Asymptotic behaviour for the Fractional Heat Equation in the Euclidean space

TL;DR

This paper studies the long-term behavior of solutions to the fractional heat equation in Euclidean space, proving convergence to the fundamental solution with rates under certain initial conditions.

Contribution

It establishes the asymptotic convergence of solutions to the fractional heat equation to the fundamental solution, including convergence rates and applications to the fractional Fokker-Planck equation.

Findings

Solutions converge to the fundamental solution as time approaches infinity.

Convergence rates are obtained for solutions with finite first moments.

Relative error convergence holds for solutions with compact support.

Abstract

We consider weak solutions of the fractional heat equation posed in the whole $n$-dimensional space, and establish their asymptotic convergence to the fundamental solution as $t\to\infty$ under the assumption that the initial datum is an integrable function, or a finite Radon measure. Convergence with suitable rates is obtained for solutions with a finite first initial moment, while for solutions with compactly supported initial data convergence in relative error holds. The results are applied to the fractional Fokker-Planck equation. Brief mention of other techniques and related equations is made.