Self-similar solution for Hardy operator

TL;DR

This paper analyzes the large-time behavior of solutions to the fractional heat equation with Hardy potentials, revealing that asymptotics are governed by a self-similar solution derived from the kernel of the Feynman-Kac semigroup.

Contribution

It introduces a self-similar solution framework for the fractional heat equation with Hardy potentials, connecting asymptotics to the kernel of the Feynman-Kac semigroup.

Findings

Asymptotic behavior is governed by a self-similar solution.

The self-similar solution is obtained as a limit of the kernel.

Results apply to subcritical and critical Hardy potentials.

Abstract

We describe the large-time asymptotics of solutions to the heat equation for the fractional Laplacian with added subcritical or even critical Hardy-type potential. The asymptotics is governed by a self-similar solution of the equation, obtained as a normalized limit at the origin of the kernel of the corresponding Feynman-Kac semigroup.