Numerical quenching for a semilinear parabolic equation with a potential and general nonlinearities
DOI:
https://doi.org/10.4067/S0716-09172008000300004Keywords:
Semidiscretizations, semilinear parabolic equation, quenching, numerical quenching time, convergence, semidiscretizaciones, ecuaciones parabólicas semilineales, tiempo numérico de quenching.Abstract
This paper concerns the study of the numerical approximation a semilinear parabolic equation subject to Neumann boundary conditions and positive initial data. We find some conditions under which the solution of a semidiscrete form of the above problem quenches in a fi- nite time and estimate its semidiscrete quenching time. We also prove that the semidiscrete quenching time converges to the real one when the mesh size goes to zero. A similar study has been also investigated taking a discrete form of the above problem. Finally, we give some numerical experiments to illustrate our analysis.References
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[2] A. Acker and B. Kawohl, Remarks on quenching, Nonl. Anal. TMA, 13, pp. 53-61, (1989).
[3] T. K. Boni, Extinction for discretizations of some semilinear parabolic equations, C. R. Acad. Sci. Paris, S´ er. I, 333, pp. 795-800, (2001).
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[6] J. S. Guo and B. Hu, The profile near quenching time for the solution of a singular semilinear heat equation, Proc. Edin. Math. Soc., 40, pp. 437-456, (1997).
[7] J. Guo, On a quenching problem with Robin boundary condition, Nonl. Anal. TMA, 17, pp. 803-809, (1991).
[8] V. A. Galaktionov and J. L. Vázquez, Continuation of blowup solutions of nonlinear heat equations in several space dimensions, Comm. Pure Appl. Math., 50, pp. 1-67, (1997).
[9] V. A. Galaktionov and J. L. Vázquez, The problem of blow-up in nonlinear parabolic equations, Discrete Contin. Systems A, 8, pp. 399-433, (2002).
[10] M. A. Herrero and J. J. L. Velazquez, Generic behaviour of one dimensional blow up patterns, Ann. Scuola Norm. Sup. di Pisa, XIX, pp. 381-950, (1992).
[11] C. M. Kirk and C. A. Roberts, 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).
[12] H. A. Levine, Quenching, nonquenching and beyond quenching for solutions of some parabolic equations, Annali Math. Pura Appl., 155), pp. 243-260, (1990).
[13] K. W. Liang, P. Lin and R. C. E. Tan, Numerical solution of quenching problems using mesh-dependent variable temporal steps, Appl. Numer. Math., 57, pp. 791-800, (2007).
[14] K. W. Liang, P. Lin, M. T. Ong and R. C. E. Tan, A splitting moving mesh method for reaction-diffusion equations of quenching type, J. Comput. Phys., 215, pp. 757-777, (2006).
[15] T. Nakagawa, Blowing up on the finite difference solution to ut= uxx+u2, Appl. Math. Optim., 2, pp. 337-350, (1976).
[16] D. Nabongo and T. K. Boni, Quenching for semidiscretization of a heat equation with a singular boundary condition, Asympt. Anal., 59, pp. 27-38, (2008).
[17] D. Nabongo and T. K. Boni, Quenching time of solutions for some nonlinear parabolic equations, An. St. Univ. Ovidius Constanta, 16, pp. 87-102, (2008).
[18] D. Nabongo and T. K. Boni, Quenching for semidiscretization of a semilinear heat equation with Dirichlet and Neumann boundary condition, Comment. Math. Univ. Carolinae, 49, pp. 463-475, (2008).
[19] D. Nabongo and T. K. Boni, Numerical quenching for a semilinear parabolic equation, Math. Modelling and Anal., To appear.
[20] M. H. Protter and H. F. Weinberger, Maximum principles in differential equations, Prentice Hall, Englewood Cliffs, NJ, (1967).
[21] Q. Sheng and A. Q. M. Khaliq, Adaptive algorithms for convectiondiffusion-reaction equations of quenching type, Dyn. Contin. Discrete Impuls. Syst. Ser. A, Math. Anal., 8, pp. 129-148, (2001).
[22] Q. Sheng and A. Q. M. Khaliq, A compound adaptive approach to degenerate nonlinear quenching problems, Numer. Methods PDE, 15, pp. 29-47, (1999).
[23] W. Walter, Differential-und Integral-Ungleichungen, Springer, Berlin, (1964).
[2] A. Acker and B. Kawohl, Remarks on quenching, Nonl. Anal. TMA, 13, pp. 53-61, (1989).
[3] T. K. Boni, Extinction for discretizations of some semilinear parabolic equations, C. R. Acad. Sci. Paris, S´ er. I, 333, pp. 795-800, (2001).
[4] T. K. Boni, On quenching of solutions for some semilinear parabolic equations of second order, Bull. Belg. Math. Soc., 7, pp. 73-95, (2000).
[5] M. Fila, B. Kawohl and H. A. Levine, Quenching for quasilinear equations, Comm. Part. Diff. Equat., 17, pp. 593-614, (1992).
[6] J. S. Guo and B. Hu, The profile near quenching time for the solution of a singular semilinear heat equation, Proc. Edin. Math. Soc., 40, pp. 437-456, (1997).
[7] J. Guo, On a quenching problem with Robin boundary condition, Nonl. Anal. TMA, 17, pp. 803-809, (1991).
[8] V. A. Galaktionov and J. L. Vázquez, Continuation of blowup solutions of nonlinear heat equations in several space dimensions, Comm. Pure Appl. Math., 50, pp. 1-67, (1997).
[9] V. A. Galaktionov and J. L. Vázquez, The problem of blow-up in nonlinear parabolic equations, Discrete Contin. Systems A, 8, pp. 399-433, (2002).
[10] M. A. Herrero and J. J. L. Velazquez, Generic behaviour of one dimensional blow up patterns, Ann. Scuola Norm. Sup. di Pisa, XIX, pp. 381-950, (1992).
[11] C. M. Kirk and C. A. Roberts, 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).
[12] H. A. Levine, Quenching, nonquenching and beyond quenching for solutions of some parabolic equations, Annali Math. Pura Appl., 155), pp. 243-260, (1990).
[13] K. W. Liang, P. Lin and R. C. E. Tan, Numerical solution of quenching problems using mesh-dependent variable temporal steps, Appl. Numer. Math., 57, pp. 791-800, (2007).
[14] K. W. Liang, P. Lin, M. T. Ong and R. C. E. Tan, A splitting moving mesh method for reaction-diffusion equations of quenching type, J. Comput. Phys., 215, pp. 757-777, (2006).
[15] T. Nakagawa, Blowing up on the finite difference solution to ut= uxx+u2, Appl. Math. Optim., 2, pp. 337-350, (1976).
[16] D. Nabongo and T. K. Boni, Quenching for semidiscretization of a heat equation with a singular boundary condition, Asympt. Anal., 59, pp. 27-38, (2008).
[17] D. Nabongo and T. K. Boni, Quenching time of solutions for some nonlinear parabolic equations, An. St. Univ. Ovidius Constanta, 16, pp. 87-102, (2008).
[18] D. Nabongo and T. K. Boni, Quenching for semidiscretization of a semilinear heat equation with Dirichlet and Neumann boundary condition, Comment. Math. Univ. Carolinae, 49, pp. 463-475, (2008).
[19] D. Nabongo and T. K. Boni, Numerical quenching for a semilinear parabolic equation, Math. Modelling and Anal., To appear.
[20] M. H. Protter and H. F. Weinberger, Maximum principles in differential equations, Prentice Hall, Englewood Cliffs, NJ, (1967).
[21] Q. Sheng and A. Q. M. Khaliq, Adaptive algorithms for convectiondiffusion-reaction equations of quenching type, Dyn. Contin. Discrete Impuls. Syst. Ser. A, Math. Anal., 8, pp. 129-148, (2001).
[22] Q. Sheng and A. Q. M. Khaliq, A compound adaptive approach to degenerate nonlinear quenching problems, Numer. Methods PDE, 15, pp. 29-47, (1999).
[23] W. Walter, Differential-und Integral-Ungleichungen, Springer, Berlin, (1964).
Published
2017-04-06
How to Cite
[1]
T. K. Boni and T. K. Kouakou, “Numerical quenching for a semilinear parabolic equation with a potential and general nonlinearities”, Proyecciones (Antofagasta, On line), vol. 27, no. 3, pp. 259-287, Apr. 2017.
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