Concentration in an advection-diffusion model with diffusion coefficient depending on the past trajectory (Q6614207)

The present paper studied a drift-diffusion equation modelling the dynamics of a population of macrophage cells within the arterial wall in the presence of lipids. Lipids occupy a given and fixed region of the arterial wall. When macrophage cells pass through this region, they internalise lipids, causing them to grow and slow down. In this equation, the unknown function depends on the spatial variable and also on an additional structural variable: the quantity of lipid ingested. The diffusion coefficient depends on this additional variable and the drift acts on this variable through a power law coupled to a spatial localisation function. This leads to a problem whose mathematical novelty is the dependence of the diffusion coefficient on the past trajectory. The authors addressed the question of whether or not this advection-diffusion equation can lead to concentration and eventually blow-up in finite time, if diffusion is not strong enough to prevent cells from being trapped in lipid dense regions. That is the conclusions of global existence and blow-up of solutions to this model were figured out.