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Higher-order evolution inequalities involving convection and Hardy-Leray potential terms in a bounded domain
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 66 / Issue 2 / May 2023
- Published online by Cambridge University Press:
- 05 May 2023, pp. 366-390
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- Article
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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 formwhere
\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*}
$\mu_1\in \mathbb{R}$ 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.
