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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 643897, 6 pages
http://dx.doi.org/10.1155/2014/643897
Research Article

Positive Solutions of a Nonlinear Parabolic Partial Differential Equation

School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China

Received 13 March 2014; Accepted 19 May 2014; Published 28 May 2014

Academic Editor: Dragos-Patru Covei

Copyright © 2014 Chengbo Zhai and Shunyong Li. 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. E. Acerbi and G. Mingione, “Regularity results for stationary electro-rheological fluids,” Archive for Rational Mechanics and Analysis, vol. 164, no. 3, pp. 213–259, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. S. Antontsev and S. Shmarev, “Blow-up of solutions to parabolic equations with nonstandard growth conditions,” Journal of Computational and Applied Mathematics, vol. 234, no. 9, pp. 2633–2645, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. L. Diening, P. Harjulehto, P. Hästö, and M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, vol. 2017 of Lecture Notes in Mathematics, Springer, Heidelberg, Germany, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. B. Hu and H.-M. Yin, “Semilinear parabolic equations with prescribed energy,” Rendiconti del Circolo Matematico di Palermo. Serie II, vol. 44, no. 3, pp. 479–505, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. J. P. Pinasco, “Blow-up for parabolic and hyperbolic problems with variable exponents,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 3-4, pp. 1094–1099, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States, Birkhauser Advanced Texts, Berlin, Germany, 2007.
  7. R. Ferreira, A. de Pablo, M. Pérez-LLanos, and J. D. Rossi, “Critical exponents for a semilinear parabolic equation with variable reaction,” Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, vol. 142, no. 5, pp. 1027–1042, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. W. Gao and Y. Han, “Blow-up of a nonlocal semilinear parabolic equation with positive initial energy,” Applied Mathematics Letters, vol. 24, no. 5, pp. 784–788, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. M. Ishiwata and T. Suzuki, “Positive solution to semilinear parabolic equation associated with critical Sobolev exponent,” Nonlinear Differential Equations and Applications, vol. 20, no. 4, pp. 1553–1576, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. W. Liu and M. Wang, “Blow-up of the solution for a p-Laplacian equation with positive initial energy,” Acta Applicandae Mathematicae, vol. 103, no. 2, pp. 141–146, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  11. C. V. Pao and W. H. Ruan, “Positive solutions of quasilinear parabolic systems with Dirichlet boundary condition,” Journal of Differential Equations, vol. 248, no. 5, pp. 1175–1211, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. X. Wu, B. Guo, and W. Gao, “Blow-up of solutions for a semilinear parabolic equation involving variable source and positive initial energy,” Applied Mathematics Letters, vol. 26, no. 5, pp. 539–543, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. D. J. Guo and V. Lakshmikantham, “Coupled fixed points of nonlinear operators with applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 11, no. 5, pp. 623–632, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. D. J. Guo, “Fixed points of mixed monotone operators with applications,” Applicable Analysis, vol. 31, no. 3, pp. 215–224, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. C. Zhai and L. Zhang, “New fixed point theorems for mixed monotone operators and local existence-uniqueness of positive solutions for nonlinear boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 382, no. 2, pp. 594–614, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. C. Zhai and M. Hao, “Fixed point theorems for mixed monotone operators with perturbation and applications to fractional differential equation boundary value problems,” Nonlinear Analysis: Theory, Methods and Applications, vol. 75, no. 4, pp. 2542–2551, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  17. L. C. Evans, Partial Differential Equations, American Mathematical Society, 1998.
  18. Q. X. Ye and Z. Y. Li, Introduction of Reaction-Diffusion Equations, Science Press, Beijing, China, 1994, (Chinese). View at Zentralblatt MATH