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 ∫ _R² u₀(x)\, dx = 8π ∫ 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 u₀^* u 0 ∗ with mass 8π 8 π such that for any initial condition u₀ u 0 sufficiently close to u₀^* u 0 ∗ and mass 8π 8 π , the solution u ( x , t ) of ( $$*$$ ∗ ) is globally defined and blows-up in infinite time. As t→ +∞ t → + ∞ it has the approximate profile aligned u(x,t) ≈ 1/λ ²(t) U( x-ξ (t)/λ (t) ) , U(y)= 8/(1+|y|²)², aligned u ( x , t ) ≈ 1 λ 2 ( t ) U x - ξ ( t ) λ ( t ) , U ( y ) = 8 ( 1 + | y | 2 ) 2 , where λ (t) ≈ c√log t λ ( t ) ≈ c log t , ξ (t)→ q ξ ( t ) → q for some $$c>0$$ c > 0 and q∈ R² q ∈ R 2 . This result affirmatively answers the nonradial stability conjecture raised in Ghoul and Masmoudi (Commun Pure Appl Math 71:1957–2015, 2018).