I am interested in the **existence** of a **vector valued** solution $y = y(x, t) \in\mathbb{R}^n$ to a system of $2n$ equations: there are **twice more equations than unknowns**. More precisely:

Let $A$ and $B$ be

matrix valuedfunctions $A, B \in C^1([0, L]\times [0, T]; \mathbb{R}^{n \times n})$.Does the system\begin{equation} \begin{aligned} \partial_x y(x, t) = A(x, t) y(x, t) \\ \partial_t y(x, t) = B(x, t) y(x, t) \end{aligned} \qquad \text{in } [0, L]\times [0, T] \end{equation} with initial and boundary conditions (with $g \in C^1([0, T]; \mathbb{R}^n)$ and $y_0 \in C^1([0, L]; \mathbb{R}^n)$) \begin{equation} y(x, 0) = y_0(x) \quad \text{for }x \in [0, L]\\ y(0, t ) = g(t) \quad \text{ for }t \in [0, T] \end{equation}have a solution $y \in C^1([0, L]\times [0, T]; \mathbb{R}^n)$?

(Here, $\mathbb{R}^{n \times n}$ denotes the square matrices of size $n \times n$ with real coefficients.)

Writing the equations in integral form, $y^1$ is solution to $\partial_x y^1 = A y^1$ if and only if

\begin{align*}
y^1(x,t) =& y^1(0, 0) + \int_0^t \partial_t y^1(0, s) ds \\
&+ \int_0^x \partial_x y^1(\xi, 0) d\xi + \int_0^x \int_0^t ((\partial_t A) y^1 + A \partial_t y^1) d\xi ds
\end{align*}
while $y^2$ is solution to $\partial_t y^2 = B y^2$ if and only if
\begin{align*}
y^2(x,t) =& y^2(0, 0) + \int_0^x \partial_x y^2(\xi, 0) d\xi \\
&+ \int_0^t \partial_t y^2(0, s) ds + \int_0^x \int_0^t ((\partial_x B) y^2 + B \partial_x y^2) d\xi ds.
\end{align*}

Assume that the following

compatibility conditionshold: \begin{align*} \partial_t A + A B = \partial_x B + B A,\qquad \text{in } [0, L]\times [0, T] \end{align*} and $g(0) = y_0(0)$.

In this case, if $y$ is solution to both systems, then the expressions for $y^1$ and $y^2$ coincide and are equal to \begin{align*} y(x,t) =& f(0) + \int_0^x \frac{\mathrm{d}}{\mathrm{d}x}y_0(x) d\xi + \int_0^t \frac{\mathrm{d}}{\mathrm{d}t}g(s) ds \\ &+ \int_0^x \int_0^t ((\partial_x B) y + B A y) d\xi ds \end{align*}

But how can we show that there

exists$y$ solution toboth$\partial_x y = A y$and$\partial_t y = By$? Which method(s) would you suggest? Do we need to add assumptions to obtain existence?

Also asked on Mathematics Stack Exchange: https://math.stackexchange.com/questions/3384407/uniqueness-but-no-existence