Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 75, pp. 1-26.

Title: Nonexistence results for hyperbolic type inequalities involving the 
       Grushin operator in exterior domains
       
Authors: Mohamed Jleli (King Saud Univ., Riyadh, Saudi Arabia)
         Bessem Samet  (King Saud Univ., Riyadh, Saudi Arabia)

Abstract:
 We study the hyperbolic type differential inequality
 $$
 u_{tt}(t,x,y)-\mathcal{L}_\ell u(t,x,y)\geq  |u(t,x,y)|^p,\quad
 (t,x,y)\in (0,\infty)\times D_1\times D_2
 $$
 under the boundary conditions
 $$\displaylines{
 u(t,x,y) \geq f(x),\quad (t,x,y)\in (0,\infty)\times \partial D_1\times D_2,\cr
 u(t,x,y) \geq g(y),\quad (t,x,y)\in (0,\infty)\times D_1\times \partial D_2,
 }$
 where $p>1$, $D_k=\{z\in \mathbb{R}^{N_k}: |z|\geq 1\}$, $k=1,2$, $N_k\geq 2$,
 $f\in L^1(\partial D_1)$, $g\in L^1(\partial D_2)$, and $\mathcal{L}_\ell$,
 $\ell\in \mathbb{R}$, is  the Grushin operator
 $$
 \mathcal{L}_\ell u=\Delta_x u+ |x|^{2\ell} \Delta_y u.
 $$
 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.