Well-Posedness of a Mathematical Model for Alzheimer's Disease

We consider the existence and uniqueness of solutions of an initial boundary value problem for a coupled system of PDEs arising in a model for Alzheimer's disease. Apart from reaction diffusion equations, the system contains a transport equation in a bounded interval for a probability measure which is related to the malfunctioning of neurons. The main ingredients to prove existence are the method of characteristics for the transport equation, a priori estimates for solutions of the reaction diffusion equations, a variant of the classical contraction theorem, and the Wasserstein metric for the part concerning the probability measure. We stress that all hypotheses on the data are not suggested by mathematical artifacts, but are naturally imposed by modeling considerations. In particular the use of a probability measure is natural from a modeling point of view. The nontrivial part of the analysis is the suitable combination of the various mathematical tools, which is not quite routine and requires various technical adjustments.