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

Juan Davila Bonczos, Manuel Del Pino, Jean Dolbeault, Monica Musso, Juncheng Wei

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Perhaps the most classical diffusion model for chemotaxis is the Keller–Segel system (Figure presented.) We consider the critical mass case ∫R 2u 0(x)dx=8π, 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 8π such that for any initial condition u 0 sufficiently close to u 0 and mass 8π, the solution u(x, t) of (∗) is globally defined and blows-up in infinite time. As t→+∞ it has the approximate profile (Formula presented.) where λ(t)≈clogt, ξ(t)→q for some c>0 and q∈R 2. This result affirmatively answers the nonradial stability conjecture raised in Ghoul and Masmoudi (Commun Pure Appl Math 71:1957–2015, 2018).

Original languageEnglish
Article number61
Number of pages154
JournalArchive for Rational Mechanics and Analysis
Issue number4
Publication statusAcceptance date - 6 Jun 2024

Data Availability Statement

Data sharing not applicable.

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