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We consider the Cauchy-Dirichlet problem for semilinear wave equations in athree space dimensional domain exterior to a bounded and non-trapping obstacle.We obtain a detailed estimate for the lower bound of the lifespan of classicalsolutions when the size of initial data tends to zero, in a similar spirit tothat 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.

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