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Analysis and numerical simulations of a chemotaxis model of aggregation of microglia in Alzheimer’s disease. (English) Zbl 1277.35205

Summary: We study the wellposedness in scales of Hilbert spaces \(E^{\alpha},\alpha\in\mathbb{R}\) defined by the noncoupled system of a chemotaxis model of aggregation of microglia in Alzheimer’s disease for a perturbated analytic semigroup, which decays exponentially in the large time 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 are proved. Thus, via a bootstrap argument the solutions to the problem are shown to be classical solutions. Furthermore, under natural conditions on the coupled elliptic system, 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.

MSC:

35K51 Initial-boundary value problems for second-order parabolic systems
35K57 Reaction-diffusion equations
92C17 Cell movement (chemotaxis, etc.)
35A35 Theoretical approximation in context of PDEs
35K90 Abstract parabolic equations
47D06 One-parameter semigroups and linear evolution equations
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References:

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