Let $u(t) \in \mathbb{R}^n$. Are there existence results for the ODE
$$C(t)u'(t) = A(t)u(t) + f(t)$$ where $A(t), C(t) \in L^\infty(0,T;\mathbb{R}^{n\times n})$, $f(t) \in L^2(0,T;\mathbb{R}^n).$
In addition, $C(t)$ is a Gram matrix and is invertible and positive-semi definite (with constant depending on $t$).
What kind of existence results are available? My problem is with the matrix $C(t)$. Yes it is invertible so I can multiply through by its inverse, but the inverse may not be in $L^2$ or $L^\infty$.