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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

Guangyu Xu 1, and  Jun Zhou 1,2,,

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.

Keywords: Non-Newton polytropic filtration equation, global existence, blow-up.
Mathematics Subject Classification: 35B33, 35K50, 35K55, 35K63.
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 & Applied Analysis, 2018, 17 (5) : 1805-1820. doi: 10.3934/cpaa.2018086
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

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H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer New York, 2010.  Google Scholar

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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

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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

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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

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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

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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

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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

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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