# Finite time blowup of solutions to semilinear wave equation in an exterior domain

## TL;DR

This paper investigates the finite time blowup of solutions to a semilinear wave equation in an exterior domain, providing sharp lifespan estimates for small initial data, especially in four dimensions, using specialized linear wave solutions.

## Contribution

It establishes sharp upper bounds for the lifespan of blowup solutions in exterior domains, extending known results from the Cauchy problem to these settings.

## Key findings

- Lifespan bounds match those of the Cauchy problem in rf3d domain.
- Sharp estimates are confirmed for the case N=4.
- Utilizes special solutions to linear wave equations with Dirichlet boundary conditions.

## Abstract

We consider the initial-boundary value problem of semilinear wave equation with nonlinearity $|u|^p$ in exterior domain in $\mathbf{R}^N$ $(N\geq 3)$. Especially, the lifespan of blowup solutions with small initial data are studied. The result gives upper bounds of lifespan which is essentially the same as the Cauchy problem in $\mathbf{R}^N$. At least in the case $N=4$, their estimates are sharp in view of the work by Zha--Zhou (2015). The idea of the proof is to use special solutions to linear wave equation with Dirichlet boundary condition which are constructed via an argument based on Wakasa--Yordanov.