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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

Kaouther Ammar 1, and  Philippe Souplet 2,

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.
Keywords: Liouville-type theorems., universal bounds, nonnegative weak solutions, blow-up estimates, Porous medium equation.
Mathematics Subject Classification: Primary: 35K59, 35B45, 35B44, 35B53; Secondary: 35K65, 35B4.
Citation: Kaouther Ammar, Philippe Souplet. Liouville-type theorems and universal bounds for nonnegative solutions of the porous medium equation with source. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 665-689. doi: 10.3934/dcds.2010.26.665
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