  We study the asymptotic behavior of the pluriclosed flow in the case of
left-invariant Hermitian structures on Lie groups. We prove that solutions on
2-step nilpotent Lie groups and on almost-abelian Lie groups converge, after a
suitable normalization, to self-similar solutions of the flow. Given that the
spaces are solvmanifolds, an unexpected feature is that some of the limits are
shrinking solitons. We also exhibit the first example of a homogeneous manifold
on which a geometric flow has some solutions with finite extinction time and
some that exist for all positive times.
