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

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: \(N_1=N_2=2\); \(N_1=2\),  \(N_2\geq 3\); \(N_1\geq 3\), \(N_2=2\); \(N_1,N_2\geq 3\). For more information see https://ejde.math.txstate.edu/Volumes/2021/75/abstr.html