Spectral Analysis for Singularity Formation of the Two Dimensional Keller–Segel System

Charles Collot, Tej Eddine Ghoul, Nader Masmoudi, Van Tien Nguyen

Research output: Contribution to journalArticlepeer-review


We analyse an operator arising in the description of singular solutions to the two-dimensional Keller-Segel problem. It corresponds to the linearised operator in parabolic self-similar variables, close to a concentrated stationary state. This is a two-scale problem, with a vanishing thin transition zone near the origin. Via rigorous matched asymptotic expansions, we describe the eigenvalues and eigenfunctions precisely. We also show a stability result with respect to suitable perturbations, as well as a coercivity estimate for the non-radial part. These results are used as key arguments in a new rigorous proof of the existence and refined description of singular solutions for the Keller–Segel problem by the authors [8]. The present paper extends the result by Dejak, Lushnikov, Yu, Ovchinnikov and Sigal [11]. Two major difficulties arise in the analysis: this is a singular limit problem, and a degeneracy causes corrections not being polynomial but logarithmic with respect to the main parameter.

Original languageEnglish (US)
Article number5
JournalAnnals of PDE
Issue number1
StatePublished - Jun 2022


  • Blowup profile
  • Blowup solution
  • Construction
  • Keller–Segel system
  • Spectral analysis
  • Stability

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics
  • Geometry and Topology
  • Mathematical Physics
  • General Physics and Astronomy


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