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 \(\Omega \subset \mathbb {R}^n\) $\mathrm{\Omega }\subset {\mathbb{R}}^n$ with \(n\ge 1\) $n\ge 1$ of the following form \(\begin{aligned} \left\{ \begin{aligned}&\partial _tm_i-\kappa \Delta m_i+|m_i|^{\gamma -2}m_i=(\partial _{x_i}p)^2,\\&-\nabla \cdot [\textbf{m}\nabla p]=S, \end{aligned} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}& {\partial }_t{m}_i-\kappa \mathrm{\Delta }{m}_i+{|m_i|}^{\gamma -2}{m}_i={({\partial }_{x_i}p)}^2,\hfill \\ \hfill & -\mathrm{\nabla }·[\mathbf{m}\mathrm{\nabla }p]=S,\hfill \end{array}\right)\end{array}$$ with \(\textbf{m}=\textrm{diag} (m_1,\cdots ,m_n)\) $\mathbf{m}=\text{diag}(m_1,\cdots ,m_n)$ , the diffusivity \(\kappa >0\) $\kappa >0$ , the metabolic exponent \(\gamma \ge 2\) $\gamma \ge 2$ and the given function S. When \(\kappa =0\) $\kappa =0$ , 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., \(\kappa >0\) $\kappa >0$ , then for general large data the corresponding initial-boundary value problem possesses a global weak solution.