## Abstract

Perhaps the most classical diffusion model for chemotaxis is the Keller–Segel system (Figure presented.) We consider the critical mass case ∫R
_{2}u
_{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 language | English |
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Article number | 61 |

Number of pages | 154 |

Journal | Archive for Rational Mechanics and Analysis |

Volume | 248 |

Issue number | 4 |

DOIs | |

Publication status | Acceptance date - 6 Jun 2024 |