# Global existence and scattering for a class of nonlinear fourth-order Schr\"odinger equation below the energy space

## TL;DR

This paper proves global existence and scattering for a class of nonlinear fourth-order Schrödinger equations below the energy space using the I-method and interaction Morawetz inequality.

## Contribution

It introduces a novel combination of the I-method and interaction Morawetz inequality to establish global well-posedness and scattering below the energy space for certain nonlinear Schrödinger equations.

## Key findings

- Global well-posedness established for specified nonlinearities.
- Scattering results proved in sub-energy Sobolev spaces.
- Applicable for dimensions 5 to 11 with certain nonlinearity ranges.

## Abstract

In this paper, we consider a class of nonlinear fourth-order Schr\"odinger equation, namely \[ \left\{ \begin{array}{rcl} i\partial_t u +\Delta^2 u &=&-|u|^{\nu-1} u, \quad 1+ \frac{8}{d}<\nu <1+\frac{8}{d-4},\\ u(0)&=&u_0 \in H^\gamma(\mathbb{R}^d), \quad 5 \leq d \leq 11. \end{array} \right. \] Using the $I$-method combined with the interaction Morawetz inequality, we establish the global well-posedness and scattering in $H^\gamma(\mathbb{R}^d)$ with $\gamma(d,\nu)<\gamma<2$ for some value $\gamma(d,\nu)>0$.