Scattering theory for the defocusing fourth-order Schrödinger equation

Abstract

In this paper, we study the global well-posedness and scattering theory for the defocusing fourth-order nonlinear Schrödinger equation (FNLS) in dimensions . We prove that if the solution u is apriorily bounded in the critical Sobolev space, that is, with all if p is an even integer or otherwise, then u is global and scatters. We will give a uniform way to treat the energy-subcritical, energy-critical and energy-supercritical FNLS by making use of the strategy derived from concentration compactness ideas, and we are able to overcome the logarithmic blowup in the double Duhamel trick in dimension eight by exploiting the refined dispersive estimate which is in sharp contrast to the Schrödinger equation.