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

On global existence for semilinear wave equations with spacedependent critical damping