The aggregation equation with power-law kernels: ill-posedness, mass concentration and similarity solutions (Q543372)

The author studies the ill-posedness, mass concentration and similarity of the aggregation equation with power-law kernels. For the equation with biological relevant potential, the author shows it is ill-posed in the critical Lebesgue space in the sense that there exist initial data in the critical space such that the unique measure-valued solution leaves the space immediately. Then the author studies the equation with more general power-law kernels, and proves a conjecture posed in [\textit{A. L. Bertozzi, T. Laurent} and \textit{J. Rosado}, Commun. Pure Appl. Math. 64, No. 1, 45--83 (2011; Zbl 1218.35075)]. Finally the author characterizes all the ``first kind'' radially symmetric similarity solutions in the higher dimensional case.