A numerical study of linearized Keller-Segel operator in self-similar variables

This report is devoted to a numerical study of the Keller-Segel model in self-similar variables. We first parametrize the set of solutions in terms of the mass parameter M ∈ (0, 8π) and consider the asymptotic regimes for M small or M close to 8π. Next we introduce the linearized operator and study its spectrum using various shooting methods: we determine its kernel, the spectrum among radial functions and use a decomposition into spherical harmonics to study the other eigenvalues. As a result, we numerically observe that the spectral gap of the linearized operator is independent of M and equal to 1, which is compatible with known results in the limiting regime corresponding to M → 0+, and with recent theoretical results obtained by the authors. We also compute other eigenvalues, which allows to state several claims on various refined asymptotic expansions of the solutions in the large time regime.