Localization of solutions of exterior domain problems for the porous media equation with radial symmetry

The paper concerns the radially-symmetric Cauchy-Dirichlet and Cauchy-Neumann problems for the porous media equation in the domain comprising the spatial variable and the temporal variable $t$ in the exterior of the unit ball in ${\mathbb{R}}^n$ and the bounded interval $(0,T)$ respectively. The subject of study is the behaviour of solutions when the initial data are compactly supported and the boundary data become unbounded as $t \uparrow T$. Necessary and sufficient conditions for localization, estimates of the size of the blow-up set, and a number of related results are obtained.