Generalized Sobolev Inequalities and Asymptotic Behaviour in Fast Diffusion and Porous Medium Problems

In this paper we prove a new family of inequalities which is intermediate between the classical Sobolev inequalities and the Gross logarithmic Sobolev inequality by the minimization of a well choosen functional and the use of recent uniqueness results for the ground state of the corresponding nonlinear scalar eld equation, which allows us to identify the optimal constants. This result is then applied to the equation u t = u m in IR N for m 2 N?1 N ; 11 (fast diiusion) and m > 1 (porous medium), thus giving an exponential rate of decay for the relative entropy to the stationary solution of a rescaled problem and describing the intermediate asymptotics in the L 1 (IR N)-norm. Radial symmetry { Uniqueness { Fast diiusion { Porous medium { Time-dependent rescaling { Relative entropy { Optimal rate of decay { Intermediate asymptotics