Asymmetric cell division is one of the fundamental processes to create cell diversity in the early stage of embryonic development. During this process, the polarity formation in the cell membrane has been considered as a key process by which the entire polarity formation in the cytosol is controlled, and it has been extensively studied in both experiments and mathematical models. Nonetheless, a mathematically rigorous analysis of the polarity formation in the asymmetric cell division has been little explored, particularly for bulk-surface models. In this article, we deal with polarity models proposed for describing the PAR polarity formation in the asymmetric cell division of a C. elegans embryo. Using a simpler but mathematically consistent model, we exhibit the long time behavior of the polarity formation of a bulk-surface cell. Moreover, we mathematically prove the existence of stable polarity solutions of the model equation in an arbitrary high-dimensional domain and analyse how the boundary position of polarity domain is determined. Our results propose that the existence and dynamics of the polarity in the asymmetric cell division can be understood universally in terms of basic mathematical structures.

不对称细胞分裂是胚胎发育早期创造细胞多样性的基本过程之一。在此过程中，细胞膜中的极性形成被认为是控制细胞质中整个极性形成的关键过程，并在实验和数学模型中得到了广泛的研究。尽管如此，对不对称细胞分裂中极性形成的数学严格分析很少被探索，特别是对于体表面模型。在本文中，我们讨论了为描述线虫不对称细胞分裂中的 PAR 极性形成而提出的极性模型胚胎。使用更简单但数学上一致的模型，我们展示了体表面电池极性形成的长时间行为。此外，我们从数学上证明了模型方程在任意高维域中稳定极性解的存在，并分析了极性域的边界位置是如何确定的。我们的结果表明，不对称细胞分裂中极性的存在和动态可以从基本的数学结构方面得到普遍理解。