Asymptotic behaviour of the porous media equation in domains with holes

Abstract

The paper deals with the asymptotic behaviour of solutions to the porous media equation, $u_t=\Delta u^m$, $m>1$, in an exterior domain, $\Omega$, which excludes one or several holes, and with zero Dirichlet data on $\partial \Omega$. When the space dimension is three or more this behaviour is given by a Barenblatt function away from the fixed boundary $\partial \Omega$ and near the free boundary. The asymptotic behaviour of the free boundary is given by the same Barenblatt function. On the other hand, if the solution is scaled according to its decay factor, away from the free boundary and close to the holes it behaves like a function whose $m$-th power is harmonic and vanishes on $\partial \Omega$. The height of such a function is determined by matching with the Barenblatt solution representing the outer behaviour. The inner and the outer behaviour can be presented in a unified way through a suitable global approximation.