Some nonexistence results for higher-order evolution inequalities in cone-like domains

Nonexistence of global (positive) solutions of semilinear higher-order evolution inequalities \begin{equation*} \frac {\partial ^k u}{\partial t^k}-\Delta u^m\ge |u|^q,\quad \frac {\partial ^k u}{\partial t^k}-\Delta u\ge |x|^\sigma u^q,\quad \frac {\partial ^ku}{\partial t^k}-\operatorname{div} (|x|^\alpha Du)\ge u^q \end{equation*} with $k=1,2,\dots$, in cone-like domains is studied. The critical exponents $q^*$ are found and the nonexistence results are proved for $1<q\le q^*$. Remark that the corresponding result for $k=1$ (parabolic problem) is sharp.