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Talk Abstract

Spike-Layer Steady States in Diffusion Systems

Spike-Layer Steady States in Diffusion Systems

**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!