  The global existence for semilinear wave equations with space-dependent
critical damping $\partial_t^2u-\Delta u+\frac{V_0}{|x|}\partial_t u=f(u)$ in
an exterior domain is dealt with, where $f(u)=|u|^{p-1}u$ and $f(u)=|u|^p$ are
in mind. Existence and non-existence of global-in-time solutions are discussed.
To obtain global existence, a weighted energy estimate for the linear problem
is crucial. The proof of such a weighted energy estimate contains an
alternative proof of energy estimates established by
Ikehata--Todorova--Yordanov [J.\ Math.\ Soc.\ Japan (2013), 183--236] but this
clarifies the precise independence of the location of the support of initial
data. The blowup phenomena is verified by using a test function method with
positive harmonic functions satisfying the Dirichlet boundary condition.
