Consider the PDE

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

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

I am wondering then if $q$ and all its derivatives are polynomially bounded and $p$ is Schwartz, too:

Does there exist a solution to this equation that decays faster than any polynomial in space $x$ at any fixed time $t>0$?

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.

As there was apparently some confusion about the meaning of this question, let me ask it again:

Fix a time $t>0$, then as a function of $x$, does the solution decay faster than any polynomial? This seems to be true in your case for example, as it is just a translation of a Schwartz function.