June  2017, 22(4): 1743-1755. doi: 10.3934/dcdsb.2017083

Instantaneous shrinking and extinction for a non-Newtonian polytropic filtration equation with orientated convection

1. 

College of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, China

2. 

School of Mathematical Sciences, South China, Normal University, Guangzhou 510631, China

Received  August 2014 Revised  June 02, 2015 Published  February 2017

Fund Project: The second author is supported by National Natural Science Foundation of China (grant 11371153).

This paper is concerned with the instantaneous shrinking and extinction for a non-Newtonian polytropic filtration equation with orientated convection
$\frac{\partial u}{\partial t}=\text{div}\left( {{\left| \nabla {{u}^{m}} \right|}^{p-2}}\nabla {{u}^{m}} \right)|-\overrightarrow{\beta }\left( x \right)\cdot \triangledown {{u}^{q}},\ \ \ \ x\in {{\mathbb{R}}^{N}},t>0$
where
$p>1, m,q>0, N≥1$
and
$\overrightarrow{β}(x)$
is a vector field defined on
$\mathbb{R}^{N}$
. Here, the orientation of the convection is specified to that with counteracting diffusion, that is
$\overrightarrow{β}(x)·(-x)≥0$
,
$x∈\mathbb{R}^N$
. Sufficient conditions are established for the instantaneous shrinking property of solutions with decayed initial datum of supports. For a certain class of initial datum, it is shown that there exists a critical time
$τ^*>0$
such that the supports of solutions are unbounded above for any
$t < τ^*$
, whilst the opposite is the case for any
$t>τ^*$
. In addition, we prove that once the supports of solutions shrink instantaneously, the solutions will vanish in finite time.
Citation: 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
References:
[1]

U. G. Abdullaev, Instantaneous shrinking of the support of solutions to a nonlinear degenerate parabolic equation(Russian), Mat. Zametki, 63 (1998), 323-331; translation in Math. Notes, 63 (1998), 285-292. doi: 10.1007/BF02317772. Google Scholar

[2]

S. N. Antontsev and S. I. Shmarev, Doubly degenerate parabolic equations with variable nonlinearity Ⅱ: Blow-up and extinction in a finite time,, Nonlinear Anal., 95 (2014), 483-498. doi: 10.1016/j.na.2013.09.027. Google Scholar

[3]

R. Aris, The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, Ⅰ, Ⅱ, , Clarendon: Oxford, 1975.Google Scholar

[4]

J. Bear, Dynamics of fluids in porous media, Soil Science, 120 (1975), 162-163. doi: 10.1097/00010694-197508000-00022. Google Scholar

[5]

M. Borelli and M. Ughi, The fast diffusion equation with strong absorption: The instantaneous shrinking phenomenon, Rend. Istit. Mat. Univ. Trieste, 26 (1994), 109-140. Google Scholar

[6]

R. Carles and C. Gallo, Finite time extinction by nonlinear damping for the Schrödinger equation, Comm. Partial Differential Equations, 36 (2011), 961-975. doi: 10.1080/03605302.2010.531074. Google Scholar

[7]

E. C. Childs, An Introduction to the Physical Basis of Soil Water Phenomena, Wiley: London, 1969.Google Scholar

[8]

L. Evans and B. F. Knerr, Instantaneous shrinking of the support of nonnegative solutions to certain nonlinear parabolic equations and variational inequalities, Ill. J. Math., 23 (1979), 153-166. Google Scholar

[9]

B. H. Gilding and R. Kersner, Instantaneous shrinking in nonlinear diffusion-convection, Proc. Amer. Math. Soc., 109 (1990), 385-394. doi: 10.1090/S0002-9939-1990-1007496-9. Google Scholar

[10]

R. G. Iagar and P. Laurençot, Positivity, decay, and extinction for a singular diffusion equation with gradient absorption, J. Funct. Anal., 262 (2012), 3186-3239. doi: 10.1016/j.jfa.2012.01.013. Google Scholar

[11]

A. S. Kalašhnikov, The nature of the propagation of perturbations in problems of non-linear heat conduction with absorption, USSR Comp. Math. Math. Phys., 14 (1974), 70-85. Google Scholar

[12]

M. Ohnuma and K. Sato, Singular degenerate parabolic equations with applications to the p-Laplace diffusion equation, Communications in Partial Differential Equations, 22 (1997), 381-411. doi: 10.1080/03605309708821268. Google Scholar

[13]

N. Su, Compactification of supports of solutions for nonlinear parabolic equations, Nonlinear Anal., 29 (1997), 347-363. doi: 10.1016/S0362-546X(96)00076-4. Google Scholar

[14]

J. L. Vázquez, The Prorous Medium Equation: Mathematical Theory, Clarendon Press: Oxford, 2007. Google Scholar

[15]

Z. Wu, J. Zhao, J. Yin and H. Li, Nonlinear Diffusion Equations, World Scientific: Singapore, 2001. doi: 10.1142/9789812799791. Google Scholar

[16]

H. Ye and J. Yin, Propagation profile for a non-Newtonian polytropic filtration equation with orientated convection, J. Math. Anal. Appl., 421 (2014), 1225-1237. doi: 10.1016/j.jmaa.2014.07.077. Google Scholar

[17]

H. YuanS. LianW. GaoX. Xu and C. Cao, Extinction and positivity for the evolution p-Laplacian equation in ${{\mathbb{R}}^{N}}$, Nonlinear Analysis, 60 (2005), 1085-1091. doi: 10.1016/j.na.2004.10.009. Google Scholar

show all references

References:
[1]

U. G. Abdullaev, Instantaneous shrinking of the support of solutions to a nonlinear degenerate parabolic equation(Russian), Mat. Zametki, 63 (1998), 323-331; translation in Math. Notes, 63 (1998), 285-292. doi: 10.1007/BF02317772. Google Scholar

[2]

S. N. Antontsev and S. I. Shmarev, Doubly degenerate parabolic equations with variable nonlinearity Ⅱ: Blow-up and extinction in a finite time,, Nonlinear Anal., 95 (2014), 483-498. doi: 10.1016/j.na.2013.09.027. Google Scholar

[3]

R. Aris, The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, Ⅰ, Ⅱ, , Clarendon: Oxford, 1975.Google Scholar

[4]

J. Bear, Dynamics of fluids in porous media, Soil Science, 120 (1975), 162-163. doi: 10.1097/00010694-197508000-00022. Google Scholar

[5]

M. Borelli and M. Ughi, The fast diffusion equation with strong absorption: The instantaneous shrinking phenomenon, Rend. Istit. Mat. Univ. Trieste, 26 (1994), 109-140. Google Scholar

[6]

R. Carles and C. Gallo, Finite time extinction by nonlinear damping for the Schrödinger equation, Comm. Partial Differential Equations, 36 (2011), 961-975. doi: 10.1080/03605302.2010.531074. Google Scholar

[7]

E. C. Childs, An Introduction to the Physical Basis of Soil Water Phenomena, Wiley: London, 1969.Google Scholar

[8]

L. Evans and B. F. Knerr, Instantaneous shrinking of the support of nonnegative solutions to certain nonlinear parabolic equations and variational inequalities, Ill. J. Math., 23 (1979), 153-166. Google Scholar

[9]

B. H. Gilding and R. Kersner, Instantaneous shrinking in nonlinear diffusion-convection, Proc. Amer. Math. Soc., 109 (1990), 385-394. doi: 10.1090/S0002-9939-1990-1007496-9. Google Scholar

[10]

R. G. Iagar and P. Laurençot, Positivity, decay, and extinction for a singular diffusion equation with gradient absorption, J. Funct. Anal., 262 (2012), 3186-3239. doi: 10.1016/j.jfa.2012.01.013. Google Scholar

[11]

A. S. Kalašhnikov, The nature of the propagation of perturbations in problems of non-linear heat conduction with absorption, USSR Comp. Math. Math. Phys., 14 (1974), 70-85. Google Scholar

[12]

M. Ohnuma and K. Sato, Singular degenerate parabolic equations with applications to the p-Laplace diffusion equation, Communications in Partial Differential Equations, 22 (1997), 381-411. doi: 10.1080/03605309708821268. Google Scholar

[13]

N. Su, Compactification of supports of solutions for nonlinear parabolic equations, Nonlinear Anal., 29 (1997), 347-363. doi: 10.1016/S0362-546X(96)00076-4. Google Scholar

[14]

J. L. Vázquez, The Prorous Medium Equation: Mathematical Theory, Clarendon Press: Oxford, 2007. Google Scholar

[15]

Z. Wu, J. Zhao, J. Yin and H. Li, Nonlinear Diffusion Equations, World Scientific: Singapore, 2001. doi: 10.1142/9789812799791. Google Scholar

[16]

H. Ye and J. Yin, Propagation profile for a non-Newtonian polytropic filtration equation with orientated convection, J. Math. Anal. Appl., 421 (2014), 1225-1237. doi: 10.1016/j.jmaa.2014.07.077. Google Scholar

[17]

H. YuanS. LianW. GaoX. Xu and C. Cao, Extinction and positivity for the evolution p-Laplacian equation in ${{\mathbb{R}}^{N}}$, Nonlinear Analysis, 60 (2005), 1085-1091. doi: 10.1016/j.na.2004.10.009. Google Scholar

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