Hey,

Is there a local existence/uniqueness theory for a system of equations of the form $$\sum_{i=1}^n a_{ij}\dot u_i(t) + b_{ij}u_i(t) = f_j(t)\qquad\text{for $j=1,...,n$}$$ where the $f_j$ are in $L^2(0,T)$, the $b_{ij}$ are in $L^\infty(0,T)$ and the $a_{ij}$ are in some Lebesgue space (not necessarily continuous). What assumptions do I need for well-posedness?

Clearly if $a_{ij} = \delta_{ij}$ there is no problem it's standard. But for general?