  We prove existence and uniqueness of solutions to the initial-boundary value
problem for the Lifshitz--Slyozov equation (a nonlinear transport equation on
the half-line), focusing on the case of kinetic rates with unbounded derivative
at the origin. Our theory covers in particular those cases with rates behaving
as power laws at the origin, for which an inflow behavior is expected and a
boundary condition describing nucleation phenomena needs to be imposed. The
method we introduce here to prove existence is based on a formulation in terms
of characteristics, with a careful analysis on the behavior near the singular
boundary. As a byproduct we provide a general theory for linear continuity
equations on a half-line with transport fields that degenerate at the boundary.
We also address both the maximality and the uniqueness of inflow solutions to
the Lifshitz--Slyozov model, exploiting monotonicity properties of the
associated transport equation.
