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?



Is there existence and uniqueness theory of this system of ODE?
If the matrix $A(t)=(a\_{ij}(t))$ is invertible, e.g. if there exists $B(t)\in L^\infty$, with
$B(t) A(t)=Id$, then you get a linear non-characteristic system of type
$$
\dot u +C(t) u=g(t),\quad C\in L^\infty.
$$
However without invertibility, you may be in deep trouble, even in the scalar case: consider
for $\nu>0$ the equation
$$
t^{\nu+1}\dot x=\nu x(t), \quad x(0)=0,
$$
which has 0 as a solution but also the smooth function $x(t)=H(t)\exp{-t^{-\nu}}$ flat at 0 since 
$$
x(t)=H(t)\exp{-t^{-\nu}},\quad \dot x(t)=H(t)\nu t^{-\nu-1}\exp{-t^{-\nu}},\quad
 t^{\nu+1}\dot x=\nu x(t).
$$
Note that when $\nu=0$, the situation is not that bad and the singularity is called a regular singularity.


