Given the equation here, I would like to ask the following relaxed question:

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 $q$ is Lipschitz and $p$ is Schwartz, too:

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