  We consider semilinear higher order (in time) evolution inequalities posed in
an exterior domain of the half-space $\mathbb{R}_+^N$, $N\geq 2$, and involving
differential operators of the form $\mathcal{L}_\lambda =-\Delta
+\lambda/|x|^2$, where $\lambda\geq -N^2/4$. A potential function of the form
$|x|^\tau$, $\tau\in \mathbb{R}$, is allowed in front of the power
nonlinearity. Under inhomogeneous Dirichlet-type boundary conditions, we show
that the dividing line with respect to existence or nonexistence is given by a
Fujita-type critical exponent that depends on $\lambda, N$ and $\tau$, but
independent of the order of the time derivative.
