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A class of nonlinear Fokker-Planck equations with superlinear drift isinvestigated in the $L^1$-supercritical regime, which exhibits a finitecritical mass. The equations have a formal Wasserstein-like gradient-flowstructure with a convex mobility and a free energy functional whose minimisingmeasure has a singular component if above the critical mass. Singularities andconcentrations also arise in the evolutionary problem and their finite-timeappearance constitutes a primary technical difficulty. This paper aims at aglobal-in-time qualitative analysis with main focus on the isotropic case,where solutions will be shown to converge to the unique minimiser of the freeenergy as time tends to infinity. A key step in the analysis consists inproperly controlling the singularity profiles during the evolution. Our studycovers the 3D Kaniadakis--Quarati model for Bose--Einstein particles, and thusprovides a first rigorous result on the continuation beyond blow-up andlong-time asymptotic behaviour for this model.

Singularities in $L^1$-supercritical Fokker-Planck equations: A qualitative analysis