Asymptotic behavior of the heat semigroup on certain Riemannian manifolds

TL;DR

This paper proves that on certain non-compact Riemannian manifolds with non-negative Ricci curvature, solutions to the heat equation with integrable initial data asymptotically resemble the heat kernel times the initial mass, extending previous results and providing counterexamples.

Contribution

It establishes asymptotic behavior of the heat semigroup without radiality assumptions on the initial data on manifolds with non-negative Ricci curvature.

Findings

Asymptotic behavior matches the heat kernel times initial mass

Results hold under Li-Yau heat kernel estimates

Counterexample shows failure in sup norm on manifolds with two Euclidean ends

Abstract

We show that, on a complete, connected and non-compact Riemannian manifold of non-negative Ricci curvature, the solution to the heat equation with $L^{1}$ initial data behaves asymptotically as the mass times the heat kernel. In contrast to the previously known results in negatively curved contexts, the radiality assumption on the initial data is not required. Similar long-time convergence results remain valid on more general manifolds satisfying the Li-Yau two-sided estimate of the heat kernel. Moreover, we provide a counterexample such that this asymptotic phenomenon fails in sup norm on manifolds with two Euclidean ends.