Title: $L^p$-asymptotic behaviour of solutions of the fractional heat equation on Riemannian symmetric spaces of noncompact type

Abstract: The main goal of this paper is to study the $L^p$-asymptotic behaviour of solutions to the heat equation and the fractional heat equations on a Riemannian symmetric space of non-compact type. For $\alpha \in (0,1]$, let $h_t^\alpha$ denotes the fractional heat kernel on a Riemannian symmetric space of non-compact type $X=G/K$ (with $h_t^1$ being the standard heat kernel of $X$). Suppose $f \in L^1(X)$ is a left $K$-invariant function. We show that if $p \in [1,2]$ then \begin{equation*} \lim_{t\to\infty}\|h_t^\alpha\|_p^{-1}\|f\ast h_t^\alpha-\what f(i \gamma_p \rho)h_t^\alpha\|_p=0, \end{equation*} where $\gamma_p=2/p-1$, $\rho$ is the half sum of positive roots and $\what f$ denotes the spherical Fourier transform of $f$. If $p\in (2,\infty]$ then \begin{equation*} \lim_{t\to\infty}\|h_t^\alpha\|_p^{-1}\|f\ast h_t^\alpha-\what f(0)h_t^\alpha\|_p=0. \end{equation*} In fact, we will show that the above results are valid for more general class of left $K$-invariant functions larger than the space of integrable left $K$-invariant functions on $X$. The above statements for symmetric spaces were earlier studied in the important special case $p=1=\alpha$, in \cite{Vaz-2, AE}.