Local existence of a solution to a free boundary problem describing migration into rubber with a breaking effect

TL;DR

This paper proves the existence and uniqueness of solutions for a one-dimensional free boundary problem modeling diffusant migration into rubber, incorporating a breaking mechanism due to hyperelastic response, which was not addressed in previous works.

Contribution

It introduces a new mathematical framework for free boundary problems with a breaking effect, establishing existence, uniqueness, and maximum boundary extent.

Findings

Proved existence and uniqueness of weak solutions.

Explicitly determined the maximum free boundary value.

Extended previous models to include breaking effects.

Abstract

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.