  Let $G/H$ be a connected, simply connected homogeneous space of a compact Lie
group $G$. We study $G$-invariant quasi-Einstein metrics on the cohomogeneity
one manifold $G/H\times (0,1)$ imposing the so-called monotypic condition on
$G/H$. We obtain estimates on the rate of blow-up for these metrics near a
singularity under a mild assumption on $G/H$. Next, we demonstrate that we can
find quasi-Einstein metrics satisfying arbitrary $G$-invariant Dirichlet
conditions.
