  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.
