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

TL;DR

This paper investigates the existence and nonexistence of solutions to higher order evolution inequalities with Hardy potential in an exterior domain, identifying a critical exponent that determines solution behavior.

Contribution

It introduces a Fujita-type critical exponent for higher order evolution inequalities with Hardy potential, extending previous results to exterior domains and variable potentials.

Findings

Critical exponent depends on $\lambda$, $N$, and $ au$

Existence or nonexistence determined by this critical exponent

Results are independent of the order of the time derivative

Abstract

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.