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

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 α ∈ (0,1], let hₜ^α denote the fractional heat kernel on a Riemannian symmetric space of non-compact type $X=G/K$ (see Section {sef1}). We show that if p ∈ [1,2] and f ∈ L¹(X) is K-bi-invariant, then \[limt→∞\|f h_t^α- f(i γ_p ρ)h_t^α\|_p/\|h_t^α\|_p=0,\] where γₚ:= 2p-1, ρ is the half sum of positive roots and f(i γₚ ρ) denotes the spherical Fourier transform of f at iγₚρ (see Section {prelim}). The above problems for symmetric spaces were studied recently by several authors for the case $p=1$ in some important cases {Vaz-2, AE, Eff1, Eff2}. However, we believe that our study for the case p ∈ (1,2] is new.