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

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,¹Th ´eodore K. Boni,¹and Diabate Nabongo²

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

2D´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 [email protected] 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 equationut uxx−ax, tfu, 0 < x < 1, t ∈ 0, T, with boundary conditions ux0, t 0,ux1, 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_xx −ax, tfu, 0< x <1, t∈0, T, u_x0, t 0, u_x1, t btg

u1, t

, t∈0, T, ux,0 u0x≥0, 0≤x≤1,

1.1

wheref : 0,∞ → 0,∞is aC¹ function,f0 0, g : 0,∞ → 0,∞is aC¹ convex function,g0 0,a ∈ C⁰0,1×R,ax, t ≥ 0 in0,1×R,a_tx, t ≤ 0 in0,1×R, b∈C¹R,bt> 0 inR,bt≥0 inR. The initial datau0 ∈C²0,1,u₀0 0,u₀1 b1gu01.

Here0, Tis the maximal time interval on which the solutionuof1.1exists. The timeT may be finite or infinite. WhereTis infinite, we say that the solutionuexists globally.

WhenT is finite, the solutionudevelops a singularity in a finite time, namely

t→limTu·, t

∞ ∞, 1.2

whereu·, t∞max_0≤x≤1|ux, t|.

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

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 boundaryx 1, the heat source has a nonlinear radition haw. Intensification of the heat source at the boundary x 1 is provided by the functionb. The functiongalso gives a dominant strength of the heat source at the boundary x1.

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 solutionuextincts 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 whereax, t 0 and bt 1. For instance in15, the above problem has been considered in the case whereax, t 0 andbt 1. In16, the authors have considered the problem1.1in the case whereax, t λ >0,bt 1,fu u^p,gu 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 in17–22similar studies concerning other parabolic problems.

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 xi ih, 0 ≤ i ≤ I, where h 1/I. We approximate the solutionuof1.1by the solutionU_ht U0t, U1t, . . . , UIt^T of the following semidiscrete equations

dU_it

dt −δ²U_it −aitf U_it

, 0≤i≤I−1, t∈ 0, T_b^h

, 2.1

dU_It

dt −δ²UIt 2 hbtg

UIt

−aItf UIt

, t∈ 0, T_b^h

, 2.2

U_i0 ϕ_i≥0, 0≤i≤I, 2.3

whereϕ_i1≥ϕi, 0≤i≤I−1,

δ²U0t 2U₁t−2U₀t

h² , δ²UIt 2U_I−1t−2U_It

h² ,

δ²U_it U_i1t−2U_it U_i−1t

h² .

2.4

Here 0, T_b^h is the maximal time interval on which Uht∞ is finite where Uht∞ max_0≤i≤IU_it. WhenT_b^h is finite, we say that the solution U_ht blows up in a finite time and the timeT_b^his called the blow-up time of the solutionUht.

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. Leta_ht∈C⁰0, T,R^I1and letV_ht∈C¹0, T,R^I1such that dV_it

dt −δ²V_it a_itVit≥0, 0≤i≤I, t∈0, T, Vi0≥0, 0≤i≤I.

3.1

Then we haveVit≥0, 0≤i≤I,t∈0, T.

Proof. LetT₀ < Tand define the vectorZ_ht e^λtV_htwhereλis large enough thata_it−λ >

0 for t ∈ 0, T0, 0 ≤ i ≤ I. Let m min_{0≤i≤I,0≤t≤T0}Zit. Since fori ∈ {0, . . . , I}, Zit is a continuous function, there existst₀ ∈0, T0such thatmZ_i0t0for a certaini₀ ∈ {0, . . . , I}.

It is not hard to see that

dZ_i0 t₀ dt lim

k→0

Z_i0 t₀

−Z_i0 t₀−k

k ≤0,

δ²Z_i0 t₀

Zi01 t0

−2Zi0

t0

Zi0−1 t0

h² ≥0 if 1≤i₀ ≤I−1, δ²Z_i0

t₀ 2Z1

t0

−2Z0

t0

h² ≥0 ifi₀0, δ²Zi0

t0

2Z_I−1 t₀

−2Z_I t₀

h² ≥0 ifi0I.

3.2

A straightforward computation reveals that

dZi0

t0

dt −δ²Z_i0 t₀

a_i0

t₀

−λ Z_i0

t₀

≥0. 3.3

We observe from 3.2 thatai0t0−λZi0t0 ≥ 0 which implies thatZi0t0 ≥ 0 because ai0t0−λ >0. We deduce thatVht≥0 fort∈0, T0and the proof is complete.

Another form of the maximum principle for semidiscrete equations is the following comparison lemma.

Lemma 3.2. LetV_ht,Uht∈C¹0, T,R^I1andf ∈C⁰R×R,Rsuch that fort∈0, T dV_it

dt −δ²Vit f Vit, t

< dU_it

dt −δ²Uit f

Uit, t

, 0≤i≤I, 3.4

V_i0< U_i0, 0≤i≤I. 3.5

Then we haveVit< Uit, 0≤i≤I,t∈0, T.

Proof. Define the vectorZ_ht U_ht−V_ht. Lett₀be the firstt∈0, Tsuch thatZ_it>0 fort∈0, t0, 0≤i≤I, butZ_i0t0 0 for a certaini₀ ∈ {0, . . . , I}. We observe that

dZ_i0 t₀ dt lim

k→0

Z_i0 t₀

−Z_i0 t₀−k

k ≤0,

δ²Z_i0 t₀

Zi01 t0

−2Zi0

t0

Zi0−1 t0

h² ≥0 if 1≤i₀ ≤I−1, δ²Z_i0

t₀ 2Z1

t0

−2Z0

t0

h² ≥0 ifi₀0, δ²Zi0

t0

2Z_I−1 t₀

−2Z_I t₀

h² ≥0 ifi0I,

3.6

which implies that dZ_i0

t₀ dt −δ²Zi0

t0

f Ui0

t0

, t0

−f Vi0

t0

, t0

≤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 solutionUh 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 operatorδ²and the second one reveals a property of the semidiscrete solution.

Lemma 4.1. LetU_h∈R^I1be such thatU_h≥0. Then we have δ²g

U_i

≥g U_i

δ²U_i for 0≤i≤I. 4.1

Proof. Apply Taylor’s expansion to obtain

g U₁

g U₀

U₁−U₀ g

U₀

U1−U0

₂

2 g

η₀ , g

U_i1 g

U_i

U_i1−U_i g

U_i

U_i1−Ui

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−Ui

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_I2

2 g

η_I ,

4.2

whereθ_i is an intermediate betweenU_i andU_i1 andη_i the one betweenU_i−1 andU_i. The first and last equalities imply that

δ²g U0

g U0

δ²U0

U₁−U₀2

h² g η0

,

δ²g UI

g UI

δ²UI

U_I−1−U_I2

h² g ηI

.

4.3

Combining the second and third equalities, we see that

δ²g Ui

g Ui

δ²Ui

U_i1−U_i2 2h² g

θi

U_i−1−U_i2 2h² g

ηi

, 1≤i≤I−1. 4.4

Use the fact thatgs≥0 fors≥0 andU_h≥0 to complete the rest of the proof.

Lemma 4.2. LetUhbe the solution of2.1–2.3. Then we have U_i1t> U_it, 0≤i≤I−1, t∈

0, T_b^h

. 4.5

Proof. Lett₀be the firstt >0 such thatU_i1t> U_itfor 0≤i≤I−1 butU_i01t0 U_i0t0 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 functionsZit U_i1t−Uitfor 0≤i≤I−1. We get

dZ_i0 t₀ dt lim

k→0

Z_i0 t₀

−Z_i0 t₀−k

k ≤0,

δ²Z_i0 t₀

Zi01 t0

−2Zi0

t0

Zi0−1 t0

h² >0 if 1≤i₀ ≤I−2, δ²Z_i0

t₀

δ²Z₀ t₀

Z1

t0

−3Z0

t0

h² >0 ifi₀ 0, δ²Zi0

t0

δ²Z_I−1 t0

Z_I−2 t₀

−3Z_I−1 t₀

h² >0 ifi0I−1,

4.6

which implies that dZ_i0

t₀ dt −δ²Zi0

t0

−ai01 t0

f Ui01t0

a_i0

t₀ f

U_i0 t₀

<0 if 0≤i₀≤I−2, dZ_i0

t₀

dt −δ²Z_i0 t₀

2 hb

t₀ g_i01

t₀

−a_i01 t₀

f U_i01t0

a_i0

t₀ f

U_i0 t₀

<0 ifi₀I−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 nodexI.

Now, we are in a position to state the main result of this section.

Theorem 4.3. LetUhbe the solution of 2.1–2.3. Suppose that there exists a positive integer A such that

δ²ϕ_i−a_i0f ϕ_i

≥0, 1≤i≤I−1, δ²ϕI−aI0f

ϕI

b0gI

ϕI

≥Ag ϕI

. 4.8

Assume that

fsgs−fsgs≥0 fors≥0. 4.9

Then the solutionUhblows up in a finite timeT_b^hand we have the following estimate

T_b^h≤ 1 A

_∞

ϕh∞

ds

gs. 4.10

Proof. Since0, T_b^his the maximal time interval on whichUht_∞ <∞, our aim is to show thatT_b^his finite and satisfies the above inequality. Introduce the vectorJhsuch that

J_it dU_it

dt , 0≤i≤I−1, J_It dU_It dt −Ag

U_It

. 4.11

A straightforward calculation gives dJi

dt −δ²Ji d dt

dUi

dt −δ²Ui

, 0≤i≤I−1, dJI

dt −δ²JI d dt

dUI

dt −δ²UI

−Ag UI

dUI

dt Aδ²g UI

.

4.12

FromLemma 4.1, we haveδ²gUI≥gUIδ²U_I which implies that dJI

dt −δ²JI ≥ d dt

dUI

dt −δ²UI

−Ag UI

dU_I dt −δ²UI

. 4.13

Using2.1, we get dJi

dt −δ²J_i ≥ −a_itf U_i

−a_itf

U_idUi

dt , 0≤i≤I−1, dJI

dt −δ²JI ≥ −a_Itf UI

−aItf UI

dUI

dt 2 hbtg

UI

2

hbtg

U_IdU_I

dt −Ag U_I

−a_Itf U_I

2 hbtg

U_I .

4.14

It follows from the fact thata_it≤0,bt≥0 anddUi/dtJiAgUithat dJI

dt −δ²JI ≥

−aItf UI

2 hbtg

UI

JIAaIt g

UI

f UI

−f UI

g UI

. 4.15

We deduce from4.9that dJ_i

dt −δ²J_i≥ −aitf U_i

J_i, 0≤i≤I−1, dJ_I

dt −δ²J_I ≥

−a_Itf U_I

2 hbtg

U_I J_I.

4.16

From4.8, we observe that

J_i0 δ²ϕ_i−a_i0f ϕ_i

≥0, 0≤i≤I−1, J_I0 δ²ϕ_I−a_I0f

ϕ_I

b0gI

ϕ_I

−Ag ϕ_I

≥0. 4.17

We deduce fromLemma 3.1thatJit ≥ 0, 0 ≤ i ≤ I, which implies that dUI/dt ≥ gUI, 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

_∞

UIt

ds

gs, 4.19

which implies that

T_b^h≤ 1 A

_∞

Uh0∞

ds

gs. 4.20

Since the quantity on the right hand side of the above inequality is finite, we deduce that the solutionUhblows up in a finite time. Use the fact thatUh0∞ϕh∞to complete the rest of the proof.

Remark 4.4. The inequality4.19implies that

T_b^h−t0≤ 1 A

_∞

Uht0∞

ds

gs if 0< t0 < T_b^h, U_it≤H

A

T_b^h−t

, 0≤i≤I,

4.21

whereHsis the inverse ofGs _∞

s dz/gz.

Remark 4.5. Ifgs s^q, thenGs s^1−q/q−1andHs 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, Twhereuis defined, the solutionUhtof2.1–

2.3approximatesu, when the mesh parameterhgoes to zero.

Theorem 5.1. Assume that1.1has a solutionu∈C^4,10,1×0, Tand the initial condition at 2.3satisfies

ϕh−uh0

∞o1 as h−→0, 5.1

whereuht ux0, t, . . . , uxI, t^T. Then, for h sufficiently small, the problem2.1–2.3has a unique solutionU_h∈C¹0, T,R^I1such that

max0≤t≤TU_ht−u_ht

∞Oϕ_h−u_h0

∞h²

ash−→0. 5.2

Proof. Letα >0 be such that

u·, t

∞≤α fort∈0, T. 5.3

The problem2.1–2.3has for eachh, a unique solutionU_h ∈ C¹0, T_b^h,R^I1. Letth ≤ min{T, T_b^h}the greatest value oft >0 such that

Uht−uht

∞<1 for t∈ 0, th

. 5.4

The relation5.1implies thatth>0 forhsufficiently small. By the triangle inequality, we obtain

U_ht_∞≤u·, t_∞U_ht−u_ht_∞ fort∈ 0, th

, 5.5

which implies that

U_ht_∞≤1α fort∈ 0, th

. 5.6

Lete_ht U_ht−u_htbe the error of discretization. Using Taylor’s expansion, we have for t∈0, th,

de_it

dt −δ²eit −aitf ξit

eit o h²

, 0≤i≤I−1, de_It

dt −δ²eIt −aItf ξIt

eIt 2 hbtg

UIt

eIt o h²

,

5.7

whereθ_Itis an intermediate value betweenU_Itand uxI, tand ξ_it the one between Uitanduxi, t. Using5.3and5.6, there exist two positive constantsKandLsuch that

de_it

dt −δ²e_it≤L e_it Kh², 0≤i≤I−1, de_It

dt −

2e_I−1t−2e_It

h² ≤ L e_It

h L eIt Kh².

5.8

Consider the functionzx, t e^M1tCx2ϕh−uh0∞Qh²whereM,C,Qare constants which will be determined later. We get

z_tx, t−z_xxx, t M1−2C−4C²x²zx, t, zx0, t 0, zx1, t 2Cz1, t, zx,0 e^Cx2ϕh−u_h0_∞Qh².

5.9

By a semidiscretization of the above problem, we may chooseM,C,Qlarge enough that d

dtz x_i, t

> δ²z x_i, t

L z

x_i, t Kh², 0≤i≤I−1, d

dtz xI, t

> δ²z xI, t

L h z

xI, t L z

xI, t Kh², z

x_i,0

> e_i0, 0≤i≤I.

5.10

It follows fromLemma 3.2that z

x_i, t

> e_it fort∈ 0, th

, 0≤i≤I. 5.11

By the same way, we also prove that z

xi, t

>−eit fort∈ 0, th

, 0≤i≤I, 5.12

which implies that

z x_i, t

> e_it fort∈ 0, th

, 0≤i≤I. 5.13

We deduce that

Uht−uht

∞≤e^MtCϕh−uh0

∞Qh² , t∈

0, th

. 5.14

Let us show thatth T. Suppose thatT > th. From5.4, we obtain 1U_h

th

−u_h

th

∞≤e^MTCϕ_h−u_h0

∞Qh²

. 5.15

Since the term on the right hand side of the above inequality goes to zero ashtends to zero, we deduce that 1≤0, which is impossible. Consequentlyth 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 problem1.1has a solution u which blows up in a finite timeTbsuch thatu∈C^4,10,1×0, Tband the initial condition at2.3satisfies

ϕh−uh0

∞o1 as h−→0. 5.16

Under the assumptions ofTheorem 4.3, the problem2.1–2.3admits a unique solutionUh which blows up in a finite timeT_b^hand we have the following relation

h→lim0T_b^hT_b. 5.17

Proof. Letε >0. There exists a positive constantNsuch that 1

A _∞

x

ds gs ≤ ε

2 forx∈N,∞. 5.18

Since the solutionublows up at the timeT_b, then there existsT₁ ∈ Tb−ε/2, T_bsuch that u·, t∞≥2Nfort∈T1, Tb. SettingT2 T1Tb/2, then we have sup_t∈0,T2|ux, t|<∞.

It follows fromTheorem 5.1that sup

t∈0,T2

U_ht−u_ht

∞≤N. 5.19

Applying the triangle inequality, we get Uh

T2

∞≥uh

T2

∞−Uh

T2

−uh

T2

∞, 5.20

which leads toUhT2∞≥N. FromTheorem 4.3,Uhtblows up at the timeT_b^h. We deduce fromRemark 4.4and5.18that

T_b−T_b^h ≤ T_b−T₂ T_b^h−T₂ ≤ ε 2 1

A _∞

UhT2∞

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 solutionU_h of2.1–2.3in the case wherebt 1. Our result is the following.

Theorem 6.1. LetUhtbe the solution of2.1–2.3. Under the assumptions ofTheorem 4.3,Uht blows up in a finite timeT_b^hand there exist two positive constantsC₁, C₂such that

H C₁

T_b^h−t

≤U_It≤H C₂

T_b^h−t

, fort∈ 0, T_b^h

, 6.1

whereHsis the inverse of the functionGs _∞

s dσ/gσ.

Proof. FromTheorem 4.3andRemark 4.4,Uhtblows up in a finite timeT_b^hand there exists a constantC₂>0 such that

UIt≤H C2

T_b^h−t

fort∈ 0, T_b^h

. 6.2

FromLemma 4.2,U_I−1 < UI. Then using2.2, we deduce thatdUI/dt ≤2/hbtgUI− a_ItfUI, which implies that dU_I/dt ≤ 2bt/hgUI. Integration this inequality over t, T_b^h, there exists a positive constantC₁such that

UIt≥H C1

T_b^h−t

fort∈ 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 constantC₀such thatsFs≤C₀and d

dtU_i−δ²U_i≤0, 0≤i≤I−1. 7.1 Assume that there exists a positive constantCsuch

Ui≤H

CT−t

, 0≤i≤I. 7.2

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

Proof. Letvx 1−x²and define Wx, t H

δvx δBT−t

for 0≤x≤1, t≥t₀, 7.3 whereδis small enough. We have

W_x0, t 0, W1, t H

δBT−t

≥u1, t, 7.4

and fort≥t₀, we get

Wx, t0 H

δvx δ

≥H2δ H 2δB

T−t0

≥H C

T−t0

≥u x, t0

. 7.5

A straightforward computation yields W_tx, t−W_xxx, t δF

Hτ

B−2−4xF

Hτ

≥δF

Hτ

B−2−4δC₀

. 7.6

This implies that there existsα >0 such that

Wtx, t−Wxxx, t≥αF

HδδBT

. 7.7

Using Taylor’s expansion, there exists a constantK >0 such that d

dtW x_i, t

−δ²W x_i, t

≥αF

HδδBT

−Kh², 0≤i≤I, 7.8

which implies that

dW xi, t

dt −δ²W x_i, t

≥0. 7.9

The maximum principle implies that

U_it≤H

δvx δB

T−t₀

fort≥t₀, 0≤i≤I. 7.10

Hence, we get

U_it≤H δvx

, 0≤i≤I. 7.11

ThereforeU_iT<∞, 0≤i≤I−1, and we have the desired result.

8. Full discretization

In this section, we consider the problem1.1in the case whereax, t 1,bt 1,fu u^p, gu u^pwithpconst>1. Thus our problem is equivalent to

u_tx, t u_xxx, t−u^px, t, 0< x <1, t∈0, T, u_x0, t 0, u_x1, t u^p1, t, t∈0, T,

ux,0 u₀x>0, 0≤x≤1,

8.1

wherep >1,u₀∈C¹0,1,u₀0 0 andu₀1 u^p₀1.

We start this section by the construction of an adaptive scheme as follows. LetIbe a positive integer and leth1/I. Define the gridx_iih, 0≤i≤Iand approximate the solution ux, t of the problem 8.1 by the solution Uⁿ_h Uⁿ₀ , Uⁿ₁ , . . . , U_I^nT of the following discrete equations

δ_tUⁿ_i δ²U_iⁿ− U_iⁿp

, 0≤i≤I−1, 8.2

δtUⁿ_I δ²Uⁿ_I − Uⁿ_I p

2 h

Uⁿ_I p

, 8.3

U_i⁰ϕi, 0≤i≤I, 8.4

wheren≥0,ϕ_i1≥ϕi, 0≤i≤I−1,

δ_tUⁿ_i Uⁿ¹_i −Uⁿ_i Δtn

, δ²Uⁿ_i Uⁿ_i1−2Uⁿ_i Uⁿ_i−1

h² , 1≤i≤I−1, δ²Uⁿ₀ 2U₁ⁿ−2Uⁿ₀

h² , δ²U_Iⁿ 2Uⁿ_I−1−2U_Iⁿ

h² .

8.5

In order to permit the discrete solution to reproduce the property of the continuous one when the timetapproaches the blow-up time, we need to adapt the size of the time step so that we takeΔtnmin{1−pτh²/3, τ/U_h^np−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. LetUⁿ_h be the solution of 8.2–8.4. Then we have

Uⁿ_i1≥Uⁿ_i , 0≤i≤I−1. 8.6

Proof. LetZⁿ_i U_i1ⁿ−Uⁿ_i , 0≤i≤I−1. We observe that

Zⁿ¹₀ −Zⁿ₀

Δtn Z₁ⁿ−3Z₀ⁿ

h² −

U₁ⁿ_p

− U₀^np

, Z_iⁿ¹−Z_iⁿ

Δtn Zⁿ_i1−2Zⁿ_i Z_i−1ⁿ

h² −

Uⁿ_i1p− U_i^np

, 1≤i≤I−2, Zⁿ¹_I−1 −Zⁿ_I−1

Δtn Zⁿ_I−2−3Z_I−1ⁿ

h² −

U_Iⁿ_p

−

Uⁿ_I−1^p 2

h U_I^np

.

8.7

Using the Taylor’s expansion, we find that

Z₀ⁿ¹ Δtn

h² Z₁ⁿ

1−3Δtn

h²

Zⁿ₀ −Δtnp ξⁿ₀ p−1

Zⁿ₀ , Z_iⁿ¹ Δtn

h² Z_i1ⁿ

1−2Δtn

h²

Zⁿ_i Δtn

h² Z_i−1ⁿ

−Δtnp ξⁿ_i p−1

Zⁿ_i , 1≤i≤I−2, Z_I−1ⁿ¹≥ Δtn

h² Z_I−2ⁿ

1−3Δtn

h²

Z_I−1ⁿ −Δtnp ξⁿ_I−1p−1

Z_I−1ⁿ,

8.8

whereξⁿ_i is an intermediate value betweenU_iⁿandUⁿ_i1. IfZⁿ_i ≤0, 0≤i≤I−1, we deduce that

Z₀ⁿ¹≥ Δtn

h² Z₁ⁿ

1−3Δtn

h² −ΔtnpUⁿ_h ^p−1

∞

Zⁿ₀ , Z_iⁿ¹≥ Δtn

h² Z_i1ⁿ

1−2Δtn

h² −ΔtnpUⁿ_h ^p−1

∞

Zⁿ_i Δtn

h² Z_i−1ⁿ, 1≤i≤I−2, Z_I−1ⁿ¹≥ Δtn

h² Z_I−2ⁿ

1−3Δtn

h² −ΔtnpU_h^np−1_∞ Z_I−1ⁿ.

8.9

Using the restrictionΔtn≤τ/U_h^np−1_∞ , we find that Z₀ⁿ¹ ≥ Δtn

h² Zⁿ₁

1−3Δtn

h² −pτ

Zⁿ₀ , Z_iⁿ¹ ≥ Δtn

h² Zⁿ_i1

1−2Δtn

h² −pτ

Zⁿ_i Δtn

h² Z_i−1ⁿ, 1≤i≤I−2, Z_I−1ⁿ¹ ≥ Δtn

h² Zⁿ_I−2

1−3Δtn

h² −pτ

Zⁿ_I−1.

8.10

We observe that 1−3Δtn/h²−pτ is nonnegative and by induction, we deduce thatZ_iⁿ≤0, 0≤i≤I−1. This ends the proof.

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

Lemma 8.2. Letaⁿ_h be a bounded vector and letV_hⁿa sequence such that

δ_tV_iⁿ−δ²V_iⁿaⁿ_i V_iⁿ≥0, 0≤i≤I, n≥0, 8.11

V_i⁰ ≥0, 0≤i≤I. 8.12

ThenV_iⁿ≥0 forn≥0, 0≤i≤IifΔtn≤h²/2aⁿ_h ∞h². Proof. IfV_hⁿ≥0 then a routine computation yields

V₀ⁿ¹≥ 2Δtn

h² V₁ⁿ

1−2Δtn

h² −Δtnaⁿ_h

∞

V₀ⁿ, V_iⁿ¹≥ Δtn

h² V_i1ⁿ

1−2Δtn

h² −Δtnaⁿ_h

∞

V_iⁿ Δtn

h² V_i−1ⁿ, 1≤i≤I−1, V_Iⁿ¹≥ 2Δtn

h² V_I−1ⁿ

1−2Δtn

h² −Δtnaⁿ_h

∞

V_Iⁿ.

8.13