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

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