I'm working on exercise 25 of Chapter 10 in Mathematical Models in Biology by Edelstein-Keshet. In the exercise we analyze the following chemotherapy model which accounts for the process of cell aging/maturation:
$$ \hspace{2cm} \frac{\partial n}{\partial t} = - v \frac{\partial n}{\partial \alpha} - \mu n \hspace{2cm} (*)$$
The variables and parameters are as follows:
- $\alpha \in [0,1]$ denotes the maturity of a cell ($0 = $ brand new cell, $1 = $ fully mature cell)
- $t$ denotes time
- $n(\alpha,t) = $ the age/maturity distribution of malignant cells at time $t$
- $v = $ the maturation rate of malignant cells (a positive constant)
- $\mu(t) = $ the malignant cell death rate at time $t$, and is given by $$ \mu(t) = \frac{K_1(t) c(t)}{K_2(t) + c(t)} $$
for some functions $K_1, K_2, c$ (which are not given). In part (b) of the exercise we are told to assume a solution of the form
$$n(\alpha,t) = N_0 e^{\beta t} h(\alpha) e^{-A(t)}$$
and asked to show that $$ A(t) = \int_{0}^{t} \mu(\tau) \, d\tau. $$
(We are not told what the function $h$ is; I'm assuming that it can be determined in the same manner as $A$.) I have made progress on finding $A(t)$, but I'm having trouble finishing it. Here's my work so far:
Differentiating the ansatz gives: \begin{align*} \frac{\partial n}{\partial t} &= N_0 h(\alpha) e^{\beta t - A(t)} \cdot (\beta - A'(t)) = n(\alpha,t) (\beta - A'(t)) \\[5pt] \frac{\partial n}{\partial \alpha} &= N_0 h'(\alpha) e^{\beta t - A(t)} = \frac{h'(\alpha)}{h(\alpha)} n(\alpha,t), \end{align*}
where the last equality is assuming that $h$ is nonzero. Then we have \begin{align*} \frac{\partial n}{\partial t}(\alpha,t) + v \frac{\partial n}{\partial \alpha}(\alpha,t) + \mu(t) n(\alpha,t) = n(\alpha,t) \left[\beta - A'(t) + v \frac{h'(\alpha)}{h(\alpha)} + \mu(t) \right]. \end{align*}
Thus, assuming that $n \not\equiv 0$, our ansatz solves $(*)$ iff \begin{align*} \beta - A'(t) + v \frac{h'(\alpha)}{h(\alpha)} + \mu(t) \equiv 0. \end{align*}
Rearranging this equation gives
\begin{align*} A'(t) - \mu(t) = \beta + v \frac{h'(\alpha)}{h(\alpha)}. \end{align*}
Now since the LHS depends only on $t$ and the RHS depends only on $\alpha$, we must have \begin{align*} A' - \mu \equiv C \equiv \beta + v \frac{h'}{h} \end{align*}
for some constant $C$. In particular, $A'(t) = \mu(t) + C$, and so \begin{align*} A(t) = A(0) + \int_{0}^{t} \mu(\tau) \,d\tau + Ct. \end{align*}
Now if I can show that $A(0) = C = 0$ then I'll be done. This is the part where I'm stuck....We are given the boundary condition $n(0,t) = 2 n(1,t)$, but it's not clear if this is helpful...Any ideas for how to proceed?