Spora motion in chemotaxis models

In this talk I will describe some systems of partial differential equations that in a suitable asymptotic limit describe the motion of "spora-like" solutions. The considered system of equations becomes in a suitable limit case the well studied Keller-Segel system of chemotaxis. Is is known that such system of equations exhibits singular solutions that develop singularities in a finite time. The problem addressed in this talk is that of the continuation of the solutions after singularity formation. Asymptotic analysis indicates that the corresponding limit solutions are made of a singular part and a regular part. The singular part consists on a set of "sporae" containing a positive mass of microorganism concentrated in a small region. Evolution equation for these sporae as well as their limit behaviour for long times will be also discussed.