  We consider partial differential equations (PDE) of drift-diffusion type in
the unit interval, supplemented by either two conservation laws or by a
conservation law and a further boundary condition. We treat two different
cases: (i) uniform parabolic problems; (ii) degenerated problems at the
boundaries. The former can be treated in a very general and complete way, much
as the traditional boundary value problems. The latter, however, bring new
issues, and we restrict our study to a class of forward Kolmogorov equations
that arise naturally when the corresponding stochastic process has either one
or two absorbing boundaries. These equations are treated by means of a uniform
parabolic regularisation, which then yields a measure solution in the vanishing
regularisation limit that is unique. Two prototypical problems from population
dynamics are treated in detail.
