We present new analytical and
numerical results for the elliptic–parabolic system of partial differential equations proposed by Hu and Cai, which models the formation of biological transport networks. The model describes the pressure field us
ing a Darcy’s type equation and the dynamics of the conductance network under pressure force effects. Randomness
in the material structure is represented by a l
inear diffusion term and conductance relaxation by an algebraic decay term. The analytical part extends the results of Haskovec et al. (2015) regard
ing the existence of weak and mild solutions to the
whole range of mean
ingful relaxation exponents. Moreover, we prove f
inite time ext
inction or break-down of solutions
in the spatially one-dimensional sett
ing for certa
in ranges of the relaxation exponent. We also construct stationary solutions for the case of vanish
ing diffusion and critical value of the relaxation exponent, us
ing a variational formulation and a penalty method.
The analytical part is complemented by extensive numerical simulations. We propose a discretization based on mixed finite elements and study the qualitative properties of network structures for various parameter values. Furthermore, we indicate numerically that some analytical results proved for the spatially one-dimensional setting are likely to be valid also in several space dimensions.