<p style='text-indent:20px;'>We consider the semilinear wave equation with time-dependent damping <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \partial_{tt}u-\Delta u +\mu (1+t)^{-\beta} \partial_t u = |u|^p, \quad (t, x)\in (0, \infty)\times D^c, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ D^c = \mathbb{R}^N\backslash D $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ D $\end{document}</tex-math></inline-formula> is the closed unit ball in <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ N\geq 2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \mu>0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ p>1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ -1<\beta<1 $\end{document}</tex-math></inline-formula>. The considered equation is investigated under the boundary conditions: <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ u(t, x) \left(\mbox{or } \frac{\partial u}{\partial n^+}(t, x)\right) = b(t)f(x)\, \, \mbox{on}\, \, (0, \infty)\times \partial D, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M8">\begin{document}$ n^+ $\end{document}</tex-math></inline-formula> is the outward (relative to <inline-formula><tex-math id="M9">\begin{document}$ D^c $\end{document}</tex-math></inline-formula>) unit normal on <inline-formula><tex-math id="M10">\begin{document}$ \partial D $\end{document}</tex-math></inline-formula>. General blow-up results are established for the considered problems. Moreover, for a certain class of functions <inline-formula><tex-math id="M11">\begin{document}$ b $\end{document}</tex-math></inline-formula>, the critical exponent in the sense of Fujita is obtained.