Electron. J. Differential Equations, Vol. 2021 (2021), No. 75, pp. 1-26.

Nonexistence results for hyperbolic type inequalities involving the Grushin operator in exterior domains

Mohamed Jleli, Bessem Samet

Abstract:
We study the hyperbolic type differential inequality

under the boundary conditions

where $p>1$, $D_k=\{z\in \mathbb{R}^{N_k}: |z|\geq 1\}$, $k=1,2$, $N_k\geq 2$, $f\in L^1(\partial D_1)$, $g\in L^1(\partial D_2)$, and $\mathcal{L}_\ell$, $\ell\in \mathbb{R}$, is the Grushin operator

We obtain sufficient conditions depending on $p$, $\ell$, $N_1$, $N_2$, $f$, and $g$, for which the considered problem admits no global weak solution. We discuss separately the four cases: $N_1=N_2=2$; $N_1=2$, $N_2\geq 3$; $N_1\geq 3$, $N_2=2$; $N_1,N_2\geq 3$.

Submitted June 25, 2021. Published September 14, 2021.
Math Subject Classifications: 35B44, 35B33, 35L10.
Key Words: Global weak solutions; hyperbolic type inequalities; exterior domain; Grushin operator.
DOI: https://doi.org/10.58997/ejde.2021.75

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Mohamed Jleli
Department of Mathematics
College of Science
King Saud University, P.O. Box 2455
Riyadh 11451, Saudi Arabia
email: jleli@ksu.edu.sa
Bessem Samet
Department of Mathematics
College of Science
King Saud University, P.O. Box 2455
Riyadh 11451, Saudi Arabia
email: bsamet@ksu.edu.sa

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