ASYMPTOTICS OF BLOWUP SOLUTIONS FOR THE AGGREGATION EQUATION

We consider the asymptotic behavior of radially symmetric solutions of the aggregation equation ut = del . (u del K * u) in R-n, for homogeneous potentials K(x) = vertical bar x vertical bar(gamma), gamma > 0. For gamma > 2, the aggregation happens in infinite time and exhibits a concentration of mass along a collapsing delta-ring. We develop an asymptotic theory for the approach to this singular solution. For gamma < 2, the solution blows up in fi nite time and we present careful numerics of second type similarity solutions for all gamma in this range, including additional asymptotic behaviors in the limits gamma -> 0(+) and gamma -> 2(-).