Global well-posedness for the logarithmically energy-supercritical Nonlinear Wave Equation with partial symmetry

TL;DR

This paper proves global well-posedness and scattering for a logarithmically energy-supercritical nonlinear wave equation with partial symmetry, extending previous radial symmetry results using weighted Morawetz and Strichartz estimates.

Contribution

It introduces new techniques for handling partial symmetry in supercritical wave equations, generalizing Tao's radial symmetry results.

Findings

Established global well-posedness and scattering under partial symmetry.

Extended the analysis to include a quantitative result for the energy-critical case.

Developed weighted estimates tailored to partial symmetry conditions.

Abstract

We establish global well-posedness and scattering results for the logarithmically energy-supercritical nonlinear wave equation, under the assumption that the initial data satisfies a partial symmetry condition. These results generalize and extend work of Tao in the radially symmetric setting. The techniques involved include weighted versions of Morawetz and Strichartz estimates, with weights adapted to the partial symmetry assumptions. In an appendix, we establish a corresponding quantitative result for the energy-critical problem.