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

Abstract. In this paper, we prove new functional inequalities of Poincaré type on the one-dimensional torus S1 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 to-wards 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 Poincare ́ type inequalities. 1. Introduction. On