Infinitely many self-similar blow-up profiles for the Keller-Segel system in dimensions 3 to 9
Van Tien Nguyen, Zhi-An Wang, Kaiqiang Zhang
TL;DR
The paper addresses finite-time blow-up in the parabolic-elliptic Keller-Segel system in dimensions $d=3$ to $9$ by constructing infinitely many backward self-similar profiles $U_n$. It employs matched asymptotic expansions and a Banach fixed-point argument to reduce the self-similar equation to a local elliptic problem in $\mathbb{R}^{d+2}$, obtaining inner and outer solutions that are glued at a small matching radius. The main contributions are the existence of infinitely many smooth radial self-similar profiles with precise near-origin and far-field behavior, their role in generating Type I blow-up solutions $u(x,t)=\frac{1}{T-t}U_n(\frac{x}{\sqrt{T-t}})$, and the demonstration of convergence to a universal profile $u^*(x)\sim \frac{1}{|x|^2}$ away from the origin as $t\to T$. This work refines the understanding of backward self-similar blow-up in higher dimensions and provides a constructive framework that may inform stability analyses and the broader blow-up classification in radial settings.
Abstract
Based on the method of matched asymptotic expansions and Banach fixed point theorem, we rigorously construct infinitely many self-similar blow-up profiles for the parabolic-elliptic Keller-Segel system \begin{equation*} \left\{\begin{array}{l} \partial_{t} u=Δu-\nabla \cdot\left(u \nabla Φ_{u}\right), \\ 0=ΔΦ_{u}+u,\\ u(\cdot,0)=u_0 \geq 0 \end{array}\quad \text{in}\ \mathbb{R}^{d},\right. \end{equation*} where $d\in \{3,\cdots,9\}$. Our findings demonstrate that the infinitely many backward self-similar profiles approximate the rescaling radial steady-state near the origin (i.e. $0<|x|\ll1$) and $\frac{2(d-2)}{|x|^2}$ at spatial infinity (i.e. $|x|\gg1$). We also establish the convergence of the self-similar blow-up solutions as time tends to the blow-up time $T>0$. Our results can give a refined description of backward self-similar profiles for all $|x|\geq 0$ rather than for $0<|x|\ll1$ or $|x|\gg1$, indicating that the blow-up point is the origin and $$ u(x,t)\sim \frac{1}{|x|^2},\ \ \ x\ne0,\ \text{as}\ t\to T. $$