# American Institute of Mathematical Sciences

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. [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. [3] R. Aris, The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, Ⅰ, Ⅱ, , Clarendon: Oxford, 1975. [4] J. Bear, Dynamics of fluids in porous media, Soil Science, 120 (1975), 162-163. doi: 10.1097/00010694-197508000-00022. [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. [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. [7] E. C. Childs, An Introduction to the Physical Basis of Soil Water Phenomena, Wiley: London, 1969. [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. [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. [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. [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. [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. [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. [14] J. L. Vázquez, The Prorous Medium Equation: Mathematical Theory, Clarendon Press: Oxford, 2007. [15] Z. Wu, J. Zhao, J. Yin and H. Li, Nonlinear Diffusion Equations, World Scientific: Singapore, 2001. doi: 10.1142/9789812799791. [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. [17] H. Yuan, S. Lian, W. Gao, X. 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.

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##### 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. [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. [3] R. Aris, The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, Ⅰ, Ⅱ, , Clarendon: Oxford, 1975. [4] J. Bear, Dynamics of fluids in porous media, Soil Science, 120 (1975), 162-163. doi: 10.1097/00010694-197508000-00022. [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. [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. [7] E. C. Childs, An Introduction to the Physical Basis of Soil Water Phenomena, Wiley: London, 1969. [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. [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. [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. [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. [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. [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. [14] J. L. Vázquez, The Prorous Medium Equation: Mathematical Theory, Clarendon Press: Oxford, 2007. [15] Z. Wu, J. Zhao, J. Yin and H. Li, Nonlinear Diffusion Equations, World Scientific: Singapore, 2001. doi: 10.1142/9789812799791. [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. [17] H. Yuan, S. Lian, W. Gao, X. 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.
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