  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.
