Given a system of partial differential equations in $n$ unknown functions $u_i(\boldsymbol x)$, $\boldsymbol x = (x_1, ..., x_m)$, $$\left\{\begin{matrix} \partial_{x_1} \boldsymbol u = \boldsymbol A_1(\boldsymbol u) \boldsymbol b_1(\boldsymbol x) \\ \vdots \\ \partial_{x_m} \boldsymbol u = \boldsymbol A_m(\boldsymbol u) \boldsymbol b_m(\boldsymbol x)\end{matrix}\right.,$$where $\boldsymbol u = (u_1, ..., u_n)$ and $\partial_{x_i} \boldsymbol u = (\partial_{x_i} u_1, ..., \partial_{x_i} u_n)^\top$ and $\boldsymbol A_i(\boldsymbol u) \in \mathbb R^{n \times m}$ is invertible. Under what conditions does this PDE have (global) unique solutions? I'm especially curious about the case when the initial data is given only in some point $\boldsymbol x_0$, i.e. $\boldsymbol u(\boldsymbol x_0) = \boldsymbol c$.
Intuitively, if $\boldsymbol A_i$ and $\boldsymbol b_i$ are sufficiently well behaved, the solution should be unique: Starting at $\boldsymbol x_0$, one can compute the local gradients of $\boldsymbol u$, and then go step-by-step in any direction. But I am unsure how to show uniqueness with common approaches like the maximum principle or energy integrals.