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We consider a class of Fokker--Planck equations with linear diffusion andsuperlinear drift enjoying a formal Wasserstein-like gradient flow structurewith convex mobility function. In the drift-dominant regime, the equations havea finite critical mass above which the measure minimising the associatedentropy functional displays a singular component. Our approach, which addressesthe one-dimensional case, is based on a reformulation of the problem in termsof the pseudo-inverse distribution function. Motivated by the structure of theequation in the new variables, we establish a general framework forglobal-in-time existence, uniqueness and regularity of monotonic viscositysolutions to a class of nonlinear degenerate (resp. singular) parabolicequations, using as a key tool comparison principles and maximum arguments. Wethen focus on a specific equation and study in more detail the regularity anddynamics of solutions. In particular, blow-up behaviour, formation ofcondensates (i.e. Dirac measures at zero) and long-time asymptotics areinvestigated. As a consequence, in the mass-supercritical case, solutions willblow up in $L^\infty$ in finite time and---understood in a generalised, measuresense---they will eventually have condensate. We further show that the singularpart of the measure solution does in general interact with the density and thatcondensates can be transient. The equations considered are motivated by a modelfor bosons introduced by Kaniadakis and Quarati (1994), which has a similarentropy structure and a critical mass if $d\ge3$.

On the singularity formation and relaxation to equilibrium in 1D Fokker-Planck model with superlinear drift