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In this paper we study the initial boundary value problem for two-dimensionalsemilinear wave equations with small data, in asymptotically Euclidean exteriordomains. We prove that if $1<p\le p_c(2)$, the problem admits almost the sameupper bound of the lifespan as that of the corresponding Cauchy problem, onlywith a small loss for $1<p\le 2$. It is interesting to see that the logarithmicincrease of the harmonic function in $2$-D has no influence to the estimate ofthe upper bound of the lifespan for $2<p\le p_c(2)$. One of the novelties isthat we can deal with the problem with flat metric and general obstacles(bounded and simple connected), and it will be reduced to the correspondingproblem with compact perturbation of the flat metric outside a ball.

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