Title: Asymptotic behavior of solutions of the fractional heat equation on Riemannian symmetric spaces of non-compact type

Abstract: The main goal of this paper is to study the 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 $h_t^\alpha$ denote the fractional heat kernel on a Riemannian symmetric space of non-compact type $X=G/K$ (see Section \ref{sef1}). We show that if $p \in [1,2]$ and $f \in L^1(X)$ is $K$-bi-invariant, then \[\lim_{t\to\infty}\frac{\|f\ast h_t^\alpha-\what f(i \gamma_p \rho)h_t^\alpha\|_p}{\|h_t^\alpha\|_p}=0,\] where $\gamma_p:=\frac 2p-1$, $\rho$ is the half sum of positive roots and $\what f(i \gamma_p \rho)$ denotes the spherical Fourier transform of $f $ at $i\gamma_p\rho$ (see Section \ref{prelim}). The above problems for symmetric spaces were studied recently by several authors for the case $p=1$ in some important cases \cite{Vaz-2, AE, Eff1, Eff2}. However, we believe that our study for the case $p \in (1,2]$ is new.