Scattering theory for the defocusing fourth-order Schr\"odinger equation

TL;DR

This paper establishes the global well-posedness and scattering for the defocusing fourth-order nonlinear Schrödinger equation in high dimensions, using concentration compactness and advanced analytical techniques to handle energy-critical and supercritical cases.

Contribution

It provides a unified approach to prove global existence and scattering for FNLS across different energy regimes, extending previous results to higher dimensions.

Findings

Proved global well-posedness and scattering for FNLS in dimensions d≥9.

Excluded finite time blowup, soliton, and cascade scenarios.

Utilized concentration compactness, Duhamel formulas, and Morawetz estimates.

Abstract

In this paper, we study the global well-posedness and scattering theory for the defocusing fourth-order nonlinear Schr\"odinger equation (FNLS) $iu_t+\Delta^2 u+|u|^pu=0$ in dimension $d\geq9$. We prove that if the solution $u$ is apriorily bounded in the critical Sobolev space, that is, $u\in L_t^\infty(I;\dot H^{s_c}_x(\R^d))$ with all $s_c:=\frac{d}2-\frac4p\geq1$ if $p$ is an even integer or $s_c\in[1,2+p)$ otherwise, then $u$ is global and scatters. The impetus to consider this problem stems from a series of recent works for the energy-supercritical and energy-subcritical nonlinear Schr\"odinger equation (NLS) and nonlinear wave equation (NLW). We will give a uniform way to treat the energy-subcritical, energy-critical and energy-supercritical FNLS, where we utilize the strategy derived from concentration compactness ideas to show that the proof of the global well-posedness and scattering is reduced to exclude the existence of three scenarios: finite time blowup; soliton-like solution and low to high frequency cascade. Making use of the No-waste Duhamel formula, we deduce that the energy or mass of the finite time blow-up solution is zero and so get a contradiction. Finally, we adopt the double Duhamel trick, the interaction Morawetz estimate and interpolation to kill the last two scenarios.