About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 927631, 5 pages
http://dx.doi.org/10.1155/2014/927631
Research Article

Non-Self-Similar Dead-Core Rate for the Fast Diffusion Equation with Dependent Coefficient

1College of Science, Xi’an University of Architecture & Technology, Xi’an 710054, China
2School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China

Received 24 February 2014; Accepted 26 May 2014; Published 9 June 2014

Academic Editor: Jaan Janno

Copyright © 2014 Liping Zhu and Zhengce Zhang. 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. Z. C. Zhang and B. Wang, “Spatial profile of the dead core for the fast diffusion equation with dependent coefficient,” International Journal of Differential Equations, vol. 2011, Article ID 751969, 9 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. C. Bandle, T. Nanbu, and I. Stakgold, “Porous medium equation with absorption,” SIAM Journal on Mathematical Analysis, vol. 29, no. 5, pp. 1268–1278, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. C. Bandle and I. Stakgold, “The formation of the dead core in parabolic reaction-diffusion problems,” Transactions of the American Mathematical Society, vol. 286, no. 1, pp. 275–293, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. L. Wang and Q. Chen, “On the dead core behavior for a semilinear heat equation,” Mathematica Applicata, vol. 10, no. 1, pp. 22–25, 1997. View at Zentralblatt MATH · View at MathSciNet
  5. X. Chen, J.-S. Guo, and B. Hu, “Dead-core rates for the porous medium equation with a strong absorption,” Discrete and Continuous Dynamical Systems B. A Journal Bridging Mathematics and Sciences, vol. 17, no. 6, pp. 1761–1774, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. J.-S. Guo, C.-T. Ling, and P. Souplet, “Non-self-similar dead-core rate for the fast diffusion equation with strong absorption,” Nonlinearity, vol. 23, no. 3, pp. 657–673, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. J.-S. Guo and P. Souplet, “Fast rate of formation of dead-core for the heat equation with strong absorption and applications to fast blow-up,” Mathematische Annalen, vol. 331, no. 3, pp. 651–667, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. J.-S. Guo and C.-C. Wu, “Finite time dead-core rate for the heat equation with a strong absorption,” The Tohoku Mathematical Journal, vol. 60, no. 1, pp. 37–70, 2008. View at MathSciNet
  9. I. Stakgold, “Reaction-diffusion problems in chemical engineering,” in Nonlinear Diffusion Problems, vol. 1224 of Lecture Notes in Mathematics, pp. 119–152, Springer, Berlin, Germany, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. J.-S. Guo, H. Matano, and C.-C. Wu, “An application of braid group theory to the finite time dead-core rate,” Journal of Evolution Equations, vol. 10, no. 4, pp. 835–855, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. M. A. Herrero and J. J. L. Velázquez, “Explosion de solutions d'équations paraboliques semilinéaires supercritiques,” Comptes Rendus de l'Académie des Sciences I. Mathématique, vol. 319, no. 2, pp. 141–145, 1994. View at Zentralblatt MATH · View at MathSciNet
  12. N. Mizoguchi, “Blow-up rate of type II and the braid group theory,” Transactions of the American Mathematical Society, vol. 363, no. 3, pp. 1419–1443, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. J.-S. Guo and B. Hu, “Blowup rate estimates for the heat equation with a nonlinear gradient source term,” Discrete and Continuous Dynamical Systems A, vol. 20, no. 4, pp. 927–937, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. Z. C. Zhang, “Gradient blowup rate for a viscous Hamilton-Jacobi equation with degenerate diffusion,” Archiv der Mathematik, vol. 100, no. 4, pp. 361–367, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. Z.-C. Zhang and B. Hu, “Boundary gradient blowup in a semilinear parabolic equation,” Discrete and Continuous Dynamical Systems A, vol. 26, no. 2, pp. 767–779, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. Z. C. Zhang and B. Hu, “Rate estimates of gradient blowup for a heat equation with exponential nonlinearity,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 12, pp. 4594–4601, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. Z. C. Zhang and Y. Li, “Blowup and existence of global solutions to nonlinear parabolic equations with degenerate diffusion,” Electronic Journal of Differential Equations, vol. 2013, no. 264, 17 pages, 2013. View at Zentralblatt MATH · View at MathSciNet
  18. Z. C. Zhang and Y. Y. Li, “Boundedness of global solutions for a heat equation with exponential gradient source,” Abstract and Applied Analysis, vol. 2012, Article ID 398049, 10 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. Z. C. Zhang and Y. Y. Li, “Gradient blowup solutions of a semilinear parabolic equation with exponential source,” Communications on Pure and Applied Analysis, vol. 12, no. 1, pp. 269–280, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. Z. C. Zhang and Z. J. Li, “A note on gradient blowup rate of the inhomogeneous Hamilton-Jacobi equations,” Acta Mathematica Scientia, vol. 33, no. 3, pp. 678–686, 2013.
  21. L. P. Zhu and Z. C. Zhang, “Rate of approach to the steady state for a diffusion-convection equation on annular domains,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 39, no. 2, 10 pages, 2012. View at MathSciNet
  22. M. S. Floater, “Blow-up at the boundary for degenerate semilinear parabolic equations,” Archive for Rational Mechanics and Analysis, vol. 114, no. 1, pp. 57–77, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. A. A. Lacey, “The form of blow-up for nonlinear parabolic equations,” Proceedings of the Royal Society of Edinburgh A. Mathematics, vol. 98, no. 1-2, pp. 183–202, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. C. P. Wang and S. N. Zheng, “Critical Fujita exponents of degenerate and singular parabolic equations,” Proceedings of the Royal Society of Edinburgh A. Mathematics, vol. 136, no. 2, pp. 415–430, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. C.-C. Wu and Z. C. Zhang, “Dead-core rates for the heat equation with a spatially dependent strong absorption,” Discrete and Continuous Dynamical Systems B. A Journal Bridging Mathematics and Sciences, vol. 18, no. 8, pp. 2203–2210, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. J.-S. Guo, C.-S. Lin, and M. Shimojo, “Blow-up for a reaction-diffusion equation with variable coefficient,” Applied Mathematics Letters, vol. 26, no. 1, pp. 150–153, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet