Existence of Finite Time Blow-Up in Keller-Segel System

Perhaps the most classical diffusion model for chemotaxis is the Keller-Segel system $\begin{array}{c}\left(\begin{array}{cc}{u}_t=& \Delta u-\nabla ·\left(u\nabla v\right)\text{in}{R}^2\times (0,T),\\ v=& {(-{\Delta }_{R^2})}^{-1}u:=\frac{1}{2\pi }{\int }_{R^2}log\frac{1}{|x-z|}u(z,t)dz,\\ & u(·,0)=u_0^{\star }\ge 0\text{in}{R}^2.\end{array}\right)\end{array}$ We show that there exists $\epsilon >0$ such that for any m satisfying $\begin{array}{c}8\pi <m\le 8\pi +\epsilon \end{array}$ and any k given points $q_1,...,q_k$ in $R^2$ there is an initial data $u_0^{\ast }$ of ( $\ast$ ) for which the solution u(x, t) blows up in finite time as $t\to T$ with the approximate profile $u(x,t)=\sum _{j=1}^k\frac{1}{{\lambda }_j^2\left(t\right)}U\left(\frac{x-{\xi }_j\left(t\right)}{{\lambda }_j\left(t\right)}\right)(1+o\left(1\right)),U\left(y\right)=\frac{8}{{(1+|y|}^2{)}^2},$ with ${\lambda }_j\left(t\right)\approx 2e^{-\frac{\gamma +2}{2}}\sqrt{T-t}{e}^{-\sqrt{\frac{|ln(T-t\left)\right|}{2}}}$ where $\gamma =0.57721...$ is the Euler-Mascheroni constant, ${\xi }_j\left(t\right)\to q_j\in R^2$ and such that ${\int }_{R^2}u(x,t)dx=km.$ This construction generalizes the existence result of the stable blow-up dynamics recently proved in [17, 18].