Renyi entropy and improved equilibration rates to self-similarity for nonlinear diffusion equations

We investigate the large-time asymptotics of nonlinear diffusion equations $u_t=\mathrm{\Delta }{u}^p$ in dimension $n\ge 1$, in the exponent interval $p > n/(n+2)$,whentheinitialdatum$u_0$ is of bounded second moment. Precise rates of convergence to the Barenblatt profile in terms of the relative R'enyi entropy are demonstrated for finite-mass solutions defined in the whole space when they are re-normalized at each time $t>0$ with respect to their own second moment. The analysis shows that the relative R'enyi entropy exhibits a better decay, for intermediate times, with respect to the standard Ralston-Newton entropy. The result follows by a suitable use of the so-called concavity of R'enyi entropy power.