Self-regulated biological transportation structures with general entropy dissipations, part Ⅰ: The 1D case

We study self-regulating processes modeling biological transportation networks as presented in [15]. In particular, we focus on the 1D setting for Dirichlet and Neumann boundary conditions. We prove an existence and uniqueness result under the assumption of positivity of the diffusivity $ D $. We explore systematically various scenarios and gain insights into the behavior of $ D $ and its impact on the studied system. This involves analyzing the system with a signed measure distribution of sources and sinks. Finally, we perform several numerical tests in which the solution $ D $ touches zero, confirming the previous hints of local existence in particular cases.