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

Part of: Miscellaneous topics - Partial differential equations Partial differential equations

Published online by Cambridge University Press:  05 May 2023

Huyuan Chen ,
Mohamed Jleli  and
Bessem Samet
Huyuan Chen
Affiliation:
Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, People’s Republic of China (chenhuyuan@yeah.net)
Mohamed Jleli
Affiliation:
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia (jleli@ksu.edu.sa; bsamet@ksu.edu.sa)
Bessem Samet
Affiliation:
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia (jleli@ksu.edu.sa; bsamet@ksu.edu.sa)
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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.

Keywords

evolution inequalitiesbounded domaindegenerate operatornonexistencepotential terms

MSC classification

Primary: 35R45: Partial differential inequalities 35B44: Blow-up 35B33: Critical exponents
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 66 , Issue 2 , May 2023 , pp. 366 - 390
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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