  We consider a class of Fokker--Planck equations with linear diffusion and
superlinear drift enjoying a formal Wasserstein-like gradient flow structure
with convex mobility function. In the drift-dominant regime, the equations have
a finite critical mass above which the measure minimising the associated
entropy functional displays a singular component. Our approach, which addresses
the one-dimensional case, is based on a reformulation of the problem in terms
of the pseudo-inverse distribution function. Motivated by the structure of the
equation in the new variables, we establish a general framework for
global-in-time existence, uniqueness and regularity of monotonic viscosity
solutions to a class of nonlinear degenerate (resp. singular) parabolic
equations, using as a key tool comparison principles and maximum arguments. We
then focus on a specific equation and study in more detail the regularity and
dynamics of solutions. In particular, blow-up behaviour, formation of
condensates (i.e. Dirac measures at zero) and long-time asymptotics are
investigated. As a consequence, in the mass-supercritical case, solutions will
blow up in $L^\infty$ in finite time and---understood in a generalised, measure
sense---they will eventually have condensate. We further show that the singular
part of the measure solution does in general interact with the density and that
condensates can be transient. The equations considered are motivated by a model
for bosons introduced by Kaniadakis and Quarati (1994), which has a similar
entropy structure and a critical mass if $d\ge3$.
