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Journal of Applied Mathematics
Volume 2012, Article ID 202137, 8 pages
http://dx.doi.org/10.1155/2012/202137
Research Article

Blow-Up Time for Nonlinear Heat Equations with Transcendental Nonlinearity

Department of Applied Mathematics, Dankook University, Anseo-Dong 29, Cheonan, Chungnam 330-714, Republic of Korea

Received 9 April 2012; Accepted 11 June 2012

Academic Editor: Julián López-Gómez

Copyright © 2012 Hee Chul Pak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. P. Baras, “Non-unicité des solutions d'une equation d'évolution non-linéaire,” Annales de la Faculté des Sciences de Toulouse, vol. 5, no. 3-4, pp. 287–302, 1983. View at Google Scholar · View at Zentralblatt MATH
  2. H. Brézis and A. Friedman, “Nonlinear parabolic equations involving measures as initial conditions,” Journal de Mathématiques Pures et Appliquées. Neuvième Série, vol. 62, no. 1, pp. 73–97, 1983. View at Google Scholar · View at Zentralblatt MATH
  3. H. Brezis, L. A. Peletier, and D. Terman, “A very singular solution of the heat equation with absorption,” Archive for Rational Mechanics and Analysis, vol. 95, no. 3, pp. 185–209, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Clarendon Press, Oxford, UK, 1988.
  5. A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ, USA, 1964.
  6. A. Haraux and F. B. Weissler, “Nonuniqueness for a semilinear initial value problem,” Indiana University Mathematics Journal, vol. 31, no. 2, pp. 167–189, 1982. View at Publisher · View at Google Scholar
  7. O. A. Ladyzhenskaya, V. A. Solonikov, and N. N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, 1968.
  8. P. Quittner, “Continuity of the blow-up time and a priori bounds for solutions in superlinear parabolic problems,” Houston Journal of Mathematics, vol. 29, no. 3, pp. 757–799, 2003. View at Google Scholar · View at Zentralblatt MATH
  9. P. Quittner, P. Souplet, and M. Winkler, “Initial blow-up rates and universal bounds for nonlinear heat equations,” Journal of Differential Equations, vol. 196, no. 2, pp. 316–339, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. F. B. Weissler, “Local existence and nonexistence for semilinear parabolic equations in Lp,” Indiana University Mathematics Journal, vol. 29, no. 1, pp. 79–102, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. D. Andreucci and E. DiBenedetto, “On the Cauchy problem and initial traces for a class of evolution equations with strongly nonlinear sources,” Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, vol. 18, no. 3, pp. 363–441, 1991. View at Google Scholar · View at Zentralblatt MATH
  12. M.-F. Bidaut-Véron, “Initial blow-up for the solutions of a semilinear parabolic equation with source term,” in Équations aux Dérivées Partielles et Applications, pp. 189–198, Gauthier-Villars, Paris, France, 1998. View at Google Scholar · View at Zentralblatt MATH
  13. S. Cano-Casanova and J. López-Gómez, “Blow-up rates of radially symmetric large solutions,” Journal of Mathematical Analysis and Applications, vol. 352, no. 1, pp. 166–174, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. S. Filippas, M. A. Herrero, and J. J. L. Velázquez, “Fast blow-up mechanisms for sign-changing solutions of a semilinear parabolic equation with critical nonlinearity,” The Royal Society of London. Proceedings. Series A, vol. 456, no. 2004, pp. 2957–2982, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. A. Friedman and B. McLeod, “Blow-up of positive solutions of semilinear heat equations,” Indiana University Mathematics Journal, vol. 34, no. 2, pp. 425–447, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. Y. Giga and R. V. Kohn, “Characterizing blowup using similarity variables,” Indiana University Mathematics Journal, vol. 36, no. 1, pp. 1–40, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. M. A. Herrero and J. J. L. Velázquez, “Explosion de solutions d'équations paraboliques semilinéaires supercritiques,” Comptes Rendus de l'Académie des Sciences, vol. 319, no. 2, pp. 141–145, 1994. View at Google Scholar · View at Zentralblatt MATH
  18. J. López-Gómez and P. Quittner, “Complete and energy blow-up in indefinite superlinear parabolic problems,” Discrete and Continuous Dynamical Systems. Series A, vol. 14, no. 1, pp. 169–186, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. J. Matos and P. Souplet, “Universal blow-up rates for a semilinear heat equation and applications,” Advances in Differential Equations, vol. 8, no. 5, pp. 615–639, 2003. View at Google Scholar · View at Zentralblatt MATH
  20. F. Merle and H. Zaag, “Refined uniform estimates at blow-up and applications for nonlinear heat equations,” Geometric and Functional Analysis, vol. 8, no. 6, pp. 1043–1085, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. F. B. Weissler, “An L blow-up estimate for a nonlinear heat equation,” Communications on Pure and Applied Mathematics, vol. 38, no. 3, pp. 291–295, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. C. Fermanian Kammerer and H. Zaag, “Boundedness up to blow-up of the difference between two solutions to a semilinear heat equation,” Nonlinearity, vol. 13, no. 4, pp. 1189–1216, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. P. Baras and L. Cohen, “Complete blow-up after T max for the solution of a semilinear heat equation,” Journal of Functional Analysis, vol. 71, no. 1, pp. 142–174, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. V. A. Galaktionov and J. L. Vazquez, “Continuation of blowup solutions of nonlinear heat equations in several space dimensions,” Communications on Pure and Applied Mathematics, vol. 50, no. 1, pp. 1–67, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. A. A. Lacey and D. Tzanetis, “Complete blow-up for a semilinear diffusion equation with a sufficiently large initial condition,” IMA Journal of Applied Mathematics, vol. 41, no. 3, pp. 207–215, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. P. Groisman and J. D. Rossi, “Dependence of the blow-up time with respect to parameters and numerical approximations for a parabolic problem,” Asymptotic Analysis, vol. 37, no. 1, pp. 79–91, 2004. View at Google Scholar · View at Zentralblatt MATH
  27. F. Merle, “Solution of a nonlinear heat equation with arbitrarily given blow-up points,” Communications on Pure and Applied Mathematics, vol. 45, no. 3, pp. 263–300, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. P. Quittner and F. Simondon, “A priori bounds and complete blow-up of positive solutions of indefinite superlinear parabolic problems,” Journal of Mathematical Analysis and Applications, vol. 304, no. 2, pp. 614–631, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. L. E. Payne, Improperly Posed Problems in Partial Differential Equations, SIAM, Philadelphia, Pa, USA, 1975.
  30. L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, USA, 1998.
  31. I. Chavel, Eigenvalues in Riemannian Geometry, vol. 115 of Pure and Applied Mathematics, Academic Press, Orlando, Fla, USA, 1984.