Consider the system

\begin{eqnarray} \partial_t p(t,x) + b(t)\partial_x p(t,x) - \partial_{xx}^2\left(\frac{\sigma(t,x)^2p(t,x)}{2\big(1+\alpha(t)\big)^2}\right) &=& 0,\quad \forall t>0, ~x>0 \\ p(0,x) &=& \rho(x),\quad \forall x>0 \\ p(t,0) &=& 0,\quad \forall t>0 \\ \alpha(t) &=& \int_0^{\infty}p(t,x)dx,\quad \forall t\ge 0, \end{eqnarray}

where $b,\sigma$ are both bounded and Lipschitz, $\inf_{(t,x)}\sigma(t,x)>0$ and $\rho:\mathbb R_+\to\mathbb R_+$ is a bounded density function, i.e.

$$\int_0^{\infty}\rho(x)dx = 1. $$

I can show the existence of the classical solution $(p,\alpha)$. Can we prove the uniqueness? Any answer, comments and references are highly appreciated.