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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?

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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.

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thanks a lot.!! – martin_e Apr 27 '13 at 15:22

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