On the stability of Type I self-similar blowups for the Keller-Segel system in three dimensions and higher

We consider the parabolic-elliptic Keller-Segel system in spatial dimensions $d\geq3$, which corresponds to the mass supercritical case. Some solutions become singular in finite time, an important example being backward self-similar solutions. Herrero et al. and Brenner et al. showed the existence of such profiles, countably many in dimensions $3\leq d \leq 9$ and at least two for $d\geq 10$. We establish that all these self-similar profiles are stable along a set of initial data with finite Lipschitz codimension equal to the number of instable eigenmodes. This extends the recent finding of Glogi\'c et al. showing the stability of the fundamental self-similar profile. We obtain additional results, such as the possibility of the solutions we construct to originate from smooth and compactly supported initial data, their convergence at blow-up time, and the Lipschitz regularity of the blow-up time. Our proof extends the approach proposed in Collot et al., based on renormalizing the solution around a modulated self-similar solution, and using a spectral gap for the linearized operator in the parabolic neighbourhood of the singularity.