Exact blow-up profiles for the parabolic-elliptic Keller-Segel system in dimensions $N\ge 3$

TL;DR

This paper determines the precise blow-up profiles for solutions of the Keller-Segel-Patlak system in dimensions three and higher, solving an open problem and advancing understanding of singularity formation in chemotaxis models.

Contribution

It introduces a novel zero number argument for nonlinear equations with unbounded coefficients and constructs backward self-similar solutions via ODE analysis.

Findings

Exact blow-up profiles for N≥3 dimensions

Solution of an open problem from 2019

Development of new zero number technique

Abstract

In this paper, we obtain the exact blow-up profiles of solutions of the Keller-Segel-Patlak system in the space with dimensions $N\ge 3$, which solves an open problem proposed by P. Souplet and M. Winkler in 2019. To establish this achievement, we develop the zero number argument for nonlinear equations with unbounded coefficients and construct a family of auxiliary backward self-similar solutions through nontrivial ODE analysis.