Singularity formed by the collision of two collapsing solitons in interaction for the 2D Keller-Segel system

Abstract

It is well-known that the two-dimensional parabolic-elliptic Keller-Segel system admits finite-time blowup solutions, which is the case if the initial density has total mass greater than $8\pi$. Several constructive examples of such solutions have been given, where for all of them a perturbed stationary state undergoes scale instability and collapses at a point, resulting in an $8\pi$-mass concentration. It was conjectured that singular solutions concentrating more than one soliton simultaneously could exist. We construct rigorously such a new blowup mechanism, where two stationary states are simultaneously collapsing and colliding, resulting in a $16\pi$-mass concentration at a single blowup point, and with a new blowup rate which corresponds to the formal prediction by Seki, Sugiyama and Velázquez. We develop, for the first time, a robust framework to rigorously construct blowup solutions that simultaneously involve the non-radial collision and concentration of several solitons, which we expect to have applications to other evolution problems.

155 pages, added comments and references