R\'enyi entropies and nonlinear diffusion equations

Since their introduction in the early sixties, the R\'enyi entropies have been used in many contexts, ranging from information theory to astrophysics, turbulence phenomena and others. In this note, we enlighten the main connections between R\'enyi entropies and nonlinear diffusion equations. In particular, it is shown that these relationships allow to prove various functional inequalities in sharp form.