Lyapunov functionals for the heat equation and sharp inequalities

The heat equation represents a powerful instrument to obtain a number of mathematical inequalities in sharp form. This may be not so well-known property goes back more or less to half a century ago, when independently from each others, researchers from information theory [22, 6] and kinetic theory [20] established a useful connection between Boltzmann’s H-functional and Fisher information exactly by means of the solution to the heat equation. In this note, we briefly discuss these original ideas, together with some new application