Perhaps the most classical diffusion model for chemotaxis is the Keller-Segel system * \(\begin{aligned} \left\{ \begin{aligned} u_t =&\; \Delta u - \nabla \cdot (u \nabla v) \quad {\quad \hbox {in } }\mathbb {R}^2\times (0,T),\\ v =&\; (-\Delta _{\mathbb {R}^2})^{-1} u := \frac{1}{2\pi } \int _{\mathbb {R}^2} \log \frac{1}{|x-z|}\,u(z,t) dz,\\&\quad \ u(\cdot ,0) = u_0^{\star } \ge 0\quad \text {in } \mathbb {R}^2. \end{aligned} \right. \end{aligned}\) We show that there exists \(\varepsilon >0\) such that for any m satisfying \(\begin{aligned} 8\pi <m\le 8\pi +\varepsilon \end{aligned}\) and any k given points \(q_{1},...,q_{k}\) in \(\mathbb {R}^{2}\) there is an initial data \(u_0^*\) of ( \(*\) ) for which the solution u(x, t) blows up in finite time as \(t\rightarrow T\) with the approximate profile \( u(x,t)=\sum _{j=1}^{k}\frac{1}{\lambda _{j}^{2}(t)}U\left( \frac{x-\xi _{j}(t)}{\lambda _{j}(t)}\right) (1+o(1)), \quad U(y)=\frac{8}{(1+|y|^{2})^{2}}, \) 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, \(\xi _{j}(t)\rightarrow q_{j}\in \mathbb {R}^{2}\) 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 [17, 18].
