International Journal of Differential Equations

International Journal of Differential Equations / 2007 / Article

Research Article | Open Access

Volume 2007 |Article ID 87696 | 16 pages | https://doi.org/10.1155/2007/87696

Uniform Blow-Up Rates and Asymptotic Estimates of Solutions for Diffusion Systems with Nonlocal Sources

Academic Editor: Nicola Bellomo
Received13 Apr 2006
Revised27 Nov 2006
Accepted19 Dec 2006
Published13 Feb 2007

Abstract

This paper investigates the local existence of the nonnegative solution and the finite time blow-up of solutions and boundary layer profiles of diffusion equations with nonlocal reaction sources; we also study the global existence and that the rate of blow-up is uniform in all compact subsets of the domain, the blow-up rate of |u(t)| is precisely determined.

References

  1. P. Souplet, “Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source,” Journal of Differential Equations, vol. 153, no. 2, pp. 374–406, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  2. K. Bimpong-Bota, P. Ortoleva, and J. Ross, “Far-from-equilibrium phenomena at local sites of reaction,” Journal of Chemical Physics, vol. 60, no. 8, pp. 3124–3133, 1974. View at: Publisher Site | Google Scholar
  3. H. Li and M. Wang, “Blow-up properties for parabolic systems with localized nonlinear sources,” Applied Mathematics Letters, vol. 17, no. 7, pp. 771–778, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  4. Q. Liu, Y. Li, and H. Gao, “Uniform blow-up rate for diffusion equations with nonlocal nonlinear source,” to appear in Nonlinear Analysis. View at: Google Scholar
  5. M. Wang, “Blow-up rate estimates for semilinear parabolic systems,” Journal of Differential Equations, vol. 170, no. 2, pp. 317–324, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  6. G. Caristi and E. Mitidieri, “Blow-up estimates of positive solutions of a parabolic system,” Journal of Differential Equations, vol. 113, no. 2, pp. 265–271, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  7. J. Furter and M. Grinfeld, “Local vs. nonlocal interactions in population dynamics,” Journal of Mathematical Biology, vol. 27, no. 1, pp. 65–80, 1989. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  8. 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: L8%20blow-up%20estimate%20for%20a%20nonlinear%20heat%20equation&author=F. B. Weissler&publication_year=1985" target="_blank">Google Scholar | Zentralblatt MATH | MathSciNet
  9. Y. Giga and R. V. Kohn, “Asymptotically self-similar blow-up of semilinear heat equations,” Communications on Pure and Applied Mathematics, vol. 38, no. 3, pp. 297–319, 1985. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  10. Z. Yang and Q. Lu, “Blow-up estimates for a non-Newtonian filtration equation,” Journal of Mathematical Research and Exposition, vol. 23, no. 1, pp. 7–14, 2003. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  11. Z. Yang, “Nonexistence of positive solutions to a quasi-linear elliptic equation and blow-up estimates for a nonlinear heat equation,” Rocky Mountain Journal of Mathematics, vol. 36, no. 4, pp. 1399–1414, 2006. View at: Google Scholar
  12. M. Wang, “Global existence and finite time blow up for a reaction-diffusion system,” Zeitschrift für Angewandte Mathematik und Physik, vol. 51, no. 1, pp. 160–167, 2000. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  13. M. Wang, “Blowup estimates for a semilinear reaction diffusion system,” Journal of Mathematical Analysis and Applications, vol. 257, no. 1, pp. 46–51, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  14. V. A. Galaktionov, S. P. Kurdyumov, and A. A. Samarskii, “A parabolic system of quasilinear equations. I,” Differential Equations, vol. 19, pp. 1558–1571, 1983. View at: Google Scholar | Zentralblatt MATH
  15. V. A. Galaktionov, S. P. Kurdyumov, and A. A. Samarskii, “A parabolic system of quasilinear equations. II,” Differential Equations, vol. 21, pp. 1049–1062, 1985. View at: Google Scholar | Zentralblatt MATH
  16. Y. Chen, “Blow-up for a system of heat equations with nonlocal sources and absorptions,” Computers & Mathematics with Applications, vol. 48, no. 3-4, pp. 361–372, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  17. M. Escobedo and M. A. Herrero, “A semilinear parabolic system in a bounded domain,” Annali di Matematica Pura ed Applicata. Serie Quarta, vol. 165, no. 1, pp. 315–336, 1993. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  18. Z. Yang and Q. Lu, “Nonexistence of positive solutions to a quasilinear elliptic system and blow-up estimates for a non-Newtonian filtration system,” Applied Mathematics Letters, vol. 16, no. 4, pp. 581–587, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  19. Z. Yang and Q. Lu, “Blow-up estimates for a quasi-linear reaction-diffusion system,” Mathematical Methods in the Applied Sciences, vol. 26, no. 12, pp. 1005–1023, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  20. F. Dickstein and M. Escobedo, “A maximum principle for semilinear parabolic systems and applications,” Nonlinear Analysis, vol. 45, no. 7, pp. 825–837, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  21. P. Pucci and J. Serrin, “The strong maximum principle revisited,” Journal of Differential Equations, vol. 196, no. 1, pp. 1–66, 2004. View at: Publisher Site | Google Scholar | MathSciNet
  22. 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet

Copyright © 2007 Zhoujin Cui and Zuodong Yang. 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.

0 Views | 0 Downloads | 0 Citations
 PDF  Download Citation  Citation
 Order printed copiesOrder