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Global existence and blowup of solutions to a singular NonNewton 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 
In this paper, we revisit the singular NonNewton 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 mountainpass 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.
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L. R. Luo and J. Zhou, Global existence and blowup to the solutions of a singular porous medium equation with critical initial energy, Boundary Value Problems, 2016 (2016), 18. 
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X. L. Wu, B. Guo and W. J. Gao, Blowup of solutions for a semilinear parabolic equation involving variable source and positive initial energy, Applied Mathematics Letters, 26 (2013), 539543. 
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X. L. Wu and W. J. Guo, Blowup of the solution for a class of porous medium equation with positive initial energy, Acta Math Sci, 33 (2013), 10241030. 
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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. 
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R. Z. Xu and J. Su, Global existence and finite time blowup for a class of semilinear pseudoparabolic equations, Journal of Functional Analysis, 264 (2013), 27322763. 
[14] 
Y. Wang, The existence of global solution and the blowup problem for some $p$laplace heat equations, Acta Math Sci, 27 (2007), 274282. 
[15] 
Z. Tan, NonNewton Filtration Equation with special medium void, Acta Math Sci, 24B (2014), 118128. 
[16] 
J. Zhou, A multidimension blowup problem to a porous medium diffusion equation with special medium void, Applied Mathematics Letters, 30 (2014), 611. 
[17] 
J. Zhou, Global existence and blowup of solutions for a nonnewton polytropic filtration system with special volumetric moisture content, Computers and Mathematics with Applications, 71 (2016), 11631172. 
show all references
References:
[1] 
M. Badiale and G. Tarantello, A sobolevhardy inequality with applications to a nonlinear elliptic equation arising in astrophysics, Archive for Rational Mechanics and Analysis, 163 (2002), 259293. 
[2] 
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer New York, 2010. 
[3] 
F. Gazzola and T. Weth, Finite time blowup and global solutions for semilinear parabolic equations with initial data at high energy level, Differential and Integral Equations, 18 (2005), 961990. 
[4] 
B. Guo and W. J. Gao, Blowup of solutions to quasilinear hyperbolic equations with $p(x,t)$laplacian and positive initial energy, Comptes Rendus Mecanique, 342 (2014), 513519. 
[5] 
A. J. Hao and J. Zhou, A new blowup condition for semilinear edge degenerate parabolic equation with singular potentials, Applicable Analysis, (2016), 112. 
[6] 
Y. Hu, J. Li and L. W. Wang, Blowup phenomena for porous medium equation with nonlinear flux on the boundary, Journal of Applied Mathematics, 2013 (2013), 15. 
[7] 
A. Khelghati and K. Baghaei, Blowup phenomena for a nonlocal semilinear parabolic equation with positive initial energy, Computers and Mathematics with Applications, 70 (2015), 896902. 
[8] 
Q. W. Li, W. J. Gao and Y. Z. Han, Global existence blow up and extinction for a class of thinfilm equation, Nonlinear Analysis Theory Methods and Applications, 147 (2016), 96109. 
[9] 
L. R. Luo and J. Zhou, Global existence and blowup to the solutions of a singular porous medium equation with critical initial energy, Boundary Value Problems, 2016 (2016), 18. 
[10] 
X. L. Wu, B. Guo and W. J. Gao, Blowup of solutions for a semilinear parabolic equation involving variable source and positive initial energy, Applied Mathematics Letters, 26 (2013), 539543. 
[11] 
X. L. Wu and W. J. Guo, Blowup of the solution for a class of porous medium equation with positive initial energy, Acta Math Sci, 33 (2013), 10241030. 
[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. 
[13] 
R. Z. Xu and J. Su, Global existence and finite time blowup for a class of semilinear pseudoparabolic equations, Journal of Functional Analysis, 264 (2013), 27322763. 
[14] 
Y. Wang, The existence of global solution and the blowup problem for some $p$laplace heat equations, Acta Math Sci, 27 (2007), 274282. 
[15] 
Z. Tan, NonNewton Filtration Equation with special medium void, Acta Math Sci, 24B (2014), 118128. 
[16] 
J. Zhou, A multidimension blowup problem to a porous medium diffusion equation with special medium void, Applied Mathematics Letters, 30 (2014), 611. 
[17] 
J. Zhou, Global existence and blowup of solutions for a nonnewton polytropic filtration system with special volumetric moisture content, Computers and Mathematics with Applications, 71 (2016), 11631172. 
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