Analysis and numerical simulations of a chemotaxis model of aggregation of microglia in Alzheimer's disease
In this paper, we study the well-posedness in scales of Hilbert spaces Eα, α ∈ ℝ defined by the non-coupled 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 ⊂ ℝ3 of the spatial average solutions. Uniform bounds in Ω × (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.