Hyperbolic inequalities with a Hardy potential singular on the boundary of an annulus

We are concerned with the study of existence and nonexistence of weak solutions for a class of hyperbolic inequalities with a Hardy potential singular on the boundary $\mathrm{\partial }{B}_1$ of the annulus $A=\left\{x\in {\mathbb{R}}^3:1<|x|\le 2\right\}$, where $\mathrm{\partial }{B}_1=\left\{x\in {\mathbb{R}}^3:|x|=1\right\}$. A singular potential function of the form $(|x|-1{)}^{-\rho }$, $\rho \ge 0$, is considered in front of the power nonlinearity. Two types of inhomogeneous boundary conditions on $(0,\mathrm{\infty })\times \mathrm{\partial }{B}_2$, $\mathrm{\partial }{B}_2=\left\{x\in {\mathbb{R}}^3:|x|=2\right\}$, are studied: Dirichlet and Neumann. We use a unified approach to show the optimal criteria of Fujita-type for each case.