American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A free boundary problem for the Fisher-KPP equation with a given moving boundary
  • CPAA Home
  • This Issue
  • Next Article
    On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities
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

[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

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

[1]

Long Wei, Zhijun Qiao, Yang Wang, Shouming Zhou. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1733-1748. doi: 10.3934/dcds.2017072
[2]

Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058
[3]

Shuyin Wu, Joachim Escher, Zhaoyang Yin. Global existence and blow-up phenomena for a weakly dissipative Degasperis-Procesi equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 633-645. doi: 10.3934/dcdsb.2009.12.633
[4]

Xiumei Deng, Jun Zhou. Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity. Communications on Pure & Applied Analysis, 2020, 19 (2) : 923-939. doi: 10.3934/cpaa.2020042
[5]

Jianbo Cui, Jialin Hong, Liying Sun. On global existence and blow-up for damped stochastic nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6837-6854. doi: 10.3934/dcdsb.2019169
[6]

Hailong Ye, Jingxue Yin. Instantaneous shrinking and extinction for a non-Newtonian polytropic filtration equation with orientated convection. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1743-1755. doi: 10.3934/dcdsb.2017083
[7]

Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843
[8]

Yue Cao. Blow-up criterion for the 3D viscous polytropic fluids with degenerate viscosities. Electronic Research Archive, 2020, 28 (1) : 27-46. doi: 10.3934/era.2020003
[9]

Shiming Li, Yongsheng Li, Wei Yan. A global existence and blow-up threshold for Davey-Stewartson equations in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1899-1912. doi: 10.3934/dcdss.2016077
[10]

Bin Li. On the blow-up criterion and global existence of a nonlinear PDE system in biological transport networks. Kinetic & Related Models, 2019, 12 (5) : 1131-1162. doi: 10.3934/krm.2019043
[11]

Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021
[12]

Lili Du, Chunlai Mu, Zhaoyin Xiang. Global existence and blow-up to a reaction-diffusion system with nonlinear memory. Communications on Pure & Applied Analysis, 2005, 4 (4) : 721-733. doi: 10.3934/cpaa.2005.4.721
[13]

Shu-Xiang Huang, Fu-Cai Li, Chun-Hong Xie. Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1519-1532. doi: 10.3934/dcds.2003.9.1519
[14]

Monica Marras, Stella Vernier Piro. On global existence and bounds for blow-up time in nonlinear parabolic problems with time dependent coefficients. Conference Publications, 2013, 2013 (special) : 535-544. doi: 10.3934/proc.2013.2013.535
[15]

Hua Chen, Huiyang Xu. Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1185-1203. doi: 10.3934/dcds.2019051
[16]

Zaihui Gan, Jian Zhang. Blow-up, global existence and standing waves for the magnetic nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 827-846. doi: 10.3934/dcds.2012.32.827
[17]

Emil Novruzov. On existence and nonexistence of the positive solutions of non-newtonian filtration equation. Communications on Pure & Applied Analysis, 2011, 10 (2) : 719-730. doi: 10.3934/cpaa.2011.10.719
[18]

Alberto Bressan, Massimo Fonte. On the blow-up for a discrete Boltzmann equation in the plane. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 1-12. doi: 10.3934/dcds.2005.13.1
[19]

Gongwei Liu. The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term. Electronic Research Archive, 2020, 28 (1) : 263-289. doi: 10.3934/era.2020016
[20]

Akmel Dé Godefroy. Existence, decay and blow-up for solutions to the sixth-order generalized Boussinesq equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 117-137. doi: 10.3934/dcds.2015.35.117

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (101)
  • HTML views (205)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences