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

TL;DR

This paper proves the global well-posedness of the defocusing cubic nonlinear wave equation outside a ball with radial data, using Fourier transform techniques and energy methods, at low regularity levels.

Contribution

It establishes the first low-regularity global well-posedness result for the semilinear wave equation with Dirichlet boundary conditions outside a ball.

Findings

Dispersive and Strichartz estimates are established for the linear wave outside the ball.

Global well-posedness is proved for solutions with Sobolev regularity s > 3/4.

The approach combines Fourier truncation with energy methods for the nonlinear problem.

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.