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

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 [

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 [

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 non- linear 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.

In this work, we are concerned with the following initial-boundary value problem:

where

Furthermore, for any

Nonlinear parabolic equations of type (1) appear in various applications. In particular they are used to des- cribe a phenomenon of thermal propagation in an absorptive medium where

The problem of determining necessary and sufficient conditions on the functions

In this section, we state the following result:

Theorem 1 Assume that

Proof: First, let us set

Multiplying Equation (1) by

On one hand thanks to regularity of functions

where

From (ii) and (iii), we deduce

Since

On the other hand, multiplying Equation (1) by

which we rewrite as

then the application

On the other hand, the increase of

Thus

and according to (vi) we obtain

This last inequality implies

Then considering (iv), we deduced

Setting

This gives after integrating

Knowing that

Finally,

In addition to the assumption of increase of F in Theorem 2.1, if we assume that

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

Indeed, for all

that gives

As

So

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

It is well known that, in general, there is no classical solution to this nonlinear parabolic equation for arbitrary choices of

In order to determine the extinction time for some

We shall call

Let

and

The cells of

A matrix of size

the matrix to identify cell types.

The idea of this numerical treatment of general domains has been suggested by Griebel et al. in [

First of all, let us give an approximation of the diffusion operator at the point

Let

Furthermore, we set

If

where

On the other hand, denoting by

noting by

the vector of the value of

which is written by the mean of the Equation (12) as

where

Similarly, given

where

From Equations (13) and (14), we deduce the approximation of the operator

where

However, it should be noted that

Considering lexicographic numerotation, we note by

Given the Equation (15), the discrete system approaching the problem (1)-(3) is rewritten by

where we have set

and where

the indices of the points

Furthermore, the initial condition is written

where

For a time step fixed

denote

where we set

It should be noticed that if the time step

Thus the extinction time is obtained using simple itrations process until the stopping criterion

is satisfied. Here

Algorithm 3.1

1. Read

2. Compute

3. Define

4. Compute the matrix

5. Set

6. Assign initial value to

7. Set

8. While

9. Compute

10. Set

11.

End while

Let

where

Equation (21) models heat propagation in medium where the solution

For our numerical experiments we have consider

We would like to numerically estimate the extinction time for solutions of problem (21)-(23) with the initial condition given by

First, for fixed accurate value

We can see in

Also, the extinction process is illustrated by

In this paper, a numerical algorithm based on Donor-cell scheme was proposed in order to compute the extinc- tion time for nonnegative solutions of some nonlinear parabolic equations on general domains. We have verified

n | 13,000 | 16,000 | 21,000 | 31,000 | 61,000 | 70,000 | 100,000 | >500,000 |
---|---|---|---|---|---|---|---|---|

T | 0.13 | 0.16 | 0.21 | 0.31 | 0.61 | 0.6123 | 0.6123 | 0.6123 |

experimentally for a class of nonlinear parabolic equations that the numerical algorithm is efficient for comput- ing the extinction time of solutions.

In the works to come, it will be better to apply the numerical algorithm to study, for example, moving boun- dary problems and extinction problems in environment.

We thank the Editor and the referee for their comments.