Asymptotic behavior of solutions to the heat equation on noncompact symmetric spaces

Comportement asymptotique des solutions de l'équation de la chaleur sur les espaces symétriques de type non-compact

Jean-Philippe Anker (1) , Effie Papageorgiou (2) , Hong-Wei Zhang (3)

1 IDP - Institut Denis Poisson
2 Department of Mathematics, University of Crete
3 Department of Mathematics [Gent/Ghent]

This paper is twofold. The first part aims to study the long-time asymptotic behavior of solutions to the heat equation on Riemannian symmetric spaces $G/K$ of noncompact type and of general rank. We show that any solution to the heat equation with bi-$K$-invariant $L^{1}$ initial data behaves asymptotically as the mass times the fundamental solution, and provide a counterexample in the non bi-$K$-invariant case. These answer problems recently raised by J.L. Vázquez. In the second part, we investigate the long-time asymptotic behavior of solutions to the heat equation associated with the so-called distinguished Laplacian on $G/K$. Interestingly, we observe in this case phenomena which are similar to the Euclidean setting, namely $L^1$ asymptotic convergence with no bi-$K$-invariance condition and strong $L^{\infty}$ convergence.

Mots clés: Noncompact symmetric space; Heat equation; Asymptotic behaviour; Long-time convergence; Distinguished Laplacian

Domaines: Mathématiques [math]

Jean-Philippe Anker, Effie Papageorgiou, Hong-Wei Zhang. Asymptotic behavior of solutions to the heat equation on noncompact symmetric spaces. 2021. ⟨hal-03459571v1⟩