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