Local existence of a solution to a free boundary problem describing migration into rubber with a breaking effect
Local existence of a solution to a free boundary problem describing migration into rubber with a breaking effect
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DOI：
10.3934/nhm.2023004
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发表时间：
2022-01
期刊：
影响因子：
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通讯作者：
K. Kumazaki;T. Aiki;A. Muntean
中科院分区：
文献类型：
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作者：
K. Kumazaki;T. Aiki;A. Muntean
我们考虑描述扩散剂在橡胶中的迁移的一维自由边界问题。在我们的设置中，自由边界表示划定扩散区域的锋面位置。该区域的生长速率由一个常微分方程来描述，该方程包含了扩散区域生长中断的影响。在这种特殊情况下，断裂机制应被视为描述对过快扩散渗透的最终超弹性响应的非耗散方式。在最近的工作中，我们考虑了类似的一类自由边界问题，模拟扩散剂在橡胶中的渗透，但没有试图处理打破或加速发生自由边界的可能性。对于简化设置，我们能够证明我们的公式的相应解的全局存在唯一性以及大时间行为。由于这里的破断效应包含在自由边界条件中，我们以前的结果不再适用。保证解存在的主要数学障碍是自由边界的非单调结构。本文建立了具有断裂效应的自由边界问题弱解的存在唯一性，并给出了自由边界所能达到的最大值。
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 should be perceived as a non-dissipative way of describing eventual 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.