We consider the asymptotic behaviour of positive solutions of the fast diffusion equation u_t = \Delta u^m in the whole Euclidea space R^d, with a precise value for the exponent m = (d − 4)/(d − 2). This case had been left open in the general study (Blanchet et al. in Arch Rat Mech Anal 191:347–385, 2009) since it requires quite different functional analytic methods, due in particular to the absence of a spectral gap for the operator generating the linearized evolution. The linearization of this flow is interpreted here as the heat flow of the Laplace–Beltrami operator of a suitabl Riemannian Manifold (R^d , g), with a metric g which is conformal to the standard Rd metric. Studying the pointwise heat kernel behaviour allows to prove suitable Gagliardo–Nirenberg inequalities associated with the generator. Such inequalities in turn allow one to study the nonlinear evolution as well, and to determine its asymptotics, which is identical to the one satisfied by the linearization. In terms of the rescaled representation, which is a nonlinear Fokker–Planck equation, the convergence rate turns out to be polynomial in time. This result is in contrast with the known exponential decay of such representation for all other values of m.

Special fast diffusion with slow asymptotics. Entropy method and flow on a Riemannian manifold

GRILLO, GABRIELE;
2010-01-01

Abstract

We consider the asymptotic behaviour of positive solutions of the fast diffusion equation u_t = \Delta u^m in the whole Euclidea space R^d, with a precise value for the exponent m = (d − 4)/(d − 2). This case had been left open in the general study (Blanchet et al. in Arch Rat Mech Anal 191:347–385, 2009) since it requires quite different functional analytic methods, due in particular to the absence of a spectral gap for the operator generating the linearized evolution. The linearization of this flow is interpreted here as the heat flow of the Laplace–Beltrami operator of a suitabl Riemannian Manifold (R^d , g), with a metric g which is conformal to the standard Rd metric. Studying the pointwise heat kernel behaviour allows to prove suitable Gagliardo–Nirenberg inequalities associated with the generator. Such inequalities in turn allow one to study the nonlinear evolution as well, and to determine its asymptotics, which is identical to the one satisfied by the linearization. In terms of the rescaled representation, which is a nonlinear Fokker–Planck equation, the convergence rate turns out to be polynomial in time. This result is in contrast with the known exponential decay of such representation for all other values of m.
2010
File in questo prodotto:
File Dimensione Formato  
ARMA 2010.pdf

Accesso riservato

: Altro materiale allegato
Dimensione 502.65 kB
Formato Adobe PDF
502.65 kB Adobe PDF   Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11311/563066
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 38
  • ???jsp.display-item.citation.isi??? 35
social impact