Many central problems in geometry, mathematical physics and biology reduce to questions regarding the behavior of solutions of nonlinear evolution equations. The global dynamical behavior of bounded solutions for large times is of significant interest. However, in many real situations, solutions develop singularities in finite time. The singularities have to be analyzed in details before attempting to extend solutions beyond their singularities or to understand their geometry in conjunction with globally bounded solutions. In this context we have been particularly interested in qualitative descriptions of blowup. A particular example in the talk will be the classical Keller-Segel system modeling biological chemotaxis processes and stellar dynamics. I will present different techniques based on spectral analysis and energy methods to study the existence and stability of blowup solutions to this problem in both L¹-critical and L¹-supercritical regimes.
The talk is based on the following joint works with C. Collot, T. Ghoul, N. Masmoudi:
 Refined description and stability for singular solutions of the 2D Keller-Segel system. Communications on Pure and Applied Mathematics, 2021. [DOI]
 Spectral analysis for singularity formation of the 2D Keller-Segel system. Annals of PDE, (to appear). [arXiv].
 Collapsing-ring blowup solutions for the Keller-Segel system in three dimensions and higher. 2021. [arXiv].