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September  2018, 17(5): 1805-1820. doi: 10.3934/cpaa.2018086

## Global existence and blow-up of solutions to a singular Non-Newton polytropic filtration equation with critical and supercritical initial energy

 1 School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China 2 College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

* Corresponding author

Received  June 2017 Revised  September 2017 Published  April 2018

Fund Project: This work is partially supported by the the Basic and Advanced Research Project of CQC-STC grant cstc2016jcyjA0018 and NSFC 11201380.

In this paper, we revisit the singular Non-Newton polytropic filtration equation, which was studied extensively in the recent years. However, all the studies are mostly concerned with subcritical initial energy, i.e., $E(u_0)<d$, where $E(u_0)$ is the initial energy and $d$ is the mountain-pass level. The main purpose of this paper is to study the behaviors of the solution with $E(u_0)≥d$ by potential well method and some differential inequality techniques.

Citation: Guangyu Xu, Jun Zhou. Global existence and blow-up of solutions to a singular Non-Newton polytropic filtration equation with critical and supercritical initial energy. Communications on Pure &amp; Applied Analysis, 2018, 17 (5) : 1805-1820. doi: 10.3934/cpaa.2018086
##### References:
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show all references

##### References:
 [1] M. Badiale and G. Tarantello, A sobolev-hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics, Archive for Rational Mechanics and Analysis, 163 (2002), 259-293.   Google Scholar [2] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer New York, 2010.  Google Scholar [3] F. Gazzola and T. Weth, Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level, Differential and Integral Equations, 18 (2005), 961-990.   Google Scholar [4] B. Guo and W. J. Gao, Blow-up of solutions to quasilinear hyperbolic equations with $p(x,t)$-laplacian and positive initial energy, Comptes Rendus Mecanique, 342 (2014), 513-519.   Google Scholar [5] A. J. Hao and J. Zhou, A new blow-up condition for semi-linear edge degenerate parabolic equation with singular potentials, Applicable Analysis, (2016), 1-12.   Google Scholar [6] Y. Hu, J. Li and L. W. Wang, Blow-up phenomena for porous medium equation with nonlinear flux on the boundary, Journal of Applied Mathematics, 2013 (2013), 1-5.   Google Scholar [7] A. Khelghati and K. Baghaei, Blow-up phenomena for a nonlocal semilinear parabolic equation with positive initial energy, Computers and Mathematics with Applications, 70 (2015), 896-902.   Google Scholar [8] Q. W. Li, W. J. Gao and Y. Z. Han, Global existence blow up and extinction for a class of thin-film equation, Nonlinear Analysis Theory Methods and Applications, 147 (2016), 96-109.   Google Scholar [9] L. R. Luo and J. Zhou, Global existence and blow-up to the solutions of a singular porous medium equation with critical initial energy, Boundary Value Problems, 2016 (2016), 1-8.   Google Scholar [10] X. L. Wu, B. Guo and W. J. Gao, Blow-up of solutions for a semilinear parabolic equation involving variable source and positive initial energy, Applied Mathematics Letters, 26 (2013), 539-543.   Google Scholar [11] X. L. Wu and W. J. Guo, Blow-up of the solution for a class of porous medium equation with positive initial energy, Acta Math Sci, 33 (2013), 1024-1030.   Google Scholar [12] Z. Q. Wu, J. X. Yin, H. L. Li and J. N. Zhao, Nonlinear diffusion equations. World Scientific Publishing Co. inc. river Edge Nj, 2001.  Google Scholar [13] R. Z. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, Journal of Functional Analysis, 264 (2013), 2732-2763.   Google Scholar [14] Y. Wang, The existence of global solution and the blowup problem for some $p$-laplace heat equations, Acta Math Sci, 27 (2007), 274-282.   Google Scholar [15] Z. Tan, Non-Newton Filtration Equation with special medium void, Acta Math Sci, 24B (2014), 118-128.   Google Scholar [16] J. Zhou, A multi-dimension blow-up problem to a porous medium diffusion equation with special medium void, Applied Mathematics Letters, 30 (2014), 6-11.   Google Scholar [17] J. Zhou, Global existence and blow-up of solutions for a non-newton polytropic filtration system with special volumetric moisture content, Computers and Mathematics with Applications, 71 (2016), 1163-1172.   Google Scholar
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