New Lipschitz estimates and long-time asymptotic behavior for porous medium and fast diffusion equations

We obtain new estimates for the solution of both the porous medium and the diffusion equations by studying the evolution of suitable Lipschitz norms. results include instantaneous regularization for all positive times,-time decay rates of the norms which are sharp and independent of the support, and new convergence results to the Barenblatt profile., we address nonlinear diffusion equations including quadratic or potentials as well. In the slow diffusion case, our strategy requires close enough to 1, while in the fast diffusion case, our results any exponent for which the problem is well-posed and mass-preserving in whole space.