  In this paper, we consider the initial-boundary value problems with several
fundamental boundary conditions (the Dirichlet/Neumann/Robin boundary
condition) for the multi-component system of semi-linear classical damped wave
equations outside a ball. By applying a test function approach with a judicious
choice of test functions, which approximates the harmonic functions being
subject to these boundary conditions on $\partial \Omega$, simultaneously we
have succeeded in proving the blow-up result in a finite time as well as in
catching the sharp upper bound of lifespan estimates for small solutions in two
and higher spatial dimensions. Moreover, such kind of these results will be
discussed in one-dimensional case at the end of this work.
