Finite-age renewal model with absorbing outflow

Fix $0<T<A$, bounded nonnegative Lipschitz rates $\mu ,\beta :[0,A]\to [0,\mathrm{\infty })$, and initial density $n_0^A\in L_{+}^1(0,A)$. The finite-age renewal model with absorbing outflow is the age transport problem on $0<a<A$ with the same renewal inflow $B_A(t)={\int }_0^A\beta (a)n^A(t,a)da$ as in Linear age-structured renewal transport problem. A cohort follows age speed $1$, retains its mortality survival $S(a,s)$ until it reaches $A$, and is then removed; no cohort enters from ages above $A$. The initial density is $n_0^A$ on $[0,A]$. A mild solution is the characteristic formula on $0<a<A$, with an initial-cohort branch for $a\ge t$, a newborn branch for $a<t$, and zero extension to $a\ge A$. Since $A>T$, no cohort born during $[0,T]$ reaches the outflow boundary.