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

TL;DR

This paper investigates the decay and scattering behavior of solutions to the defocusing semilinear wave equation in a two-dimensional exterior domain with a star-shaped obstacle, extending known results from the whole space to this setting.

Contribution

It establishes decay and scattering results for the wave equation in an exterior domain, generalizing previous findings from the entire space to obstacle-containing regions.

Findings

Potential energy decays appropriately

Solutions decay pointwise to zero

Solutions scatter in energy and Sobolev spaces

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 ${\mathbb{R}}\times{\mathbb{R}}^{2}\backslash{\mathcal K}$.