Local and global well-posedness for the critical Schrödinger-Debye system

We establish local well-posedness results for the Initial Value Problem associated to the Schrödinger-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 by B. Bidégaray, and by A. J. Corcho and F. 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 by G. Fibich and G. C. Papanicolau concerning the formation of singularities for these solutions.