Abstract and Applied Analysis

Positive Solutions of a Nonlinear Parabolic Partial Differential Equation

Chengbo Zhai and Shunyong Li

Full-text: Open access

Abstract

We deal with the existence and uniqueness of positive solutions to a class of nonlinear parabolic partial differential equations, by using some fixed point theorems for mixed monotone operators with perturbation.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 643897, 6 pages.

Dates
First available in Project Euclid: 2 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412279723

Digital Object Identifier
doi:10.1155/2014/643897

Mathematical Reviews number (MathSciNet)
MR3216070

Zentralblatt MATH identifier
07022814

Citation

Zhai, Chengbo; Li, Shunyong. Positive Solutions of a Nonlinear Parabolic Partial Differential Equation. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 643897, 6 pages. doi:10.1155/2014/643897. https://projecteuclid.org/euclid.aaa/1412279723


Export citation

References

  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States, Birkhauser Advanced Texts, Berlin, Germany, 2007.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • D. J. Guo, “Fixed points of mixed monotone operators with applications,” Applicable Analysis, vol. 31, no. 3, pp. 215–224, 1988.
  • 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.
  • 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.
  • L. C. Evans, Partial Differential Equations, American Mathematical Society, 1998.
  • Q. X. Ye and Z. Y. Li, Introduction of Reaction-Diffusion Equa-tions, Science Press, Beijing, China, 1994, (Chinese). \endinput