Well-posedness of a network transport model

We prove existence and uniqueness of solutions of a model for the progression of soluble and insoluble toxic Tau proteins on a graph of nerve cells in an Alzheimer brain. The model was recently introduced to deal with the existence of two timescales in Alzheimer's disease, a fast one for most of the involved physical and chemical mechanisms and a much slower one for the evolution of the disease. Considering the physical and chemical mechanisms as instantaneous, one obtains a quasi-static model in the slow timescale. The model combines an active transport mechanism of soluble Tau on the edges of the graph with the dynamics of Tau at the nodes.