  Recently a Dynamic-Monge-Kantorovich formulation of the PDE-based
$L^1$-optimal transport problem was presented. The model considers a diffusion
equation enforcing the balance of the transported masses with a time-varying
conductivity that volves proportionally to the transported flux. In this paper
we present an extension of this model that considers a time derivative of the
conductivity that grows as a power law of the transport flux with exponent
$\beta>0$. A sub-linear growth ($0<\beta<1$) penalizes the flux intensity and
promotes distributed transport, with equilibrium solutions that are reminiscent
of Congested Transport Problems. On the contrary, a super-linear growth
($\beta>1$) favors flux intensity and promotes concentrated transport, leading
to the emergence of steady-state "singular" and "fractal-like" configurations
that resemble those of Branched Transport Problems. We derive a numerical
discretization of the proposed model that is accurate, efficient, and robust
for a wide range of scenarios. For $\beta>1$ the numerical model is able to
reproduce highly irregular and fractal-like formations without any a-priory
structural assumption.
