On the Computation of Extinction Time for Some Nonlinear Parabolic Equations

The phenomenon of extinction is an important property of solutions for many evolutionary equations. In this paper, a numerical simulation for computing the extinction time of nonnegative solutions for some nonlinear parabolic equations on general domains is presented. The solution algorithm utilizes the Donor-cell scheme in space and Euler’s method in time. Finally, we will give some numerical experiments to illustrate our algorithm.


Introduction
There is a large number of nonlinear partial differential equations of parabolic type whose solutions for given initial data become identically nulle in finite time T.Such a phenomenon is called extinction and T is called the extinction time.For certain problems, the extinction time can be computed explicitly, but in many cases one can only know the existence of extinction time.
Since the appearance of the pioneering work of Kalashnikov [1], extinction phenomenon in nonlinear parabolic equations has been studied extensively by many authors [2] [3].Particular emphasis has been placed on the question as to the existence of extinction time [4]- [7].
Generally speaking, it is difficult to simulate extinction phenomenon accurately on general domains.Indeed, it is not at all clear if features of such a phenomenon as extinction can be well reflected in the discretized equation which approximates the original equation.In [8]- [11], some numerical schemes have been used to study the extinction phenomenon of solutions for some nonlinear parabolic equations.
In this work, we propose a numerical algorithm for computing the extinction time for nonnegative solutions of some nonlinear parabolic equations.Our motivation is to reproduce the extinction phenomenon of some nonlinear parabolic equations on general domains.This paper is organized as follows.In the next section, we present the problem model and some theoretical results.A discretization of this problem is derived in Section 3, while numerical experiments are reported in Section 4 and Section 5 is devoted to concluding remarks.

The Model Problem
In this work, we are concerned with the following initial-boundary value problem: [ [ where Ω is a 2 » bounded domain with boundary ∂Ω , 0 u , φ and F , being given functions.Furthermore, for any 0 t > and for all u defined in ] [

0,
Ω× ∞ , we will set ( )( ) ( ) Nonlinear parabolic equations of type (1) appear in various applications.In particular they are used to describe a phenomenon of thermal propagation in an absorptive medium where u stands for temperature [5].In other applications, u is a concentration and the process is described as diffusion with absorption.
The problem of determining necessary and sufficient conditions on the functions φ and F which ensure the existence of an extinction time for solutions of ( 1)-( 3) has been considered by several authors [1] [6] [7] [12].
In this section, we state the following result: Theorem 1 Assume that ( ) ( ) is a nonnegative solution of the problem (1)-( 3) where φ and F are nondecreasing, nonnegative derivatives functions and if ( ) . We have ( ) ( ) Multiplying Equation (1) by ( ) On one hand thanks to regularity of functions F , φ and u , we can write where n is the unit outward to ∂Ω , ds denotes an element of surface area, since u vanishes on ∂Ω and ( ) From (ii) and (iii), we deduce ≥ and ( ) 0 u φ′ ≥ , and according to (i), we obtain On the other hand, multiplying Equation ( 1) by u yields On the other hand, the increase of F implies that and according to (vi) we obtain Then considering (iv), we deduced , we obtain This gives after integrating Knowing that ( ) 0 0 w > .it follows ( ) 0 w t ≥ .The passage to the limit allows us to write In addition to the assumption of increase of F in Theorem 2.1, if we assume that ( ) C is a positive constant.then the following result is easily shown.Corollary 1 Suppose that the assumptions of the Theorem 2.1 are satisfied, and if (4) holds for (5) Indeed, for all ( ) u t solution of ( 1)-( 3), it comes from the assumption (4) that , and as a consequence of Theorem 2.1 □ In summary, under some assumptions we know that all nonnegative solutions of ( 1)-( 3) have extinction time as t → ∞ .We want to determine whether extinction occurs in finite time for any given φ and F .
It is well known that, in general, there is no classical solution to this nonlinear parabolic equation for arbitrary choices of φ and F .However, there are some works dealing with approximation of extinction time for solutions of (1).For example, in [13] a numerical method to approximate the solutions of (1) has been developed in the case 1 N = and in [14] an algorithm based on splitting technique was derived to compute the extinction time for solutions on a rectangular domain.
In order to determine the extinction time for some φ and F , we will derive in the next section a numerical scheme based on Donor-cell scheme.Given a sufficiently small parameter 0 >  , we would like to determine the positive real 0 T >  such that a solution ( ) u t of the problem (1)-(3) has to satisfy the above relation We shall call T  satisfying (6) as the  -extinction time.

Discretization of the Studied Domain
Let 2 Ω ⊂ » be a considered domain that we assume to be of irregular shape, we approximate Ω by a domain , , i i j j x x y y  and has center the point noted ( ) , The cells of R Ω are then divided into inner cell (which lie completely in Ω ), external cell (which lie in \ R Ω Ω) and boundary cells (which lie in a part of ∂Ω ).The problem model is then solved only in the inner cells.
A matrix of size max max i j × gives a description of the discretized domain.For example, consider three sets of indices I , B et E corresponding to the inner, boundary and external cells, we then admit to define the following matrix ∈  (7) the matrix to identify cell types.
The idea of this numerical treatment of general domains has been suggested by Griebel et al. in [15].An example of this numerical treatment is illustrated in Figure 1 and its matrix representative is given by the following (8).0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0

Spatial Discretization
First of all, let us give an approximation of the diffusion operator at the point , i j X which we rewrite as Let , i j V be the approximation of ( ) X .In the following we do apply a discretization that is similar to the one of Donor-cell scheme where the expression ( ) by a progressive finite differences scheme and ( )  by a central finite differences scheme.
Furthermore, we set ( ) and we note ( ) If x V denotes the vector of components , i j V then, one can write ( ) diag Φ the diagonal matrix whose diagonal is the vector Φ and p x D denotes forward differentiation matrix the x -direction.
On the other hand, denoting by , i j W the approached value of ( ) X and noting by ( ) the vector of the value of ( ) X , it follows through the central difference scheme, the relation which is written by the mean of the Equation ( 12) as ( ) where c x D denotes central differentiation matrix in the x -direction.
Similarly, given y W , the vector of the approached values of ( ) where p y D denotes forward differentiation matrix in the y -direction.From Equations ( 13) and ( 14), we deduce the approximation of the operator ( ) where ( ) L U is the vector of value of ( ) X .Thus, we have defined, an approximation operator L to approach the operator ( ) . However, it should be noted that Φ is a vector dependent of U .Considering lexicographic numerotation, we note by ( ) U t the vector of the values of u in points  15), the discrete system approaching the problem (1)-( 3) is rewritten by ( ) ( ) [ [ where 0 u is the continuous nonnegative function in Ω , vanishing on ∂Ω , and 0 q > .Equation (21) models heat propagation in medium where the solution u stands for temperature.
For our numerical experiments we have consider Figure 2 to be our studied domain and we have use discretization parameters We can see in Table 1 that this value is approximated by 0 0.61 T =  (25) Also, the extinction process is illustrated by Figure 4 where we can appreciate the numerical solution extinct in a finite time.

Concluding Remarks
In this paper, a numerical algorithm based on Donor-cell scheme was proposed in order to compute the extinction time for nonnegative solutions of some nonlinear parabolic equations on general domains.We have verified     experimentally for a class of nonlinear parabolic equations that the numerical algorithm is efficient for computing the extinction time of solutions.
In the works to come, it will be better to apply the numerical algorithm to study, for example, moving boundary problems and extinction problems in environment.

hΩΩ a grid of step x δ and y δ in x and y direction
whose boundary is specified by the set of boundary edges lying on gridlines.We imbed h Ω in a rectanrespectively.The set of points ( ) occupies the spatial region [ ] 1 1

Figure 1 .
Figure 1.An example of the discretization of a non rectangular domain into cells.
to approach the derivative on the set of the points , i jXwe can replace respectively by the matrices which are obtained by deleting the rows and columns corresponding to the indices of the points , Given the Equation (

»
to numerically estimate the extinction time for solutions of problem (21)-(23) with the initial condition given by -euclidian norm of the sequence n U solution of the numerical scheme (18) for various values of parameters n .Table1and Figure3clearly show that the approximation extinction time can be given by

Figure 2 .
Figure 2. Discretization of studied domain

Figure 3 .
Figure 3. Variation norm of the numerical solution.

Figure 4 .
Figure 4. Extinction phenomenon of the numerical solution.

Table 1 .
Numerical extinction time relatively to time iteration parameter n.