  We prove a global existence result for weak solutions to a one-dimensional
free boundary problem with flux boundary conditions describing swelling along a
halfline. Additionally, we show that solutions are not only unique but also
depend continuously on data and parameters. The key observation is that the
structure of our system of partial differential equations allows us to show
that the moving a priori unknown interface never disappears. As main
ingredients of the global existence proof, we rely on a local weak solvability
result for our problem, uniform estimates of the solution, integral estimates
on quantities defined at the free boundary, as well as a fine pointwise lower
bound for the position of the moving boundary. Some of the estimates are
time-independent. They allow us to explore the large time behavior of the
position of the moving boundary. The approach is specific to one-dimensional
settings.
