  We investigate the large-time asymptotics of nonlinear diffusion equations
$u_t = \Delta u^p$ in dimension $n \ge 1$, in the exponent interval $p >
n/(n+2)$, when the initial datum $u_0$ 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.
