Collot, CCCollotGhoul, TETEGhoulMasmoudi, NNMasmoudiVan Tien Nguyen2022-12-212022-12-212022-062524-5317https://scholars.lib.ntu.edu.tw/handle/123456789/626702We 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.Keller-Segel system; Blowup solution; Blowup profile; Stability; Construction; Spectral analysis; POINT DYNAMICS; MODE-STABILITY; LIMITjournal article10.1007/s40818-022-00118-52-s2.0-85126813283WOS:000770768200001https://api.elsevier.com/content/abstract/scopus_id/85126813283