On the Global Well-Posedness of a Non-Autonomous Reaction–Diffusion–Advection System Stemming from Alzheimer’s Modeling

In this paper, we study the global well-posedness of a reaction–diffusion–advection system with time-dependent coefficients that models the interaction of ATM and ApoE proteins in cells under Alzheimer’s disease. We prove the existence and uniqueness of a non-negative global solution to this system considering in particular a non-autonomous advective term by demonstrating the existence of a local-in-time solution both with an autonomous and non-autonomous approach. The non-negativity of the solution is proven in the non-autonomous case and passed to the solution of the autonomous setting, and we compute energy estimates that show that this solution is in fact global.