Is there any reference that explicitely comments on the well-posedness (existence of unique solution) for a PDE of the form?
$$\begin{cases} \frac{\partial}{\partial s}v(s,x)+\varepsilon u\frac{\partial}{\partial x}v(s,x)-\xi\frac{\partial^2}{\partial x^2}v(s,x)=f(s,x)\\ v(0,x)=g(x)& \forall x\in[a, b]\\ v(s,a)=h_a(s), v(s,b)=h_b(s) &\forall s\in[0,T] \end{cases}$$
for some functions $g,h_a,h_b$ that prescribe the boundary conditions on a truncated space. I have the impression that most books just take most equations as granted without any further thoughts on well-posedness.