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

Abstract

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 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.

References in corpus (1)

• Global well-posedness for KdV in Sobolev Spaces of negative index