Geodesic
On Necessary Conditions for the Existence of Local Solutions to Singular Nonlinear Ordinary Differential Equations and Inequalities
Matematičeskie zametki, Tome 72 (2002) no. 6, pp. 924-935

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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. 
@article{MZM_2002_72_6_a12,
     author = {J. Hay},
     title = {On {Necessary} {Conditions} for the {Existence} of {Local} {Solutions} to {Singular} {Nonlinear} {Ordinary} {Differential} {Equations} and {Inequalities}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {924--935},
     publisher = {mathdoc},
     volume = {72},
     number = {6},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2002_72_6_a12/}
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J. Hay. On Necessary Conditions for the Existence of Local Solutions to Singular Nonlinear Ordinary Differential Equations and Inequalities. Matematičeskie zametki, Tome 72 (2002) no. 6, pp. 924-935. http://geodesic.mathdoc.fr/item/MZM_2002_72_6_a12/