Characteristic formula and Volterra solution for age renewal transport

The old-cohort branch in the Statement follows the characteristic $a(s)=x+s$ and survival $S(x,t)$ from [F1]; the newborn branch starts from the boundary value $B(t-a)$ at time $t-a$ and follows survival $S(0,a)$. These are the characteristic branches in the definition of a mild solution. We use them to construct a candidate after solving the Volterra equation; no pointwise differentiability of a weak solution is assumed.

The functions $K$ and $G$ are continuous. Indeed, $K$ is a product of Lipschitz functions; for $s,t\in [0,T]$, the integrand multiplier in $G$ changes by at most $(\mathrm{Lip}(\beta )+\mathrm{\parallel }\beta {\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }\mu {\mathrm{\parallel }}_{\mathrm{\infty }})\mathrm{\mid }t-s\mathrm{\mid }$, since $\mathrm{\mid }{e}^{-u}-e^{-v}\mathrm{\mid }\le \mathrm{\mid }u-v\mathrm{\mid }$ for $u,v\ge 0$. Multiplication by $n_0\in L^1$ and integration gives continuity of $G$.

Substitute the two branches into $B(t)={\int }_0^{\mathrm{\infty }}\beta (a)n(t,a)da$ and split at $a=t$. In the old-cohort part set $x=a-t$; the result is exactly $G(t)+{\int }_0^t\beta (a)e^{-M(a)}B(t-a)da$.

On any interval of length $\tau$ with $\tau \mathrm{\parallel }K{\mathrm{\parallel }}_{\mathrm{\infty }}<1$, the right side of the Volterra equation is a contraction in the sup norm on continuous birth-rate functions: the difference of two outputs is bounded by $\tau \mathrm{\parallel }K{\mathrm{\parallel }}_{\mathrm{\infty }}$ times the input difference. If $\mathrm{\parallel }K{\mathrm{\parallel }}_{\mathrm{\infty }}=0$, the equation is simply $B=G$. Starting at time zero and then treating the already determined history as a fixed forcing term, finitely many such intervals cover $[0,T]$; each interval therefore has one continuous solution and the solutions agree at their shared endpoints.

Let $\varphi \in C_c^1([0,T]\times [0,\mathrm{\infty }))$ satisfy $\varphi (T,a)=0$. On an initial cohort with coordinate $x=a-t\ge 0$, the chain rule gives $\frac{d}{dt}[n_0(x)S(x,t)\varphi (t,x+t)]=n(t,x+t)({\varphi }_t+{\varphi }_a-\mu \varphi )(t,x+t)$ for almost every $t$. Integrating from $0$ to $T$ makes the terminal term zero and leaves $-n_0(x)\varphi (0,x)$; integration over $x\ge 0$ gives the initial-data contribution to the weak identity.

For a newborn cohort with birth time $s=t-a\ge 0$, the chain rule gives $\frac{d}{da}[B(s)S(0,a)\varphi (s+a,a)]=n(s+a,a)({\varphi }_t+{\varphi }_a-\mu \varphi )(s+a,a)$. Integrating from $a=0$ to $T-s$ gives $-B(s)\varphi (s,0)$ because $\varphi (T,T-s)=0$; integrating over $s\in [0,T]$ gives the renewal-boundary contribution.

When $n_0\ge 0$, $G,K\ge 0$. Picard iteration from the zero function on the first interval preserves nonnegativity and converges to its unique fixed point; the same argument applies on each continuation interval because its known-history forcing is nonnegative. Thus $B\ge 0$, and the characteristic formula gives $n\ge 0$.

The formula has finite $L^1$ norm because its old-cohort contribution is at most $\mathrm{\parallel }{n}_0{\mathrm{\parallel }}_1$ and its newborn contribution is at most ${\int }_0^{t}B(s)ds$. On the old branch, continuity in time follows from translation continuity of $n_0$ in $L^1$ and dominated convergence for the survival multiplier. On the newborn branch, continuity follows from continuity and boundedness of $B$ and $S$, with the interval of newly entering cohorts having length tending to zero. Thus $n\in C([0,T];L^1)$.

The old-cohort region $a\ge t$ and newborn region $a<t$ partition the time-age domain up to a measure-zero diagonal. Adding their integrals from steps 2.3 and 2.4, then multiplying by $-1$, gives exactly the weak identity in [F1] with initial term $\int n_0(x)\varphi (0,x)dx$ and boundary term ${\int }_0^{T}B(s)\varphi (s,0)ds$. Step 2.1 gives $B(t)={\int }_0^{\mathrm{\infty }}\beta (a)n(t,a)da$, and step 4.1 gives $n\in C([0,T];L^1)$ and nonnegativity. Thus the characteristic formula is a weak, hence mild, solution.

Any mild solution has a birth rate satisfying the Volterra equation in step 2.1. Uniqueness from step 2.2 forces that same birth rate, and the characteristic formula in step 1.1 then forces the same density.