Consider the PDE

$$\partial_t f(x,t) = \langle Bq(x), \nabla \rangle f(t,x) + p(x),$$

with Schwartz initial data $f(0,x) = f_0(x) \in \mathscr S(\mathbb R^n).$ 

I am wondering then if $q,p$ and all its derivatives are polynomially bounded.

Does there exist a solution to this equation that decays faster than any polynomial? This sounds plausible to me, but I am not sure how one argues for such an equation. I assume it must be a classical question.