  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.
