# On nonexistence and existence of positive global solutions to heat equation with a potential term on Riemannian manifolds

## TL;DR

This paper investigates conditions under which positive solutions to the heat equation with a potential exist or do not exist on Riemannian manifolds, providing sharp criteria based on volume growth of geodesic balls.

## Contribution

It introduces a natural sharp condition involving geodesic ball volume to determine the existence or nonexistence of solutions, advancing understanding in geometric analysis.

## Key findings

- Sharp volume-based condition for nonexistence of solutions
- Criteria applicable to a wide class of Riemannian manifolds
- Enhanced understanding of heat equation behavior with potentials

## Abstract

We reinvestigate nonexistence and existence of global positive solutions to heat equation with a potential term on Riemannian manifolds. Especially, we give a very natural sharp condition only in terms of the volume of geodesic ball to obtain nonexistence results.