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.