An advanced pantograph-type partial differential equation, supplemented with initial and boundary conditions, arises in a model of asymmetric cell division. Methods for solving such problems are limited owing to functional (nonlocal) terms. The separation of variables entails an eigenvalue problem that involves a nonlocal ordinary differential equation. We discuss plausible eigenvalues that may yield nontrivial solutions to the problem for certain choices of growth and division rates of cells. We also consider the asymmetric division of cells with linear growth rate which corresponds to “exponential growth” and exponential rate of cell division, and show that the solution to the problem is a certain Dirichlet series. The distribution of the first moment of the biomass is shown to be unimodal.

中文翻译：

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呈指数增长的不对称细胞分裂

一个先进的受电弓型偏微分方程，辅以初始和边界条件，出现在不对称细胞分裂模型中。由于功能（非局部）术语，解决此类问题的方法受到限制。变量的分离需要一个涉及非局部常微分方程的特征值问题。我们讨论了合理的特征值，这些特征值可能会为细胞生长和分裂速率的某些选择产生非平凡的解决方案。我们还考虑了细胞的不对称分裂，其线性增长率对应于“指数增长”和细胞分裂的指数率，并表明该问题的解决方案是某个狄利克雷级数。生物量的第一时刻的分布被证明是单峰的。