Size transport with a nonlocal division kernel

Fix a finite horizon $T>0$ and a nondegenerate fixed size interval $[0,X]$ with $X>0$. Let $g,\mu ,\gamma :[0,X]\to [0,\mathrm{\infty })$ be bounded Lipschitz functions, with $g(x)\ge g_{\min }>0$. Here $g$ is size growth, $\mu$ is ordinary mortality, and $\gamma$ is the rate at which a dividing parent is removed. For each parent size $y\in [0,X]$, let $K_y$ be a finite nonnegative Borel measure of daughter sizes, supported on $[0,y]$. Assume $y\mapsto K_y(E)$ is measurable for every Borel set $E\subseteq [0,X]$, and that the number moment $k_0(y):=K_y([0,y])$ is uniformly bounded; the size moment $k_1(y):={\int }_{[0,y]}x{K}_y(dx)$ is then finite.

Because $K_y$ may have atoms, represent the population by a finite nonnegative Borel measure ${\nu }_t$ on $[0,X)$ rather than requiring a density. Given an initial measure ${\nu }_0$, a weak solution is a curve $t\mapsto {\nu }_t$ that is continuous when tested against continuous functions on $[0,X]$ vanishing at $X$ and satisfies, for every $\varphi \in C^1([0,T]\times [0,X])$ with $\varphi (t,X)=0$,

${\int }_{[0,X)}\varphi (T,x){\nu }_T(dx)-{\int }_{[0,X)}\varphi (0,x){\nu }_0(dx)={\int }_0^T\text{\hspace{0.17em}⁣}{\int }_{[0,X)}({\varphi }_t+g{\varphi }_x-(\mu +\gamma )\varphi )(t,x){\nu }_t(dx)dt+{\int }_0^T\text{\hspace{0.17em}⁣}{\int }_{[0,X)}\text{\hspace{0.17em}⁣}({\int }_{[0,y]}\varphi (t,x)K_y(dx)){\nu }_t(dy)dt\mathrm{.}$

This is the measure-valued transport equation with daughter production ${\int }_{[0,X)}{K}_y(dx){\nu }_t(dy)$, no incoming source at $x=0$, and absorbing outflow at $x=X$. The test condition $\varphi (t,X)=0$ encodes removal when a characteristic reaches the upper boundary.

Let ${\mathrm{\Pi }}_{[0,X]}(x):=\min \{X,\max \{0,x\}\}$ and extend the growth field to $\tilde{g}(x):=g({\mathrm{\Pi }}_{[0,X]}(x))$ on $\mathbb{R}$. This is a bounded globally Lipschitz vector field. By Size characteristics, positivity, and discrete number and size-moment balances ↗, its IVP has a unique solution ${\chi }_y$ on $[0,T]$ from every initial size $y\in [0,X]$. A model characteristic is ${\chi }_y$ restricted to $[0,\min \{T,{\tau }_y\}]$, where ${\tau }_y$ is its first hitting time of $X$ (taking ${\tau }_y=+\mathrm{\infty }$ if there is no hit on $[0,T]$, and ${\tau }_y=0$ if $y=X$). On this restricted interval, $\tilde{g}=g$. For $y<X$ the restricted interval is nondegenerate, and the characteristic is a first-order IVP solution in the sense of First-order systems, initial value problems, and solutions on intervals. For $y=X$ the restriction is only the stopped path on $\{0\}$; no derivative on that singleton is asserted, while the extended path ${\chi }_X$ remains an IVP solution on $[0,T]$. The interval is fixed; moving boundaries, zero growth and discontinuous coefficients are outside this definition.