Global well-posedness of the defocusing nonlinear wave equation outside of a ball with radial data for $3<p<5$
Guixiang Xu, Pengxuan Yang, Zhuohui You
TL;DR
This work proves global well-posedness for the defocusing nonlinear wave equation outside a ball in 3D with radial data in the super-conformal range 3<p<5. The authors fuse distorted Fourier truncation (à la Bourgain), global-in-time endpoint Strichartz estimates for radial data, and energy methods, augmented by a radial Sobolev inequality, to handle the nonlinear evolution in a Dirichlet exterior domain. The result extends prior cubic-case and energy-subcritical analyses to 3<p<5, providing explicit admissible Sobolev regularity and a quantitative growth bound for the solution in $\dot H^s_D(Ω)$ over long times. This advances understanding of NLW dynamics in exterior domains under radial symmetry and contributes techniques applicable to exterior-domain dispersive problems with non-conformal nonlinearities.
Abstract
We continue the study of the Dirichlet boundary value problem of nonlinear wave equation with radial data in the exterior $Ω= \mathbb{R}^3\backslash \bar{B}(0,1)$. 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 $\left(\dot H^{s}_{D}(Ω) \cap L^{p+1}(Ω) \right)\times \dot H^{s-1}_{D}(Ω)$ 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.