One-dimensional Barenblatt-type solutions and related inequalities

We present and discuss connections between the problem of trend to equilibrium for one-dimensional Fokker–Planck equations, and one-dimensional functional inequalities of the type of Poincaré and Wirtinger, with weight, for probability densities in the form of one-dimensional Barenblatt solutions of the porous medium equation. It it also shown that Hardy’s classical one-dimensional inequality can be obtained resorting to the differential expression of the Cauchy-type density, as given by a related Fokker–Planck equation.