Hello all!

I appreciate it if you can share any thoughts that you may have regarding how to solve the following PDE:

\frac{\partial U(z,t)}{\partial t}+(1-z)\frac{\partial U(z,t)}{\partial z}=(\frac{1}{z}-1)\left(U(z,t)-U(0,t)\right)

Initial condition: U(z,0)=z^{K}

U(0,t) arises due to a boundary condition, but it is unknown and should be derived by solving the PDE itself (details are irrelevant).

A typical strategy would be using Method of Characteristics to tackle this. But U(0,t) can't be dealt with.

Specifically, I am stuck when integrating both sides of the following subsidiary equation due to U(0,t):
\frac{dU(z,t)}{U(z,t)-U(0,t)}=(\frac{1}{z}-1)dt

I greatly appreciate it if you could share any thoughts you might have regarding how I can proceed to solve this through this method or any other method.
Thanks!