Global well - posedness for the defocusing, cubic, nonlinear wave equation in three dimensions for radial initial in $\dot{H}^{s} \times \dot{H}^{s - 1}$, $s > \frac{1}{2}$

TL;DR

This paper proves global well-posedness for the defocusing cubic nonlinear wave equation in three dimensions with radial initial data in certain Sobolev spaces above the critical regularity, using the I-method and Strichartz estimates.

Contribution

It extends global well-posedness results to initial data in Sobolev spaces with regularity above the critical level for radial solutions.

Findings

Global well-posedness established for s > 1/2

Uses I-method and long-time Strichartz estimates

Applicable to radial initial data in specified Sobolev spaces

Abstract

In this paper we study the defocusing, cubic nonlinear wave equation in three dimensions with radial initial data. The critical space is $\dot{H}^{1/2} \times \dot{H}^{-1/2}$. We show that if the initial data is radial and lies in $(\dot{H}^{s} \times \dot{H}^{s - 1}) \cap (\dot{H}^{1/2} \times \dot{H}^{-1/2})$ for some $s > \frac{1}{2}$, then the cubic initial value problem is globally well - posed. We use the I - method and the long time Strichartz estimates. This method is quite similar to the method used in [D2].