Absence of solutions of differential inequalities and systems of hyperbolic type in conic domains

We establish conditions sufficient for the absence of global solutions of semilinear hyperbolic inequalities and systems in conic domains of the Euclidean space $\mathbb R^N$.
We consider a model problem in a cone $K$: that given by the inequality
$$ \dfrac{\partial^2u}{\partial t^2}-\Delta u\geqslant |u|^q, \qquad (x,t)\in K\times(0,\infty), $$
The proof is based on the test-function method developed by Veron, Mitidieri, Pokhozhaev, and Tesei.