Entropy-energy inequalities and improved convergence rates fornonlinear parabolic equations
José Antonio Carrillo, Jean Dolbeault, Ivan Gentil, Ansgar Jüngel · Discrete and Continuous Dynamical Systems - B · 2006
In this paper, we prove new functional inequalities of Poincaré typeon the one-dimensional torus $S^1$ and explore their implicationsfor the long-time asymptotics of periodic solutions of nonlinearsingular or degenerate parabolic equations of second and fourthorder. We generically prove a global algebraic decay of an entropy functional, faster than exponential for short times, andan asymptotically exponential convergence of positive solutionstowards their average. The asymptotically exponential regime isvalid for a larger range of parameters for all relevant cases ofapplication: porous medium/fast diffusion, thin film and logarithmicfourth order nonlinear diffusion equations. The techniques areinspired by direct entropy-entropy production methods and based onappropriate Poincaré type inequalities.