On the periodic Schr\"odinger-Debye equation

We study local and global well-posedness of the initial value problem for the Schr\"odinger-Debye equation in the \emph{periodic case}. More precisely, we prove local well-posedness for the periodic Schr\"odinger-Debye equation with subcritical nonlinearity in arbitrary dimensions. Moreover, we derive a new \emph{a priori} estimate for the $H^1$ norm of solutions of the periodic Schr\"odinger-Debye equation. A novel phenomena obtained as a by-product of this \emph{a priori} estimate is the global well-posedness of the periodic Schr\"odinger-Debye equation in dimensions $1,2$ and 3 \emph{without} any smallness hypothesis of the $H^1$ norm of the initial data in the ``focusing'' case.