Higher-order evolution inequalities involving convection and Hardy-Leray potential terms in a bounded domain

Abstract

We consider a class of nonlinear higher-order evolution inequalities posed in $(0,\infty)\times B_1\backslash\{0\}$ , subject to inhomogeneous Dirichlet-type boundary conditions, where B1 is the unit ball in $\mathbb{R}^N$ . The considered class involves differential operators of the form \begin{equation*} \mathcal{L}_{\mu_1,\mu_2}=-\Delta +\frac{\mu_1}{|x|^2}x\cdot \nabla +\frac{\mu_2}{|x|^2},\qquad x\in \mathbb{R}^N\backslash\{0\}, \end{equation*} where $\mu_1\in \mathbb{R}$ and $\mu_2\geq -\left(\frac{\mu_1-N+2}{2}\right)^2$ . Optimal criteria for the nonexistence of weak solutions are established. Our study yields naturally optimal nonexistence results for the corresponding class of elliptic inequalities. Notice that no restriction on the sign of solutions is imposed.