Nonradial stability of self-similar blowup to Keller-Segel equation in three dimensions

TL;DR

This paper extends the understanding of the Keller-Segel system by proving the nonradial stability of a self-similar blow-up solution in three dimensions, using advanced spectral and perturbative analysis techniques.

Contribution

It develops a novel approach to establish nonradial stability for self-similar blow-up solutions, overcoming challenges posed by nonlocal linearized operators.

Findings

Confirmed nonradial stability of the blow-up solution

Developed a localization method for nonlocal operators

Provided spectral analysis techniques applicable to similar problems

Abstract

In three dimensions, the parabolic-elliptic Keller-Segel system exhibits a rich variety of singularity formations. Notably, it admits an explicit self-similar blow-up solution whose radial stability, conjectured more than two decades ago in [Brenner-Constantin-Kadanoff-Schenkel-Venkataramani, 1999], was recently confirmed by [Glogi\'c-Sch\"orkhuber, 2024]. This paper aims to extend the radial stability to the nonradial setting, building on the finite-codimensional stability analysis in our previous work [Li-Zhou, 2024]. The main input is the mode stability of the linearized operator, whose nonlocal nature presents essential challenges for the spectral analysis. Besides a quantitative perturbative analysis for the high spherical classes, we adapt in the first spherical class the wave operator method of [Li-Wei-Zhang, 2020] for the fluid stability to localize the operator and remove the known unstable mode simultaneously. Our method provides localization beyond the partial mass variable and is independent of the explicit formula of the profile, so it potentially sheds light on other linear nonlocal problems.