Fast diffusion equations: Matching large time asymptotics by relative entropy methods

Pages: 701 - 716, Volume 4, Issue 3, September 2011

doi:10.3934/krm.2011.4.701       Abstract        References        Full Text (519.8K)       Related Articles

Jean Dolbeault - Ceremade (UMR CNRS no. 7534), Université Paris Dauphine, Place de Lattre de Tassigny, 75775 Paris Cédex 16, France (email)
Giuseppe Toscani - Department of Mathematics at the University of Pavia, via Ferrata 1, 27100 Pavia, Italy (email)

Abstract: A non self-similar change of coordinates provides improved matching asymptotics of the solutions of the fast diffusion equation for large times, compared to already known results, in the range for which Barenblatt solutions have a finite second moment. The method is based on relative entropy estimates and a time-dependent change of variables which is determined by second moments, and not by the scaling corresponding to the self-similar Barenblatt solutions, as it is usually done.

Keywords:  Fast diffusion equation, porous media equation, Barenblatt solutions, Hardy-Poincaré inequalities, large time behaviour, second moment, asymptotic expansion, intermediate asymptotics, sharp rates, optimal constants.
Mathematics Subject Classification:  Primary: 35B40; Secondary: 35K55, 39B62.

Received: May 2010;      Revised: June 2011;      Published: August 2011.

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