Global well-posedness and nonlinear stability of a chemotaxis system modeling multiple sclerosis

TL;DR

This paper proves the global existence and uniform boundedness of solutions for a chemotaxis system modeling multiple sclerosis, establishes nonlinear stability under certain conditions, and demonstrates pattern formation through numerical simulations.

Contribution

It provides the first proof of global well-posedness and stability for this chemotaxis model related to multiple sclerosis, including numerical analysis of pattern formation.

Findings

Global existence of strong solutions in any dimension

Solutions are uniformly bounded in time

Turing patterns emerge with large chemotaxis

Abstract

We consider a system of reaction-diffusion equations including chemotaxis terms and coming out of the modeling of multiple sclerosis. The global existence of strong solutions to this system in any dimension is proved, and it is also shown that the solution is bounded uniformly in time. Finally, a nonlinear stability result is obtained when the chemotaxis term is not too big. We also perform numerical simulations to show the appearance of Turing patterns when the chemotaxis term is large.