Higher order evolution inequalities with Hardy potential in the exterior of a half-ball

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.