Geodesic
On the Blow-up of Solutions to Nonlinear Initial–Boundary Value Problems
Informatics and Automation, Function theory and nonlinear partial differential equations, Tome 260 (2008), pp. 213-226 
S. I. Pokhozhaev. On the Blow-up of Solutions to Nonlinear Initial–Boundary Value Problems. Informatics and Automation, Function theory and nonlinear partial differential equations, Tome 260 (2008), pp. 213-226. http://geodesic.mathdoc.fr/item/TRSPY_2008_260_a14/
@article{TRSPY_2008_260_a14,
     author = {S. I. Pokhozhaev},
     title = {On the {Blow-up} of {Solutions} to {Nonlinear} {Initial{\textendash}Boundary} {Value} {Problems}},
     journal = {Informatics and Automation},
     pages = {213--226},
     year = {2008},
     volume = {260},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2008_260_a14/}
}
TY  - JOUR
AU  - S. I. Pokhozhaev
TI  - On the Blow-up of Solutions to Nonlinear Initial–Boundary Value Problems
JO  - Informatics and Automation
PY  - 2008
SP  - 213
EP  - 226
VL  - 260
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2008_260_a14/
LA  - ru
ID  - TRSPY_2008_260_a14
ER  - 
%0 Journal Article
%A S. I. Pokhozhaev
%T On the Blow-up of Solutions to Nonlinear Initial–Boundary Value Problems
%J Informatics and Automation
%D 2008
%P 213-226
%V 260
%U http://geodesic.mathdoc.fr/item/TRSPY_2008_260_a14/
%G ru
%F TRSPY_2008_260_a14

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

We consider the problem of nonexistence (blow-up) of solutions of nonlinear evolution equations in the case of a bounded (with respect to the space variables) domain. Following the method of nonlinear capacity based on the application of test functions that are optimal (“characteristic”) for the corresponding nonlinear operators, we obtain conditions for the blow-up of solutions to nonlinear initial–boundary value problems. We also show by examples that these conditions are sharp in the class of problems under consideration.

[1] Bellout H., Benachour S., Titi E. S., “Finite-time singularity versus global regularity for hyper-viscous Hamilton–Jacobi-like equations”, Nonlinearity, 16:6 (2003), 1967–1989 | DOI | MR | Zbl

[2] Galaktionov V. A., Pokhozhaev S. I., “Otsutstvie globalnykh reshenii nelineinykh smeshannykh zadach”, DAN, 412:4 (2007), 444–447 | MR

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

[4] Pokhozhaev S. I., “Suschestvenno nelineinye emkosti, indutsirovannye differentsialnymi operatorami”, DAN, 357:5 (1997), 592–594 | MR | Zbl

[5] Pokhozhaev S. I., “Normalnaya razreshimost nelineinykh uravnenii v ravnomerno vypuklykh banakhovykh prostranstvakh”, Funkts. analiz i ego pril., 3:2 (1969), 80–84 | MR

[6] Pokhozhaev S. I., “O slabo nelineinykh giperbolicheskikh sistemakh”, Tr. Mosk. mat. o-va, 23, 1970, 61–76 | MR | Zbl

[7] Sveshnikov A. G., Alshin A. B., Korpusov M. O., Pletner Yu. D., Lineinye i nelineinye uravneniya sobolevskogo tipa, Fizmatlit, M., 2007, 734 pp. | Zbl