In this lecture, we investigate the large-time asymptotics of nonlinear diffusion equations ut = ∆up in dimension n ≥ 1, in the exponent interval p > n/(n + 2), where the Barenblatt profile is of bounded second moment. Precise rates of convergence to self-similarity in terms of the relative Renyi entropy are demonstrated for solutions of finite mass, 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 [1,2]. The analysis shows that the relative Renyi entropy exhibits a better decay, for intermediate times, with respect to the standard Ralston-Newman entropy usually used in this context. The result follows from the so-called concavity of Renyi entropy power, a new property for Renyi entropies, recently proven in [3].

*1- G. Toscani, A central limit theorem for solutions of the porous medium equation, J. Evol. Equ. 5, 185-203 (2005) 2- J.A. Carrillo, M. DiFrancesco, and G. Toscani, Intermediate asymptotics beyond homogeneity and self-similarity: long time behavior for ut = ∆Φ(u), Arch. Rational Mech. Anal. 180, 127–149 (2006)3- G. Savarè and G. Toscani, The concavity of Renyi entropy power, IEEE Transactions on Information Theory, 60 (5) 2687-2693 (2014)*