Existence and uniqueness for matrix ODE $X' = f(X) A X$, with $f$ linear

Question

This is probably a rather trivial question, but nevertheless I think I need help.

I am interested in studying the following initial value problem in $X \colon (a,b) \to \mathbb{R}^{m}$:

\begin{equation} \begin{cases} X' = f(X) A X\\ X(0)=x \end{cases} \,. \end{equation} Here $f$ is a given linear function $\mathbb{R}^{m} \to \mathbb{R}$, whereas $A$ a given matrix function $I \to \mathbb{R}^{m\times m}$.

My question is: Does the solution of this problem exist for any arbitrary choice of functions $f$, $A$, and interval $(a,b)$?

I know that, if $f \equiv 1$, then we have indeed existence and uniqueness.

Answer 1

Yes, you inherit all claims of the standard existence and uniqueness theorems for local solutions.

No if you want the existence of a solution over the fixed interval $I$. The standard non-linear counter-example applies, $y'=y^2$ has the given form with $f(y)=y$ and $A=1$. Depending on the initial value, the solution can blow up inside the interval.