April  2010, 26(2): 665-689. doi: 10.3934/dcds.2010.26.665

Liouville-type theorems and universal bounds for nonnegative solutions of the porous medium equation with source

1. 

TU Berlin, Institut für Mathematik, Strasse des 17 Juni 135, MA 6-4, 10623 Berlin, Germany

2. 

Université Paris 13, CNRS UMR 7539, Laboratoire Analyse, Géométrie et Applications, 99, avenue J.-B. Clément, 93430 Villetaneuse, France

Received  February 2009 Revised  August 2009 Published  October 2009

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 RN and for solutions on a half-line.
Citation: Kaouther Ammar, Philippe Souplet. Liouville-type theorems and universal bounds for nonnegative solutions of the porous medium equation with source. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 665-689. doi: 10.3934/dcds.2010.26.665
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