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 ${\mathbb{R}}_{+}^N$, $N\ge 2$, and involving differential operators of the form ${\mathcal{L}}_{\lambda }=-\mathrm{\Delta }+\lambda \mathrm{/}\mathrm{\mid }x{\mathrm{\mid }}^2$, where $\lambda \ge -N^2\mathrm{/}4$. A potential function of the form $\mathrm{\mid }x{\mathrm{\mid }}^{\tau }$, $\tau \in \mathbb{R}$, 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.