Asymptotic behaviour of reversible chemical reaction-diffusion equations
Ivan Gentil Bogusław Zegarlinski
Kinetic & Related Models 2010, 3(3): 427-444 doi: 10.3934/krm.2010.3.427
We investigate the asymptotic behavior of a large class of reversible chemical reaction-diffusion equations with the same diffusion. In particular we prove the optimal rate in two cases : when there is no diffusion and in the classical "two-by-two" case.
keywords: Asymptotic behaviour. Reaction-diffusion equations Poincaré inequality
Entropy-energy inequalities and improved convergence rates for nonlinear parabolic equations
José A. Carrillo Jean Dolbeault Ivan Gentil Ansgar Jüngel
Discrete & Continuous Dynamical Systems - B 2006, 6(5): 1027-1050 doi: 10.3934/dcdsb.2006.6.1027
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

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