One Remark on Barely \dot{H}^{s_{p}} Supercritical Wave Equations

TL;DR

This paper extends the critical theory for 3D wave equations to certain barely supercritical cases, establishing global well-posedness under specific conditions and generalizing previous results for the critical case.

Contribution

It introduces a method to prove global well-posedness for barely supercritical wave equations using a modified critical theory and Kenig-Merle type conditions.

Findings

Global well-posedness for barely supercritical wave equations established.

Extension of critical theory to new supercritical regimes.

Conditions under which solutions remain smooth and unique.

Abstract

We prove that a good \dot{H}^{s_{p}} critical theory for the 3D wave equation \partial_{tt} u - \triangle u = -|u|^{p-1} u can be extended to prove global well-posedness of smooth solutions of at least one 3D barely \dot{H}^{s_{p}} supercritical wave equation \partial_{tt} u - \triangle u =- |u|^{p-1} u g(|u|), with g growing slowly to infinity, provided that a Kenig-Merle type condition is satisfied. This result extends those obtained for the particular case s_{p}=1.