Nonexistence of global solutions to critical semilinear wave equations in exterior domain in high dimensions

The aim of this paper is to study the long time behavior of solutions to the initial boundary value problem of critical semilinear wave equations with small data in exterior domain in high dimensions (n≥5n≥5). We prove that solutions cannot exist globally in time. The novelty is that we first use a cutoff function to transform the exterior problem to a Cauchy problem. Then we divide the time into two intervals, in one time interval we establish an energy estimate to control the error terms while in the other interval the asymptotic behavior of two test functions is used. Furthermore, we obtain the upper bound of the lifespan by following the idea of Zhou and Han (2014).