On Necessary Conditions for the Existence of Local Solutions to Singular Nonlinear Ordinary Differential Equations and Inequalities

We obtain conditions for the nonexistence of global solutions and estimates of existence time for local solutions to the problem
$$ \frac {d^ky}{dt^k} \ge a_1(t)|y|^{q_1}+a_2(t)|y|^{q_2}+\dots +a_n(t)|y|^{q_n}. $$
The proofs are based on the method of trial functions developed by Mitidieri and Pokhozhaev.