Totally discrete solution of the phenomenon of quenching for a localized heat equation

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*Corresponding author: Yoro.Gozo

International Journal of

Current Multidisciplinary Studies

Available Online at http://www.journalijcms.com

Vol. 2, Issue, 6, pp.336-346, June, 2016

RESEARCH ARTICLE

Totally discrete solution of the phenomenon of quenching for a localized heat

equation

Yoro.Gozo

*1

_{Halima.Nachid}

2

_and

_{L.B. Sobo Blin}

3

123

_{Université Nangui Abrogoua, UFR-SFA, Département de Mathématiques et Informatiques, (Côte d'Ivoire),}

A R T I C L E I N F O A B S T R A C T

Received 18 th_{, March, 2016,} Received in revised form 27 th_, April, 2016, Accepted 13 th_{, May, 2016,} Published online 28th_{,June , 2016}

In this paper we consider the following initial-boundary problem.

0

( , )

( (0, )),

( 1,1),

(0, )

( 1, )

0,

(1, )

0,

(0, )

( , 0)

( )

0,

[ 1,1]

,

t xx

x x

u x t

u

x t

f u

t

x

t

T

u

t

u

t

T

u x

x







 















_

_

_{ }



where

f

: (0, )

 

(0, )



is a

C

1 convex, non-increasing function,

0

lim

( )

,

s_ 

f s

 

₀

,

( )

ds

f s



 



for any real positif



,

u

₀



C

2

([ 1,1])



,

0

' ( 1)

0 u





and

u

' (1)

₀



0.

We find some conditions under which the solution of a

discrete form of the above problem quenches in a finite time and estimate its numerical quenching time. We also prove that the numerical quenching time converges to the real one when the mesh size goes to zero. Finally, we give some numerical results to illustrate our analysis.

Keywords:

Discretizations, Semilinear Parabolic Equation, Quenching, Numeri- cal Quenching Time, Convergence

INTRODUCTION

0

( , ) ( , ) ( (0, )), ( 1,1), (0, ), (1) ( 1, ) 0, (1, ) 0, (0, ), (2) ( , 0) ( ) 0, [ 1,1],

t xx

x x

u x t u x t f u t x t T

u t u t t T

u x u x x

    

   

    (3)

where

f

: (0, )

 

(0, )



is a

C

1 convex, non-increasing function,

0

lim

( )

,

s 

f s

 

0

_{( )}

,

ds

f s



 



2

0

([ 1,1])

u



C



0 0

0 ( ) 1, [ 1,1], ( ) 0, ( 1,1), (4) ( ) ( (0)) 0, ( 1,1), (5)

( 1) 0, (1) 0

u x x u x x

u x f u x

u u



      

    

     (6)

The particularity of the problem describes in (1) (3) is that it represents a model in physical phenomena where the reaction is driven by the temperature at a single site. This kind of phenomena is observed in biological systems and in chemical reaction di usion processes in which the reaction takes place only at some local sites. For instance, the above model is appropriate to describe:

(i) The in uence of defect structures on a catalytic surface. (ii) The temperature in a solid-fuel combustion scenario where

the heat that is input into the system is localized, say as in a laser focused on one spot in the domain.

(iii) Chemical reaction-di usion processes in which, due to e ect of catalyst, the reaction takes place only at a single site. (iv) A heat stationary source which can support an explosive

reaction. A stationary source provides a continuous supply of heat to the same environment.

IJCMS

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(v) The ignition of a combustible medium with damping, where either a heated wire or a pair of small electrodes supplies a large amount of energy to every con ned area.

For more physical motivation see [18], [19] and [20]. Here (0, T ) is the maximal time interval on which the solution u of (1) (3) exists. The time T may be nite or in nite. When T is in nite, we say that the solution u exists globally. When T is nite, the solution u develops a singularity in a nite time, namely

lim ∥u(., t)∥inf = 0, t→T where ∥u(., t)∥inf = minx∈[−1,1] u(x, t). In this last case, we say that the solution u quenches in a

nite time and the time T is called the quenching time of u.

For instance, if we take f (s) = s−p or f (s) = (ln(1 + s))−p with p = const > 0, it is not hard to see that f veri es the above hypotheses.

The theoretical study of solutions for localized semilinear parabolic equations which quench in a nite time has been the subject of investigations of many authors (see [2], [4] [9], [14], [21] and the references cited therein). Lo- cal in time existence of a classical solution has been proved and this solution is unique. In this paper, we are interested in the numerical study of the phenomenon of quenching for a localized semilinear heat equation. Under some assumptions, we show that the solution of a discrete form of (1) (3) quenches in a nite time and estimate its numerical quenching time. We also prove that the numerical quenching time goes to the real one when the mesh size goes to zero. In [13], Nabongo and Boni have found an analogous result considering a semidiscretization of the following boundary value problem

0

( , )

( , ),

(0,1),

(0, ),

(0, )

0,

(1, )

(1, ),

(0, ),

( , 0)

( )

0,

[0,1],

t xx

p

x x

u x t

u

x t x

t

T

u

t

u

t

u

t t

T

u x

x

















with p =const>0. Our work was also motived by the papers in [1], [3] and [12]. In [1] and [12], the authors have used semidiscrete and discrete forms for some parabolic equations to study the phenomenon of blow-up (we say that a solution blows up in a nite time if it reaches the value in nity in a nite time). In [3], some schemes have been used to study the phenomenon of extinction (we say that a solution extincts in a nite time if it becomes zero after a nite time for equations without singularities). Concerning the numerical study, the authors in [10], [11], [16], [22] have proposed some numerical schemes for computing the numerical solutions for parabolic problems which present a solution with one singularity. Let us notice that in these papers, the convergence of the numerical quenching time to the theoretical one has not been proved. some authors have used semidiscrete and discrete schemes to study the phenomenon of blow-up, but only the case where the reaction term f (u(0, t)) is replaced by f (u(x, t)) has been taken into account (see [9], [21], [22]).

In this paper, our aim is to build an explicit scheme in which the discrete solution reproduces the properties of the continuous one. We start by the construc- tion of an adaptive scheme as follows. Let I be a positive integer, and consider

the grid x_iih, 0 i I where h = 2/I . Approximate the solution u of (1)--(3) by the solution ( ) ( ) ( ) ( )

0 1

( , , , )

n n n n T

h I

U  U U U

of the following discrete equations δt U (n)

( ) 2 ( ) ( )

(0)

( ), 0 , (7) , 0 , (8)

n n n

t i i k

i i

U U f U i I

U i I





   

  

where n ≥ 0, k is the integer part of I /2,

( ) ( ) ( )

2 ( ) 1 1

2

( ) ( ) ( ) ( ) 2 ( ) 1 0 2 ( ) 1

0 2 2

( 1) ( ) ( )

1

2

, 1 1,

2 2 2 2

, ,

.

, 0 , 0, 0 1,

.

n n n

n i i i

i

n n n n

n n I I

I

n n

n i i

t i

n

i I i i

i i i

U U U

U i I

h

U U U U

U U

h h

U U

U

t

i I i k

h



 



   

 

 

 





 

 

 

   

 

 

 



      

 

Since the solution u quenches in a nite time T , in order to allow the discrete solution to reproduce the properties of the continuous solution, we need to adapt the size of the time step so that we take

( ) 2

inf ( )

inf

(1

)

min{

,

}, 0

1.

3 (

)

n h

n n

h

U

h

t

f

U









‖



‖

The rest of the paper is organized as follows. In the next section, we give some results which will be used later. In the third section, under some conditions, we prove that the solution of a discrete form of (1) (3) quenches in a nite time and estimate its numerical quenching time. In the fourth section, we prove the convergence of the numerical quenching time. Finally, in the last Section, we give some numerical results to illustrate our analysis.

Properties of a discrete scheme

In this section, we give some lemmas about the discrete maximum principle for localized parabolic problems and reveal certain properties concerning the discrete solution. Let us notice that the restriction on the time step ensures the positivity of the discrete solution.

The lemma below gives a property of the operator δ2 .

Lemma 2.1 Let I 1

h

U 

 be such that Uh0. Then we have

2 2

( ( ))

f U

i

f U

(

i

)

U

i

, 1

i

I







 

Proof. Applying Taylor's expansion, we nd that

2

2 2 1

2 2

1 2

(

)

( ( ))

(

)

( )

2 (

)

( )

if

1 1,

2

i i

i i i i

i i

i

U

f U

U

f

h

U

f

i

I

h

























  

2

2 2 1

2

(

)

( ( ))

(

)

I I

(

),

I I I I

U

f U

U

f

h



_





_







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complete the rest of the proof. Another property of the discrete solution is the following.

Lemma 2.2 Let

U

_h( )n be the solution of (7) (8) and suppose that

( ) ( )

1 1

1, , 0 1. n n, 0 1.

h i i i I Ui Ui i I

           

‖ ‖

Then we have ( )n₁ ( )n

,

0

1.

i i

U

_



U

  

i

I

Proof. Set

Z

_i( )n



U

_i( )_n₁



U

_i( )n

, 0

  

i

I

1.

Obviously

( ) 0

0

n

Z



because Lemma 2.2. A straightforward computation reveals that

( 1) ( ) ( ) ( ) ( )

( ) ( )

1 1

1 2

2

( ( ) ( )),1 2,

n n n n n

n n

i i i i i

i i

n

Z Z Z Z Z

f U f U i I

t h



 



  

     

 \

( 1) ( ) ( ) ( )

( ) ( )

1 1 2 1

1 2

3 ( (

)

(

))

n n n n

n n

I I I I

I I

n

Z

f U

t

h



   









Applying the mean value theorem, we arrive at

( 1) ( ) ( ) ( ) ( ) ( )

1 1

2 (1 2 2) 2 ( ) ,1 2,.

n n n n n n n n n

i i i i i i

t t t

Z Z Z Z f Z i I

h h h 



 

  



       

( 1) ( ) ( ) ( ) ( )

1 2 2

(1 3

2

)

1

(

)

,

n n n n n n n

I I I I I

t

Z

f

Z

h





  









where



_i( )n is an intermediate value between

U

_i( )n and

( ) 1

n i

U

_ and



_I( )n the one

U

_I( )n between

U

_I( )_n₁

.

and

U

_I( )n_₁

.

Since (0)

1 0 0 1

i i i

Z      i I , we indeduce by induction

that ( )n

0, 0

1

i

Z



  

i

I

and the proof is complete. The above lemma says that if the initial data of the discrete solution is nonincreasing in space, then the discrete solution is also nonincreasing in space. This property will be utilized later to show that the discrete solution attains its

minimum at the last node

x

I

.

. Another property of the operator



2 is the result below.

Lemma 2.3

Let I1

h

U 

 and suppose that Ui1Ui1 , 1  i I 1. Then

we have



2

(

iU

i

)



i U



2 i

1   

i

I

1

. Proof. A straightforward computation reveals that

2 2 1 1

2

(

)

i i

,

1

i i

U

iU

i

U

i

I

h



_



_





_{ }

_

Use the fact that

U

i₁



U

i₁

, 1

  

i

I

1

to complete the rest of the proof.

The following lemma is a discrete form of the maximum principle for localized parabolic problems.

Lemma 2.4 Let

a

_h( )n be a bounded vector and let V (n) a sequence such that

( ) 2 ( ) ( ) ( )

(0)

0, 1

,

0, (9)

0, 0

(10)

n n n n

t i i i i

i

V

a V

i

I

n

V

i

I











 



 

Then

V

_i( )n



0

for

n



0

0  

i

I

if

2

( ) 2

2

n n

h

t

a

_

h







‖

Proof. If

V

h( )n



0

then a routine computation yields

( 1) ( ) ( ) ( )

0 2 1 2 0

2 (1 2

)

.

n n n n n n

n h

t

V

t

a

V

h













 

‖

( 1) ( ) ( ) ( ) ( )

1 1

2 (1 2 2 ) 2 ,1 1,

n n n n n n n n

i i n h i i

t t t

V V t a V V i I

h h h



  

  

     ‖ ‖    

( 1) ( ) ( ) ( )

1

2 2

2 (1 2

)

.

n n n n n n

I I n h I

t

V

t

a

V

h



 









 

‖

Since

2

( ) 2

2

n n

h

t

a

_

h







‖

we see that

( ) 2

1 2

n n

n h

t

a

h







 

‖

is nonnegative. From (10), we

deduce by induction that

V

h( )n



0

which ends the proof. A direct consequence of the above result is the following comparison lemma. Its proof is straightforward.

Lemma 2.5 Suppose that

a

_h( )n and

b

_h( )n are two vectors such that

a

_h( )n is bounded. Let

V

_h( )n and

W

_h( )n two sequences such that

( )n 2 ( )n ( )n ( )n ( )n ( )n 2 ( )n ( )n ( )n ( )n

tVi Vi a Vi i bi tWi Wi a Wi i bi

       

1  

i

I n

,



0

(0) (0)

, 0

i i

V



W

 

i

I

.Then

( ) ( )

0, 0

n n

i i

V



W

for n



 

i

I

if

2

( ) 2

2

n n

h

t

a

_

h







‖

Now, let us end this section with a property of the operator



_t

Lemma 2.6 Let

U

( )n



be such that

U

( )n



0 for n



0

Then we have



t

f U

(

( )n

)



f U

(

( )n

)



t

U

( )n

,

n



0 Proof.

From Taylor's expansion, we find that

( 1) ( ) 2

( ) ( ) ( )

(

)

( )

(

)

(

)

(

),

2

n n

n n n n

t t

n

U

f U

U

f

t



















where



( )n

.

is an intermediate value between

U

( )n and

(n 1)

U



. Use the fact that ( )n

0

0 U



for n



and

( )

0

0 f



s



for s



to complete the rest of the proof.

The lemma below reveals some properties of the discrete solution.

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( )n ( )n _, ₀ _, ( )n _0, ₀ _{1. (11)}

i I i i

U U i I U i k



      

Proof. Introduce the vector

V

_h( )n defined as follows

( ) ( ) ( )

, 0

,

0

n n n

i i I i

V



U



U

_

 

i

I

n



A routine calculation reveals that

( 1) ( ) ( )

0

2

2 1

(1 2

2

)

0

,

0,.

n

t

n n

t

n n

V

n

h













( 1) ( ) ( ) ( )

1 1

2 (1 2 2) 2 , 1 1, 0

n n n n n n n

i i i i

t t t

V V V V i I n

h h h



 

  

       

( 1) ( ) ( )

1

2 2

2 (1 2

)

,

0,

n n n n n

I I I

t

V

n

h

 











(0)

0,

0

i

V



 

i

I

Using an argument of recursion, we easily note that

( )

0,

0

n h

V



n



and the first part of the lemma is proved. In order to prove the second one, we proceed as follows. Set

( ) ( ) ( )

1

, 0

1.

n n n

i i i

W



U





U

 

i

k



We remark that

( ) ( )

( ) 1 0

0 2

3 ,

0 (12)

n n n t

W

n

h









On the other hand, it is easy to check that

U

_k( )n_₁



U

_k( )n if

I

is odd, and

U

_k( )n_₁



U

_k( )n_₁ if

I

is even. This implies that

( ) ( )

1 2

2 2 ( )

1 _{( )} _{( )}

1 2

2

2 if

is odd

3 if

is even.

n n

k k

n

k _n _n

k k

W

I

h

W

I

h



    

 





 









Obviously

( ) 2 ( )

, 0

2,

0 (13)

n n

t

W

i

W

i

k

n







  



.

Making use of the above relations, we arrive at

( 1) ( ) ( )

0 2 1

(1 3

2

)

0

,

0,.

n

t

n n

t

n n

W

n

h













( 1) ( ) ( ) ( )

1 1

2 (1 2 2) 2 ,1 2, 0,

n n n n n n n

i i i i

t t t

W W W W i k n

h h h



 

  

       

( 1) ( ) ( )

1 2 2 (1 3 2 ) 1, 0 if is even,

n n n n n

k k k

t t

W W W n I

h h



  

 

   

( 1) ( ) ( )

1 2 2 (1 2 2 ) 1, 0 if is odd,

n n n n n

k k k

t t

W W W n I

h h           (0)

0, 1

1.

i

W



 

i

k



We deduce by induction that

( )

0, 1

1,

0

n i

W



 

i

k



n



. This completes the proof.

Numerical Quenching solutions

In this section, under some assumptions, we show that the solution of the discrete problem quenches in a nite time and

estimate its numerical quenching time.We need the following lemmas.

Lemma 3.1 Let a and b be two positive numbers such that b < 1. Then, we have

0 0

1 .

(

)

( )

ln( )

( )

n

a

n n

ab

a

d

f ab

f a

b

f



 









Proof.. We observe that

1 1 1 1 0 0 0

(

)

(

)

(

)

x x n

n n

x _n x _n n

n n

ab dx

ab

dx

f ab

        



_



_



,

because

f s

( )

is nonincreasing for

s



0

. We deduce

that 1 1 0 0 0

(

)

(

)

( )

(

)

x n n

n n

ab dx

ab

a

ab

f ab

f a

f ab

      



_

 



_



.

On the other hand, by a change of variables, we see

that

0 0

1 (

)

ln( )

( )

x

a

x

ab dx

d

f ab

b

f





 



, which implies that

0 0

1 .

(

)

( )

ln( )

( )

n _a

n n

ab

a

d

f ab

f a

b

f













_

This ends the proof.

To handle the phenomenon of quenching for discrete equations, we need the following definition.

Denition 3.1

We say that the solution

( ) ( )

inf

of (7) (9)quenches in a finite time if 0 for 0

n n

h h

U  ‖U ‖  n

but

1 ( )

inf

0

lim

0 and

lim

.

n

n t

h h i

n n

i

U

T

t

    



_

  

‖

The number

T

_ht

is called the numerical quenching time of

( )n h

U

.

The following theorem reveals that the discrete solution

U

_h( )n . of (7)-(8) quenches in a finite time under some hypotheses.

Theorem 3.1 Let

U

_h( )n be the solution of (7)-(8) and assume thath

1

1, , 0 1. (14)

h i i i I

      

‖ ‖

Suppose that there exists a constant

A



(0,1]

such that

2

( )

( ), 1

(15)

i

f

i

Aihf

i

i I

 





 



 

Then

U

_h( )n quenches in a finite time

T

_ht which satisfies the following estimate

inf

0 inf

,

(

)

(1

)

( )

h inf t h h h

ds

T

f

ln

f s



 



















‖ ‖

Where 2 inf inf

(1

)

(

)

min{

, }

3

h

h f

A











 

‖ ‖

(5)

Proof.

Introduce the vector

J

( ) ( ) ( )

(

),

0

n n n

i t i i

J





U



Aihf U

 

i

I

. A straightforward computation yields





( ) 2 ( ) ( ) 2 ( ) ( ) 2 ( )

( ) ( ( )), 1 1,

n n n n n n

tJi Ji t tUi Ui Aih tf Ui Ah if Ui i I

           





( ) 2 ( ) ( ) 2 ( ) ( ) 2 ( )

(

)

(

)

n n n n n n

t I

J

I t t

U

I

U

I

A f U

t I

A f U

I









 













Using (7), we arrive at

( ) 2 ( ) ( ) 2 ( )

(1

)

(

)

( (

)),1

1,

n n n n

t i

J

i

Aih f U

t i

Ah

if U

i

i I







 







  

( ) 2 ( ) ( ) 2 ( )

(1

)

(

)

(

).

n n n n

t

J

I

J

I

A

t

f U

I

A

f U

I







  







It follows from Lemmas 2.1, 2.2, 2.3 and 2.6 that

( ) 2 ( ) ( ) ( ) ( ) 2 ( )

(1

) (

)

(

) (

),1

,

n n n n n n

t i

J

i

Aih f U

i t

U

i

Ahif U

i

U

i

i I







 











 

which implies that

( ) 2 ( ) ( ) ( ) ( ) ( ) 2 ( )

(

) (

)

(

)( (

)

(

)),1

.

n n n n n n n

t i

J

i

f U

i t

U

i

Ahif U

i t

U

i

U

i

i I























 

Due to (7), we deduce that

( ) 2 ( ) ( ) ( )

(

)

, 1

.

n n n n

t

J

i

J

i

f U

i

J

i

I







 



 

Obviously, we have ₀( )n

0 J



and from (14), we see that

(0)

0.

h

J



It follows from Lemma 2.4 that

( )

0,

n h

J



which implies that

( ) ( )

(

n

)

(

n

)

0.

t

U

I

Af U

I







From Lemma 2.2, we have

( ) ( ) inf

.

n n

I h

U



‖

U

‖

Consequently, we get

( 1) ( ) ( )

inf inf

(

inf

).

n n n

h h n h

U



U

A t f

U



 

‖

‖ ‖

‖

This

inequality reveals that the sequence

‖

U

_h( )n

‖

_inf

,

is nonincreasing. By induction, we obtain

( ) (0)

inf inf inf

.

n

h h h

U



U





‖

‖ ‖

‖ ‖ ‖

Thus the following holds

( ) 2

inf inf

( )

inf inf

(

)

(1 )

(

)

min{

, }

. (16)

3

n

h h

n n

h h

f U

h f

A t

A

U















‖ ‖



‖ ‖



‖ ‖

Consequently, we get

( 1) ( )

inf inf

(1

), (17)

n n

h h

U





U







‖

‖ ‖ ‖

and by iteration, we arrive at

( ) (0)

inf inf

(1

)

inf

(1

) . (18)

n n n

h h h

U



U

















‖ ‖ ‖ ‖

‖ ‖

Since the term on the right hand side of the above equality goes

to zero as

n

approaches infinity, we conclude that

‖

U

_h( )n

‖



tends to zero as

n

approaches infinity. Now, let us estimate the numerical quenching time. We observe that

1

inf inf

0

inf inf

(1

)

(1

)

.

(

(1

) )

(

(1

) )

x x

n

h h

x _n x

n h h

dx

f

















   

















‖ ‖

_{‖ ‖}



‖ ‖

_{‖ ‖}

Since

(1



)

x

(1



)

n1

.









for

x



( ,

n n



1)

and

( )

s

f s

is

a nondecreasing function for the positive values of

s

,

we see that 1 1 inf inf 1 inf inf

(1

)

(1

)

(

(1

) )

(

(1

)

x n n h h x n n h h

dx

f

















  













‖ ‖

_{‖ ‖}

‖ ‖

_{‖ ‖}

We deduce that

1 inf inf 1 0 0 inf inf

(1

)

(1

)

(

(1

) )

(

(1

)

x n h h x n n h h

dx

f

















    

















‖ ‖

_{‖ ‖}

‖ ‖

_{‖ ‖}

,

which implies that

inf inf inf

0

inf inf inf

(1

)

(1

)

(

)

(

(1

) )

(

(1

) )

x n

h h h

x n

n

h h h

dx

f

















  



















‖ ‖

.

By a change of variable, it is not hard to see that

0 0

(1

)

1 (

(1

) )

(1

)

( )

h inf x h inf x h inf

dx

ds

f

ln

f s















 







‖ ‖

_{‖ ‖}



‖ ‖ It

follows that

0 ₀

(1

)

1 (

(1

) )

(

)

(1

)

( )

h inf

n

h inf h inf

n n

h inf h inf

ds

f

ln

f s













 

















‖ ‖

On the other hand, using the restriction on the time step, we

have

( )

0 0 ( )

inf

.

(

)

n

h inf

n n n n

h

U

t

f

U



   





 

‖

Due to (18) and the fact that the function

( )

s

f s

is

nondecreasing for the positive values of

s

,

we arrive at

inf

0 0

inf

(1

)

.

(

(1

) )

n h

n n n n

h

t

f









   









 





‖ ‖

We deduce that

inf

0 ₀

inf

.

(

)

(1

)

( )

h h n n h

ds

t

f

ln

f s









 















‖ ‖

Use the

fact that the quantity on the right hand side of the above inequality is finite to complete the rest of the proof.

Remark 3.1 From (17), we deduce by induction that

( ) ( )

inf inf

(1

)

n q n q

h h

U











‖

‖ ‖

‖

and we see that

( ) inf 0 ( )

inf

,

(

)

n

t h

h q n q n n n

h

U

T

t

f

U



  

 



 



 

‖

which implies

that ( ) ( ) inf inf 0 ( ) ( ) inf inf

(1

)

(1

)

.

(

(1

) )

(

(1

) )

q n q q n

t h h

h q n q q n q n q n

h h

U

T

t

f U



      





  

 





‖ ‖

Reasoning as above, we deduce that

( ) inf

( ) ₀

inf

.

(

)

(1

)

( )

q h

q

U

t h

h q q

h

U

ds

T

t

f

U

ln

f s

(6)

Since

2

inf

(1

)

(

)

'

min{

, }

3

h

h f

A













‖ ‖

if we take

2

h





, we get

2

inf inf

(1

) (

)

(

)

min{

,1}

min{

,1}.

'

3

4

h h

h f

f

A



A











‖ ‖



‖ ‖

Therefore, there exists a positive constant

C

such that

'

C





. On the other hand, use Taylor's

expansion to obtain

ln

(1





')





'



o

( ')



which implies that

( ')

(1

(1))

.

(1

')

'

1 (1)

C

o

ln

o



















Hence,

we deduce that

2 (1

')

C

ln









. Consequently ( )

( ) inf

( ) ₀

inf

2 (

)

( )

n inf h

q

U

t h

h q q

h

U

ds

T

t

C

f

U

f s









‖



_

‖ ‖

‖

In the sequel, we take





h

2

.

Convergence of the quenching time

In this section, under some conditions, we show that the solution of the discrete problem quenches in a nite time and its numerical quenching time goes to the real one when the mesh size tends to zero.Firstly, let us prove the convergence of our scheme by the following.

Theorem 4.1

Suppose that the problem (1)-(3) has a solution

4,2

([0,1] [0,

])

u



C



T





such that

[0, ] inf

min

_t_ _T__

‖

u

(., )

t

‖







0

. With





(0, ).

T

Assume that the initial data at (8) verifies

(0)

(1) as

0. (19)

h

u

h

o

h











‖

Then the problem (7)--(8) has a solution

U

_h( )n for

h

sufficiently small,

0 

n



J

and the following estimate holds

( ) 2

0

max

 n J

‖

U

hn



u t

h n

( )

‖





O

(

‖



h



u

h

(0)

‖



 

h

t

n

) as

h



0

where

J

is such that

1

0

J

n n

t

T







 





and

1

0

n

n j

j

t





_



.

Proof. For each h, the problem (7)--(8) has a solution

U

_h( )n . Let

N



J

be the greatest value of n such that

( )

for

(20)

2

n

h h n

U



u t







n N



‖

We know that

N



1

because of (18). Applying the triangle inequality, we have

( ) ( )

inf

( )

for

. (21)

2

n n

h h n inf h h n

U



u t



U u t









n N



‖ ‖‖ ‖ ‖

‖

Since

u



C

4,2

,

taking the derivative in

x

on both sides of (1) and due to the fact that

u

_x ,

u

_xt vanish at

x



1

, we observe that

u

_xxx also vanishes at

x



1

. Applying Taylor's expansion, we discover that

2 2

( , )

( , ), 1

12

xx i n i n xxx i n

h

u

x t





u x t



u

x t



 

i

I

To establish the above equality for

i



I

, we have used the fact that

u

_x and

u

_xxx vanish at

x



1

. We deduce that for

,1

,

n



N

 

i

I

2 2

( , )

( ( , ))

( , )

12

2

n

t i n i n i n xxxx i n tt i n

t

h

u x t

f u x t

u

x t

u x t



















Let ( )n ( )n

( )

h h h n

e



U



u t

be the error of discretization. From the mean value theorem, we get for $

n



N

,1

 

i

I

,

2 ( ) 2 ( ) ( ) ( )

(

)

( , )

12

2

n n n n n

t i i i i xxxx i n tt i n

t

h

e

f

e

u

x t

u x t























where



_i( )n is an intermediate value between

( , )

_i _n

u x t

and

U

_i( )n Since

u

_xxxx

( , )

x t

_i _n ,

u x t

_tt

( , )

are bounded, there exists a positive constant

M

such that

( ) 2 ( ) ( ) ( ) 2

(

)

, 1

,

. (22)

n n n n

t i

e

i

f

i

e

i

Mh M t

n

i I n N















 

 



Set

( )

2 K



f





and introduce the vector

V

( 1)

( ) 2

(

(0)

), 0

.

n

K t

n

i h h n

V

e





u

Mh

M t

i I n N





‖



‖





 



A straightforward computation gives

( ) 2 ( ) ( ) ( ) 2

(0) (0)

(

)

, 1

,

, (23)

,0

. (24)

n n n n

t i i i i n

i i

V

f

V

Mh

M t

i I n N

V

e

i I















 

 





 

We observe that

(

f





i( )n

)

is bounded from above by

K

. It follows from Comparison Lemma 2.5 that

V

_h( )n



e

_h( )n . By the same way, we also prove that

V

_h( )n

 

e

_h( )n , which implies that

( 1)

( ) 2

( )

K tn

(

(0)

),

. (25)

n

h h n h h n

U

u t

e





u

Mh M t n N

 









 



‖

Let us show that

N



J

. Suppose that

N



J

. If we replace

n

by

N

in (27) and use (21), we find that

( ) ( 1) 2

( )

(

(0)

)

2

N K T

h h N h h n

U

u t

e

u

Mh

M t







 



‖



‖



‖



‖





Since the term on the right hand side of the second inequality

goes to zero as

h

goes to zero, we deduce that

0

2 



, which