Hot spots of solutions to the heat equation with inverse square potential

TL;DR

This paper studies the long-term behavior of hot spots in solutions to the heat equation with a radially decaying inverse square potential, revealing their connection to harmonic functions and classifying their asymptotic behavior.

Contribution

It classifies the large-time behavior of hot spots for the heat equation with inverse square potential, extending previous work and linking hot spot dynamics to harmonic functions.

Findings

Hot spots tend to stabilize or move predictably over time.

The behavior of hot spots is closely related to harmonic functions associated with the potential.

The classification depends on the decay rate and structure of the potential.

Abstract

We investigate the large time behavior of the hot spots of the solution to the Cauchy problem for the heat equation with a potential $\partial_t u-\Delta u+V(|x|)u=0$, where $V=V(r)$ decays quadratically as $r\to\infty$. In this paper, based on the arguments in [K. Ishige and A. Mukai, preprint (arXiv:1709.00809)], we classify the large time behavior of the hot spots of $u$ and reveal the relationship between the behavior of the hot spots and the harmonic functions for $-\Delta+V$.