The radial defocusing energy-supercritical cubic nonlinear wave equation in R^{1+5}

TL;DR

This paper proves that for the energy-supercritical defocusing cubic nonlinear wave equation in five dimensions, a bound in the critical space guarantees global solutions and scattering, using a frequency localized Morawetz inequality.

Contribution

It introduces a novel application of frequency localized Morawetz inequalities to establish global well-posedness and scattering in a supercritical setting.

Findings

A priori bounds imply global well-posedness.

A priori bounds imply scattering.

Frequency localized Morawetz inequality is effective in supercritical regimes.

Abstract

In this work, we consider the energy-supercritical defocusing cubic nonlinear wave equation in dimension d=5 for radially symmetric initial data. We prove that an a priori bound in the critical space implies global well-posedness and scattering. The main tool that we use is a frequency localized version of the classical Morawetz inequality, inspired by recent developments in the study of the mass and energy critical nonlinear Schr\"odinger equation.