Improved intermediate asymptotics for the heat equation

Bartier, Jean-Philippe; Blanchet, Adrien; Dolbeault, Jean; Escobedo, Miguel

Bartier, Jean-Philippe; Blanchet, Adrien; Dolbeault, Jean; Escobedo, Miguel (2011), Improved intermediate asymptotics for the heat equation, Applied Mathematics Letters, 24, 1, p. 76-81. http://dx.doi.org/10.1016/j.aml.2010.08.020

Type

Article accepté pour publication ou publié

External document link

http://arxiv.org/abs/0908.2226v1

Date

2011

Journal name

Applied Mathematics Letters

Volume

Number

Publisher

Elsevier

Pages

76-81

Abstract (EN)

This letter is devoted to results on intermediate asymptotics for the heat equation. We study the convergence towards a stationary solution in self-similar variables. By assuming the equality of some moments of the initial data and of the stationary solution, we get improved convergence rates using entropy/entropy-production methods. We establish the equivalence of the exponential decay of the entropies with new, improved functional inequalities in restricted classes of functions. This letter is the counterpart in a linear framework of a recent work on fast diffusion equations; see Bonforte et al. (2009) [18]. The results extend to the case of a Fokker–Planck equation with a general confining potential.

Subjects / Keywords

Intermediate asymptotics; Self-similar variables; Ornstein–Uhlenbeck equation; Entropy; Poincaré inequality; Logarithmic Sobolev inequality