Global well-posedness of the defocusing, cubic nonlinear wave equation outside of the ball with radial data

Abstract

We consider the defocusing, cubic nonlinear wave equation with zero Dirichlet boundary value in the exterior $\Omega = \mathbb{R}^3\backslash \bar{ B}(0,1)$. We make use of the distorted Fourier transform in \cite{LiSZ:NLS, Taylor:PDE:II} to establish the dispersive estimate and the global-in-time (endpoint) Strichartz estimate of the linear wave equation outside of the ball with radial data. As an application, we combine the Fourier truncation method as those in \cite{Bourgain98:FTM, GallPlan03:NLW, KenigPV00:NLW} with the energy method to show global well-posedness of radial solution to the defoucusing, cubic nonlinear wave equation outside of a ball in the Sobolev space $\left(\dot H^{s}_{D}(\Omega) \cap L^4(\Omega) \right)\times \dot H^{s-1}_{D}(\Omega)$ with $s>3/4$. To the best of the author's knowledge, it is first result about low regularity of semilinear wave equation with zero Dirichlet boundary value on the exterior domain.