Abstract
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.
An exact expression of positive periodic solution for a first-order singular equation