There are a number of good resources for theorems and proofs on the existence and uniqueness of solutions to linear parabolic PDEs, but I cannot find any clearly written results and proofs for PDEs of the following form:
$$Lu := \sum_{i,j} a_{ij}(x,t)\frac{\partial^2u}{\partial x_i \partial x_j} + \sum_{i}b_i(x,t) \frac{\partial u}{\partial x_i} + c(x,t)u(x,t) + d(x,t)\frac{\partial u}{\partial t} = f(x,t)$$
on some open rectangle $D$ defined by $x_i \in (a_i,b_i), t \in (t_1,t_2]$ with some boundary condition $u(t, x) = g(x,t)$ for $(x,t) \in \partial D$. In particular, I'm looking for results where $d \equiv 1$ (instead of the parabolic $d \equiv -1$). Does anyone know of a reference with such results (preferably one whose proofs are not too awful to read)?