Energy scattering for a class of the defocusing inhomogeneous nonlinear Schrödinger equation

Abstract

In this paper, we consider a class of the defocusing inhomogeneous nonlinear Schrödinger equation \[ i\partial_t u + Δu - |x|^{-b} |u|^αu = 0, \quad u(0)=u_0 \in H^1, \] with $b,\alpha >0$. We firstly study the decaying property of global solutions for the equation when $0<\alpha <{\alpha }^{\star }$ where ${\alpha }^{\star }=\frac{4-2b}{d-2}$ for $d\ge 3$. The proof makes use of an argument of Visciglia. We next use this decay to show the energy scattering for the equation in the case ${\alpha }_{\star }<\alpha <{\alpha }^{\star }$, where ${\alpha }_{\star }=\frac{4-2b}{d}$.

18 pages. arXiv admin note: text overlap with arXiv:1710.01392

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