I'm taking an introductory course on mathematical modelling in biology and am completely at a loss with this question. We are considering the PDE for a population of cells $n$ with age $a$ over time $t$:
$$\frac{\partial n}{\partial t}+(1+\beta a)\frac {\partial n}{\partial a}=-\mu n$$
with conditions $n(a,0) = f(a)$ and $n(0,t)=2n(1,t)$
I'm meant to seek a separable solution of the form $n(a,t) = e^{(\gamma t)} N(a)$ and then derive an expression for the unique value of $\mu=\mu(\beta)$ such that the population tends to a non-trivial, time-independent solution as $t$ goes to infinity. But surely then $\gamma$ needs to be negative which will lead to 0? And from solving this PDE and substituting in the conditions I found that $\gamma$ cannot be a sum of a negative real number and some linear multiple of $1/t$ which would fulfil the conditions, since $\gamma$ is only a function of $\beta$ and $\mu$ (which is also a function of $\beta$).
Am I missing something obvious? Thanks for the help!