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

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 $\gamma(d,\nu)>0$.