Finite time blow-up in some models of chemotaxis

We consider a class of models of chemotactic bacterial populations, introduced by Keller-Segel. For those models, we investigate the possibility of chemotactic collapse, in other words, the possibility that in finite time the population of predators aggregates to form a delta-function. To study this phenomenon, we construct self-similar solutions, which may or may not blow-up (in finite time), depending on the relative strength of three mechanisms in competition: (i) the chemotactic attraction of bacteria towards regions of high concentration in substrate (ii) the rate of consumption of the substrate by the bacteria and (iii) (possibly) the diffusion of bacteria. The solutions we construct are radially symmetric, and therefore have no relation with the classical traveling wave solutions. Our scaling can be justified by a dimensional analysis. We give some evidence of numerical stability.