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Perhaps the most classical diffusion model for chemotaxis is the Keller-Segelsystem \begin{equation}\tag{$\ast$} \label{ks0} \left\{ \begin{aligned} u_t=&\; \Delta u - \nabla \cdot(u \nabla v) \quad in {\mathbbR}^2\times(0,\infty),\\ v =&\; (-\Delta_{\R^2})^{-1} u := \frac 1{2\pi}\int_{R^2} \, \log \frac 1{|x-z|}\,u(z,t)\, dz, \\ & \qquad\ u(\cdot ,0) = u_0\geq 0\quad\hbox{in } R^2. \end{aligned} \right. \end{equation} We consider the{\em critical mass case} $\int_{R^2} u_0(x)\, dx = 8\pi$ which corresponds tothe exact threshold between finite-time blow-up and self-similar diffusiontowards zero. We find a radial function $u_0^*$ with mass $8\pi$ such that forany initial condition $u_0$ sufficiently close to $u_0^*$ the solution $u(x,t)$of \equ{ks0} is globally defined and blows-up in infinite time. As $t\to+\infty$ it has the approximate profile $$ u(x,t) \approx \frac 1{\lambda^2}\ch{U}\left (\frac {x-\xi(t)}{\lambda(t)} \right ), \quad \ch{U}(y)=\frac{8}{(1+|y|^2)^2}, $$ where $\lambda(t) \approx \frac c{\sqrt{\log t}}, \\xi(t)\to q $ for some $c>0$ and $q\in \R^2$. This result answers affirmativelythe nonradial stability conjecture raised in \cite{g}.

Existence and stability of infinite time blow-up in the Keller-Segel system