$L^p$ asymptotics for the heat equation on symmetric spaces for non-symmetric solutions

TL;DR

This paper investigates the long-term behavior of solutions to the heat equation on symmetric spaces, revealing how the $L^p$ asymptotics depend on initial data and symmetry, and providing explicit formulas for the mass function.

Contribution

It introduces a new $L^p$-dependent mass function for non-symmetric initial data on symmetric spaces and characterizes the asymptotic behavior of heat solutions, extending previous results.

Findings

The $L^p$ norm of the difference between the solution and the scaled heat kernel tends to zero as time goes to infinity.

Different expressions for the mass function are derived for $1 extless p extless 2$ and $2 extless p extless \infty$ cases.

Explicit formulas for the mass function are provided under bi-$K$-invariance assumptions.

Abstract

The main goal of this work is to study the $L^p$-asymptotic behavior of solutions to the heat equation on arbitrary rank Riemannian symmetric spaces of non-compact type $G/K$ for non-bi-$K$ invariant initial data. For initial data $u_0$ compactly supported or in a weighted $L^1(G/K)$ space with a weight depending on $p\in [1, \infty]$, we introduce a mass function $M_p(u_0)(\cdot)$, and prove that if $h_t$ is the heat kernel on $G/K$, then $$\|h_t\|_p^{-1}\,\|u_0\ast h_t \, - \,M_p(u_0)(\cdot)\,h_t\|_p \rightarrow 0 \quad \text{as} \quad t\rightarrow \infty.$$ Interestingly, the $L^p$ heat concentration leads to completely different expressions of the mass function for $1\leq p <2$ and $2\leq p\leq \infty$. If we further assume that the initial data are bi-$K$-invariant, then our mass function boils down to the constant $\int_{G/K}u_0$ in the case $p=1$, and more generally to $\mathcal{H}{u_0}(i\rho(2/p-1))$ if $1\leq p<2$, and to $\mathcal{H}{u_0}(0)$ if $2\leq p \leq \infty$. Thus we improve upon results by V\'azquez, Anker et al, Naik et al, clarifying the nature of the problem.