  We investigate gradient flows of some homogeneous functionals in R^N ,
arising in the Lagrangian approximation of systems of self-interacting and
diffusing particles. We focus on the case of negative homogeneity. In the case
of strong self-interaction, the functional possesses a cone of negative energy.
It is immediate to see that solutions with negative energy at some time become
singular in finite time, meaning that a subset of particles concentrate at a
single point. Here, we establish that all solutions become singular in finite
time for the class of functionals under consideration. The paper is completed
with numerical simulations illustrating the striking non linear dynamics when
initial data have positive energy.
