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2008 | 62 | 1 | 37-48
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Continuity of the quenching time in a semilinear parabolic equation

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In this paper, we consider the following initial-boundary value problem [...] where Ω is a bounded domain in RN with smooth boundary ∂Ω, p > 0, Δ is the Laplacian, v is the exterior normal unit vector on ∂Ω. 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 data u0. Finally, we give some numerical results to illustrate our analysis.
Opis fizyczny
  • Institut National Polytechnique, Houphouët-Boigny de Yamoussoukro, BP 1093 Yamoussoukro Côte d'Ivoire
  • Département de Mathmatiques et Informatiques, Université d'Abobo-Adjamé, UFR-SFA, 02 BP 801 Abidjan 02 Côte d'Ivoire
  • 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 (1998), 399-414.[Crossref]
  • Acker, A., Walter, W., The quenching problem for nonlinear parabolic differential equations, Ordinary and partial differential equations (Proc. Fourth Conf., Univ. Dundee, Dundee, 1976), Lecture Notes in Math., Vol. 564, Springer, Berlin, 1976, 1-12.
  • Acker, A., Kawohl, B., Remarks on quenching, Nonlinear Anal. 13 (1989), 53-61.
  • Bandle, C., Braumer, C. M., Singular perturbation method in a parabolic problem with free boundary, BAIL IV (Novosibirsk, 1986), Boole Press Conf. Ser., 8, Boole, Dún Laoghaire, 1986, 7-14.
  • Baras, P., Cohen, L., Complete blow-up after Tmaxfor the solution of a semilinear heat equation, J. Funct. Anal. 71 (1987), 142-174.[Crossref]
  • Boni, T. K., Extinction for discretizations of some semilinear parabolic equations, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), 795-800.
  • Boni, T. K., On quenching of solutions for some semilinear parabolic equations of second order, Bull. Belg. Math. Soc. 7 (2000), 73-95.
  • Cortazar, C., del Pino, M. and Elgueta, M., On the blow-up set for ut = Δum + um, m > 1, Indiana Univ. Math. J. 47 (1998), 541-561.
  • 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 Linéare 19 (2002), 927-960.
  • Deng, K., Levine, H. A, On the blow-up of ut at quenching, Proc. Amer. Math. Soc. 106 (1989), 1049-1056.
  • Deng, K., Xu, M., Quenching for a nonlinear diffusion equation with singular boundary condition, Z. Angew. Math. Phys. 50 (1999), 574-584.[Crossref]
  • Fermanian Kammerer, 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 (2000), 195-237.
  • Fila, M., Kawohl, B. and Levine, H. A., Quenching for quasilinear equations, Comm. Partial Differential Equations 17 (1992), 593-614.
  • Fila, M., Levine, H. A., Quenching on the boundary, Nonlinear Anal. 21 (1993), 795-802.
  • Friedman, A., McLeod, B., Blow-up of positive solutions of nonlinear heat equations, Indiana Univ. Math. J., 34 (1985), 425-477.[Crossref]
  • Galaktionov, V. A., Boundary value problems for the nonlinear parabolic equation ut = Δuσ+1 + uβ, Differential Equations 17 (1981), 551-555.
  • Galaktionov, V. A., Kurdjumov, S. P., Mihailov, A. P. and Samarskii, A. A., On unbounded solutions of the Cauchy problem for the parabolic equation ut = ∇(uσ∇u) + uβ, (Russian) Dokl. Akad. Nauk SSSR 252 (1980), no. 6, 1362-1364.
  • Galaktionov, V. A., Vazquez, J. L., Continuation of blow-up solutions of nonlinear heat equations in several space dimensions, Comm. Pure Appl. Math., 50 (1997), 1-67.
  • Galaktionov, V. A., Vazquez, J. L., The problem of blow-up in nonlinear parabolic equation, Current developments in partial differential equations (Temuco, 1999). Discrete Contin. Dyn. Syst. 8 (2002), 399-433.
  • Guo, J., On a quenching problem with Robin boundary condition, Nonlinear Anal. 17 (1991), 803-809.
  • Groisman, P., Rossi, J. D., Dependence of the blow-up time with respect to parameters and numerical approximations for a parabolic problem, Asymptot. Anal. 37 (2004), 79-91.
  • 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. Partial Differential Equations 28 (2003), 737-744.
  • Herrero, M. A., Velázquez, J. J. L., Generic behaviour of one-dimensional blow up patterns, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1992), no. 3, 381-450.
  • Kawarada, H., On solutions of initial-boundary problem for ut = uxx + 1/(1 - u), Publ. Res. Inst. Math. Sci. 10 (1974/75), 729-736.[Crossref]
  • Kirk, C. M., 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 (2003), 343-356.
  • Ladyzenskaya, A., Solonnikov, V. A. and Ural'ceva, N. N., Linear and quasilinear equations parabolic type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R. I., 1967.
  • Levine, H. A., The phenomenon of quenching: a survey, Trends in the theory and practice of nonlinear analysis (Arlington, Tex., 1984), North-Holland Math. Stud., 110, North-Holland, Amsterdam, 1985, 275-286.
  • Levine, H. A., The quenching of solutions of linear parabolic and hyperbolic equations with nonlinear boundary conditions, SIAM J. Math. Anal. 14 (1983), 1139-1152.
  • Levine, H. A., Quenching, nonquenching and beyond quenching for solution of some parabolic equations, Ann. Math. Pura Appl. (4) 155 (1989), 243-260.
  • Merle, F., Solution of a nonlinear heat equation with arbitrarily given blow-up points, Comm. Pure Appl. Math. 45 (1992), 293-300.
  • Nakagawa, T., Blowing up of the finite difference solution to ut = uxx + u2, Appl. Math. Optim. 2 (1975/76), 337-350.[Crossref]
  • Phillips, D., Existence of solution of quenching problems, Appl. Anal. 24 (1987), 253-264.[Crossref]
  • Protter, M. H., Weinberger, H. F., Maximum Principles in Differential Equations, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1967.
  • Quittner, P., Continuity of the blow-up time and a priori bounds for solutions in superlinear parabolic problems, Houston J. Math. 29 (2003), no. 3, 757-799 (electronic).
  • Sheng, Q., Khaliq, A. Q. M., A compound adaptive approach to degenerate nonlinear quenching problems, Numer. Methods Partial Differential Equations, 15 (1999), 29-47.
  • Walter, W, Differential- und Integral-Ungleichungen und ihre Anwendung bei Abschätzungs- und Eindeutigkeits-problemen, (German) Springer Tracts in Natural Philosophy, Vol. 2, Springer-Verlag, Berlin-New York, 1964.
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