On the periodic Schr\"odinger-Debye equation

TL;DR

This paper investigates the well-posedness of the periodic Schr"odinger-Debye equation, establishing local and global results in various dimensions and deriving new a priori estimates that remove smallness constraints in certain cases.

Contribution

It proves local well-posedness for subcritical nonlinearities and introduces a new a priori estimate enabling global well-posedness without small data assumptions in low dimensions.

Findings

Global well-posedness in dimensions 1, 2, and 3 without smallness conditions

New a priori estimate for the $H^1$ norm of solutions

Local well-posedness for subcritical nonlinearities in arbitrary dimensions

Abstract

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.