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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 714248, 18 pages
http://dx.doi.org/10.1155/2011/714248
Research Article

A Two-Species Cooperative Lotka-Volterra System of Degenerate Parabolic Equations

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

Received 13 December 2010; Revised 13 February 2011; Accepted 24 February 2011

Academic Editor: Elena Braverman

Copyright © 2011 Jiebao Sun et al. 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. A. Cui, “Global asymptotic stability in n-species cooperative system with time delays,” Systems Science and Mathematical Sciences, vol. 7, no. 1, pp. 45–48, 1994.
  2. D. Hu and Z. Zhang, “Four positive periodic solutions to a Lotka-Volterra cooperative system with harvesting terms,” Nonlinear Analysis: Real World Applications, vol. 11, no. 2, pp. 1115–1121, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. K. I. Kim and Z. Lin, “A degenerate parabolic system with self-diffusion for a mutualistic model in ecology,” Nonlinear Analysis: Real World Applications, vol. 7, no. 4, pp. 597–609, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. Y. Lou, T. Nagylaki, and W.-M. Ni, “On diffusion-induced blowups in a mutualistic model,” Nonlinear Analysis: Theory, Methods & Applications, vol. 45, no. 3, pp. 329–342, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. Z. Lin, J. Liu, and M. Pedersen, “Periodicity and blowup in a two-species cooperating model,” Nonlinear Analysis: Real World Applications, vol. 12, no. 1, pp. 479–486, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, NY, USA, 1992.
  7. Z. Y. Lu and Y. Takeuchi, “Permanence and global stability for cooperative Lotka-Volterra diffusion systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 19, no. 10, pp. 963–975, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. J. Sun, B. Wu, and D. Zhang, “Asymptotic behavior of solutions of a periodic diffusion equation,” Journal of Inequalities and Applications, vol. 2010, Article ID 597569, 12 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. S. Ahmad and A. C. Lazer, “Asymptotic behaviour of solutions of periodic competition diffusion system,” Nonlinear Analysis: Theory, Methods & Applications, vol. 13, no. 3, pp. 263–284, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. A. Tineo, “Existence of global coexistence state for periodic competition diffusion systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 19, no. 4, pp. 335–344, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. K. Gopalsamy, “Global asymptotic stability in a periodic Lotka-Volterra system,” Journal of the Australian Mathematical Society. Series B, vol. 27, no. 1, pp. 66–72, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. J. López-Gómez, “Positive periodic solutions of Lotka-Volterra reaction-diffusion systems,” Differential and Integral Equations, vol. 5, no. 1, pp. 55–72, 1992. View at Zentralblatt MATH
  13. J. J. Morgan and S. L. Hollis, “The existence of periodic solutions to reaction-diffusion systems with periodic data,” SIAM Journal on Mathematical Analysis, vol. 26, no. 5, pp. 1225–1232, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. C. V. Pao, “Periodic solutions of parabolic systems with nonlinear boundary conditions,” Journal of Mathematical Analysis and Applications, vol. 234, no. 2, pp. 695–716, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. C. Tian and Z. Lin, “Periodic solutions of reaction diffusion systems in a half-space domain,” Nonlinear Analysis: Real World Applications, vol. 9, no. 3, pp. 811–821, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. K. J. Brown and P. Hess, “Positive periodic solutions of predator-prey reaction-diffusion systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 16, no. 12, pp. 1147–1158, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. J.-B. Sun, C.-H. Jin, and Y.-Y. Ke, “Existence of non-trivial nonnegative periodic solutions for a nonlinear diffusion system,” Northeastern Mathematical Journal, vol. 23, no. 2, pp. 167–175, 2007. View at Zentralblatt MATH
  18. J.-B. Sun, “Asymptotic bounds for solutions of a periodic reaction diffusion system,” Applied Mathematics E-Notes, vol. 10, pp. 128–135, 2010.
  19. J. Yin and Y. Wang, “Asymptotic behaviour of solutions for nonlinear diffusion equation with periodic absorption,” in Partial Differential Equations and Their Applications (Wuhan, 1999), pp. 305–308, World Scientific, River Edge, NJ, USA, 1999. View at Zentralblatt MATH
  20. A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ, USA, 1964.
  21. O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, vol. 23 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 1967.
  22. Z. Q. Wu, J. X. Yin, and C. P. Wang, Introduction to Elliptic and Parabolic Equations, Scientific Publications, Beijing, China, 2003.
  23. Z. Wu, J. Zhao, J. Yin, and H. Li, Nonlinear Diffusion Equations, World Scientific, River Edge, NJ, USA, 2001.
  24. E. DiBenedetto, “Continuity of weak solutions to a general porous medium equation,” Indiana University Mathematics Journal, vol. 32, no. 1, pp. 83–118, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. M. M. Porzio and V. Vespri, “Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations,” Journal of Differential Equations, vol. 103, no. 1, pp. 146–178, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. P. Hess, M. A. Pozio, and A. Tesei, “Time periodic solutions for a class of degenerate parabolic problems,” Houston Journal of Mathematics, vol. 21, no. 2, pp. 367–394, 1995. View at Zentralblatt MATH