Solvability for a reaction-diffusion system modeling biological transportation network

The aim of this paper is to investigate the initial-boundary value problem of a possibly degenerate reaction-diffusion system over Omega subset of R-n with n >= 1 of the following form {partial derivative(t)m(i )- kappa Delta m(i )+ |m(i)|(gamma-2)m(i) = (partial derivative x(i)p)(2), -del & sdot;[m del p] = S, with m = diag(m(1),& ctdot;,m(n)), the diffusivity kappa > 0, the metabolic exponent gamma >= 2 and the given function S. When kappa = 0, this system was introduced by Haskovec, Kreusser and Markowich as a continuous version of the discrete Hu-Cai model for biological transport networks. In this work, our result asserts that whenever the random fluctuations of the conductance in the medium were considered, i.e., kappa > 0, then for general large data the corresponding initial-boundary value problem possesses a global weak solution.