Abstract:
The Keller-Segel equation is a nonlocal PDE modeling the
collective motion of cells attracted by a self-emitted chemical
substance. When this equation is set up in 2D with a degenerate
diffusion term, it is known that solutions exist globally in
time, but their long-time behavior remain unclear. In a joint
work with J. Carrillo, S. Hittmeir and B. Volzone, we prove that
all stationary solutions must be radially symmetric up to a
translation, and use this to show convergence towards the
stationary solution as the time goes to infinity. I will also
discuss another joint work with K. Craig and I. Kim, where we
let the power of degenerate diffusion go to infinity in the 2D
Keller-Segel equation, so it becomes an aggregation equation
with a constraint on the maximum density. We will show that if
the initial data is a characteristic function, the solution will
converge to the characteristic function of a disk as the time
goes to infinity with certain convergence rate. |