$1^{st}$ order PDE in population system

Question

Here is the age-structured continuous population partial differential equation: \begin{equation} \left\{ \begin{array}{lcl} \frac{\partial p(a,t)}{\partial a}+\frac{\partial p(a,t)}{\partial t} = -m(a)p(a,t),\\ p(0,t)=\varphi (t)=\beta (t)\int^{a_2}_{a_1} h(a,t)k(a,t)p(a,t)da,\\ p(a,0)=p_{0}(a). \end{array} \right. \end{equation} It is an non-homogeneous first order linear equation with a non-homogeneous boundary term. And Olsder solved this equation system by using characteristic lines in 1975, however, I could not find any reference/ solutions to that. All I know is that the solution looks like:
\begin{equation} p(a,t) = \begin{cases} p_{0}(a-t)e^{-\int^{a}_{a-t}m(x)dx}, & \mbox{ } 0\le t\le a,\mbox{} \\ \varphi (t-a)e^{-\int^{a}_{0}m(x)dx}, & \mbox{} a\le t.\mbox{ } \end{cases} \end{equation} Can anyone explain to me why and how to get this solution using characteristic line? Or anyone has the G.J. Olsder's solution?
Thank!

Answer 1

Hint: try to write $$\frac{\partial p(a,t)}{\partial a}+\frac{\partial p(a,t)}{\partial t} $$ as $dp(a(t), t) / dt$ for a certain function $a(t)$.