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

TL;DR

This paper introduces new Lipschitz estimates for porous medium and fast diffusion equations, demonstrating regularization, decay rates, and convergence to Barenblatt profiles, applicable to a broad class of nonlinear diffusion problems.

Contribution

It provides novel Lipschitz norm estimates that improve understanding of regularization and long-time behavior in nonlinear diffusion equations, including cases with potentials.

Findings

Instantaneous regularization for all positive times

Sharp decay rates of Lipschitz norms independent of initial support

Convergence to Barenblatt profiles for a wide range of exponents

Abstract

We obtain new estimates for the solution of both the porous medium and the fast 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 the initial support, and new convergence results to the Barenblatt profile. Moreover, we address nonlinear diffusion equations including quadratic or bounded potentials as well. In the slow diffusion case, our strategy requires exponents close enough to 1, while in the fast diffusion case, our results cover any exponent for which the problem is well-posed and mass-preserving in the whole space.