Asymptotics of the fast diffusion equation via entropy estimates

Grillo, Gabriele; Vazquez, Juan-Luis; Blanchet, Adrien; Bonforte, Matteo; Dolbeault, Jean

Grillo, Gabriele; Vazquez, Juan-Luis; Blanchet, Adrien; Bonforte, Matteo; Dolbeault, Jean (2009), Asymptotics of the fast diffusion equation via entropy estimates, Archive for Rational Mechanics and Analysis, 191, 2, p. 347-385. http://dx.doi.org/10.1007/s00205-008-0155-z

Type

Article accepté pour publication ou publié

External document link

http://hal.archives-ouvertes.fr/hal-00142404/en/

Date

2009

Journal name

Archive for Rational Mechanics and Analysis

Volume

191

Number

Publisher

Springer

Pages

347-385

Abstract (EN)

We consider non-negative solutions of the fast diffusion equation u_t=\Delta u^m with m\in(0,1), in the Euclidean space R^d, d \ge 3, and study the asymptotic behavior of a natural class of solutions, in the limit corresponding to t\to\infty for m \ge mc=(d-2)/d, or as t approaches the extinction time when m < mc. For a class of initial data we prove that the solution converges with a polynomial rate to a self-similar solution, for t large enough if m \ge mc, or close enough to the extinction time if m < mc. Such results are new in the range m \le mc where previous approaches fail. In the range mc

Subjects / Keywords

Fast diffusion equation; self-similar solutions; asymptotic behavior; free energy methods; Hardy-Poincaré inequalities