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

We investigate the large-time asymptotics of nonlinear diffusion equations in dimension n ≥ 1, in the exponent interval p > n/(n + 2), when the initial datum 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, in the range p > max ((n − 1)/n, n/(n + 2)), the relative Rényi entropy exhibits a better decay, for intermediate times, with respect to the standard Ralston- Newman entropy. The result follows by a suitable use of sharp Gagliardo-Nirenberg-Sobolev inequalities, and their information-theoretical proof, known as concavity of Rényi entropy power.