  We study the J-flow on Kahler surfaces when the Kahler class lies on the
boundary of the open cone for which global smooth convergence holds, and
satisfies a nonnegativity condition. We obtain a C^0 estimate and show that the
J-flow converges smoothly to a singular Kahler metric away from a finite number
of curves of negative self-intersection on the surface. We discuss an
application to the Mabuchi energy functional on Kahler surfaces with ample
canonical bundle.
