  We prove an existence and uniqueness result for solutions to nonlinear
diffusion equations with degenerate mobility posed on a bounded interval for a
certain density $u$. In case of \emph{fast-decay} mobilities, namely mobilities
functions under a Osgood integrability condition, a suitable coordinate
transformation is introduced and a new nonlinear diffusion equation with linear
mobility is obtained. We observe that the coordinate transformation induces a
mass-preserving scaling on the density and the nonlinearity, described by the
original nonlinear mobility, is included in the diffusive process. We show that
the rescaled density $\rho$ is the unique weak solution to the nonlinear
diffusion equation with linear mobility. Moreover, the results obtained for the
density $\rho$ allow us to motivate the aforementioned change of variable and
to state the results in terms of the original density $u$ without prescribing
any boundary conditions.
