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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Abstract

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
Volume248
Issue number4
DOIs
Publication statusPublished - 31 Aug 2024

Data Availability Statement

Data sharing not applicable.

Funding

J. Dávila has been supported by a Royal Society Wolfson Fellowship, UK. M. del Pino has been supported by the Royal Society Research Professorship Grant RP-R1-180114 and by the ERC/UKRI Horizon Europe Grant ASYMEVOL, EP/Z000394/1. J. Dolbeault thanks the ANR Project Conviviality # ANR-23-CE40-0003 for partial support. M. Musso has been supported by EPSRC research Grant EP/T008458/1. We are grateful to Federico Buseghin for his valuable comments and corrections.

FundersFunder number
European Research Council
Royal SocietyRP-R1-180114
UKRI Horizon EuropeEP/Z000394/1
French National Research AgencyANR-23-CE40-0003
Engineering and Physical Sciences Research CouncilEP/T008458/1

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