Let us consider the equation: \begin{align*} (\partial_t - \Delta - b(t,x) \partial_x) u(t,x)& = f(t,x) \\ u(0,x) & = u_0 \end{align*} defined on the whole real line (so in one dimension - but this is only for simplicity) and with a possibly finite time horizon, so say on $[0,T]\times \mathbb{R}.$ Suppose that $b$ and $f$ have polynomial growth, say $$ |u_0(x)|/(1+|x|)^a + |b(t,x)|/ (1 + |x|)^a + |f(t,x)|/(1 + |x|)^a \le C $$ for some $a \ge 0.$

My question is simple: is the solution $u$ to this equation also of polynomial growth?

I am interested in any kind of approach to this question, say through semigroups, operators, Fourier analysis or simple constructions. It seems like a natural question, so I almost expect this to have a well-known answer. We can add any kind of regularity conditions on $b,f$ and $u_0.$ We may even look at the case $ u_0 = 0.$ The only real constraint is the polynomial growth of $b$ and of $f$.

My approach: I have used Schauder estimates in weighted spaces, and this guarantees that the solution is of exponential growth. Also looking at the fundamental solution to this equation has so far not delivered me any better than exponential estimates.

It might even be that the solution can't be expected to grow better than exponential.