##\displaystyle \frac{\partial u(x,t)}{\partial t}-p(x) \frac{\partial^2 u(x)}{\partial x^2}-\frac{\partial p(x)}{\partial x}\frac{\partial u(x)}{\partial x}=f(x,t)##.
With periodic boundary conditions: ##u(x,t)=u(x+2\pi,t)##. The right hand side is also a periodic function of space, and it can...