ETNA
Electronic Transactions on Numerical Analysis.
Volume 30, pp. 247-257, 2008.
Copyright  2008, Kent State University.
ISSN 1068-9613.
Kent State University
http://etna.math.kent.edu
NUMERICAL BLOW-UP SOLUTIONS FOR SOME SEMILINEAR HEAT EQUATIONS
FIRMIN K. N’GOHISSE AND THEODORE K. BONI Abstract. This paper concerns the study of the numerical approximation for the following initial-boundary
value problem,
"!#$&%' (!&)*+(
(!&)*!,&%-.
+$/012345 *,6 .78
:9
9
and ;
. We give some conditions under which the solution of a semidiscrete form of the above
where
problem blows up in a finite time and estimate its semidiscrete blow-up time. Under some assumptions, we also
show that the semidiscrete blow-up time converges to the continuous blow-up time when the mesh size goes to zero.
Finally, we give some numerical results to illustrate our analysis.
Key words. semidiscretizations, discretizations, semilinear heat equations, semidiscrete blow-up time
AMS subject classifications. 35B40, 35K65, 65M06
1. Introduction. In this paper, we consider the following initial-boundary value problem for semilinear heat equation of the form
(1.1)
(1.2)
(1.3)
<2=>?<@@BADC <@FAG<IH+JKEMLON&PIJRQS1JUTVLON&PWJXYS1J
E
< @ N&PIJTSZ>[PWJ\<-NQJTSZ>]PIJ\TVLON&PIJXYS
J
<-N&E^J_PSZ>[<2`aNbE)S1JKEcLed PWJfQ1g J
which models the temperature distribution of a large number of physical phenomenon from
physics, chemistry and biology. The term < ` H represents nonlinear heat generation with hMi Q
and C is a positive parameter. Here < ` LMj Nd PWJfQfg8S , <)`k N&PSZ>[P , < ` NQSZ>]P , N&PWJXYS is the maximal time interval on which l <-N&E^JTS lm is finite where l <'NbE^JTS lRm >onqpsr `Rt)@tvuw <-N&E^JTS w .
The time X may be finite or infinite. When X is infinite, we say that the solution < exists
globally. When X is finite, the solution < develops a singularity in a finite time, namely
x5y
=&z|n { l <'NbE}JTS l m >]~[
In this case, we say that the solution < blows up in a finite time and the time X is called the
blow-up time of the solution < .
The theoretical study of blow-up of solutions for semilinear parabolic equations has been
the subject of investigations of many authors; see [2, 5, 7, 9, 10], and the references cited
therein.
The authors have proved that under some assumptions, the solution of (1.1)–(1.3) blows
up in a finite time and the blow-up time is estimated.
Let us notice that if we consider the semilinear heat equation
<2=>€<@R@FAG<WH/J\EcL‚ƒJKTVLON&PIJXYS
J
Received January 8, 2008. Accepted for publication May 22, 2008. Published online on September 12, 2008.
Recommended
by O. Widlund.
Universite d’Abobo-Adjame, UFR-SFA, Departement de Mathematiques et Informatiques, 02 BP 801 Abidjan 02, Cote d’Ivoire (firmingoh@yahoo.fr).
Institut National Polytechnique Houphouet-Boigny de Yamoussoukro, BP 1093 Yamoussoukro, Cote d’Ivoire
(theokboni@yahoo.fr).
247
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248
F. K. N’GOHISSE AND T. K. BONI
with boundary conditions
<-N&E^JTS„>[PWJ\EcL‚…JKT†LeN&PWJXYS1J
and initial data
<-N&E^J_PSZ>[<2`N&E)S‡L 3J
where ˆ>Š‰REML‚‹'ŒŽ w E w+ Q‘ , …Ž>Š‰REML‚‹'Œe w E w >’Q‘ , the radial symmetric solutions are
solutions of (1.1)–(1.3) with C >’Q .
In this paper, we are interested in the numerical study using a semidiscrete form of (1.1)–
(1.3). Let “ be a positive integer and define the grid E”'>€•—– , P˜™•„˜ “ , where –>ŠQsš “ . Ap{
proximate the solution < of (1.1)–(1.3) by the solution ›„œ N&TSY>N › `aN&TS
J › usN&TS
JRffRJ ›Zž N&TSS
of the following semidiscrete equations
Ÿ `aNbTS
›
>ˆN—Q†A S— s¡ › `aNbTS^A › ` H NbTS
J¢T†LeN&PWJX £ œ S
J
Ÿ T
(1.4)
C
Ÿ ” NbTS
›
>] ¡ › ”NbTS}A C s¤ › ”N&TS^A › ” H N&TS
J¥Q˜¦•„˜ “§ QJ\T†LON(PIJX £ œ S
J
Ÿ T
(1.5)
•—–
œ
(1.6)
›ž NbTS„>]PIJ\TVLeN(PIJX £ S
J
(1.7)
› ”N(PSZ>]¨'”J©Pª˜¦•Z˜ “ J
where
su NbTS §
s¡ › ` N&TS„>« ›
–
”
s
u
&
N
T
S
s¡ › ”N&TS„> › ¤
” u N&TS
s¤ › ”N&TS„> › ¤
–
`NbTS
« ›
J
¡
§ « › ” NbTS^A › ”&¬vu‘NbTS ­
J Q®˜¦•„˜
– ¡
§O› ” N&TS J­Q˜¦•„˜
“,§ Q
œ
Here, N(PIJX £ S is the maximal time interval on which lR› œ NbTS lfm
lR›„œ N&TS l m
Ҥ
QJ
is finite where
>¥¯°I± w › ”NbTS w 
`ft”bt ž
œ
œ
When X £ is finite, we say that the solution › œ N&TS blows up in a finite time and the time X £
&
N
T
S
is called the blow-up time of the solution ›†œ
. We give some conditions under which the
solution of (1.4)–(1.7) 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. A similar study has been undertaken in [1, 4, 6, 8]. In [1, 8], the
authors have considered the equation (1.1) for C >²P with Dirichlet boundary conditions
and nonnegative initial data. Numerical methods for heat equations with nonlinear boundary
conditions have been described in [4, 6]. In the same way in [2], 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).
The paper is organized as follows. In the Section 2, we give some properties concerning our scheme. In Section 3, under some conditions, we prove that the solution of the
semidiscrete problem blows up in a finite time and estimate its semidiscrete blow-up time. In
Section 4, we study the convergence of the semidiscrete blow-up time. Finally, in Section 5
we report on some numerical experiments using several discretisations of (1.1)–(1.3).
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NUMERICAL BLOW-UP SOLUTION FOR HEAT EQUATIONS
249
2. Properties of the semidiscrete scheme. In this section, we give some results about
the discrete maximum principle. The following lemma is a discrete form of the maximum
principle.
`
u
ž u
ž u
L EMMA 2.1. Let ³Iœ N&TS:Lej Nd PWJXBg J‹ ¤ S and let ´)œ N&TSFLŽj Nd PWJXBg J‹ ¤ S such that
for TVLŽN(PIJXYS ,
Ÿ `/NbTS
´
Ÿ T
§ N—Q†A C S s¡ ´ `aN&TS^A ³ `aNbTS ´ `/NbTS‡µ€PIJ
Ÿ ” NbTS
´
Ÿ T
§ ¡ ´ ”_NbTS § •—C – s¤ ´ ”N&TS^A ³ ”_NbTS ´ ”¶N&TSVµ¦PWJ·Q|˜¦•Z˜ “|§ QJ
(2.1)
´Wž NbTSVµ€PIJ
´ ”_N&PSVµ€PIJKP˜¦•Z˜
“ 
Then we have ´ ” N&TS‡µ¸P , Pª˜¦•Z˜ “ , TVLŽN(y$PIº+J» XYS .
`ft”bt ž = ¼+½ `R¾ {¿.À ´ ” NbTS . Since for •ÁL‰PWJffR1J “  ,
Proof. Let X `? X and let ¹ >
b
N
T
S
T
”
´
is a continuous function, there exists ` LÂd PWJX ` g such that ¹ > ´ ” ¿aNbT ` S for a certain
• `ªL¸‰PWJfRf1J “  . If • `,> “ , we have a contradiction because of (2.1). When • ` is between P
and “|§ Q , we observe that
Ÿ ”5¿aNbT.`S
”$¿NbT.`sS §O´ ”$¿N&T.` §OÄ S
´
> x$y n ´
˜¦PWJ
Ÿ T
(2.2)
à z`
Ä
”$¿ uNbT.`S § ´ ”5¿NbT.`S}A ´ ”5¿f¬vu‘NbT.`sS
s¡ ´ ”5¿aNbT.`SZ> ´ ¤
«
µ¦P if Q|˜™• `|˜ “|§ QJ
(2.3)
– ¡
´ ” ¿ ¤ u NbT ` S §O´ ” ¿N&T ` S µ€P
(2.4) ¤ ´ ”5¿aNbT.`SZ>
if Q|˜¸• `|˜ “|§ QJ
–
u‘NbT.`sS § ´ `/NbT.`S
«
¡ ´ ” ¿aNbT ` SZ> « ´
µ¦P if • ` >[PW
(2.5)
– ¡
=
Define the vector ńœ NbTS†>[ÆsÇ ´2œ NbTS where È is such that ³ ”$¿N&T.`S §OÈMi P . A straightforward
computation reveals that
Ÿ ”$¿NbT.`sS
Å
Ÿ T
§ N—Q†A C S— ¡ Å ” ¿‘NbT ` S^A]N ³ ” ¿NbT ` S §OÈ S Å ” ¿N&T ` SVµ€P if • ` >]PIJ
(2.6)
Ÿ ”$¿N&T.`S
Å
Ÿ T
(2.7)
§ s¡ Å ”5¿NbT.`S § •—C – ¤ Å ”$¿‘NbT.`sS^A]N ³ ”$¿NbT.`S §OÈ S Å ”$¿N&T.`SVµ€P
if Q˜¦• ` ˜ “§ Q
We observe from (2.2)–(2.5) that
Ÿ ”$¿N&T.`S
Å
˜€PIJK s¡ Å ” ¿NbT ` SVµ¸P
Ÿ T
³aÉ
Ÿ
¤ Å ” ¿NbT ` S‡µ¸PW
Using (2.6)–(2.7), we arrive at N ³ ” ¿NbTS §[È S Å ” ¿N&T ` SµoP , which implies that Å ” ¿N&T ` S3µP .
Therefore ´ ” ¿NbT ` SZ> ¹ µ¸P and we have the desired result.
Another version of the discrete maximum
principle
is the following
comparison lemma.
u
`
ž u
L EMMA 2.2. Let ´)œ N&TS , ›Zœ NbTS‡LMj Nd PWJXBg J‹ ¤ S and Ê LMj N&‹ÂˋVJ‹:S such that for
T†LeN&PWJXYS
Ÿ
(2.8)
(2.9)
Ÿ ` NbTS
´ /
› ` N&TS
Ÿ T
Ÿ T
§ NQ†A C S s¡ ´ `aN&TS^A Ê N ´ `aN&TS
JTS 
§
Ÿ ” NbTS
´
Ÿ T
§ ¡ ´ ”_NbT S § •—C – s¤ ´ ”_bN TS^A Ê N ´ ”N&TS
JTS
N—Q†A
C
S— s¡ › `N&TS^A Ê N › `aNbTS1JTS
J
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250
F. K. N’GOHISSE AND T. K. BONI
(2.10)
(2.11)
Ÿ
› ” N&TS
Ÿ T
§ s¡ › ” NbTS § •—C – ¤ › ” N&TS^A Ê N › ” NbTS1JTS
J­Q|˜¦•Z˜¦• §
´2ž NbTS  ›ž NbTS
J
´ ”_N&PS  › ”N&PaS
J©P˜™•†˜ “ 

QJ
Then we have ´)œ NbTS  ›Zœ NbTS , t LŽN(PIJXYS

Proof. Define the vector ņœ N&TSª> ›„œ N&TS §[´2œ NbTS . Let T.` be the first T i P such that
b
N
T
ÅZœ S i P for T‡LGd PIJT—`RS , but Å ”$¿N&T.`S„>]P for a certain • `LŽ‰PWJfRf1J “  . If • `Y> “ , we have a
contradiction because of (2.10). If • ` is between P and “§ Q , we observe that
Ÿ ”$¿N&T.`S
”$¿‘N&T.`S §GÅ ”$¿‘N&T.` §eÄ S
Å
> x$y n Å
˜¸PW
Ÿ T
à z`
Ä
”$¿ u‘N&T.`S § Å ”5¿‘N&T.`SÌA Å ”$¿f¬vusN&T.`S
s¡ Å ”5¿‘N&T.`SZ> Å ¤
«
µ€P if Q®˜¦• `|˜ “§ QJ
– ¡
” ¿ u N&T ` S §GÅ ” ¿N&T ` S
¤ Å ”5¿‘N&T.`SZ> Å ¤
µ€P if Q|˜¦• `®˜ “§ QJ
–
u NbT ` S § Å ` NbT ` S
¡ Å ” ¿‘N&T ` SZ>« Å
«
µ¸P if • ` >?PWJ
– ¡
which implies that
Ÿ ”$¿NbT.`sS
Å
S A Ê N › ” ¿NbT ` S JT ` S O
Ÿ T
§ s¡ Å ” ¿NbT ` S § • `C – ¤ Å ” ¿‘N&T ` ^
§ Ê
Ÿ ” ¿N&T ` S
Å
Q A —S ¡ Å ” ¿‘N&T ` ^
S A Ê N › ” ¿NbT ` S JT ` S O
Ÿ T
§ N—†
§ Ê
C
N ´ ” ¿NbT ` S
JT ` SV˜¦P
if
Q|˜¦• ` ˜
N ´ ” ¿NbT ` S
JT ` SV˜¦P
if • ` >[PW
Ҥ
QJ
But these inequalities contradict (2.8) and (2.9).
To finish this section, let us state
a result on the operators ¡ and ¤ .
u
ž
M
L
‹
¤ such that › œ µ€P . Then we have
L EMMA 2.3. Let › œ
¬vu
¡ › ” H µ hv› ” H
¡ › ” JÍQ|˜¸•„˜ “|§ QJ
¬vu
H
H
s¤ › ” µ hv› ”
s¤ › ” J­Q|˜¦•Z˜ “|§ Q
Proof. Using Taylor’s expansion, we get
N
QS ¬
¬Ìu
h N hq§ QS ”H ¬ ¡ J·Q|˜¦•Z˜
H” ¡ A[N ”&¬vu
¡ › ” H > h› ” H
¡ › ” A]N › ” u §O› ” S ¡ h hq§
›
§G› ” S ¡
¤
– ¡
– ¡
Î
Ï
« QS
«
N
¬Ìu
¬
H” ¡ J
s¡ › ` H > h› ` H
s¡ › `†AÂN › u §O› `S—¡ h hq§
– ¡
Î
N
QS
¬Ìu
¬
H ” ¡ J­Q|˜¸•„˜
¤ › ” H > h› ” H
s¡ › ”2A]N › ” u §O› ”(S—¡ h hq§
“|§ QJ
¤
– ¡
Ð
«
where Î ” and Ð ” are intermediate values between › ” and › ” ¤ u , Ï ” is an intermediate value
between › ” and › ”&¬vu . Use the fact that ›„œ is nonnegative to complete the rest of the proof.
3. Semidiscrete blow-up solutions. In this section under some assumptions, we show
that the solution of the semidiscrete problem blows up in a finite time and estimate its semidiscrete blow-up time.
T HEOREM 3.1. Suppose that there exists a positive integer A such that the initial data at
(1.7) satisfies
N—Q†A
C
S s¡R¨-`VA™¨ H` µ¦ÑY¨ H` J
“|§
QJ
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NUMERICAL BLOW-UP SOLUTION FOR HEAT EQUATIONS
(3.1)
C ¤ ¨ ” A™¨ H ” µ¦ÑF¨ H” J­Q|˜™•„˜
•—–
¡ ¨ ” A
Q
“|§
œ
Then the solution › œ NbTS of (1.4)–(1.7) blows up in a finite time X £ and we have the following
estimate
u
¬
Q l ¨ œ)l H
m
X £œ ˜

Ñ N hƒ§ QS
œ
Proof. Since &N PWJX £ S be the maximal time interval on which lR› œ NbTS lfm is finite, our aim
ϣ
X
is to show that
is finite and obeys the above inequality. Introduce the vector Ò œ such that
Ÿ ”
›
>
”
Ÿ T § Ñ › ” H J©P˜¦•Z˜ “ 
Ò
A straightforward calculation gives
Ÿ Ÿ `
Ÿ `
›
Ò
Ÿ T § N—Q†A C S ¡ Ò ` > Ÿ T N Ÿ T § N—Q†A C S— ¡ › ` S
Ÿ ”
Ÿ Ÿ ”
Ò
›
¡
>
”
Ÿ T §
Ÿ T N Ÿ T § ¡ › ”S § Ñ v
Ò
h ›
From Lemma 2.3, we have Ÿ
Ÿ `
Ò
Ÿ T § N—Q†A C S— ¡ Ò ` µ Ÿ
Ÿ ”
Ÿ
Ò
Ÿ T § s¡ Ò ” µ Ÿ
¬vu
¡ › ”H µ h › ”H
Ÿ `
N Ÿ›
N QVA
T
T §
Ÿ ”
N Ÿ › § s¡ › ”
T
T
Use (1.4)–(1.5) to obtain
Ÿ `
Ò
Ÿ T § NQ†A C S s¡ Ò `|µ h ›
Ÿ ”
Ò
Ÿ T § ¡ Ò ” µ h ›
`H
”H
¬Ìu
¬Ìu
`
T
A™Ñ,N—Q‡A
C
S ¡ › ` H J
¡ › ”H ­
J Q|˜™•†˜ “,§ Q
¡ › ” which implies that
¬vu Ÿ › `
S— ¡ › ` S § Ñ hv› ` H
N Ÿ
N—QVA S ¡ › ` S
J
C
C
T §
¬Ìu Ÿ › ”
S § Ñ h› ” H
N Ÿ
s¡ › ” S
­
J Q|˜¸•„˜ “|§ Q
T §
Ò `aJ
Ò ” A
¬Ìu Ÿ ›
H
Ñ
`
Ÿ
§
h ›
Ÿ
¬vu › ”
A ÑY ”H
Ÿ T G
C N& s¤
—• –
Ÿ
¬vu
› ”
H
s¤ › ” S1J­Q|˜™•†˜
Ÿ T § Ñ hv› ”
“|§
Q
H
Taking into account the expression of Ò œ NbTS , we get ¤ Ò ” >[ ¤,Ó1Ó Ôa= Õ § ÑY ¤ › ” , which¬vimplies
u
H
¤ Ó1Ôa= Õ µ€ ¤ Ò ”fA3Ñ › ” H
¤ › ”.
that ¤ Ó1Ó Ôa= Õ >] ¤ Ò ”RA3ÑF ¤ › ” . From
Lemma
2.3,
we
arrive
at
¬Ìu
Ó
H
Ñ
'
”
¦
µ
”
1
Ó
a
Ô
Õ
¤
¤
¤
”
=
Therefore, we get
›
›
Ò and due to (3.1), we discover that
Ó §
Ÿ ”
¬vu
Ò
Ÿ T § s¡ Ò ” § •—C – ¤ Ò ”µ hv› ” H
Ò ”J­Q˜¦•„˜ “§ Q
Obviously, Ò ž NbTSZ>]P and the hypotheses on the initial data ensure that Ò œ N(PSVµ€P . It follows
H
œ
from Lemma 2.1 that Ò/œ N&TSBµ?P for TBLON&PWJX £ S , which implies that Ó1Ó Ôa= Õ µ?Ñ › ” , Pq˜€•†˜ “ .
This estimation may be rewritten as follows
Ÿ ”
› µ€Ñ Ÿ T
J©P˜¸•„˜
“ 
›|” Ö
œ
Integrating this inequality over NbT
JX £ S , we find that
(3.2)
X £ œ § TV˜
Q N › ” NbTSS u ¬ H
J
Ñ
N hq§ QS
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F. K. N’GOHISSE AND T. K. BONI
which implies that
u¶¬
Q l ¨ 2
œ l m H 
Ñ N hƒ§ Q S
X £œ ˜
œ
Therefore X £ is finite and the proof is complete.
R EMARK 3.2. The inequalities (3.2) imply that
Q l›Zœ N&T.`S l u
¬ H
m
X £œ § T ` ˜
if
Ñ
N h3§ QS
and there exists a constant j i P such that
› ”_NbTSV˜
j
NbX £ œ § T SG1Ø Ù ×
×
P 
J©P˜¸•„˜
T `| X £ œ
“ 
T HEOREM 3.3. Let › œ NbTS be the solution of (1.4)–(1.7). Then we have
u
¬
¨
H
X £ œ µ l œ)l m

N hĤ QS
Proof. Let • ` be such that › ”5¿NbTS„>?nqpsr+`ft”bt ž-› ”NbTS . It is not hard to see that
› ” ¿ ¤ u NbTS § « › ” ¿‘N&TS^A › ” ¿ ¬Ìu NbTS ˜€P
if Q|˜¸• `˜ “,§ QJ
– ¡
” ¿ u NbTS §G› ” ¿N&TS
˜¦P if Q|˜™• ` ˜ “|§ QJ
¤ › ” ¿N&TSZ> › ¤
–
«
u‘NbTS § › `NbTS
¡ › ” ¿N&TSZ>o« ›
«
˜¦P if • ` >]PI
– ¡
Ÿ T
Ó1ÔaÕ ¿
Ó1ÔaÕ ¿
H
We deduce that Ó = ˜ › ”$¿ , which implies that Ô Ø ¿ ˜
. Integrating this inequality over
Õ
N&PWJX £ œ S , we obtain
N › ” ¿N(PSS u¶¬ H
X £œ µ

N hĤ QS
Use the fact that › ” ¿N&PS†> l ¨ œ lfm to complete the rest of the proof.
s¡ › ”$¿N&TSZ>
4. Convergence of the semidiscrete blow-up time. In this section, under some assumptions, we show that the blow-up time for the solution of the semidiscrete problem converges to the blow-up time for the solution of the continuous problem when the mesh size
tends to zero. In order to prove this result, firstly we show that for each fixed time interval d PWJXBg where the solution < of (1.1)–(1.3) is defined, the solution ›†œ N&TS of (1.4)–(1.7)
approximates <}J when the mesh parameter – goes to zero by the following
theorem.
¾u
T HEOREM 4.1. Assume that (1.1)–(1.3) has a solution <MLcjÚ Nd PWJfQfgÌˎd PIJXBg8S and the
initial condition at (1.7) satisfies
l ¨ œ§ < œ N&PS l m >?Û+NQS as –ÜÝPIJ
{
where < œ NbTSƒ>ÞNb<'NbEW`aJTS1Jf$5$J<'NbE ž Ju TSS . Then, u for h sufficiently small, the problem (1.4)–
ž
(1.7) has a unique solution › œ LMj Nd PWJXBg J‹ ¤ S such that
(4.1)
`fnq
t)= pst)r { l› œ NbTS § < œ NbTS lfm
>]ßN l ¨ œ § < œ N(PS lRm
A¸–)S
as
–3ÜÝPW
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NUMERICAL BLOW-UP SOLUTION FOR HEAT EQUATIONS
Proof. Since <cL‚j Ú
(4.2)
¾u
, there exist positive constants à and á
¬vu
˜ á 
l < l m ˜ à J h N à A?QS0H
The problem (1.4)–(1.7) has for each –ÌJ a unique solution ›†œ L‚j
the greatest value of T i P such that
l› œ NbTS § < œ NbTS lfm
(4.3)

Q
such that
u Nd PIJX œ S
J_‹ ž u S
¤ . Let T1N(–)S
Ö
TVLeN(PIJT1N(–)SS

for
$y º ‰ T1N(–)S
JX
The relation (4.1) implies that T1N(–)S i P for – sufficiently small. Let T¶âN(–)SV>ãn
.
By the triangle inequality, we obtain
lR›„œ N&TS l m
˜
l < œ NbTS l m
A
lR›Zœ NbTS § < œ N&TS l m
for
T†LeN&PWJT â N(–)SS1J
which implies that
l›Zœ NbTS l m
(4.4)
˜]Q†A à
TVLeN&PIJT â N(–)SS

for
Let Æ œ NbTS> ›Zœ NbTS § < œ N&TS be the error of discretization. Since <ÂLÂj Ú
expansion, we have for T†LeN&PWJT¶â‘N0–2SS ,
Ÿ ÆR`aNbTS
¬vu
H
Æ`aNbTS^A™Û+N(–)S
J
Ÿ T
§ N—Q‡A C S s¡fÆ`N&TSZ> h2ä `
Ÿ Æ ” NbTS
¬vu
H
Ɣ_NbTS^A™Û+N(–)S
J
Ÿ T
§ s¡RƔNbTS § •—C – ¤ ƔN&TSZ> h2ä ”
¾u J
using Taylor’s
where ä ” is an intermediate value between › ” N&TS and <'NbE)”JTS . Using (4.2) and (4.4), there
exists a positive constant á such that
Ÿ Æ ` NbTS
Ÿ T
§ NQ†A C S ¡ Æ`N&TS‡˜ á w Æ`N&TS w A á –^J
Ÿ Ɣ_NbTS
Ÿ T
§ s¡fÆ ” NbTS § •—C – ¤ Æ ” N&TS‡˜ á w Æ ” N&TS w A á –^JÍQ®˜¦•„˜ “§ Q
Introduce the vector ņœ N&TS such that
uç8=
N l ¨ œ § < œ N(PS lRm
Å ” NbTS„>]Æå$æ ¤
A straightforward calculation reveals that
Ÿ ` N&TS
Å
Ÿ T
§ NQ†A C S— s¡ Å `N&TS ¦
i á
Ÿ ” N&TS
Å
Ÿ T
§ ¡ Å ” NbTS § •—C – ¤ Å ” N&TS ¦
i á
&
N
T
S
ÅZž
i Æ ž
A
á
–)S
JKP˜™•†˜
“ 
w Å `aNbTS w A á
–^J
w Å ” NbTS w A á
NbTS

–^J­Q˜¸•†˜ “,§ QJ
It follows from Lemma 2.2 that ÅZœ NbTS i Æ œ N&TS for TFL?N&PWJTâ‘N(–)SS . In the same way, we also
prove that Å œ N&TS i]§ Æ œ NbTS for TVLŽN(PIJT_â‘N0–2SS , which implies that
uç8=
N l ¨ œ § < œ N&PaS lfm A á –2S1JKT†LeN&PWJT â N(–)SS

l› œ NbTS § < œ NbTS lfm ˜¦Æå$æ ¤
Let us show that T_â‘N0–2S†>€X . Suppose that X
QF>
l›Zœ NbT1N(–)SS § < œ N&T1N(–)SS l m
i T1N0–2S . From (4.3), we obtain
uçb{
˜¸Æå$æ ¤
N l ¨ œ§ < œ N(PS l m A á
–)S

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254
F. K. N’GOHISSE AND T. K. BONI
Since the term on the right hand side of the above inequality tends to zero as – goes to zero,
we deduce that Q¦˜èP , which is impossible. Consequently T¶âN(–)Sc>ÝX , and the proof is
complete.
Now, we are in a position to prove the main theorem of this section.
T HEOREM 4.2. Suppose that¾ u the problem (1.1)–(1.3) has a solution < which blows up in
a finite time X £ such that <cLMj Ú Nd PWJfQ1g}ˎd PIJX £ SS and the initial data at N 1.7S satisfies
l ¨ œ§ < œ N&PaS l m
>[Û+N—QS
as
–ƒÜ²PW
Under the assumptions of Theorem 3.1, the problem (1.4)–(1.7) has a solution › œ NbTS which
œ
blows up in a finite time X £ and
x5y n X £ œ >[X £ 
œ z`
P . There exists a positive constant N such that
Q E u¶¬ H
˜ é  ~
(4.5)
for EcLed êcJ¶A|~¸S

Ñ N hƒ§ QS
«
X}u such that w X^u § X £ w ˜ìë and l <-N&E^JTS l m µ
Since < blows up at the time X £ , then there
¡
{ exists
{sí
ê for T¦L²d X^usJX £ g . Letting X
>
× ¤ ¡ , we see that ¯°I± = ¼+½ `f¾ {î—À w <-N&E^JTS wq ~ . It
¡
«
follows from Theorem 4.1 that the problem (1.4)–(1.7) has a solution ›„œ NbTS which obeys
¯°I± = ¼+½ `f¾ { î À w ›Zœ NbTS § < œ NbTS w m ˜*ê . Applying the triangle inequality, we get lR›Vœ NbTS l m µ
l < œ N&TS l m §l›Zœ NbTS § < œ N&TS l m , which leads to l›„œ NbTS l m µïê for TeLðd PWJX ¡ g . From
Theorem 3.1, ›„œ NbTS blows up at the time X £ œ . We deduce from Remark 3.2 and (4.5) that
Q lR› œ NbX S l u
¬ H
¡ m
˜ é J
w X £œ § X £ w ˜ w X £œ § X w A w X § X £ w ˜ é A
¡
¡
Ñ
N hq§ QS
«
Proof. Let éªi
and we have the desired result.
5. Numerical experiments. In this section, we study the phenomenon of blow-up, using full discrete schemes (explicit and implicit)
of (1.1)–(1.3). Firstly, we approximate the
ç
å Œ >ˆN ` Œ J u Œ JfRf1J Œ S {
solution < of (1.1)–(1.3) by the solution › œ
of the following explicit
›
›
› ž
scheme
uç
ç
åŒ ¤
åŒ
ç
ç
› `
§O› `
>ˆN—Q†A S s¡ › ` å Œ A]N › ` å Œ S0HIJ
ñ T
(5.1)
C
ç
ç
uç Œ
ç
åŒ ¤
åŒ
ç
ç
ӌ Πu
”å Œ
”
›
§O› ”
å” Œ A C N › ¤ §G›
]
>
s
¡
S^A[N › ” å Œ S H/J­Q|˜¦•Z˜ “|§ QJ
ñ T
›
(5.2)
•—–
–
Œ
ç
å Œ >]PIJ
(5.3)
› ž
`¶ç
å
(5.4)
› ” > « P„ò1ó¯fNb•—–Zô S1J©P˜™•†˜ “ J
«
where É µ¦P ,
ç
ñ T >€n y5º ‰ – ¡ J_–2¡
å Œ u
¬ H J
l
›
Œ
œ l m
A
C
ç«
ç
ç
uå Œ § › `å Œ
«
s¡ › ` å Œ > « ›
J
– ¡
ç
ç
ç
Πu
ç
” å Œ u § › ” å Œ A › ”bå ¬Ì
›
Œ
å
«
¤
¡ › ”
>
J­Q|˜™•†˜ “§ Q
– ¡
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NUMERICAL BLOW-UP SOLUTION FOR HEAT EQUATIONS
255
Let us notice that the restriction on the time step guarantees the positivity of the discrete
solution.
ç
åŒ
Secondly, approximate the solution < of (1.1)–(1.3) by the solution › œ of the following
implicit scheme
uç
ç
åŒ ¤
åŒ
uç
ç
› `
§G› `
>ˆN—Q†A S— s¡ › ` å Œ ¤
A[N › ` å Œ S H/J
ñ T
(5.5)
C
uç
u—ç
uç Œ
ç
åŒ ¤
åŒ ¤
åŒ ¤
åŒ
u—ç
ç
”
› ”¤ u
§G› ”
›
§G› ”
å” Œ ¤
]
>
s
¡
U
A
C
N
S}A[N › ” å Œ S H+J
ñ T
(5.6)
›
•.–
–
Œ
Q®˜¦•„˜ “§ QJ
u—ç
åŒ ¤
>]PIJ
(5.7)
› ž
`_ç
å
(5.8)
› ” > « PZòfó¯1N&•.– ô S
J©P˜¸•„˜ “ J
«
ç
ñ T >– ¡
å Œ u¶¬ H 
lR› œ l m
The above equations may be rewritten in the following form
where Œ
ñ T
ñ T
ç
ç
Œ S ` å Œ ¤ uç
Œ å Œ ¤ uç >
å Œ A ñ T N ` å Œ S0HIJ
—
N
†
Q
A
(5.9)
§ « – ¡ › u
› `
Œ ›
« – ¡ ›
ñ T
ñ T
ñ T
ñ T
ñ T
Œ ”&å ¬vŒ u¤ u—ç A]N—Q†A
ΠA C
Œ S ” å Œ ¤ uç
ΠA C
Œ S ” å Œ u¤ u—ç
N
§ – ¡ ›
§
(5.10)
« – ¡
•—– ¡ ›
– ¡
•—– ¡ › ¤
ç
ç
å Œ A ñ T N ” å Œ S0HIJÍQ|˜¸•Z˜
>
› ”
“|§ QJ
Œ ›
uç
åŒ ¤
>
PI
› ž
(5.11)
The equalities (5.9)–(5.11) lead us to the linear system below
ç
u—ç
ç
Ñ åŒ ´ åŒ ¤
>]õ å Œ J
œ
œ
œ
ç
åŒ
where Ñ œ is a “ Ë “ tridiagonal matrix defined as follows
ö÷ `
þü ý
P
ùfùRù
P
`
÷ ³
ý
C
÷ ú u
ý
P
R
ù
f
ù
ù
u
u
÷
ý
³
C
÷
ý
ç
÷
ý
..
.. ..
..
P
Ñ åŒ >
.
.
.
.
œ
..
.. ..
¬
ø ...
ÿ
.
.
.
C ž ¡
P
ùfùfùûP
ú ž ¬vu ³/ž ¬vu
with
ñ T
Œ N—QVA S1J
Y
`
Š
>
†
Q
A
³
« – ¡
C
ñ T
ñ T
Œ N—QVA SÌA C
Œ JK•->’QJfRf1J
Š
>
†
Q
A
”
³
Ҥ QJ
« – ¡
C
•—– ¡
ñ T
ñ T
ñ T
Œ NQ†A S1J
ΠA C
Œ S1JK•'>’QJR5$5J
` > §
” > § N
“ § « J
« – ¡
« – ¡
C
C
C
•—– ¡
ñ T
Œ JK•'>’QJRfffJ
ú1”-> §
“|§ QJ
– ¡
õ ” Œ > › ” Œ A ñ T N › ” Œ S H J©P˜¸•„˜ “|§ QJ
Œ
uç
u—ç
u—ç
u—ç {
Œ
åŒ ¤
å
å Œ ¤ JRfffJ å Œ ¬Ì¤ u
¤
>
J

`
u
´ œ
›
›
› ž
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256
F. K. N’GOHISSE AND T. K. BONI
It is not hard to see that
N(Ñ Œ S ”5” i PIJ¥N(Ñ Œ S”
œ
œ

P
” • > J¥N(Ñ Œ S ”5” i
œ
w N(Ñ Œ S ” w 
œ
These inequalities imply that the linear system has a unique solution for É µÝP and the
discrete solution is nonnegative. For the proof, see for instance [3]
We need the following definition.
ç
åŒ
D EFINITION 5.1. We say that
the solution › œ of (5.1)–(5.4) or (5.5)–(5.8) blows up in
ç
x5y
åŒ
¤ m ñ
a finite time if n Œ z ¤ mãl› œ l >A|~ and the series Œ ` T Œ ç converges. The quantity
åŒ
¤ m ` ñ T
Œ is called the numerical blow-up time of the solution › œ .
Œ
In the Tables 5.1–5.6, we present the numerical blow-up times, values of É J the CPU
times and the orders of the approximations corresponding
to meshes of 16, 32, 64, 128. We
¬vu
Œ ` ñ T
X Π>
take for the numerical
blow-up
time
which
is computed at the first time
u
¬Ìu
ñ
. The order(s) of the method is computed from
when T Œ > w X Œ ¤ § X ŒÌw ˜ÂQRP
x ó NNbX
œ § X ¡ œ Sš+N&X ¡ œ § X œ SS 
®>
x ó 2N S
«
`¶ç
å
For the numerical values, we take, › ” > « PZòfó¯1N ¡ •—–2S , h > « J and >– ¡ .
First case: C >[P
TABLE 5.1
Numerical blow-up times, numbers of iterations, CPU times (seconds), and orders of the approximations obtained with the explicit Euler method.
I
16
32
64
128
%
0.056343
0.056231
0.056210
0.055203
n
7191
27359
103908
405086
"! 16
69
560
7052
s
2.31
1.68
TABLE 5.2
Numerical blow-up times, numbers of iterations, CPU times (seconds), and orders of the approximations obtained with the first implicit Euler method.
I
16
32
64
128
%
0.056376
0.056240
0.056213
0.056207
n
7191
27360
103908
406009
"! 20
113
1460
20746
s
2.22
1.85
Second case: C >’Q
TABLE 5.3
Numerical blow-up times, numbers of iterations, CPU times (seconds), and orders of the approximations
obtained with the explicit Euler method.
I
16
32
64
128
%
0.067710
0.066986
0.066527
0.06630
n
7406
28264
107691
409301
"! 13
70s
594
7203
s
0.76
1.02
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NUMERICAL BLOW-UP SOLUTION FOR HEAT EQUATIONS
257
TABLE 5.4
Numerical blow-up times, numbers of iterations, CPU times (seconds), and orders of the approximations
obtained with the first implicit Euler method.
I
16
32
64
128
Third case: C >
%
0.067802
0.067008
0.066532
0.066303
n
7407
28265
107692
409302
"! 18
109
1506
21000
s
0.84
1.05
«
TABLE 5.5
Numerical blow-up times, numbers of iterations, CPU times (seconds), and orders of the approximations obtained with the explicit Euler method.
I
16
32
64
128
%
0.090365
0.089372
0.087243
0.086658
n
7765
29778
114065
435971
"! 15
71
630
8580
s
0.85
0.95
TABLE 5.6
Numerical blow-up times, numbers of iterations, CPU times (seconds) and orders of the approximations
obtained with the first implicit Euler method.
I
16
32
64
128
%
0.090575
0.0884238
0.087226
0.086636
n
7767
29780
114067
435972
"! 19
115
1718
23051
s
0.85
1.07
REFERENCES
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[2] T. K. B ONI , On the blow-up and asymptotic behavior of solutions to a nonlinear parabolic equation of second
order, Asympt. Anal., 55 (1999), pp. 202–224.
[3] T. K. B ONI , Extinction for discretizations of some semilinear parabolic equations, C. R. Acad. Sci. Paris Sér.
I Math., 333 (2001), pp. 795–800.
[4] R. G. D UR ÁN , J. I. E TCHEVERRY, AND J. D. R OSSI , Numerical approximation of a parabolic problem with
a nonlinear boundary condition, Discrete Contin. Dyn. Syst., 4 (1998), pp. 497–506.
[5] A. F RIEDMAN AND M. M CLEOD , Blow-up of positive solutions of semilinear heat equations, Indiana Univ.
Math. J., 34 (1984), pp. 202–224.
[6] R. F ERREIRA , P. G ROISMAN , AND J. D. R OSSI , Numerical blow-up for a nonlinear problem with a nonlinear boundary condition, Math. Models Methods Appl. Sci., 12 (2002), pp. 461–483.
[7] H. A. L EVINE , The role of critical exponents in blow-up Theorems, SIAM Rev., 32 (1990), pp. 262–288.
[8] T. N AKAGAWA , Blowing up on the finite difference solution to ‡ , Appl. Math. Optim., 2 (1976),
pp. 337–350.
[9] S. O HTA AND A. K ANEKO , Critical exponent of blow-up for semilinear heat equation on a product domain,
J. Sect. IA. Math., 40 (1993), pp. 635–650.
[10] F. B. W EISSLER , Single point blow-up of semilinear initial value problem, J. Differential Equations, 55
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