FINITE-TIME BLOW-UP IN SOME MODELS OF CHEMOTAXIS

We consider a class of models of chemotactic bacterial populations, in troduced by Keller-Segel. For those models, we investigate the possibi lity of chemotactic collapse, in other words, the possibility that in finite time the population of predators aggregates to form a delta-fun ction. To study this phenomenon, we construct self-similar solutions, which may or may not blow-up (in finite time), depending on the relati ve strength of three mechanisms in competition: (i) the chemotactic at traction of bacteria towards regions of high concentration in substrat e (ii) the rate of consumption of the substrate by the bacteria and (i ii) (possibly) the diffusion of bacteria. The solutions we construct a re radially symmetric, and therefore have no relation with the classic al traveling wave solutions. Our scaling can be justified by a dimensi onal analysis. We give some evidence of numerical stability.