Non-existence of global solutions for higher-order evolution inequalities in unbounded cone-like domains

We use the test function method developed by Mitidieri and Pohozaev to get a priori estimates and non-existence results for semi-linear “higher-order evolution inequalities” in unbounded cone-like domains. As a model we consider the problem in a cone K with the positive initial-boundary conditions
$$ \frac{\partial^ku}{\partial t^k}-\Delta u\ge|u|^q, \quad k=1,2,\dots; \quad u_{|\partial K\times[0,\infty)}\ge0, \quad \frac{\partial^{k-1}u}{\partial t^{k-1}}\biggl|_{t=0}\ge0, $$
where $\Delta$ denotes the Laplace operator.