Global well-posedness of the defocusing nonlinear wave equation outside of a ball with radial data for $3<p<5$

TL;DR

This paper proves the global well-posedness of the defocusing nonlinear wave equation outside a ball with radial data for 3<p<5, extending previous results and employing advanced Fourier and Strichartz techniques.

Contribution

It extends the global well-posedness results to the case 3<p<5 for the nonlinear wave equation outside a ball with radial data, incorporating new radial Sobolev inequality methods.

Findings

Established global well-posedness for 3<p<5

Extended previous cubic nonlinearity results

Utilized radial Sobolev inequality for super-conformal cases

Abstract

We continue the study of the Dirichlet boundary value problem of nonlinear wave equation with radial data in the exterior $\Omega = \mathbb{R}^3\backslash \bar{B}(0,1)$. We combine the distorted Fourier truncation method in \cite{Bourgain98:FTM}, the global-in-time (endpoint) Strichartz estimates in \cite{XuYang:NLW} with the energy method in \cite{GallPlan03:NLW} to prove the global well-posedness of the radial solution to the defocusing, energy-subcriticial nonlinear wave equation outside of a ball in $\left(\dot H^{s}_{D}(\Omega) \cap L^{p+1}(\Omega) \right)\times \dot H^{s-1}_{D}(\Omega)$ with $1-\frac{(p+3)(1-s_c)}{4(2p-3)}<s<1$, $s_c=\frac{3}{2}-\frac{2}{p-1} $, which extends the result for the cubic nonlinearity in \cite{XuYang:NLW} to the case $3<p<5$. Except from the argument in \cite{XuYang:NLW}, another new ingredient is that we need make use of the radial Sobolev inequality to deal with the super-conformal nonlinearity in addition to the Sobolev inequality.