Consider a linear first order PDE: $$\sum_i A_i(x)\frac \partial{\partial x_i}f(x) = B(x) f(x)$$ $f:\Omega\subset\mathbb{R}^n\to \mathbb{R}^m$ where $\Omega$ is bounded and $A_i,B :\Omega\to Hom(\mathbb{R}^n, \mathbb{R}^m)$ are smooth.
I am looking for references about the regularity of the solutions and in particular to the effect of $B$.
More specifically, I am interested to the case where the PDE takes the form $$\frac \partial {\partial t} f(t,x) + \sum_i A_i(x)\frac\partial{\partial x_i}f(t,x) = B(x) f(t,x)$$
Is there a systematic treatment for these equations (like Elliptic regularity for elliptic PDEs)?
Suppose that we can prove that if $B= 0$ then the solutions are smooth what happens then when $B\neq 0$?