Consider the following linear partial differential equation:
\begin{align} & C_{il}(x) {\partial f_i \over \partial x_l} + {\partial f_i \over \partial x_m} = D_{ik}(x) y_k \\ & E_{sl}(x) {\partial y_s \over \partial x_l} + {\partial y_s \over \partial x_m} + F^s_{qj}(x) {\partial f_q \over \partial x_j} = G_{sk}(x) y_k \end{align}
$i,q = 1, ..., n_1$, $s,k = 1, ..., n_2$, $l = 1, ..., m-1$, $j = 1, ..., m$, summation over $l$, $k$, $q$ and $j$ is implicit. No summation over $i$ or $s$. $x = (x_1, ..., x_m)$, $f_i, y_s : [-a, a]^m \rightarrow \mathbb{R}$.
$C_{il}, G_{sk}, F^s_{qj}, E_{sl}(x), D_{ik} : [-a, a]^m \rightarrow \mathbb{R}$ are $C^r$ (for some $r \geq 1$) functions.
Lets assume $f_i$ and $y_s$ satisfy an initial condition of zero at $x_m = 0$: $f_i (x_1, ...,x_{m-1}, 0) =0$, $y_s (x_1, ...,x_{m-1}, 0) =0$ for $(x_1, ...,x_{m-1}) \in [-a, a]^{m-1}$. The question is as follows:
The linear system of PDE above obviously has a solution of zero based on an initial condition of zero. Is this solution unique? Or can it have a non-trivial differentiable solution based on an initial condition of zero? Is there any theorem that would guarantee the uniqueness of a zero solution for linear systems of PDE of the form above based on an initial condition of zero?
Having a non-trivial solution based on an initial condition of zero would lead to some unsatisfactory results. For example if the above linear system of PDE has a non-trivial solution then a constant multiple of that solution would also be a solution based on an initial condition of zero, therefore this would lead to an infinite family of solutions based on an initial condition of zero. Or if we consider the discretization of the above linear system of PDE and want to solve it numerically at each discretization hyperplane in the $x_m$ direction parallel to the initial condition hyperplane we will obtain a zero value for the functions $y_s$ and $f_i$. Therefore it looks like that a non-trivial solution cannot be captured by the discretization of the above PDE.
I am looking for a more rigorous statement that would guarantee that the above system of PDE has a unique solution of zero based on an initial condition of zero. Need it for a paper that I am working on.