Let $\vec u\colon \mathbb{R}^n\times \mathbb{R}\to \mathbb{R}^k$ be at least $C^1$- smooth vector valued function. Assume they satisfy a first order linear equation $$\partial_t \vec u(x,t)=\sum_{j=1}^n A_j(x,t) \partial_{x_j}\vec u(x,t),$$ where $A_j$ are matrix valued functions.

Assume in addition $$u(x,0)=0.$$

Under what assumptions on the regularity of $\vec u, A_j$ (weaker than real analiticity, say infinite smoothness) one can guarantee that $u$ vanishes identically?

A reference would be helpful.