Numerical Quenching solutions of Localized Semilinear Parabolic Equation with a Variable Reaction

N'Guessan Koffi; Diabate Nabongo

N'Guessan Koffi UFR SED, Alassane Ouattara University of Bouake, 02 BP 801 Abidjan 02 Cote d'Ivoire
Diabate Nabongo UFR SED, Alassane Ouattara University of Bouake, 02 BP 801 Abidjan 02 Cote d'Ivoire

Keywords: semidiscretizations, localized semilinear parabolicequation, semidiscrete quenching time, convergence.

Abstract

In this paper, we study the semidiscrete approximation for the following initial-boundary value problem

and l=1/2. We prove, under suitable conditions on p(x) and initial datum, that the semidiscrete solution quenches in a finite time and estimate its semidiscrete quenching time. We also establish the convergence of the semidiscrete quenching time to the theoretical one when the mesh size tendsto zero. Finally, we give some numerical experiments for a best illustration ofour analysis.

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References

L. Abia, J. C. López-Marcos and J. Martinez, On the blow-up time convergence of semidiscretizations of reaction-diffusion equations, Appl. Numer. Math., 26 (1998), 399-414.

T. K. Boni, Extinction for discretizations of some semilinear parabolic equations, C.R.A.S, Serie I, 333 (2001), 795-800.

T. K. Boni, On quenching of solutions for some semilinear parabolic equations of second order, Bull. Belg. Math. Soc., 7 (2000), 73-95.

K. Bimpong-Bota, P. Ortoleva and J. Ross, Far- from equilibrium phenomenon at local sites of reactions, J. Chem. Physic., 60 (1974), 3124-3133.

J. M. Chadam and H. M. Yin, A diffusion equation with localized chemical reactions, Proc. Edinb. Math. Soc., 37 (1994), 101-118.

K. Deng and C. A. Roberts , Quenching for a diffusive equation with a concentrated singularity, Diff. Int. Equat., 10 (1997), 369-379.

K. Deng, Dynamical behavior of solution of a semilinear parabolic equation with nonlocal singularity, SIAM J. Math. Anal., 26 (1995), 98-111.

R. Ferreira, A. De Pablo, M. Pérez-Llanos and J. D. Rossi, Critical exponents for a semilinear parabolic equation with variable reaction, Submitted.

A. Friedman, Partial differential equation of parabolic type, Prentice all, Englewood Cliffs, NJ, (1964).

H. A. Levine, Quenching, nonquenching and beyond quenching for solution of some semilinear parabolic equations, Annali Math. Pura Appl.,

(1990), 243-260.

T. Nakagawa, Blowing up on the finite difference solution to ut = uxx + u2, Appl. Math. Optim., 2 (1976), 337-350.

D. Nabongo and T. K. Boni , Numerical quenching for semilinear parabolic equations, Math. Model. and Anal., 13(4) (2008), 521-538.

D. Nabongo and T. K. Boni , Numerical quenching solutions of localized semilinear parabolic equation, Bolet. de Mate., Nueva Serie, 14(2) (2007), 92-109.

P. Ortoleva and J. Ross, Local structures in chemical reactions with heterogeneous catalysis, J. Chem. Phys., 56 (1972), 4397-4452.

L. Wamg and Q. Chen, The asymptotic behavior of blow-up solution of localized nonlinear equations, J. Math. Anal. Appl., 200 (1996), 315-321.