  In this paper we first derive several results concerning the $L^p$ spectrum
of arithmetic locally symmetric spaces whose $\Q$-rank equals one. In
particular, we show that there is an open subset of $\C$ consisting of
eigenvalues of the $L^p$ Laplacian if $p <2$ and that corresponding
eigenfunctions are given by certain Eisenstein series. On the other hand, if
$p>2$ there is at most a discrete set of real eigenvalues of the $L^p$
Laplacian. These results are used in the second part of this paper in order to
show that the dynamics of the $L^p$ heat semigroups for $p<2$ is very different
from the dynamics of the $L^p$ heat semigroups if $p\geq 2$.
