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

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.