Well-posedness and finite-time extinction of a PDE-ODE spatial-network model with anisotropic diffusion

We study a system of reaction-diffusion equations posed on a bounded domain composed of subdomains separated by a connected network with a metric graph structure. The reaction-diffusion dynamics with anisotropic diffusion on the graph edges are coupled to well-mixed ODE dynamics occurring at the vertices by junction conditions, and to similar PDE dynamics occurring on adjacent subdomains through Robin-like boundary conditions. The resulting PDE-ODE system can be used in epidemiological and ecological settings to study population movement in between cluster centers along road-like structures and into the surrounding continuum. We employ a semi-Galerkin approximation to establish the well-posedness of weak solutions to the PDE-ODE system, and examine further properties such as regularity, boundedness and finite-time extinction.