Entropy-energy inequalities and improved convergence rates fornonlinear parabolic equations

José A. Carrillo; Jean Dolbeault; Ivan Gentil; Ansgar Jüngel

doi:10.3934/dcdsb.2006.6.1027

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2006, Volume 6, Issue 5: 1027-1050. Doi: 10.3934/dcdsb.2006.6.1027

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Entropy-energy inequalities and improved convergence rates for nonlinear parabolic equations

1.
ICREA-Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra
2.
Ceremade (UMR CNRS 7534), Université Paris Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris, Cédex 16
3.
Ceremade, UMR CNRS 7534, Université Paris-Dauphine, Place du Maréchal De Lattre de Tassigny, 75775 PARIS Cedex 16
4.
Fachbereich Physik, Mathematik und Informatik, Universität Mainz, Staudingerweg 9, 55099 Mainz

Received: August 31, 2005

Revised: February 28, 2006

Published: August 2006

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Abstract

In this paper, we prove new functional inequalities of Poincaré type on the one-dimensional torus $S^1$ and explore their implications for the long-time asymptotics of periodic solutions of nonlinear singular or degenerate parabolic equations of second and fourth order. We generically prove a global algebraic decay of an entropy functional, faster than exponential for short times, and an asymptotically exponential convergence of positive solutions towards their average. The asymptotically exponential regime is valid for a larger range of parameters for all relevant cases of application: porous medium/fast diffusion, thin film and logarithmic fourth order nonlinear diffusion equations. The techniques are inspired by direct entropy-entropy production methods and based on appropriate Poincaré type inequalities.

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Mathematics Subject Classification: MSC (2000): 35K35, 35K55, 35G25, 35B40, 26D10,35B45, 76A20, 76D08, 76S05.

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