Scattering theory below energy for the cubic fourth-order Schr\"{o}dinger equation

TL;DR

This paper establishes global existence and scattering results for the cubic fourth-order Schrödinger equation in low regularity spaces by developing new interaction Morawetz estimates and extending existing methods.

Contribution

It introduces an alternative approach to derive interaction Morawetz estimates and extends their applicability to lower regularity and higher dimensions.

Findings

Proves global existence and scattering in low regularity spaces

Develops a new interaction Morawetz estimate

Extends the dimension range for the estimates

Abstract

We investigate the global existence and scattering for the cubic fourth-order Schr\"{o}dinger equation $iu_t+\Delta^2u+|u|^2u=0$ in the low regularity space $H^s(\R^n)$ with $s<2$. We provide an alternative approach to obtain a new interaction Morawetz estimate and extend the range of the dimension of the interactive estimate in Pausader \cite{P08} by modifying a tensor product method appeared in \cite{CGT}. We combine interaction Morawetz estimates, energy increments for the I-method to prove the result.