Blow-up for the Porous Media Equation with Source Term and Positive Initial Energy1

We study the Cauchy]Dirichlet problem for the porous media equation with nonlinear source term in a bounded subset of Rn. The problem describes the propagation of thermal perturbations in a medium with a nonlinear heat-conduction coefficient and a heat source depending on the temperature. The aim of the paper is to extend the unstable set to a part of the positive energy region, a phenomenon which was known only for linear conduction.