Hot spots of solutions to the heat equation with inverse square 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 $-\Delta+V$.