Well-posedness of a mathematical model for Alzheimer's disease

TL;DR

This paper proves the existence and uniqueness of solutions for a complex PDE system modeling Alzheimer's disease, combining reaction-diffusion, transport equations, and probability measures with advanced mathematical techniques.

Contribution

It introduces a novel mathematical framework for Alzheimer's modeling, integrating probability measures with PDEs and establishing well-posedness under realistic assumptions.

Findings

Existence and uniqueness of solutions are established.

The model incorporates biologically relevant probability measures.

Advanced mathematical tools are combined to handle the coupled PDE system.

Abstract

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.