Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities

Matteo Bonforte (1) , Jean Dolbeault (2) , Gabriele Grillo (3) , Juan-Luis Vázquez (1)

1 Departamento de Matemáticas
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 Dipartimento di Matematica "Giuseppe Peano" [Torino]

The goal of this note is to state the optimal decay rate for solutions of the nonlinear fast diffusion equation and, in self-similar variables, the optimal convergence rates to Barenblatt self-similar profiles and their generalizations. It relies on the identification of the optimal constants in some related Hardy-Poincaré inequalities and concludes a long series of papers devoted to generalized entropies, functional inequalities and rates for nonlinear diffusion equations.

Mots clés: Fast diffusion equation; porous media equation; Barenblatt solutions; Hardy-Poincaré inequalities; large time behaviour; asymptotic expansion; intermediate asymptotics; sharp rates; optimal constants

Domaines: Equations aux dérivées partielles [math.AP]

Matteo Bonforte, Jean Dolbeault, Gabriele Grillo, Juan-Luis Vázquez. Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities. Proceedings of the National Academy of Sciences of the United States of America, 2010, 107 (38), pp.16459-16464. ⟨10.1073/pnas.1003972107⟩. ⟨hal-00404718⟩