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    Modelling and long-time behaviour for phase transitions with entropy balance and thermal memory conductivity
September  2006, 6(5): 1027-1050. doi: 10.3934/dcdsb.2006.6.1027

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

José A. Carrillo 1, , Jean Dolbeault 2, , Ivan Gentil 3, and  Ansgar Jüngel 4,

1. 

ICREA-Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain
2. 

Ceremade (UMR CNRS 7534), Université Paris Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris, Cédex 16, France
3. 

Ceremade, UMR CNRS 7534, Université Paris-Dauphine, Place du Maréchal De Lattre de Tassigny, 75775 PARIS Cedex 16, France
4. 

Fachbereich Physik, Mathematik und Informatik, Universität Mainz, Staudingerweg 9, 55099 Mainz, Germany

Received  September 2005 Revised  March 2006 Published  June 2006

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.
Keywords: Poincare inequality, higher-order nonlinear PDEs, entropy production, entropy-entropy production method, Sobolev estimates, entropy, fast diffusion equation, parabolic equations, Logarithmic Sobolev inequality, long-time behavior, thin film equation., porous media equation.
Mathematics Subject Classification: MSC (2000): 35K35, 35K55, 35G25, 35B40, 26D10,35B45, 76A20, 76D08, 76S0.
Citation: José A. Carrillo, Jean Dolbeault, Ivan Gentil, Ansgar Jüngel. Entropy-energy inequalities and improved convergence rates for nonlinear parabolic equations. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 1027-1050. doi: 10.3934/dcdsb.2006.6.1027
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