I have come across the following easy-looking problem that is driving me mad.

I have a family of measures (on the real line $\mathbb R$) $\{\mu_t\}_{t>0}$ which is uniformly bounded (the measures being possibly signed). I know that $\mu_0 = 0$ and that for every $\varphi \in C_c^\infty([0,+\infty) \times \mathbb R)$ it holds $$ \iint_{(0,+\infty) \times \mathbb R} \left[ \partial_t \varphi(t,x)- \sqrt[3]{x} \partial_x \varphi(t,x) \right] d\mu_t(x)dt = 0. $$

Question.Does $\mu_t \equiv 0$ for every $t>0$?

At the beginning I thought it was an easy consequence of the method of characteristics, but the point is that the (1d-)vector field $f(x) = -\sqrt[3]{x}$ is *not* Lipschitz, so the standard theory does not apply. Nevertheless, I know that the characteristics of $f$ are unique for every initial data (for $t>0$, am I right?). Do you know any references showing the extension of the method of characteristics to this setting?