Lower bound of the lifespan of solutions to semilinear wave equations in an exterior domain

We consider the Cauchy-Dirichlet problem for semilinear wave equations in a three space dimensional domain exterior to a bounded and non-trapping obstacle. We obtain a detailed estimate for the lower bound of the lifespan of classical solutions when the size of initial data tends to zero, in a similar spirit to that of the works of John and H\"ormander where the Cauchy problem was treated. We show that our estimate is sharp at least for some special case.