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

Hailong Ye; Jingxue Yin

doi:10.3934/dcdsb.2017083

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2017, Volume 22, Issue 4: 1743-1755. Doi: 10.3934/dcdsb.2017083

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

Hailong Ye^1, and
Jingxue Yin^2,

1.
College of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, China
2.
School of Mathematical Sciences, South China, Normal University, Guangzhou 510631, China

Author Bio: E-mail address: ye2006hailong@yeah.net; E-mail address: yjx@scnu.edu.cn

Received: August 2014

Revised: June 2015

Published: August 2017

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

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Abstract

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.

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Mathematics Subject Classification: Primary:35K55;Secondary:35B99.

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References

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