The functional partial differential equation (FPDE) for cell division, $$ \begin{align*} &\frac{\partial}{\partial t}n(x,t) +\frac{\partial}{\partial x}(g(x,t)n(x,t))\\ &\quad = -(b(x,t)+\mu(x,t))n(x,t)+b(\alpha x,t)\alpha n(\alpha x,t)+b(\beta x,t)\beta n(\beta x,t), \end{align*} $$is not amenable to analytical solution techniques, despite being closely related to the first-order partial differential equation (PDE) $$ \begin{align*} \frac{\partial}{\partial t}n(x,t) +\frac{\partial}{\partial x}(g(x,t)n(x,t)) = -(b(x,t)+\mu(x,t))n(x,t)+F(x,t), \end{align*} $$which, with known $F(x,t)$, can be solved by the method of characteristics. The difficulty is due to the advanced functional terms $n(\alpha x,t)$ and $n(\beta x,t)$, where $\beta \ge 2 \ge \alpha \ge 1$, which arise because cells of size x are created when cells of size $\alpha x$ and $\beta x$ divide.The nonnegative function, $n(x,t)$, denotes the density of cells at time t with respect to cell size x. The functions $g(x,t)$, $b(x,t)$ and $\mu (x,t)$ are, respectively, the growth rate, splitting rate and death rate of cells of size x. The total number of cells, $\int _{0}^{\infty }n(x,t)\,dx$, coincides with the $L^1$ norm of n. The goal of this paper is to find estimates in $L^1$ (and, with some restrictions, $L^p$ for $p>1$) for a sequence of approximate solutions to the FPDE that are generated by solving the first-order PDE. Our goal is to provide a framework for the analysis and computation of such FPDEs, and we give examples of such computations at the end of the paper.

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细胞分裂的函数微分方程模型的近似解的估计

细胞分裂的功能偏微分方程（FPDE），$$ \begin{align*} &\frac{\partial}{\partial t}n(x,t) +\frac{\partial}{\partial x}(g(x,t)n(x,t ))\\ &\quad = -(b(x,t)+\mu(x,t))n(x,t)+b(\alpha x,t)\alpha n(\alpha x,t) +b(\beta x,t)\beta n(\beta x,t), \end{align*} $$尽管与一阶偏微分方程 (PDE) 密切相关，但不适用于解析求解技术$$ \begin{align*} \frac{\partial}{\partial t}n(x,t) +\frac{\partial}{\partial x}(g(x,t)n(x,t) ) = -(b(x,t)+\mu(x,t))n(x,t)+F(x,t), \end{align*} $$其中，已知$F(x,t)$，可以通过特征的方法求解。困难在于高级功能术语$n(\alpha x,t)$和$n(\beta x,t)$， 在哪里$\beta \ge 2 \ge \alpha \ge 1$，这是因为细胞大小X当大小的单元格创建$\阿尔法x$和$\beta x$除。非负函数，$n(x,t)$, 表示时间的细胞密度吨关于单元格大小X. 功能$g(x,t)$,$b(x,t)$和$\mu (x,t)$分别是大小细胞的生长率、分裂率和死亡率X. 细胞总数，$\int _{0}^{\infty }n(x,t)\,dx$, 与$L^1$规范n. 本文的目标是在$L^1$（并且，有一些限制，$L^p$为了$p>1$) 对于通过求解一阶 PDE 生成的 FPDE 的一系列近似解。我们的目标是为此类 FPDE 的分析和计算提供一个框架，并在本文末尾给出此类计算的示例。