Rényi Entropy and Improved Equilibration Rates to Self-similarity for Nonlinear Diffusion Equations

We investigate the large-time asymptotics of nonlinear diffusion equations ut = ∆u p in dimension n ≥ 1, in the exponent interval p > n/(n+ 2), when the initial datum u0 is of bounded second moment. Precise rates of convergence to the Barenblatt profile in terms of the relative Rényi 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, as proposed in [15, 45]. 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 RalstonNewman entropy. The result follows by a suitable use of sharp Gagliardo-Nirenberg-Sobolev inequalities considered in [30], and their information-theoretical proof [43], known as concavity of Rényi entropy power. 2000 AMS subject classification. 35K55, 35K60, 35K65, 35B40.