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 $$ \begin{eqnarray} \partial_t p(t,x) + b(t)\partial_x p(t,x) - \partial_{xx}^2\left(\frac{\sigma(t,x)^2p(t,x)}{2\big(1+\alpha(t)\big)^2}\right) &=& 0, & \forall t>0, ~x>0 \\ p(0,x) &=& \rho(x), &\forall x>0 \\ p(t,0) &=& 0, & \forall t>0 \\ \alpha(t) &=& \int\limits_0^{\infty}p(t,x)dx, & \forall t\ge 0, \end{eqnarray} $$ 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.
