Suppose that $X_0:M^n \rightarrow \mathbb{R}^{n+1}$ is a smooth embedding of a compact hypersurface. I'm given a scalar function $\eta \in C^{\infty}\left( M \times [0, T) \right)$ and I'd like to solve the problem
$$ \frac{dX}{dt}(x,t) = \eta(x,t)\vec{N}(x,t) $$ $$ X(\cdot, 0) = X_0 $$ where
$\vec{N}(x,t)$ is the unit normal vector determined by the embedding $$X( \cdot, t) \rightarrow \mathbb{R}^{n+1} $$
What conditions does $\eta$ have to satisfy in order to guarantee that the solution exists, even for some short time?
If everything in sight is analytic, Cauchy-Kovalevskaya should do the trick, but can this be weakened to smooth?