Local and global well-posedness for the critical Schrödinger-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 x H^l, with s and l satisfying max{0, s − 1} ≤ l ≤ min{2s, s + 1}. In particular, these include the energy space H^1 × 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 × 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.