On Analysis and Numerical Simulations of a Chemotaxis Model of Aggregation of Microglia in Alzheimer's Disease

Abstract

In this paper, we study the wellposedness in scales of Hilbert spaces $E^{\alpha},\alpha\in\mathbb{R}$ defined by the noncoupled system partial differential operator of a chemotaxis model of aggregation of microglia in Alzheimer's disease for a perturbated analytic semigroup, which decays exponentially in the large time asymptotic dynamics of the problem to a finite dimensional set $K\subset \mathbb{R}^{3}$ of the spatial average solutions. Uniform bounds in $\Omega\times (0,T)$ of solutions and gradient solutions to the system of equations are proved. Thus via a bootstrap argument solutions to the problem are shown to be classical solutions. Furthermore, under natural conditions on the coupled elliptic system quasilinear differential operator, we prove the existence of a fundamental solution or evolution operator for the model equations in cited function spaces. In conclusion numerical simulation results are provided.