Modeling, analysis, computation and application of kinetic equations

Long time behavior of solutions to the 2D Keller-Segel equation with degenerate diffusion

Yao Yao

Georgia Tech

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.