  We consider the existence and uniqueness of solutions of an initial-boundary
value problem for a coupled system of PDE's 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 artefacts, but are naturally imposed by
modelling considerations. In particular the use of a probability measure is
natural from a modelling 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.
