  We consider a one-dimensional free boundary problem describing the migration
of diffusants into rubber. In our setting, the free boundary represents the
position of the front delimitating the diffusant region. The growth rate of
this region is described by an ordinary differential equation that includes the
effect of breaking the growth of the diffusant region. In this specific
context, the breaking mechanism is assumed to be the hyperelastic response to a
too fast diffusion penetration. In recent works, we considered a similar class
of free boundary problems modeling diffusants penetration in rubbers, but
without attempting to deal with the possibility of breaking or accelerating the
occurring free boundaries. For simplified settings, we were able to show the
global existence and uniqueness as well as the large-time behavior of the
corresponding solutions to our formulations. Since here the breaking effect is
contained in the free boundary condition, our previous results are not anymore
applicable. The main mathematical obstacle in ensuring the existence of a
solution is the non-monotonic structure of the free boundary. In this paper, we
establish the existence and uniqueness of a weak solution to the free boundary
problem with breaking effect and give explicitly the maximum value that the
free boundary can reach.
