Local and Global Well-Posedness for the Critical Schrodinger-Debye System

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 $H^s\times H^{\mathrm{\ell }}$, with $s$ and $\mathrm{\ell }$ satisfying $\max {0, s-1} \le \ell \le \min{2s, s+1}$\mathrm{.}Inparticular,theseincludetheenergyspace$H^1\times L^2$. 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.