Calculus of Variations and Geometric Measure Theory

N. David - F. Santambrogio

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

created by santambro on 03 Aug 2023
modified on 04 Jul 2024

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Inserted: 3 aug 2023
Last Updated: 4 jul 2024

Year: 2023

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.

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