Low-regularity global well-posedness for the defocusing cubic wave equation outside a ball with radial initial data

Abstract

We study the three-dimensional defocusing cubic nonlinear wave equation with homogeneous Dirichlet boundary condition in the exterior of the unit ball, $Ω=\R^3\backslash\overline{B(0,1)}$, for radial initial data. For every $\frac12\leq s\leq\frac34$, we establish global well-posedness in $\dot H^s_{D,rad}(Ω)\times\dot H^{s-1}_{D,rad}(Ω)$, including the critical endpoint $s=\frac12$, without an additional $L^4$ assumption on the initial displacement. The proof uses the distorted Fourier transform associated with the Dirichlet Laplacian to establish radial Sobolev embeddings, including the quantitative estimate $\|f\|_{L^p(Ω)}\leq C\sqrt p\,\|f\|_{\dot H_D^{1/2}(Ω)}$ for $4\leq p<\infty$, with $C$ independent of $p$. We combine a nonlinear Fourier truncation argument with the critical Strichartz estimate to construct a global nonlinear component with high-frequency-truncated initial data. For the remainder, we construct a local energy solution in an affine energy space and derive a logarithmic energy inequality that prevents finite-time energy blow-up. Persistence of the original Sobolev regularity, uniqueness, and local Lipschitz dependence on each finite time interval complete the proof.