  In many industrial applications, rubber-based materials are routinely used in
conjunction with various penetrants or diluents in gaseous or liquid form. It
is of interest to estimate theoretically the penetration depth as well as the
amount of diffusants stored inside the material. In this framework, we prove
the global solvability and explore the large time-behavior of solutions to a
one-phase free boundary problem with nonlinear kinetic condition that is able
to describe the migration of diffusants into rubber. The key idea in the proof
of the large time behavior is to benefit of a contradiction argument, since it
is difficult to obtain uniform estimates for the growth rate of the free
boundary due to the use of a Robin boundary condition posed at the fixed
boundary.
