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

TL;DR

This paper investigates the long-time asymptotic behavior of heat equation solutions on noncompact symmetric spaces, revealing conditions under which solutions resemble fundamental solutions and exploring phenomena similar to Euclidean spaces.

Contribution

It provides a comprehensive analysis of asymptotic behaviors for heat equations on noncompact symmetric spaces, including cases with bi-$K$-invariant data and the distinguished Laplacian, answering recent open problems.

Findings

Solutions with bi-$K$-invariant $L^{1}$ data behave like the fundamental solution asymptotically.

Counterexamples are provided for non bi-$K$-invariant initial data.

Similar Euclidean phenomena are observed for the distinguished Laplacian case.

Abstract

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. V\'azquez. 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.