A model for the diffusion of Aβ-amyloids in the brain: existence, uniqueness, and qualitative analysis

We introduce a PDE model for the dynamics and aggregation of A β -amyloids in a small neighbourhood of a few neurons, in relation to their role in the early stage of Alzheimer's disease. We analyse the well-posedness of the associated semilinear system of heat equations with nonlinear, time-dependent boundary conditions: local existence, uniqueness and global boundedness of positive Ct,x0 solutions are established via a semigroup approach. We further perform a linear stability analysis for the monomer equilibrium u⁎=(m⁎,0,…,0). We show that this equilibrium is stable only under very strong assumptions on the parameters b,β,γ. Several numerical simulations show how oligomers affect the nearby neurons, and tend to move from dying neurons to healthy ones.