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 RN(N≥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 RN. At least in the case N=4, their estimates are sharp in view of the work by Zha–Zhou [21]. 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 [15].