Let $U\subset\mathbb{R}^n$ be open bounded, $T>0$.
Given the parabolic PDE $$\partial_tf+a\partial_xf+b\partial_{xx}f = g \qquad (1)$$ I'm interested in the initial and boundary conditions that make it well-posed or ill-posed. One set of initial condition to make it well-posed is that (1) holds on $U\times(0,T]$ while $f=0\text{ on } \partial U\times [0,T]$ and $u=h$ on $U\times \{t=0\}$ (or on $U\times\{t=T\}$).