Existence and Stability of Infinite Time Blow-Up in the Keller–Segel System

Perhaps the most classical diffusion model for chemotaxis is the Keller–Segel system

We consider the critical mass case \(\int _{{\mathbb {R}}^2} u_0(x)\, \textrm{d}x = 8\pi \), which corresponds to the exact threshold between finite-time blow-up and self-similar diffusion towards zero. We find a radial function \(u_0^*\) with mass \(8\pi \) such that for any initial condition \(u_0\) sufficiently close to \(u_0^*\) and mass \(8\pi \), the solution u(x, t) of (\(*\)) is globally defined and blows-up in infinite time. As \(t\rightarrow +\infty \) it has the approximate profile

$$\begin{aligned} u(x,t) \approx \frac{1}{\lambda ^2(t)} U\left( \frac{x-\xi (t)}{\lambda (t)} \right) , \quad U(y)= \frac{8}{(1+|y|^2)^2}, \end{aligned}$$

where \(\lambda (t) \approx \frac{c}{\sqrt{\log t}}\), \(\xi (t)\rightarrow q\) for some \(c>0\) and \(q\in {\mathbb {R}}^2\). This result affirmatively answers the nonradial stability conjecture raised in Ghoul and Masmoudi (Commun Pure Appl Math 71:1957–2015, 2018).