  We consider a drift-diffusion model, with an unknown function depending on
the spatial variable and an additional structural variable, the amount of
ingested lipid. The diffusion coefficient depends on this additional variable.
The drift acts on this additional variable, with a power-law coefficient of the
additional variable and a localization function in space. It models the
dynamics of a population of macrophage cells. Lipids are located in a given
region of space; when cells pass through this region, they internalize some
lipids. This leads to a problem whose mathematical novelty is the dependence of
the diffusion coefficient on the past trajectory. We discuss global existence
and blow-up of the solution.
