Abstract
We prove universal bounds for nonnegative weak solutions of the porous medium equation with source u(t) - Delta u(m) = u(p) where 1 < m < p. These bounds imply initial and final blow-up rate estimates, as well as a priori estimates or decay rates for global solutions. We consider both radial and nonradial solutions, and in the radial case we cover all Sobolev-subcritical values of p/m, which is the best possible range. Our bounds are proved as a consequence of Liouville-type theorems for entire solutions and doubling and rescaling arguments. In this connection, we use known Liouville-type theorems for radial solutions, along with some new Liouville-type theorems that are here established for nonradial solutions in R(N) and for solutions on a half-line.