  This paper is devoted to the study of the evolution of positively curved
metrics on the Wallach spaces $SU(3)/T_{\max}$, $Sp(3)/Sp(1)\times Sp(1)\times
Sp(1)$, and $F_4/Spin(8)$. We prove that for all Wallach spaces, the normalized
Ricci flow evolves all generic invariant Riemannian metrics with positive
sectional curvature into metrics with mixed sectional curvature. Moreover, we
prove that for the spaces $Sp(3)/Sp(1)\times Sp(1)\times Sp(1)$ and
$F_4/Spin(8)$, the normalized Ricci flow evolves all generic invariant
Riemannian metrics with positive Ricci curvature into metrics with mixed Ricci
curvature. We also get similar results for some more general homogeneous
spaces.
