Asymptotics of blowup solutions for the aggregation equation (Q423839)

The authors consider the asymptotic behavior of radially symmetric solutions of the aggregation equation \(u_{t}=\nabla\cdot(u\nabla{K}\ast{u})\) in \(\mathbb{R}^{n}\) for homogeneous potentials \(K(x)=|x|^{\gamma}\), \(\gamma>0\). For \(\gamma>2\), the aggregation happens in infinite time and exhibits a concentration of mass along a collapsing \(\delta\)-ring. The authors develop an asymptotic theory for the approach to this singular solution. For \(\gamma<2\), the solution blows up in finite time and the authors present careful numerical investigations of second-type similarity solutions for all \(\gamma\) in this range, including additional asymptotic behaviors in the limits \(\gamma\to0^{+}\) and \(\gamma\to2^{-}\).