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 convectionis a vector field defined on R N . Here, the orientation of the convection is specified to that with counteracting diffusion, that is − → β (x) · (−x) ≥ 0, x ∈ 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.2010 Mathematics Subject Classification. Primary: 35K55; Secondary: 35B99.