An optimal Liouville-type theorem for radial entire solutions of the porous medium equation with source

We consider nonnegative (continuous) weak solutions of the porous medium equation with source u(t) - Delta u(m) = u(p), with p > m > 1. We address the question of existence of nontrivial entire solutions, that is, solutions defined for all x is an element of R(n) and t is an element of R. Such solutions do exist for critical and supercritical p (positive bounded stationary Solutions). Our main result asserts that for subcritical p there are no bounded radial entire solutions u not equivalent to 0. This parabolic Liouville-type theorem is the first of its kind for reaction-diffusion equations involving porous medium operators. On the other hand, it will be the main tool in the Study of universal bounds for global and nonglobal Solutions in the forthcoming article [K. Ammar, Ph. Souplet, Liouville-type results and universal bounds for positive solutions of the porous Medium equation with source, in preparation]. The proof is based on intersection-comparison arguments. A key step is to first show the positivity of possible bounded radial entire solutions. Among other auxiliary results, we establish pointwise gradient estimates of possible independent interest. (C) 2008 Elsevier Inc. All rights reserved.