Continuity of the quenching time in a semilinear heat equation with Neumann boundary condition

Firmin K. N'gohisse; Théodore K. Boni

doi:10.33993/jnaat391-921

Authors

Firmin K. N'gohisse Université d'Abobo-Adjamé, Côte d'Ivoire
Théodore K. Boni Institut National Polytechnique Houphouet-Boigny de Yamoussoukro, Côte d'Ivoire

DOI:

https://doi.org/10.33993/jnaat391-921

Keywords:

quenching, semilinear parabolic equation, numerical quenching time

Abstract views: 252

Abstract

This paper concerns the study of a semilinear parabolic equation subject to Neumann boundary conditions and positive initial datum. 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 initial datum. Finally, we give some numerical results to illustrate our analysis.

Downloads

Download data is not yet available.

References

Abia, L. M., López-Marcos, J. C. and Martínez J., On the blow-up time convergence of semidiscretizations of reaction-diffusion equations, Appl. Numer. Math., 26, pp. 399-414, 1998. https://doi.org/10.1016/s0168-9274(97)00105-0 DOI: https://doi.org/10.1016/S0168-9274(97)00105-0

Acker, A. and Walter, W., The quenching problem for nonlinear parabolic differential equations, Proc. Fourth Conf., Univ. Dundee, (1976), Lecture Notes in Math., Springer-Verlag, 564, pp. 1-12, 1976. https://doi.org/10.1007/bfb0087321 DOI: https://doi.org/10.1007/BFb0087321

Acker, A. and Kawohl, B., Remarks on quenching, Nonl. Anal. TMA, 13, pp. 53-61, 1989. https://doi.org/10.1016/0362-546x(89)90034-5 DOI: https://doi.org/10.1016/0362-546X(89)90034-5

Bandle, C. and Braumer, C. M., Singular perturbation method in a parabolic problem with free boundary, BAIL IV (Novosibirsk 1986), Boole Press Conf. Ser., Boole Dún Laoghaire, 8, pp. 7-14, 1986.

Baras, P. and Cohen, L., Complete blow-up after T_{max} for the solution of a semilinear heat equation, J. Funct. Anal., 71, pp. 142-174, 1987. https://doi.org/10.1016/0022-1236(87)90020-6 DOI: https://doi.org/10.1016/0022-1236(87)90020-6

Boni, T. K., Extinction for discretizations of some semilinear parabolic equations, C. R. Acad. Sci. Paris, Sér. I, Math., 333, pp. 795-800, 2001. https://doi.org/10.1016/s0764-4442(01)02078-x DOI: https://doi.org/10.1016/S0764-4442(01)02078-X

Boni, T. K., On quenching of solutions for some semilinear parabolic equations of second order, Bull. Belg. Math. Soc., 26, pp. 73-95, 2000. DOI: https://doi.org/10.36045/bbms/1103055721

Boni, T. K. and N'gohisse, F. K., Continuity of the quenching time in a semilinear parabolic equation, An. Univ. Mariae Curie Sklodowska, LXII pp. 37-48, 2008. DOI: https://doi.org/10.2478/v10062-008-0004-4

Cortazar, C., del Pino, M. and Elgueta, M., On the blow-up set for ut=Δum+um, m>1, Indiana Univ. Math. J., 47, pp. 541-561, 1998. DOI: https://doi.org/10.1512/iumj.1998.47.1399

Cortazar, C., del Pino, M. and Elgueta, M., Uniqueness and stability of regional blow-up in a porous-medium equation, Ann. Inst. H. Poincaré Anal. Non. Linéaire, 19, pp. 927-960, 2002. https://doi.org/10.1016/s0294-1449(02)00107-5 DOI: https://doi.org/10.1016/s0294-1449(02)00107-5

Deng, K. and Levine, H. A., On the blow-up of u_{t} at quenching, Proc. Amer. Math. Soc., 106, pp. 1049-1056, 1989. https://doi.org/10.1090/s0002-9939-1989-0969520-0 DOI: https://doi.org/10.1090/S0002-9939-1989-0969520-0

Deng, K. and Xu, M., Quenching for a nonlinear diffusion equation with singular boundary condition, Z. Angew. Math. Phys., 50, pp. 574-584, 1999. https://doi.org/10.1007/s000330050167 DOI: https://doi.org/10.1007/s000330050167

Fermanian, K. C., Merle, F. and Zaag, H., Stability of the blow-up profile of nonlinear heat equations from the dynamical system point of view, Math. Ann., 317, pp. 195-237, 2000. DOI: https://doi.org/10.1007/s002080000096

Fila, M., Kawohl, B. and Levine, H. A., Quenching for quasilinear equations, Comm. Part. Diff. Equat., 17, pp. 593-614, 1992.

Fila, M. and Levine, H. A., Quenching on the boundary, Nonl. Anal. TMA, 21, pp. 795-802, 1993. https://doi.org/10.1016/0362-546x(93)90124-b DOI: https://doi.org/10.1016/0362-546X(93)90124-B

Friedman, A., Partial Differential Equation of parabolic type, Prentice-Hall, Englewood chiffs, 1969.

Friedman, A. and McLeod, B., Blow-up of positive solutions of nonlinear heat equations, Indiana Univ. Math. J., 34, pp. 425-477, 1985. https://doi.org/10.1512/iumj.1985.34.34025 DOI: https://doi.org/10.1512/iumj.1985.34.34025

Galaktionov, V. A., Boundary value problems for the nonlinear parabolic equation ut=Δuσ+1+uβ, Diff. Equat., 17, pp. 551-555, 1981.

Galaktionov V. A. and Vazquez, J. L., Continuation of blow-up solutions of nonlinear heat equations in several space dimensions, Comm. Pure Appl. Math., 50, pp. 1-67, 1997. https://doi.org/10.1002/(SICI)1097-0312(199701)50:1<1::AID-CPA1>3.0.CO;2-H DOI: https://doi.org/10.1002/(SICI)1097-0312(199701)50:1<1::AID-CPA1>3.0.CO;2-H

Galaktionov, V. A. and Vazquez, J. L., The problem of blow-up in nonlinear parabolic equation, current developments in PDE (Temuco,1999), Discrete contin. Dyn. Syst., 8, pp. 399-433, 2002. https://doi.org/10.3934/dcds.2002.8.399 DOI: https://doi.org/10.3934/dcds.2002.8.399

Galaktionov, V. A., Miklhailov, S. P. K. and and Samarskii A. A., Unbounded solutions of the Cauchy problem for the parabolic equation ut=∇(uσ∇u)+uβ, Soviet Phys. Dokl., 25, pp. 458-459, 1980.

Guo, J., On a quenching problem with Robin boundary condition, Nonl. Anal. TMA, 17, pp. 803-809, 1991. https://doi.org/10.1016/0362-546x(91)90154-s DOI: https://doi.org/10.1016/0362-546X(91)90154-S

Groisman, P. and Rossi, J. D., Dependance of the blow-up time with respect to parameters and numerical approximations for a parabolic problem, Asympt. Anal., 37, pp. 79-91, 2004.

Groisman, P., Rossi, J. D. and Zaag, H., On the dependence of the blow-up time with respect to the initial data in a semilinear parabolic problem, Comm. Part. Diff. Equat., 28, pp. 737-744, 2003. https://doi.org/10.1081/pde-120020494 DOI: https://doi.org/10.1081/PDE-120020494

Kawarada, H., On solutions of initial-boundary problem for u_{t}=u_{xx}+1/(1-u), Publ. Res. Inst. Math. Sci., 10, pp. 729-736, 1975. https://doi.org/10.2977/prims/1195191889 DOI: https://doi.org/10.2977/prims/1195191889

Kirk, C. M. and Roberts, C. A., A review of quenching results in the context of nonlinear volterra equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10, pp. 343-356, 2003.

Ladyzenskaya A., Solonnikov V. A. and Ural'ceva, N. N., Linear and quasilinear equations parabolic type, Trans. Math. Monogr., 23, AMS, Providence, RI, 1967.

Levine, H. A., The phenomenon of quenching: a survey, Trends in the Theory and Practice of Nonlinear Analysis, North-Holland, Amsterdam, 110, pp. 275-286, 1985. https://doi.org/10.1016/s0304-0208(08)72720-8 DOI: https://doi.org/10.1016/S0304-0208(08)72720-8

Levine, H. A., The quenching of solutions of linear parabolic and hyperbolic equations with nonlinear boundary conditions, SIAM J. Math. Anal., 14, pp. 1139-1152, 1983. https://doi.org/10.1137/0514088 DOI: https://doi.org/10.1137/0514088

Levine, H. A., Quenching, nonquenching and beyond quenching for solution of some parabolic equations, Ann. Math. Pura Appl., 155, pp. 243-260, 1989. https://doi.org/10.1007/bf01765943 DOI: https://doi.org/10.1007/BF01765943

Merle, F., Solution of a nonlinear heat equation with arbitrarily given blow-up points, Comm. Pure Appl. Math., 45, pp. 263-300, 1992. https://doi.org/10.1002/cpa.3160450303 DOI: https://doi.org/10.1002/cpa.3160450303

Nakagawa, T., Blowing up on the finite difference solution to u_{t}=u_{xx}+u², Appl. Math. Optim., 2, pp. 337-350, 1976. https://doi.org/10.1007/bf01448176 DOI: https://doi.org/10.1007/BF01448176

Phillips, D., Existence of solutions of quenching problems, Appl. Anal., 24, pp. 253-264, 1987. https://doi.org/10.1080/00036818708839668 DOI: https://doi.org/10.1080/00036818708839668

Protter M. H. and Weinberger, H. F., Maximum principles in differential equations, Prentice Hall, Inc., Englewood Cliffs, NJ, 1967.

Pao, C. V., Nonlinear parabolic and elliptic equations, Plenum Press, New York, London, 1992. DOI: https://doi.org/10.1007/978-1-4615-3034-3

Quittner, P., Continuity of the blow-up time and a priori bounds for solutions in superlinear parabolic problems, Houston J. Math., 29, pp. 757-799, 2003 (electronic).

Shang, Q. and Khaliq, A. Q. M., A compound adaptive approach to degenerate nonlinear quenching problems, Numer. Meth. Part. Diff. Equat., 15, pp. 29-47, 1999. https://doi.org/10.1002/(SICI)1098-2426(199901)15:1<29::AID-NUM2>3.0.CO;2-L DOI: https://doi.org/10.1002/(SICI)1098-2426(199901)15:1<29::AID-NUM2>3.0.CO;2-L

Walter, W., Differential-und Integral-Ungleichungen, und ihre Anwendung bei Abschätzungs-und Eindeutigkeit-problemen, (German) Springer, Berlin, 2, 1964. DOI: https://doi.org/10.1007/978-3-662-35247-2