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

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