SIAM Journal on Mathematical Analysis, Volume 53, Issue 1, Page 1214-1238, January 2021.
The amplification of an external signal is a key step in direction sensing of biological cells. We consider a simple model for the response to a time-depending signal, which was previously proposed by the last three authors. The model consists of a bulk-surface reaction-diffusion model. We prove that in a suitable asymptotic limit the system converges to a bulk-surface parabolic obstacle--type problem. For this model and a reduction to a nonlocal surface equation we show an $L^1$-contraction property and, in the case of time-constant signals, the stability of stationary states.

中文翻译：

---

细胞极化模型中的抛物线自由边界问题

SIAM数学分析杂志，第53卷，第1期，第1214-1238页，2021
年1月。外部信号的放大是生物细胞方向感测的关键步骤。我们考虑了一个对时间相关信号的响应的简单模型，该模型先前由后三位作者提出。该模型由体表反应扩散模型组成。我们证明了在适当的渐近极限下，系统收敛到一个体表面抛物线型障碍物问题。对于该模型以及对非局部表面方程的简化，我们显示出$ L ^ 1 $的收缩特性，在时间常数信号的情况下，显示出稳态的稳定性。