Linear age-structured renewal transport problem

Fix a finite horizon $T>0$. Let mortality and fertility be bounded Lipschitz functions $\mu ,\beta :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$, and let $n_0\in L^1(0,\mathrm{\infty })$ be nonnegative. The linear age-structured renewal transport problem is

$n_t(t,a)+n_a(t,a)=-\mu (a)n(t,a),a>0,0<t\le T,$

with initial density $n(0,a)=n_0(a)$ and inflow condition

$n(t,0)=B(t),B(t):={\int }_0^{\mathrm{\infty }}\beta (a)n(t,a)da\mathrm{.}$

Here $a$ is age, $B$ is the birth rate, and the characteristic age path solves the first-order initial value problem $a^{\mathrm{\prime }}(s)=1$ in the sense of First-order systems, initial value problems, and solutions on intervals, hence $a(s)=a(0)+s$. Define survival from age $a$ over duration $s\ge 0$ by

$S(a,s):=\exp \text{\hspace{0.17em}⁣}(-{\int }_a^{a+s}\mu (r)dr)\mathrm{.}$

A weak solution on $[0,T]$ is a nonnegative $n\in C([0,T];L^1(0,\mathrm{\infty }))$ with continuous $B:[0,T]\to [0,\mathrm{\infty })$ satisfying $B(t)={\int }_0^{\mathrm{\infty }}\beta (a)n(t,a)da$ and, for every $\varphi \in C_c^1([0,T]\times [0,\mathrm{\infty }))$ with $\varphi (T,a)=0$,

$-{\int }_0^T\text{\hspace{0.17em}⁣}{\int }_0^{\mathrm{\infty }}n(t,a)({\varphi }_t+{\varphi }_a-\mu (a)\varphi )dadt={\int }_0^{\mathrm{\infty }}{n}_0(a)\varphi (0,a)da+{\int }_0^{T}B(t)\varphi (t,0)dt\mathrm{.}$

A mild solution is a weak solution that also satisfies, for almost every $a\ge 0$,

$n(t,a)=\{\begin{array}{ll}{n}_0(a-t)S(a-t,t),& a\ge t,\\ B(t-a)S(0,a),& 0\le a<t,\end{array}$

The formula follows the age characteristics $a^{\mathrm{\prime }}(s)=1$ and their mortality survival. No delay equation or finite-age cutoff is identified with this model by definition.