Michel Rascle, University of Nice
We consider a few models of chemotactism, inspired from the classical Keller-Segel model, namely
where denotes the gradient, u the concentration in predators, s the concentration in substrate. I will discuss a few different examples in which the population of predators concentrates in finite time to a delta-function : aggregation. I will mainly focus on cartoons, where the diffusion is neglected. In the unstable case (the + case), aggregation is linked with a severe pathology in the structure of the underlying system of conservation laws, which is typically mixed type. However, I will show how one can mathematically solve the problem, and even describe such a singular solution after aggregation. Such solutions seem to be numerically very stable!