  In this work, we investigate the influence of general damping and potential
terms on the blow-up and lifespan estimates for energy solutions to power-type
semilinear wave equations. The space-dependent damping and potential functions
are assumed to be critical or short range, spherically symmetric perturbation.
The blow up results and the upper bound of lifespan estimates are obtained by
the so-called test function method. The key ingredient is to construct special
positive solutions to the linear dual problem with the desired asymptotic
behavior, which is reduced, in turn, to constructing solutions to certain
elliptic "eigenvalue" problems.
