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

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 $$\Vert h_t{\Vert }_p^{-1}\Vert u_0\ast h_t-M_p(u_0)(\cdot )h_t{\Vert }_p\to 0\text{as}t\to \infty .$$ Interestingly, the $L^p$ heat concentration leads to completely different expressions of the mass function for $1\le p<2$ and $2\le p\le \infty$. 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 $1\le p<2$, and to $\mathcal{H}{u}_0(0)$ if $2\le p\le \infty$. Thus we improve upon results by Vázquez, Anker et al, Naik et al, clarifying the nature of the problem.