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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 289245, 8 pages
http://dx.doi.org/10.1155/2014/289245
Research Article

Blow-Up Analysis for a Quasilinear Parabolic Equation with Inner Absorption and Nonlinear Neumann Boundary Condition

School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China

Received 24 February 2014; Revised 11 April 2014; Accepted 11 April 2014; Published 30 April 2014

Academic Editor: Zhi-Bo Huang

Copyright © 2014 Zhong Bo Fang and Yan Chai. 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. J. Bebernes and D. Eberly, Mathematical Problems from Combustion Theory, vol. 83 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989. View at MathSciNet
  2. C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, NY, USA, 1992. View at MathSciNet
  3. J. L. Vazquez, The Porous Medium Equations: Mathematical Theory, Oxford University Press, Oxford, UK, 2007.
  4. J. Filo, “Diffusivity versus absorption through the boundary,” Journal of Differential Equations, vol. 99, no. 2, pp. 281–305, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. H. A. Levine and L. E. Payne, “Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time,” Journal of Differential Equations, vol. 16, pp. 319–334, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. B. Straughan, Explosive Instabilities in Mechanics, Springer, Berlin, Germany, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  7. A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov, Blow-Up in Quasilinear Parabolic Equations, vol. 19 of de Gruyter Expositions in Mathematics, Walter de Gruyter, Berlin, Germany, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  8. P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-Up, Global Existence and Steady States, Birkhäuser Advanced Texts, Birkhäuser, Basel, Switzerland, 2007. View at MathSciNet
  9. C. Bandle and H. Brunner, “Blowup in diffusion equations: a survey,” Journal of Computational and Applied Mathematics, vol. 97, no. 1-2, pp. 3–22, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. V. A. Galaktionov and J. L. Vázquez, “The problem of blow-up in nonlinear parabolic equations,” Discrete and Continuous Dynamical Systems, vol. 8, no. 2, pp. 399–433, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. H. A. Levine, “The role of critical exponents in blowup theorems,” SIAM Review, vol. 32, no. 2, pp. 262–288, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. H. A. Levine, “Nonexistence of global weak solutions to some properly and improperly posed problems of mathematical physics: the method of unbounded Fourier coefficients,” Mathematische Annalen, vol. 214, pp. 205–220, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. L. E. Payne, G. A. Philippin, and P. W. Schaefer, “Blow-up phenomena for some nonlinear parabolic problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 10, pp. 3495–3502, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. M. Marras and S. Vernier Piro, “On global existence and bounds for blow-up time in nonlinear parabolic problems with time dependent coefficients,” Discrete and Continuous Dynamical Systems, vol. 2013, pp. 535–544, 2013. View at Google Scholar
  15. Y. Li, Y. Liu, and C. Lin, “Blow-up phenomena for some nonlinear parabolic problems under mixed boundary conditions,” Nonlinear Analysis: Real World Applications, vol. 11, no. 5, pp. 3815–3823, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. Y. Li, Y. Liu, and S. Xiao, “Blow-up phenomena for some nonlinear parabolic problems under Robin boundary conditions,” Mathematical and Computer Modelling, vol. 54, no. 11-12, pp. 3065–3069, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. C. Enache, “Blow-up phenomena for a class of quasilinear parabolic problems under Robin boundary condition,” Applied Mathematics Letters, vol. 24, no. 3, pp. 288–292, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. J. Ding, “Global and blow-up solutions for nonlinear parabolic equations with Robin boundary conditions,” Computers & Mathematics with Applications, vol. 65, no. 11, pp. 1808–1822, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  19. Y. Liu, “Blow-up phenomena for the nonlinear nonlocal porous medium equation under Robin boundary condition,” Computers & Mathematics with Applications, vol. 66, no. 10, pp. 2092–2095, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  20. L. E. Payne, G. A. Philippin, and S. Vernier Piro, “Blow-up phenomena for a semilinear heat equation with nonlinear boundary conditon, I,” Zeitschrift für Angewandte Mathematik und Physik, vol. 61, no. 6, pp. 999–1007, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  21. L. E. Payne, G. A. Philippin, and S. Vernier Piro, “Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, II,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 4, pp. 971–978, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. Y. Liu, “Lower bounds for the blow-up time in a non-local reaction diffusion problem under nonlinear boundary conditions,” Mathematical and Computer Modelling, vol. 57, no. 3-4, pp. 926–931, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  23. Y. Liu, S. Luo, and Y. Ye, “Blow-up phenomena for a parabolic problem with a gradient nonlinearity under nonlinear boundary conditions,” Computers & Mathematics with Applications, vol. 65, no. 8, pp. 1194–1199, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  24. Z. B. Fang, R. Yang, and Y. Chai, “Lower bounds estimate for the blow-up time of a slow diffusion equation with nonlocal source and inner absorption,” Mathematical Problems in Engineering, vol. 2014, Article ID 764248, 6 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet