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

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