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

Abstract

We consider the initial-boundary value problem of semilinear wave equation with nonlinearity $|u{|}^p$ in exterior domain in ${\mathbf{R}}^N$ $(N\ge 3)$. 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.

13 pages