Asymptotic behavior of the heat semigroup on certain Riemannian manifolds

Comportement asymptotique du semi-groupe de la chaleur sur certaines variétés riemanniennes

Alexander Grigor'Yan (1) , Effie Papageorgiou (2) , Hong-Wei Zhang (3)

1 SFB 705 - Fakultät für Mathematik [Bielefeld]
2 Department of Mathematics, University of Crete
3 Department of Mathematics [Gent/Ghent]

We show that, on a complete, connected and non-compact Riemannian manifold of non-negative Ricci curvature, the solution to the heat equation with L1 initial data behaves asymptotically as the mass times the heat kernel. In contrast to the previously known results in negatively curved contexts, the radial assumption is not required. Moreover, we provide a counterexample such that this asymptotic phenomenon fails in sup norm on manifolds with two Euclidean ends.

Domaines: Géométrie différentielle [math.DG]; Equations aux dérivées partielles [math.AP]

Alexander Grigor'Yan, Effie Papageorgiou, Hong-Wei Zhang. Asymptotic behavior of the heat semigroup on certain Riemannian manifolds. 2022. ⟨hal-03666655⟩