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 ${\Bbb R}^n$ and the bounded interval (0,T), respectively. The subject of study is the behavior 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 allied results are obtained.