  We establish local well-posedness results for the Initial Value Problem
associated to the Schr\"odinger-Debye system in dimensions $N=2, 3$ for data in
$H^s\times H^{\ell}$, with $s$ and $\ell$ satisfying $\max \{0, s-1\} \le \ell
\le \min\{2s, s+1\}$. In particular, these include the energy space $H^1\times
L^2$. Our results improve the previous ones obtained in \cite{Bidegaray1},
\cite{Bidegaray2} and \cite{Corcho-Linares}. Moreover, in the critical case
(N=2) and for initial data in $H^1\times L^2$, we prove that solutions exist
for all times, thus providing a negative answer to the open problem mentioned
in \cite{Fibich-Papanicolau} concerning the formation of singularities for
these solutions.
