Global existence and scattering for a class of nonlinear fourth-order Schrödinger equation below the energy space

Abstract

In this paper, we consider a class of nonlinear fourth-order Schrödinger equation, namely \[ \left\{ \begin{array}{rcl} i\partial_t u +Δ^2 u &=&-|u|^{ν-1} u, \quad 1+ \frac{8}{d}<ν<1+\frac{8}{d-4},\\ u(0)&=&u_0 \in H^γ(\mathbb{R}^d), \quad 5 \leq d \leq 11. \end{array} \right. \] 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 $\gamma (d,\nu )>0$.

25 pages, revised version, to appear in Nonlinear Analysis. arXiv admin note: text overlap with arXiv:1706.06517

References in corpus (2)

• Global wellposedness and scattering for the focusing energy-critical nonlinear Schrodinger equations of fourth order in the radial case
• Global wellposedness and scattering for the defocusing energy-critical nonlinear Schrodinger equations of fourth order in dimensions $d\ge 9$