  In this paper we investigate regularity properties of weak solutions to a PDE
system that arises in the study of biological transport networks. The system
consists of a possibly singular elliptic equation for the scalar pressure of
the underlying biological network coupled to a diffusion equation for the
conductance vector of the network. There are several different types of
nonlinearities in the system. Of particular mathematical interest is a term
that is a polynomial function of solutions and their partial derivatives and
this polynomial function has degree three. That is, the system contains a cubic
nonlinearity. Only weak solutions to the system have been shown to exist. The
regularity theory for the system remains fundamentally incomplete. In
particular, it is not known whether or not weak solutions develop
singularities. In this paper we obtain a partial regularity theorem, which
gives an estimate for the parabolic Hausdorff dimension of the set of possible
singular points.
