Infinitely many self-similar blow-up profiles for the Keller-Segel system in dimensions 3 to 9

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. $0<|x|\ll 1$) and $\frac{2(d-2)}{|x{|}^2}$ at spatial infinity (i.e. $|x|\gg 1$). 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 $|x|\ge 0$ rather than for $0<|x|\ll 1$ or $|x|\gg 1$, indicating that the blow-up point is the origin and $$u(x,t)\sim \frac{1}{|x{|}^2},x\ne 0,\text{as}t\to T.$$