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

Abstract

AbstractPerhaps the most classical diffusion model for chemotaxis is the Keller–Segel system We consider the critical mass case $$\int _{{\mathbb {R}}^2} u_0(x)\, \textrm{d}x = 8\pi $$ ∫ R 2 u 0 ( x ) d x = 8 π , which corresponds to the exact threshold between finite-time blow-up and self-similar diffusion towards zero. We find a radial function $$u_0^*$$ u 0 ∗ with mass $$8\pi $$ 8 π such that for any initial condition $$u_0$$ u 0 sufficiently close to $$u_0^*$$ u 0 ∗ and mass $$8\pi $$ 8 π , the solution u(x, t) of ($$*$$ ∗ ) is globally defined and blows-up in infinite time. As $$t\rightarrow +\infty $$ t → + ∞ it has the approximate profile $$\begin{aligned} u(x,t) \approx \frac{1}{\lambda ^2(t)} U\left( \frac{x-\xi (t)}{\lambda (t)} \right) , \quad U(y)= \frac{8}{(1+|y|^2)^2}, \end{aligned}$$ u ( x , t ) ≈ 1 λ 2 ( t ) U x - ξ ( t ) λ ( t ) , U ( y ) = 8 ( 1 + | y | 2 ) 2 , where $$\lambda (t) \approx \frac{c}{\sqrt{\log t}}$$ λ ( t ) ≈ c log t , $$\xi (t)\rightarrow q$$ ξ ( t ) → q for some $$c>0$$ c > 0 and $$q\in {\mathbb {R}}^2$$ q ∈ R 2 . This result affirmatively answers the nonradial stability conjecture raised in Ghoul and Masmoudi (Commun Pure Appl Math 71:1957–2015, 2018).