Improved intermediate asymptotics for the heat equation

TL;DR

This paper improves the understanding of how solutions to the heat equation converge to stationary states by establishing faster convergence rates through entropy methods and new inequalities, extending to Fokker-Planck equations.

Contribution

It introduces improved convergence rates for the heat equation's intermediate asymptotics using entropy methods and establishes new functional inequalities, extending results to Fokker-Planck equations.

Findings

Enhanced convergence rates under moment matching conditions

Equivalence between entropy decay and functional inequalities

Extension of results to Fokker-Planck equations with confining potentials

Abstract

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.