Lifespan estimates for $2$-dimensional semilinear wave equations in asymptotically Euclidean exterior domains

In this paper we study the initial boundary value problem for two-dimensional semilinear wave equations with small data, in asymptotically Euclidean exterior domains. We prove that if $1<ple p_c(2)$, the problem admits almost the same upper bound of the lifespan as that of the corresponding Cauchy problem, only with a small loss for $1<ple 2$. It is interesting to see that the logarithmic increase of the harmonic function in $2$-D has no influence to the estimate of the upper bound of the lifespan for $2<ple p_c(2)$. One of the novelties is that we can deal with the problem with flat metric and general obstacles (bounded and simple connected), and it will be reduced to the corresponding problem with compact perturbation of the flat metric outside a ball.