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.

Voir la notice de l'article provenant de la source Math-Net.Ru

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/}
}
TY  - JOUR
AU  - J. Hay
TI  - On Necessary Conditions for the Existence of Local Solutions to Singular Nonlinear Ordinary Differential Equations and Inequalities
JO  - Matematičeskie zametki
PY  - 2002
SP  - 924
EP  - 935
VL  - 72
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2002_72_6_a12/
LA  - ru
ID  - MZM_2002_72_6_a12
ER  - 
%0 Journal Article
%A J. Hay
%T On Necessary Conditions for the Existence of Local Solutions to Singular Nonlinear Ordinary Differential Equations and Inequalities
%J Matematičeskie zametki
%D 2002
%P 924-935
%V 72
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2002_72_6_a12/
%G ru
%F MZM_2002_72_6_a12
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/

[1] Mitidieri E., Pokhozhaev S. I., Apriornye otsenki i otsutstvie reshenii nelineinykh differentsialnykh uravnenii i neravenstv v chastnykh proizvodnykh, Tr. MIAN, 234, Nauka, M., 2001

[2] Khei Dzh., “O neobkhodimykh usloviyakh suschestvovaniya globalnykh reshenii nelineinykh obyknovennykh differentsialnykh neravenstv vysokogo poryadka”, Differents. uravneniya, 38:3 (2002), 344–350 | MR

[3] Khei Dzh., “O neobkhodimykh usloviyakh suschestvovaniya globalnykh reshenii sistem nelineinykh obyknovennykh differentsialnykh uravnenii i neravenstv vysokogo poryadka”, Differents. uravneniya (to appear)

[4] Kiguradze I. T., Chanturiya T. A., Asimptoticheskie svoistva reshenii neavtonomnykh obyknovennykh differentsialnykh uravnenii, Nauka, M., 1990 | Zbl

[5] Kiguradze I. T., Nachalnye i kraevye zadachi dlya sistem obyknovennykh differentsialnykh uravnenii, T. I, Metsniereba, Tbilisi, 1997 | Zbl

[6] Konkov A. A., “O resheniyakh neavtonomnykh obyknovennykh differentsialnykh uravnenii”, Izv. RAN. Ser. matem., 65:2 (2001), 81–125 | MR

[7] Mitidieri E., Pokhozhaev S. I., “Otsutstvie globalnykh polozhitelnykh reshenii dlya kvazilineinykh ellipticheskikh neravenstv”, Dokl. RAN, 359:4 (1998), 456–460 | MR | Zbl

[8] Mitidieri E., Pokhozhaev S. I., “Otsutstvie polozhitelnykh reshenii dlya kvazilineinykh ellipticheskikh zadach v $\mathbb R^N$”, Tr. MIAN, 227, Nauka, M., 1999, 192–222 | MR | Zbl

[9] Mitidieri E., Pokhozhaev S. I., “Otsutstvie polozhitelnykh reshenii dlya sistem kvazilineinykh ellipticheskikh uravnenii i neravenstv v $\mathbb R^N$”, Dokl. RAN, 366:1 (1999), 13–17 | MR | Zbl

[10] Mitidieri E., Pokhozhaev S. I., “Otsutstvie slabykh reshenii dlya nekotorykh vyrozhdennykh i singulyarnykh giperbolicheskikh zadach v $\mathbb R_+^{n+1}$”, Tr. MIAN, 232, Nauka, M., 2001, 248–267 | MR | Zbl

[11] Levine H. A., “The role of critical exponents in blow-up theorems”, SIAM Reviews, 32 (1990), 371–386

[12] Laptev G. G., “Otsutstvie globalnykh polozhitelnykh reshenii sistem polulineinykh ellipticheskikh neravenstv v konusakh”, Izv. RAN. Ser. matem., 64 (2000), 107–124 | MR | Zbl

[13] Laptev G. G., “Ob otsutstvii reshenii odnogo klassa singulyarnykh polulineinykh differentsialnykh neravenstv”, Tr. MIAN, 232, Nauka, M., 2001, 223–235 | MR

[14] Laptev G. G., “Some nonexistence results for higher–order evolution inequalities in cone–like domains”, Electron. Res. Announc. Amer. Math. Soc., 7 (2001), 87–93 | DOI | MR | Zbl

[15] Laptev G. G., “Ob otsutstvii globalnykh reshenii parabolicheskikh uravnenii tipa poristoi sredy”, ZhVMiMF, 42:8 (2002), 1191–1199 | MR

[16] Laptev G. G., “Absence of solutions to evolution-type differential inequalities in the exterior of a ball”, Russ. J. Math. Phys., 9:2 (2002), 180–187 | MR

[17] Laptev G. G., “Ob otsutstvii reshenii ellipticheskikh differentsialnykh neravenstv v konicheskikh oblastyakh”, Matem. zametki, 71:6 (2002), 855–866 | MR