¹Institut National Polytechnique Houphouët-Boigny de Yamoussoukro, Abidjan, Côte d'Ivoire. Email: theokboni@yahoo.fr
²Université d'Abobo-Adjamé, UFR-SFA, Abidjan, Côte d'Ivoire. Email: kkthibaut@yahoo.fr

In this paper, we consider a semilinear heat equation with a potential subject to Neumann boundary conditions and positive initial data. Under some assumptions, we show that the solution of the above problem quenches in a finite time and estimate its quenching time. We also prove the continuity of the quenching time as a function of the potential and the initial data. Finally, we give some numerical results to illustrate our analysis.

Key words: Quenching, semilinear heat equation, numerical quenching time.

En este trabajo consideramos una ecuación semilineal de calor con potencial, sujeta a condiciones de Neumann de frontera y datos iniciales positivos. Bajo ciertos supuestos mostramos que la solución de dicha ecuación se apaga en tiempo finito y estimamos el tiempo en que lo hace. También probamos la continuidad del tiempo de extinción en función del potencial y de los datos iniciales. Finalmente damos algunos resultados numéricos que ilustran nuestro análisis.

Palabras clave: Apagamiento, ecuación de calor semilineal, tiempo de apagamiento numérico.

Texto completo disponible en PDF

References

[1] Abia, L., López-Marcos, J. & Martínez, J., `On the blow-up time convergence ofsemidiscretizations of reaction-diffusion equations´, Appl. Numer. Math. 26, (1998), 399-414.         [ Links ]

[2] Acker, A. & Kawohl, B., `Remarks on quenching´, Nonl. Anal. TMA 13, (1989), 53-61.         [ Links ]

[3] Acker, A. & Walter, W., The quenching problem for nonlinear parabolic differential equations, Springer Verlag, New York, 1989. Lecture Notes in Math., 564.         [ Links ]

[4] Bandle, C. & Braumer, C., Singular perturbation method in a parabolic problem with free boundary, `BAIL IVth Conference, Boole Press Conf.´, (1986), 8, Novosibirsk, p. 7-14.         [ Links ]

[5] Baras, P. & Cohen, L., `Complete blow-up after T_max for the solution of a semilinear heat equation´, J. Funct. Anal. 71, (1987), 142-174.         [ Links ]

[6] Boni, T., `On quenching of solutions for some semilinear parabolic equations of second order´, Bull. Belg. Math. Soc. 7, (2000), 73-95.         [ Links ]

[7] Boni, T., `Extinction for discretizations of some semilinear parabolic equations´, C. R. Acad. Sci. Paris 333, (2001), 795-800.         [ Links ]

[8] Boni, T. & N'gohisse, F., Continuity of the quenching time in a semilinear heat equation. An. Univ. Mariae Curie Sklodowska. To appear..         [ Links ]

[9] Cortazar, C., Pino, M. d. & Elgueta, M., `On the blow-up set for u_tΔ u^m+u^m, m > 1´, Indiana Univ. Math. J. 47, (1998), 541-561.         [ Links ]

[10] Cortazar, C., Pino, M. d. & Elgueta, M., `Uniqueness and stability of regional blow-up in a porous-medium equation´, Ann. Inst. H. Poincaré Anal. Non Lineaire 19, (2002), 927-960.         [ Links ]

[11] Deng, K. & Levine, H., `On the blow-up of u_t at quenching´, Proc. Amer. Math. Soc. 106, (1989), 1049-1056.         [ Links ]

[12] Deng, K. & Xu, M., `Quenching for a nonlinear diffusion equation with singular boundary condition´, Z. Angew. Math. Phys. 50, (1999), 574-584.         [ Links ]

[13] Fermanian, C., Merle, F. & H. Zaag,, `Stability of the blow-up profile of nonlinear heat equations from the dynamical system point of view´, Math. Ann. 317, (2000), 195-237.         [ Links ]

[14] Fila, M., , B. K. & Levine, H., `Quenching for quasilinear equations´, Comm. Part. Diff. Equat. 17, (1992), 593-614.         [ Links ]

[15] Fila, M. & Levine, H., `Quenching on the boundary´, Nonl. Anal. TMA 21, (1993), 795-802.         [ Links ]

[16] Friedman, A. & McLeod, B., `Blow-up of positive solutions of nonlinear heat equations´, Indiana Univ. Math. J. 34, (1985), 425-477.         [ Links ]

[17] Galaktionov, V., `Boundary value problems for the nonlinear parabolic equation u_tΔ u^σ+1+u^β´, Differ. Equat. 17, (1981), 551-555.         [ Links ]

[18] Galaktionov, V. & Vazquez, J., `Continuation of blow-up solutions of nonlinear heat equations in several space dimensions´, Comm. Pure Appl. Math. 50, (1997), 1-67.         [ Links ]

[19] Galaktionov, V. & Vazquez, J., `The problem of blow-up in nonlinear parabolic equation´, Discrete Contin. Dyn. Syst. 8, (2002), 399-433.         [ Links ]

[20] Groisman, P. & Rossi, J., `Dependance of the blow-up time with respect to parameters and numerical approximations for a parabolic problem´, Asympt. Anal. 37, (2004), 79-91.         [ Links ]

[21] Groisman, P., Rossi, J. & Zaag, H., `On the dependence of the blow-up time with respect to the initial data in a semilinear parabolic problem´, Comm. Part. Differential Equations 28, (2003), 737-744.         [ Links ]

[22] Guo, J., `On a quenching problem with Robin boundary condition´, Nonl. Anal. TMA 17, (1991), 803-809.         [ Links ]

[23] Herreros, M. & Vazquez, J., `Generic behaviour of one-dimensional blow up patterns´, Ann. Scuola. Norm. Sup. Pisa Cl. Sci. 19, 3 (1992), 381-450.         [ Links ]

[24] Kawarada, H., `On solutions of initial-boundary problem for u_tu_xx+1/(1-u)´, Pul. Res. Inst. Math. Sci. 10, (1975), 729-736.         [ Links ]

[25] Kirk, C. & Roberts, C., `A review of quenching results in the context of nonlinear volterra equations´, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 10, (2003), 343-356.         [ Links ]

[26] Ladyzenskaya, A., Solonnikov, V. & Ural'ceva, N. N., Linear and quasililear equations parabolic type, `Trans. Math. Monogr.´, Vol. 23, AMS, Providence, RI, 1968.         [ Links ]

[27] Levine, H., `The quenching of solutions of linear parabolic and hyperbolic equations with nonlinear boundary conditions´, SIAM J. Math. Anal. 14, (1983), 1139-1152.         [ Links ]

[28] Levine, H., Trends in the Theory and Practice of Nonlinear Analysis, North Holland, Amsterdam, (1985), chapter The phenomenon of quenching: a survey. 275-286.         [ Links ]

[29] Levine, H., `Quenching, nonquenching and beyond quenching for solution of some parabolic equations´, Ann. Math. Pura Appl. 155, (1989), 243-260.         [ Links ]

[30] Merle, F., `Solution of a nonlinear heat equation with arbitrarily given blow-up points´, Comm. Pure Appl. Math. 45, (1992), 293-300.         [ Links ]

[31] Nabongo, D. & Boni, T., `Quenching time of solutions for some nonlinear parabolic equations´, An. St. Univ. Ovidius Constanta, Ser. Math. 16, (2008), 87-102.         [ Links ]

[32] Nakagawa, T., `Blowing up on the finite difference solution to u_tu_xx+u²´, Appl. Math. Optim. 2, (1976), 337-350.         [ Links ]

[33] Phillips, D., `Existence of solution of quenching problems´, Appl. Anal. 24, (1987), 253-264.         [ Links ]

[34] Protter, M. & Weinberger, H., Maximum principles in differential equations, Prentice Hall, Englewood Cliffs, NJ, 1967.         [ Links ]

[35] Quittner, P., `Continuity of the blow-up time and a priori bounds for solutions in superlinear parabolic problems´, Houston J. Math. 29, 3 (2003), 757-799.         [ Links ]

[36] Shang, Q. & Khaliq, A., `A compound adaptive approach to degenerate nonlinear quenching problems´, Numer. Methods Partial Differential Equations 15, (1999), 29-47.         [ Links ]

[37] V. A. Galaktionov, S. K. & Samarskii, A., `Unbounded solutions of the Cauchy problem for the parabolic equation u_t\nabla(u^σ\nabla u)+u^β´, Soviet Phys. Dokl. 25, (1980), 458-459.         [ Links ]

[38] Walter, W., Differential-und Integral-Ungleichungen, Springer Verlag, Berlin, 1964.         [ Links ]

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX: