Global existence and nonexistence of the non-Newtonian polytropic filtration equations coupled with nonlinear boundary conditions

Abstract

In this article, we deal with the global existence and nonexistence of solutions to the non-Newtonian polytropic filtration equations coupled with nonlinear boundary conditions. By constructing various kinds of sub- and super-solutions and using the basic properties of M-matrix, we give the necessary and sufficient conditions for global existence of nonnegative solutions. The critical curve of Fujita type is conjectured with the aid of some new results, which extend the recent results of Zheng, Song, and Jiang [Critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux, J. Math. Anal. Appl. 298 (2004), pp. 308–324], Zhou and Mu [Critical curve for a non-Newtonian polytropic filtration system coupled via nonlinear boundary flux, Nonlinear Anal. 68 (2008), pp. 1–11], and Zhou and Mu [Algebraic criteria for global existence or blow-up for a boundary coupled system of nonlinear diffusion equations, Appl. Anal. 86 (2007), pp. 1185–1197] to more general equations.

Keywords:

• global existence curve
• degenerate parabolic systems
• M-matrix
• nonlinear boundary flux
• blow-up

AMS Subject Classifications::

• 35K55
• 35K65
• 35B40

Acknowledgements

The authors thank the anonymous referees for their suggestions and comments on the original manuscript. This work was partially supported by NSF of China (10771226), Innovative Talent Training Project of the Third Stage of ‘211 Project’ of Chongqing University (Project No. S-09110) and Foundation Project of China Yangtze Normal University.