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

Abstract

In this paper we prove a universal bound for nonnegative radial solutions of the p-Laplace equation with nonlinear source u t = div ( | ∇ u | p − 2 ∇ u ) + u q , 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 .