Higher-order evolution inequalities with Hardy potential on the Kor\'{a}nyi ball

TL;DR

This paper investigates higher-order evolution inequalities with Hardy potential on the Korányi ball, establishing nonexistence results and analyzing critical behaviors depending on the potential parameter.

Contribution

It provides a general nonexistence theorem for the problem and characterizes sharp conditions and critical behaviors for specific weight functions.

Findings

Nonexistence results for the evolution inequality under certain conditions.

Identification of three distinct critical behaviors based on the potential parameter.

Sharpness of nonexistence results in the case of power-type weight functions.

Abstract

We consider a higher order in (time) semilinear evolution inequality posed on the Kor\'{a}nyi ball under an inhomogeneous Dirichlet-type boundary condition. The problem involves an inverse-square potential $\lambda/|\xi|_\mathbb{H}^2$, where $\lambda \geq -(Q-2)^2/4$ and a general weight function $V$ depending on the space variable in front of the power nonlinearity. We first establish a general nonexistence result for the considered problem. Next, in the special case $V(\xi):=|\xi|_\mathbb{H}^a$, $a\in \mathbb{R}$, we prove the sharpness of our nonexistence result and show that the problem admits three different critical behaviors according to the value of the parameter $\lambda$.