I asked the following question at Math Stackexchange a while ago here but did not get a correct answer.

Let $f(x,t)$ and $G(x,t)$ be smooth functions from $\mathbb R^2\to\mathbb R$.

The PDE $$\dfrac{\partial}{\partial t}f(x,t)=2f(x,t) \dfrac{\partial}{\partial x}G(x,t)+G(x,t)\dfrac{\partial}{\partial x}f(x,t)$$ applies on all of $\mathbb R^2$. Furhermore, let us impose the condition $$f(x,0)=0, \forall x\in \mathbb R$$

Is it necessarily true that $f(x,t)=0$ for all $(x,t)\in\mathbb R^2$?

I will comment that it is easy to show this is true if $f$ is assumed to be analytic, but it seems rather difficult if I don't have this assumption. Through some basic PDE tricks (method of characteristics, etc) it is possible to show a local version of this result: that for a fixed $x$, there is a small $\epsilon(x)$ such that $f(x,t)=0$ for all $0<t<\epsilon(x)$, but this is not good enough for my purposes.