}*0

*ϕ*

*≥0, 0≤*

_{i}*i*≤

*I,*2.3

where*ϕ** _{i1}*≥

*ϕ*

*i*, 0≤

*i*≤

*I*−1,

*δ*^{2}*U*0t 2U_{1}t−2U_{0}t

*h*^{2} *,* *δ*^{2}*U**I*t 2U* _{I−1}*t−2U

*t*

_{I}*h*^{2} *,*

*δ*^{2}*U** _{i}*t

*U*

*t−2U*

_{i1}*t*

_{i}*U*

*t*

_{i−1}*h*^{2} *.*

2.4

Here 0, T_{b}* ^{h}* is the maximal time interval on which U

*h*t∞ is finite where U

*h*t∞ max

_{0≤i≤I}

*U*

*t. When*

_{i}*T*

_{b}*is finite, we say that the solution*

^{h}*U*

*t blows up in a finite time and the time*

_{h}*T*

_{b}*is called the blow-up time of the solution*

^{h}*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∈

*C*

^{0}0, T,R

^{I1}*and letV*

*t∈*

_{h}*C*

^{1}0, T,R

^{I1}*such that*

*dV*

*t*

_{i}*dt* −*δ*^{2}*V** _{i}*t

*a*

*tV*

_{i}*i*t≥0, 0≤

*i*≤

*I, t*∈0, T,

*V*

*i*0≥0, 0≤

*i*≤

*I.*

3.1

*Then we haveV**i*t≥*0, 0*≤*i*≤*I,t*∈0, T.

*Proof. LetT*_{0} *< T*and define the vector*Z** _{h}*t

*e*

^{λt}*V*

*twhere*

_{h}*λ*is large enough that

*a*

*t−λ >*

_{i}0 for *t* ∈ 0, T0, 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, T0such that*mZ*_{i}_{0}t0for a certain*i*_{0} ∈ {0, . . . , I}.

It is not hard to see that

*dZ*_{i}_{0}
*t*_{0}
*dt* lim

*k→*0

*Z*_{i}_{0}
*t*_{0}

−*Z*_{i}_{0}
*t*_{0}−*k*

*k* ≤0,

*δ*^{2}*Z*_{i}_{0}
*t*_{0}

*Z**i*01
*t*0

−2Z*i*0

*t*0

*Z**i*0−1
*t*0

*h*^{2} ≥0 if 1≤*i*_{0} ≤*I*−1,
*δ*^{2}*Z*_{i}_{0}

*t*_{0}
2Z1

*t*0

−2Z0

*t*0

*h*^{2} ≥0 if*i*_{0}0,
*δ*^{2}*Z**i*0

*t*0

2Z_{I−1}*t*_{0}

−2Z_{I}*t*_{0}

*h*^{2} ≥0 if*i*0*I.*

3.2

A straightforward computation reveals that

*dZ**i*0

*t*0

*dt* −*δ*^{2}*Z*_{i}_{0}
*t*_{0}

*a*_{i}_{0}

*t*_{0}

−*λ*
*Z*_{i}_{0}

*t*_{0}

≥0. 3.3

We observe from 3.2 thata*i*0t0−*λZ**i*0t0 ≥ 0 which implies that*Z**i*0t0 ≥ 0 because
*a**i*0t0−*λ >*0. We deduce that*V**h*t≥0 for*t*∈0, T0and 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∈

*C*

^{1}0, T,R

^{I1}*andf*∈

*C*

^{0}R×R,R

*such that fort*∈0, T

*dV*

*t*

_{i}*dt* −*δ*^{2}*V**i*t *f*
*V**i*t, t

*<* *dU** _{i}*t

*dt* −*δ*^{2}*U**i*t *f*

*U**i*t, t

*,* 0≤*i*≤*I,* 3.4

*V** _{i}*0

*< U*

*0, 0≤*

_{i}*i*≤

*I.*3.5

*Then we haveV**i*t*< U**i*t, 0≤*i*≤*I,t*∈0, T.

*Proof. Define the vectorZ** _{h}*t

*U*

*t−*

_{h}*V*

*t. Let*

_{h}*t*

_{0}be the first

*t*∈0, Tsuch that

*Z*

*t*

_{i}*>*0 for

*t*∈0, t0, 0≤

*i*≤

*I, butZ*

_{i}_{0}t0 0 for a certain

*i*

_{0}∈ {0, . . . , I}. We observe that

*dZ*_{i}_{0}
*t*_{0}
*dt* lim

*k→*0

*Z*_{i}_{0}
*t*_{0}

−*Z*_{i}_{0}
*t*_{0}−*k*

*k* ≤0,

*δ*^{2}*Z*_{i}_{0}
*t*_{0}

*Z**i*01
*t*0

−2Z*i*0

*t*0

*Z**i*0−1
*t*0

*h*^{2} ≥0 if 1≤*i*_{0} ≤*I*−1,
*δ*^{2}*Z*_{i}_{0}

*t*_{0}
2Z1

*t*0

−2Z0

*t*0

*h*^{2} ≥0 if*i*_{0}0,
*δ*^{2}*Z**i*0

*t*0

2Z_{I−1}*t*_{0}

−2Z_{I}*t*_{0}

*h*^{2} ≥0 if*i*0*I,*

3.6

which implies that
*dZ*_{i}_{0}

*t*_{0}
*dt* −*δ*^{2}*Z**i*0

*t*0

*f*
*U**i*0

*t*0

*, t*0

−*f*
*V**i*0

*t*0

*, t*0

≤0. 3.7

But this inequality contradicts3.4and the proof is complete.

**4. Semidiscrete blow-up solutions**

In this section under some assumptions, we show that the solution*U**h* of2.1–2.3blows
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

^{I1}*be such thatU*

*≥*

_{h}*0. Then we have*

*δ*

^{2}

*g*

*U*_{i}

≥*g*^{}
*U*_{i}

*δ*^{2}*U*_{i}*for 0*≤*i*≤*I.* 4.1

*Proof. Apply Taylor’s expansion to obtain*

*g*
*U*_{1}

*g*
*U*_{0}

*U*_{1}−*U*_{0}
*g*^{}

*U*_{0}

*U*1−*U*0

_{2}

2 *g*^{}

*η*_{0}
*,*
*g*

*U*_{i1}*g*

*U*_{i}

*U** _{i1}*−

*U*

_{i}*g*

^{}

*U*_{i}

*U** _{i1}*−

*U*

*i*

_{2}

2 *g*^{}

*θ*_{i}

*,* 1≤*i*≤*I*−1,
*g*

*U*_{i−1}*g*

*U*_{i}

*U** _{i−1}*−

*U*

_{i}*g*

^{}

*U*_{i}

*U** _{i−1}*−

*U*

*i*

_{2}

2 *g*^{}

*η*_{i}

*,* 1≤*i*≤*I*−1,
*g*

*U*_{I−1}*g*

*U*_{I}

*U** _{I−1}*−

*U*

_{I}*g*

^{}

*U*_{I}

*U** _{I−1}*−

*U*

_{I}_{2}

2 *g*^{}

*η*_{I}*,*

4.2

where*θ** _{i}* is an intermediate between

*U*

*and*

_{i}*U*

*and*

_{i1}*η*

*the one between*

_{i}*U*

*and*

_{i−1}*U*

*. The first and last equalities imply that*

_{i}*δ*^{2}*g*
*U*0

*g*^{}
*U*0

*δ*^{2}*U*0

*U*_{1}−*U*_{0}2

*h*^{2} *g*^{}
*η*0

*,*

*δ*^{2}*g*
*U**I*

*g*^{}
*U**I*

*δ*^{2}*U**I*

*U** _{I−1}*−

*U*

*2*

_{I}*h*^{2} *g*^{}
*η**I*

*.*

4.3

Combining the second and third equalities, we see that

*δ*^{2}*g*
*U**i*

*g*^{}
*U**i*

*δ*^{2}*U**i*

*U** _{i1}*−

*U*

_{i}_{2}2h

^{2}

*g*

^{}

*θ**i*

*U** _{i−1}*−

*U*

_{i}_{2}2h

^{2}

*g*

^{}

*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 have
*U** _{i1}*t

*> U*

*t, 0≤*

_{i}*i*≤

*I*−1, t∈

0, T_{b}^{h}

*.* 4.5

*Proof. Lett*_{0}be the first*t >*0 such that*U** _{i1}*t

*> U*

*tfor 0≤*

_{i}*i*≤

*I*−1 but

*U*

_{i}_{0}

_{1}t0

*U*

_{i}_{0}t0 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*

*t−*

_{i1}*U*

*i*tfor 0≤

*i*≤

*I*−1. We get

*dZ*_{i}_{0}
*t*_{0}
*dt* lim

*k→*0

*Z*_{i}_{0}
*t*_{0}

−*Z*_{i}_{0}
*t*_{0}−*k*

*k* ≤0,

*δ*^{2}*Z*_{i}_{0}
*t*_{0}

*Z**i*01
*t*0

−2Z*i*0

*t*0

*Z**i*0−1
*t*0

*h*^{2} *>*0 if 1≤*i*_{0} ≤*I*−2,
*δ*^{2}*Z*_{i}_{0}

*t*_{0}

*δ*^{2}*Z*_{0}
*t*_{0}

*Z*1

*t*0

−3Z0

*t*0

*h*^{2} *>*0 if*i*_{0} 0,
*δ*^{2}*Z**i*0

*t*0

*δ*^{2}*Z*_{I−1}*t*0

*Z*_{I−2}*t*_{0}

−3Z_{I−1}*t*_{0}

*h*^{2} *>*0 if*i*0*I*−1,

4.6

which implies that
*dZ*_{i}_{0}

*t*_{0}
*dt* −*δ*^{2}*Z**i*0

*t*0

−*a**i*01
*t*0

*f*
*U**i*01t0

*a*_{i}_{0}

*t*_{0}
*f*

*U*_{i}_{0}
*t*_{0}

*<*0 if 0≤*i*_{0}≤*I*−2,
*dZ*_{i}_{0}

*t*_{0}

*dt* −*δ*^{2}*Z*_{i}_{0}
*t*_{0}

2
*hb*

*t*_{0}
*g*_{i}_{0}_{1}

*t*_{0}

−*a*_{i}_{0}_{1}
*t*_{0}

*f*
*U*_{i}_{0}_{1}t0

*a*_{i}_{0}

*t*_{0}
*f*

*U*_{i}_{0}
*t*_{0}

*<*0 if*i*_{0}*I*−1.

4.7

But this contradicts2.1-2.2and 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*

*δ*^{2}*ϕ** _{i}*−

*a*

*0f*

_{i}*ϕ*

_{i}≥0, 1≤*i*≤*I*−1,
*δ*^{2}*ϕ**I*−*a**I*0f

*ϕ**I*

*b0g**I*

*ϕ**I*

≥*Ag*
*ϕ**I*

*.* 4.8

*Assume that*

*fsg*^{}s−*f*^{}sgs≥*0 fors*≥0. 4.9

*Then the solutionU**h**blows up in a finite timeT*_{b}^{h}*and we have the following estimate*

*T*_{b}* ^{h}*≤ 1

*A*

_{∞}

ϕ*h*∞

*ds*

*gs.* 4.10

*Proof. Since*0, T_{b}* ^{h}*is the maximal time interval on whichU

*h*t

_{∞}

*<*∞, our aim is to show that

*T*

_{b}*is finite and satisfies the above inequality. Introduce the vector*

^{h}*J*

*h*such that

*J** _{i}*t

*dU*

*t*

_{i}*dt* *,* 0≤*i*≤*I*−1, *J** _{I}*t

*dU*

*t*

_{I}*dt*−

*Ag*

*U** _{I}*t

*.* 4.11

A straightforward calculation gives
*dJ**i*

*dt* −*δ*^{2}*J**i* *d*
*dt*

*dU**i*

*dt* −*δ*^{2}*U**i*

*,* 0≤*i*≤*I*−1,
*dJ**I*

*dt* −*δ*^{2}*J**I* *d*
*dt*

*dU**I*

*dt* −*δ*^{2}*U**I*

−*Ag*^{}
*U**I*

*dU**I*

*dt* *Aδ*^{2}*g*
*U**I*

*.*

4.12

FromLemma 4.1, we have*δ*^{2}*gU**I*≥*g*^{}U*I*δ^{2}*U** _{I}* which implies that

*dJ*

*I*

*dt* −*δ*^{2}*J**I* ≥ *d*
*dt*

*dU**I*

*dt* −*δ*^{2}*U**I*

−*Ag*^{}
*U**I*

dU_{I}*dt* −*δ*^{2}*U**I*

*.* 4.13

Using2.1, we get
*dJ**i*

*dt* −*δ*^{2}*J** _{i}* ≥ −a

^{}

*tf*

_{i}*U*

_{i}−*a** _{i}*tf

^{}

*U*_{i}*dU**i*

*dt* *,* 0≤*i*≤*I*−1,
*dJ**I*

*dt* −*δ*^{2}*J**I* ≥ −a^{}* _{I}*tf

*U*

*I*

−*a**I*tf^{}
*U**I*

*dU**I*

*dt* 2
*hb*^{}tg

*U**I*

2

*hbtg*^{}

*U*_{I}*dU*_{I}

*dt* −*Ag*^{}
*U*_{I}

−*a** _{I}*tf

*U*

_{I} 2
*hbtg*

*U*_{I}*.*

4.14

It follows from the fact that*a*^{}* _{i}*t≤0,

*b*

^{}t≥0 and

*dU*

*i*

*/dtJ*

*i*

*AgU*

*i*that

*dJ*

*I*

*dt* −*δ*^{2}*J**I* ≥

−*a**I*tf^{}
*U**I*

2
*hbtg*^{}

*U**I*

*J**I**Aa**I*t
*g*^{}

*U**I*

*f*
*U**I*

−*f*^{}
*U**I*

*g*
*U**I*

*.*
4.15

We deduce from4.9that
*dJ*_{i}

*dt* −*δ*^{2}*J** _{i}*≥ −a

*i*tf

^{}

*U*

_{i}*J*_{i}*,* 0≤*i*≤*I*−1,
*dJ*_{I}

*dt* −*δ*^{2}*J** _{I}* ≥

−*a** _{I}*tf

^{}

*U*

_{I} 2
*hbtg*^{}

*U*_{I}*J*_{I}*.*

4.16

From4.8, we observe that

*J** _{i}*0

*δ*

^{2}

*ϕ*

*−*

_{i}*a*

*0f*

_{i}*ϕ*

_{i}≥0, 0≤*i*≤*I*−1,
*J** _{I}*0

*δ*

^{2}

*ϕ*

*−*

_{I}*a*

*0f*

_{I}*ϕ*_{I}

*b0g**I*

*ϕ*_{I}

−*Ag*
*ϕ*_{I}

≥0. 4.17

We deduce fromLemma 3.1that*J**i*t ≥ 0, 0 ≤ *i* ≤ *I, which implies that* *dU**I**/dt* ≥ *gU**I*,
0≤*i*≤*I. Obviously we have*

*dU*_{I}*g*

*U** _{I}* ≥

*A dt.*4.18

Integrating this inequality overt, T_{b}* ^{h}*, we arrive at

*T*_{b}* ^{h}*−

*t*≤ 1

*A*

_{∞}

*U**I*t

*ds*

*gs,* 4.19

which implies that

*T*_{b}* ^{h}*≤ 1

*A*

_{∞}

U*h*0∞

*ds*

*gs.* 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 thatU*h*0∞ϕ*h*∞to complete the rest
of the proof.

*Remark 4.4. The inequality*4.19implies that

*T*_{b}* ^{h}*−

*t*0≤ 1

*A*

_{∞}

U*h*t0∞

*ds*

*gs* if 0*< t*0 *< T*_{b}^{h}*,*
*U** _{i}*t≤

*H*

*A*

*T*_{b}* ^{h}*−

*t*

*,* 0≤*i*≤*I,*

4.21

where*Hs*is the inverse of*Gs *_{∞}

*s* dz/gz.

*Remark 4.5. Ifgs s** ^{q}*, then

*Gs s*

^{1−q}

*/q*−1and

*Hs q*−1s

^{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 interval0, Twhere*u*is defined, the solution*U**h*tof2.1–

2.3approximates*u, when the mesh parameterh*goes to zero.

* Theorem 5.1. Assume that*1.1

*has a solutionu*∈

*C*

^{4,1}0,1×0, T

*and the initial condition at*2.3

*satisfies*

*ϕ**h*−*u**h*0

∞*o1* *as* *h*−→0, 5.1

*whereu**h*t ux0*, t, . . . , ux**I**, t*^{T}*. Then, for h suﬃciently small, the problem*2.1–2.3*has a*
*unique solutionU** _{h}*∈

*C*

^{1}0, T,R

^{I1}*such that*

max0≤t≤T*U** _{h}*t−

*u*

*t*

_{h}∞*Oϕ** _{h}*−

*u*

*0*

_{h}∞*h*^{2}

*ash*−→0. 5.2

*Proof. Letα >*0 be such that

*u·, t*

∞≤*α* for*t*∈0, T. 5.3

The problem2.1–2.3has for each*h, a unique solutionU** _{h}* ∈

*C*

^{1}0, T

_{b}*,R*

^{h}*. Let*

^{I1}*th*≤ min{T, T

_{b}*}the greatest value of*

^{h}*t >*0 such that

*U**h*t−*u**h*t

∞*<*1 for *t*∈
0, th

*.* 5.4

The relation5.1implies that*th>*0 for*h*suﬃciently small. By the triangle inequality, we
obtain

*U** _{h}*t

_{∞}≤

*u·, t*

_{∞}

*U*

*t−*

_{h}*u*

*t*

_{h}_{∞}for

*t*∈ 0, th

*,* 5.5

which implies that

*U** _{h}*t

_{∞}≤1

*α*for

*t*∈ 0, th

*.* 5.6

Let*e** _{h}*t

*U*

*t−*

_{h}*u*

*tbe the error of discretization. Using Taylor’s expansion, we have for*

_{h}*t*∈0, th,

*de** _{i}*t

*dt* −*δ*^{2}*e**i*t −a*i*tf^{}
*ξ**i*t

*e**i*t *o*
*h*^{2}

*,* 0≤*i*≤*I*−1,
*de** _{I}*t

*dt* −*δ*^{2}*e**I*t −a*I*tf^{}
*ξ**I*t

*e**I*t 2
*hbtg*^{}

*U**I*t

*e**I*t *o*
*h*^{2}

*,*

5.7

where*θ** _{I}*tis an intermediate value between

*U*

*tand*

_{I}*ux*

*I*

*, t*and

*ξ*

*t the one between*

_{i}*U*

*i*tand

*ux*

*i*

*, t. Using*5.3and5.6, there exist two positive constants

*K*and

*L*such that

*de** _{i}*t

*dt* −*δ*^{2}*e** _{i}*t≤

*L*

*e*

*t*

_{i}*Kh*

^{2}

*,*0≤

*i*≤

*I*−1,

*de*

*t*

_{I}*dt* −

2e* _{I−1}*t−2e

*t*

_{I}*h*^{2} ≤ *L* *e** _{I}*t

*h* *L* *e**I*t *Kh*^{2}*.*

5.8

Consider the function*zx, t e*^{M1tCx}^{2}^{}ϕ*h*−*u**h*0∞Qh^{2}where*M,C,Q*are constants
which will be determined later. We get

*z** _{t}*x, t−

*z*

*x, t M1−2C−4C*

_{xx}^{2}

*x*

^{2}zx, t,

*z*

*x*0, t 0,

*z*

*x*1, t 2Cz1, t,

*zx,*0

*e*

^{Cx}^{2}ϕ

*h*−

*u*

*0*

_{h}_{∞}

*Qh*

^{2}.

5.9

By a semidiscretization of the above problem, we may choose*M,C,Q*large enough that
*d*

*dtz*
*x*_{i}*, t*

*> δ*^{2}*z*
*x*_{i}*, t*

*L* *z*

*x*_{i}*, t* *Kh*^{2}*,* 0≤*i*≤*I*−1,
*d*

*dtz*
*x**I**, t*

*> δ*^{2}*z*
*x**I**, t*

*L*
*h* *z*

*x**I**, t* *L* *z*

*x**I**, t* *Kh*^{2}*,*
*z*

*x*_{i}*,*0

*> e** _{i}*0, 0≤

*i*≤

*I.*

5.10

It follows fromLemma 3.2that
*z*

*x*_{i}*, t*

*> e** _{i}*t for

*t*∈ 0, th

*,* 0≤*i*≤*I.* 5.11

By the same way, we also prove that
*z*

*x**i**, t*

*>*−e*i*t for*t*∈
0, th

*,* 0≤*i*≤*I,* 5.12

which implies that

*z*
*x*_{i}*, t*

*>* *e** _{i}*t for

*t*∈ 0, th

*,* 0≤*i*≤*I.* 5.13

We deduce that

*U**h*t−*u**h*t

∞≤*e*^{MtC}*ϕ**h*−*u**h*0

∞*Qh*^{2}
*,* *t*∈

0, th

*.* 5.14

Let us show that*th T*. Suppose that*T > th. From*5.4, we obtain
1*U*_{h}

*th*

−*u*_{h}

*th*

∞≤*e*^{MTC}*ϕ** _{h}*−

*u*

*0*

_{h}∞*Qh*^{2}

*.* 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*th 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 timeT*

*b*

*such*

*thatu*∈

*C*

^{4,1}0,1×0, T

*b*

*and the initial condition at*2.3

*satisfies*

*ϕ**h*−*u**h*0

∞*o1* *as* *h*−→0. 5.16

*Under the assumptions ofTheorem 4.3, the problem*2.1–2.3*admits a unique solutionU**h* *which*
*blows up in a finite timeT*_{b}^{h}*and we have the following relation*

*h→*lim0*T*_{b}^{h}*T*_{b}*.* 5.17

*Proof. Letε >*0. There exists a positive constant*N*such that
1

*A*
_{∞}

*x*

*ds*
*gs* ≤ *ε*

2 for*x*∈N,∞. 5.18

Since the solution*u*blows up at the time*T** _{b}*, then there exists

*T*

_{1}∈ T

*b*−

*ε/2, T*

*such that u·, t∞≥2Nfor*

_{b}*t*∈T1

*, T*

*b*. Setting

*T*2 T1

*T*

*b*/2, then we have sup

_{t∈0,T}_{2}

_{}|ux, t|

*<*∞.

It follows fromTheorem 5.1that sup

*t∈0,T*2

*U** _{h}*t−

*u*

*t*

_{h}∞≤*N.* 5.19

Applying the triangle inequality, we get
*U**h*

*T*2

∞≥*u**h*

*T*2

∞−*U**h*

*T*2

−*u**h*

*T*2

∞*,* 5.20

which leads toU*h*T2∞≥*N. From*Theorem 4.3,*U**h*tblows up at the time*T*_{b}* ^{h}*. We deduce
fromRemark 4.4and5.18that

*T** _{b}*−

*T*

_{b}*≤*

^{h}*T*

*−*

_{b}*T*

_{2}

*T*

_{b}*−*

^{h}*T*

_{2}≤

*ε*2 1

*A*
_{∞}

U*h*T2∞

*ds*

*gs* ≤*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}* of2.1–2.3in the case
where

*bt*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 timeT*_{b}^{h}*and there exist two positive constantsC*_{1}*, C*_{2}*such that*

*H*
*C*_{1}

*T*_{b}* ^{h}*−

*t*

≤*U** _{I}*t≤

*H*

*C*

_{2}

*T*_{b}* ^{h}*−

*t*

*,* *fort*∈
0, T_{b}^{h}

*,* 6.1

*whereHsis the inverse of the functionGs *_{∞}

*s* dσ/gσ.

*Proof. From*Theorem 4.3andRemark 4.4,*U**h*tblows up in a finite time*T*_{b}* ^{h}*and there exists
a constant

*C*

_{2}

*>*0 such that

*U**I*t≤*H*
*C*2

*T*_{b}* ^{h}*−

*t*

for*t*∈
0, T_{b}^{h}

*.* 6.2

FromLemma 4.2,*U*_{I−1}*< U**I*. Then using2.2, we deduce that*dU**I**/dt* ≤2/hbtgU*I*−
*a** _{I}*tfU

*I*, which implies that

*dU*

_{I}*/dt*≤ 2bt/hgU

*I*. Integration this inequality over t, T

_{b}*, there exists a positive constant*

^{h}*C*

_{1}such that

*U**I*t≥*H*
*C*1

*T*_{b}* ^{h}*−

*t*

for*t*∈
0, T_{b}^{h}

*,* 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**C_{0}*such thatsF*^{}s≤*C*_{0}*and*
*d*

*dtU** _{i}*−

*δ*

^{2}

*U*

*≤0, 0≤*

_{i}*i*≤

*I*−1. 7.1

*Assume that there exists a positive constantCsuch*

*U**i*≤*H*

*CT*−*t*

*,* 0≤*i*≤*I.* 7.2

*Then the numerical blow-up set isB*{1}.

*Proof. Letvx *1−*x*^{2}and define
*Wx, t H*

*δvx δBT*−*t*

for 0≤*x*≤1, t≥*t*_{0}*,* 7.3
where*δ*is small enough. We have

*W** _{x}*0, t 0,

*W1, t H*

*δBT*−*t*

≥*u1, t,* 7.4

and for*t*≥*t*_{0}, we get

*Wx, t*0 *H*

*δvx δ*

≥*H2δ H*
2δB

*T*−*t*0

≥*H*
*C*

*T*−*t*0

≥*u*
*x, t*0

*.* 7.5

A straightforward computation yields
*W** _{t}*x, t−

*W*

*x, t*

_{xx}*δF*

*Hτ*

*B*−2−4xF^{}

*Hτ*

≥*δF*

*Hτ*

*B*−2−4δC_{0}

*.* 7.6

This implies that there exists*α >*0 such that

*W**t*x, t−*W**xx*x, t≥*αF*

*HδδBT*

*.* 7.7

Using Taylor’s expansion, there exists a constant*K >*0 such that
*d*

*dtW*
*x*_{i}*, t*

−*δ*^{2}*W*
*x*_{i}*, t*

≥*αF*

*HδδBT*

−*Kh*^{2}*,* 0≤*i*≤*I,* 7.8

which implies that

*dW*
*x**i**, t*

*dt* −*δ*^{2}*W*
*x*_{i}*, t*

≥0. 7.9

The maximum principle implies that

*U** _{i}*t≤

*H*

*δvx δB*

*T*−*t*_{0}

for*t*≥*t*_{0}*,* 0≤*i*≤*I.* 7.10

Hence, we get

*U** _{i}*t≤

*H*

*δvx*

*,* 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 problem1.1in the case where*ax, t *1,*bt *1,*fu u** ^{p}*,

*gu u*

*with*

^{p}*p*const

*>*1. Thus our problem is equivalent to

*u** _{t}*x, t

*u*

*x, t−*

_{xx}*u*

*x, t, 0*

^{p}*< x <*1, t∈0, T,

*u*

*0, t 0,*

_{x}*u*

*1, t*

_{x}*u*

*1, t,*

^{p}*t*∈0, T,

*ux,*0 *u*_{0}x*>*0, 0≤*x*≤1,

8.1

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
*ux, t* of the problem 8.1 by the solution *U*^{n}* _{h}* U

^{n}

_{0}

*, U*

^{n}

_{1}

*, . . . , U*

_{I}^{n}

*of the following discrete equations*

^{T}*δ*_{t}*U*^{n}_{i}*δ*^{2}*U*_{i}^{n}−
*U*_{i}^{n}*p*

*,* 0≤*i*≤*I*−1, 8.2

*δ**t**U*^{n}_{I}*δ*^{2}*U*^{n}* _{I}* −

*U*

^{n}

_{I}*p*

2
*h*

*U*^{n}_{I}*p*

*,* 8.3

*U*_{i}^{0}*ϕ**i**,* 0≤*i*≤*I,* 8.4

where*n*≥0,*ϕ** _{i1}*≥

*ϕ*

*i*, 0≤

*i*≤

*I*−1,

*δ*_{t}*U*^{n}_{i}*U*^{n1}* _{i}* −

*U*

^{n}

*Δt*

_{i}*n*

*,*
*δ*^{2}*U*^{n}_{i}*U*^{n}* _{i1}*−2U

^{n}

_{i}*U*

^{n}

_{i−1}*h*^{2} *,* 1≤*i*≤*I*−1,
*δ*^{2}*U*^{n}_{0} 2U_{1}^{n}−2U^{n}_{0}

*h*^{2} *,* *δ*^{2}*U*_{I}^{n} 2U^{n}* _{I−1}*−2U

_{I}^{n}

*h*^{2} *.*

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
takeΔt*n*min{1−*pτ*h^{2}*/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^{n}_{h}*be the solution of* 8.2–8.4. Then we have

*U*^{n}* _{i1}*≥

*U*

^{n}

_{i}*,*0≤

*i*≤

*I*−1. 8.6

*Proof. LetZ*^{n}_{i}*U*_{i1}^{n}−*U*^{n}* _{i}* , 0≤

*i*≤

*I*−1. We observe that

*Z*^{n1}_{0} −*Z*^{n}_{0}

Δt*n* *Z*_{1}^{n}−3Z_{0}^{n}

*h*^{2} −

*U*_{1}^{n}_{p}

−
*U*_{0}^{n}^{p}

*,*
*Z*_{i}^{n1}−*Z*_{i}^{n}

Δt*n* *Z*^{n}* _{i1}*−2Z

^{n}

_{i}*Z*

_{i−1}^{n}

*h*^{2} −

*U*^{n}_{i1}*p*−
*U*_{i}^{n}^{p}

*,* 1≤*i*≤*I*−2,
*Z*^{n1}* _{I−1}* −

*Z*

^{n}

_{I−1}Δt*n* *Z*^{n}* _{I−2}*−3Z

_{I−1}^{n}

*h*^{2} −

*U*_{I}^{n}_{p}

−

*U*^{n}_{I−1}* ^{p}*
2

*h*
*U*_{I}^{n}^{p}

*.*

8.7

Using the Taylor’s expansion, we find that

*Z*_{0}^{n1} Δt*n*

*h*^{2} *Z*_{1}^{n}

1−3Δt*n*

*h*^{2}

*Z*^{n}_{0} −Δt*n**p*
*ξ*^{n}_{0} *p−1*

*Z*^{n}_{0} *,*
*Z*_{i}^{n1} Δt*n*

*h*^{2} *Z*_{i1}^{n}

1−2Δt*n*

*h*^{2}

*Z*^{n}* _{i}* Δt

*n*

*h*^{2} *Z*_{i−1}^{n}

−Δt*n**p*
*ξ*^{n}_{i}*p−1*

*Z*^{n}_{i}*,* 1≤*i*≤*I*−2,
*Z*_{I−1}^{n1}≥ Δt*n*

*h*^{2} *Z*_{I−2}^{n}

1−3Δt*n*

*h*^{2}

*Z*_{I−1}^{n} −Δt*n**p*
*ξ*^{n}_{I−1}*p−1*

*Z*_{I−1}^{n}*,*

8.8

where*ξ*^{n}* _{i}* is an intermediate value between

*U*

_{i}^{n}and

*U*

^{n}

*. If*

_{i1}*Z*

^{n}

*≤0, 0≤*

_{i}*i*≤

*I*−1, we deduce that

*Z*_{0}^{n1}≥ Δt*n*

*h*^{2} *Z*_{1}^{n}

1−3Δt*n*

*h*^{2} −Δt*n**pU*^{n}_{h}^{p−1}

∞

*Z*^{n}_{0} *,*
*Z*_{i}^{n1}≥ Δt*n*

*h*^{2} *Z*_{i1}^{n}

1−2Δt*n*

*h*^{2} −Δt*n**pU*^{n}_{h}^{p−1}

∞

*Z*^{n}* _{i}*
Δt

*n*

*h*^{2} *Z*_{i−1}^{n}*,* 1≤*i*≤*I*−2,
*Z*_{I−1}^{n1}≥ Δt*n*

*h*^{2} *Z*_{I−2}^{n}

1−3Δt*n*

*h*^{2} −Δt*n**pU*_{h}^{n}^{p−1}_{∞}
*Z*_{I−1}^{n}*.*

8.9

Using the restrictionΔt*n*≤*τ/U*_{h}^{n}^{p−1}_{∞} , we find that
*Z*_{0}^{n1} ≥ Δt*n*

*h*^{2} *Z*^{n}_{1}

1−3Δt*n*

*h*^{2} −*pτ*

*Z*^{n}_{0} *,*
*Z*_{i}^{n1} ≥ Δt*n*

*h*^{2} *Z*^{n}_{i1}

1−2Δt*n*

*h*^{2} −*pτ*

*Z*^{n}* _{i}*
Δt

*n*

*h*^{2} *Z*_{i−1}^{n}*,* 1≤*i*≤*I*−2,
*Z*_{I−1}^{n1} ≥ Δt*n*

*h*^{2} *Z*^{n}_{I−2}

1−3Δt*n*

*h*^{2} −*pτ*

*Z*^{n}_{I−1}*.*

8.10

We observe that 1−3Δt*n**/h*^{2}−*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^{n}_{h}*be a bounded vector and letV*_{h}^{n}*a sequence such that*

*δ*_{t}*V*_{i}^{n}−*δ*^{2}*V*_{i}^{n}*a*^{n}_{i}*V*_{i}^{n}≥0, 0≤*i*≤*I, n*≥0, 8.11

*V*_{i}^{0} ≥0, 0≤*i*≤*I.* 8.12

*ThenV*_{i}^{n}≥*0 forn*≥*0, 0*≤*i*≤*Iif*Δt*n*≤*h*^{2}*/2*a^{n}* _{h}* ∞

*h*

^{2}.

*Proof. IfV*

_{h}^{n}≥0 then a routine computation yields

*V*_{0}^{n1}≥ 2Δt*n*

*h*^{2} *V*_{1}^{n}

1−2Δt*n*

*h*^{2} −Δt*n**a*^{n}_{h}

∞

*V*_{0}^{n}*,*
*V*_{i}^{n1}≥ Δt*n*

*h*^{2} *V*_{i1}^{n}

1−2Δt*n*

*h*^{2} −Δt*n**a*^{n}_{h}

∞

*V*_{i}^{n}
Δt*n*

*h*^{2} *V*_{i−1}^{n}*,* 1≤*i*≤*I*−1,
*V*_{I}^{n1}≥ 2Δt*n*

*h*^{2} *V*_{I−1}^{n}

1−2Δt*n*

*h*^{2} −Δt*n**a*^{n}_{h}

∞

*V*_{I}^{n}*.*

8.13