ASYMPTOTICS OF BLOWUP SOLUTIONS FOR THE AGGREGATION EQUATION

摘要

We consider the asymptotic behavior of radially symmetric solutions of the aggregation equation u_t = ▽ • (u▽K * u) in R~n, for homogeneous potentials K(x) = |x|~γ, γ > 0. For γ > 2, the aggregation happens in infinite time and exhibits a concentration of mass along a collapsing δ-ring. We develop an asymptotic theory for the approach to this singular solution. For γ < 2, the solution blows up in finite time and we present careful numerics of second type similarity solutions for all 7 in this range, including additional asymptotic behaviors in the limits γ → 0~+ and γ → 2~-.