A universal bound for radial solutions of the quasilinear parabolic equation with p-Laplace operator

In this paper we prove a universal bound for nonnegative radial solutions of the p-Laplace equation with nonlinear source ut=div(|∇u|p−2∇u)+uq, where p>2 and q>p−1. This bound implies initial and final blowup rate estimates, as well as a priori estimate or decay rate for global solutions. Our bound is proved as a consequence of Liouville-type theorems for entire solutions and doubling and rescaling arguments. In this connection, we use a known Liouville-type theorem for radial solutions, along with a new Liouville-type theorem that is here established for nontrivial solutions in R.