  Motivated by recent physics papers describing rules for natural network
formation, we study an elliptic-parabolic system of partial differential
equations proposed by Hu and Cai. The model describes the pressure field thanks
to Darcy's type equation and the dynamics of the conductance network under
pressure force effects with a diffusion rate $D$ representing randomness in the
material structure. We prove the existence of global weak solutions and of
local mild solutions and study their long term behaviour. It turns out that, by
energy dissipation, steady states play a central role to understand the pattern
capacity of the system. We show that for a large diffusion coefficient $D$, the
zero steady state is stable. Patterns occur for small values of $D$ because the
zero steady state is Turing unstable in this range; for $D=0$ we can exhibit a
large class of dynamically stable (in the linearized sense) steady states.
