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

TL;DR

This paper rigorously constructs a new type of finite-time blowup solution for the 2D Keller-Segel system where two solitons collide and concentrate mass simultaneously, revealing a novel singularity formation mechanism.

Contribution

It introduces a robust framework for constructing solutions with multiple colliding solitons, advancing understanding of complex blowup phenomena in chemotaxis models.

Findings

Constructed solutions with 16π mass concentration at a single point

Demonstrated simultaneous collision and collapse of two solitons

Validated formal predictions of new blowup rates

Abstract

It is well-known that the two-dimensional Keller-Segel system admits finite time blowup solutions, which is the case if the initial density has a total mass greater than $8\pi$ and a finite second moment. Several constructive examples of such solutions have been obtained, where for all of them a perturbed stationary state undergoes scale instability and collapses at a point, resulting in a $8\pi$-mass concentration. It was conjectured that singular solutions concentrating simultaneously more than one solitons 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 Vel\'azquez. We develop for the first time a robust framework to construct rigorously such blowup solutions involving simultaneously the non-radial collision and concentration of several solitons, which we expect to find applications to other evolution problems.