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On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities
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 |
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.
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.
|
| [2] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer New York, 2010. |
| [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.
|
| [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.
|
| [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.
|
| [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.
|
| [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.
|
| [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.
|
| [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.
|
| [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.
|
| [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.
|
| [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 blow-up for a class of semilinear pseudo-parabolic equations, Journal of Functional Analysis, 264 (2013), 2732-2763.
|
| [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.
|
| [15] |
Z. Tan,
Non-Newton Filtration Equation with special medium void, Acta Math Sci, 24B (2014), 118-128.
|
| [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.
|
| [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.
|
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.
|
| [2] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer New York, 2010. |
| [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.
|
| [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.
|
| [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.
|
| [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.
|
| [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.
|
| [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.
|
| [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.
|
| [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.
|
| [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.
|
| [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 blow-up for a class of semilinear pseudo-parabolic equations, Journal of Functional Analysis, 264 (2013), 2732-2763.
|
| [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.
|
| [15] |
Z. Tan,
Non-Newton Filtration Equation with special medium void, Acta Math Sci, 24B (2014), 118-128.
|
| [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.
|
| [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.
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