<i>L</i> 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 $$ \begin{align*} &amp;\|h_t\|_p^{-1}\,\|u_0\ast h_t \, - \,M_p(u_0)(\cdot)\,h_t\|_p \rightarrow 0 \quad \textrm{as} \quad t\rightarrow \infty.\end{align*} $$ Interestingly, the $L^{p}$ heat concentration leads to completely different expressions of the mass function for $1\leq p &amp;lt;2$ 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 $1\leq p&amp;lt;2$, and to $\mathcal{H}{u_{0}}(0)$ if $2\leq p \leq \infty $. Thus, we improve upon results by Vázquez, Anker et al., and Naik et al., clarifying the nature of the problem.