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

Abstract

We study the defocusing semilinear wave equation in ${\mathbb{R}}\times{\mathbb{R}}^2\backslash{\mathcal K}$ 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 ${\mathbb{R}}\times{\mathbb{R}}^{2}\backslash{\mathcal K}$.