On the uniqueness for the heat equation on complete Riemannian manifolds

TL;DR

This paper establishes new uniqueness results for solutions to the heat equation on complete Riemannian manifolds, including for $L^p$ solutions with $0<p<1$, and improves previous $L^1$ uniqueness results by relaxing curvature conditions.

Contribution

It introduces novel uniqueness theorems for heat equation solutions on manifolds, extending the range of $L^p$ spaces and weakening curvature assumptions.

Findings

Proves uniqueness of $L^p$ solutions for $0<p<1$.

Weakens curvature conditions needed for $L^1$ uniqueness.

Enhances previous results by P. Li on heat equation uniqueness.

Abstract

We prove some uniqueness result for solutions to the heat equation on Riemannian manifolds. In particular, we prove the uniqueness of $L^p$ solutions with $0< p< 1$, and improves the $L^1$ uniqueness result of P. Li by weakening the curvature assumption.