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

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