The global behaviors for defocusing wave equations in two dimensional exterior region

We study the defocusing semilinear wave equation in $\mathbb{R}\times {\mathbb{R}}^2\mathrm{\backslash }\mathcal{K}$ with the Dirichlet boundary condition, where $\mathcal{K}$ is a star-shaped obstacle with smooth boundary. We first show that the potential energy of the solution will decay appropriately. Based on it, we show that the solution also pointwisely decays to $0$. Finally, we show that the solution scatters both in energy space and the critical Sobolev space. In general, we show that most of the conclusions obtained in \cite{MR4395159}, which hold on ${\mathbb{R}}^{1+2}$, remain valid on $\mathbb{R}\times {\mathbb{R}}^2\mathrm{\backslash }\mathcal{K}$.