We consider non-negative solutions of the fast diffusion equation in the Euclidean space R^d, and study the asymptotic behavior of a natural class of solutions foir large times, where "large" means that t is approaching infinity in case the parameter m appearing in the equation is close to one, or that t approaches the finite extinction time in case the diffusion is "very fast". For a class of initial data, we prove that the solution converges with a polynomial rate to a self-similar solution. Such results are new in the "very fast" case, whereas when m is close to one we improve on known results. A fundamental role in this studied is played by suitable new functional inequalities which are related to the spectral properties of the linearized generator of the evolution considered, after suitable rescaling.

Asymptotics of the fast diffusion equation via entropy estimates

GRILLO, GABRIELE;
2009

Abstract

We consider non-negative solutions of the fast diffusion equation in the Euclidean space R^d, and study the asymptotic behavior of a natural class of solutions foir large times, where "large" means that t is approaching infinity in case the parameter m appearing in the equation is close to one, or that t approaches the finite extinction time in case the diffusion is "very fast". For a class of initial data, we prove that the solution converges with a polynomial rate to a self-similar solution. Such results are new in the "very fast" case, whereas when m is close to one we improve on known results. A fundamental role in this studied is played by suitable new functional inequalities which are related to the spectral properties of the linearized generator of the evolution considered, after suitable rescaling.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11311/562408
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