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

We continue the study of the Dirichlet boundary value problem of nonlinear wave equation with radial data in the exterior $\mathrm{\Omega }={\mathbb{R}}^3\mathrm{\backslash }\bar{B}(0,1)$. 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 $({\dot{H}}_D^s(\mathrm{\Omega })\cap L^{p+1}(\mathrm{\Omega }))\times {\dot{H}}_D^{s-1}(\mathrm{\Omega })$ 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.