Improved intermediate asymptotics for the heat equation

Jean-Philippe Bartier (1) , Adrien Blanchet (2) , Jean Dolbeault (1) , Miguel Escobedo (3)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 GREMAQ - Groupe de recherche en économie mathématique et quantitative
3 Departamento de Matemáticas [Bilbao]

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-Dolbeault-Grillo-Vazquez]. Results extend to the case of a Fokker-Planck equation with a general confining potential.

Mots clés: Heat equation; Fokker-Planck equation; Ornstein-Uhlenbeck equation; intermediate asymptotics; self-similar variables; stationary solutions; large time behavior; rate of convergence; entropy; PoincarŽ inequality; logarithmic Sobolev inequality; interpolation inequalities

Domaines: Equations aux dérivées partielles [math.AP]

Jean-Philippe Bartier, Adrien Blanchet, Jean Dolbeault, Miguel Escobedo. Improved intermediate asymptotics for the heat equation. Applied Mathematics Letters, 2011, 24 (1), pp.76 - 81. ⟨10.1016/j.aml.2010.08.020⟩. ⟨hal-00409935⟩