Existence of finite time blow-up in Keller-Segel system

Perhaps the most classical diffusion model for chemotaxis is the Keller-Segel system \begin{equation} \begin{cases} u_{t} =Δu - \nabla \cdot(u \nabla v) \ \ \ \text{in } \mathbb{R}^2\times(0,T),\\[5pt] v = (-Δ_{\mathbb{R}^2})^{-1} u := \displaystyle\frac {1}{2π} \displaystyle\int_{\mathbb{R}^2} \log \frac {1}{|x-z|}u(z,t) dz, \ \ \ \ \ \ \ \ \ (\star)\\[5pt] u(\cdot ,0) = u_{0}^{\star} \ge 0 \ \ \ \text{in } \mathbb{R}^2. \end{cases} \end{equation} We show that there exists $\epsilon >0$ such that for any $m$ satisfying $8\pi <m\le 8\pi +\epsilon$ and any $k$ given points $q_1,\mathrm{.}\mathrm{.}\mathrm{.},q_k$ in ${\mathbb{R}}^2$ there is an initial data $u_0^{\ast }$ of $(\star )$ for which the solution $u(x,t)$ blows-up in finite time as $t\to T$ with the approximate profile $$u(x,t)=\sum _{j=1}^k\frac{1}{{\lambda }_j^2(t)}U\left(\frac{x-{\xi }_j(t)}{{\lambda }_j(t)}\right)(1+o(1)),U(y)=\frac{8}{(1+\mathrm{\mid }y{\mathrm{\mid }}^2{)}^2},$$ with ${\lambda }_j(t)\approx 2e^{-\frac{\gamma +2}{2}}\sqrt{T-t}{e}^{-\sqrt{\frac{\mathrm{\mid }\ln (T-t)\mathrm{\mid }}{2}}}$ where $\gamma =\mathrm{0.57721...}$ is the Euler-Mascheroni constant, ${\xi }_j(t)\to q_j\in {\mathbb{R}}^2$ and such that ${\int }_{{\mathbb{R}}^2}u(x,t)dx=km\mathrm{.}$ This construction generalizes the existence result of the stable blow-up dynamics recently proved in \cite{CGMN1,CGMN2}.