Geodesic
Instantaneous blow-up of classical solutions to the Cauchy problem for the Khokhlov–Zabolotskaya equation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 7, pp. 1170-1175 
M. O. Korpusov; S. G. Mikhailenko. Instantaneous blow-up of classical solutions to the Cauchy problem for the Khokhlov–Zabolotskaya equation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 7, pp. 1170-1175. http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_7_a7/
@article{ZVMMF_2017_57_7_a7,
     author = {M. O. Korpusov and S. G. Mikhailenko},
     title = {Instantaneous blow-up of classical solutions to the {Cauchy} problem for the {Khokhlov{\textendash}Zabolotskaya} equation},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1170--1175},
     year = {2017},
     volume = {57},
     number = {7},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_7_a7/}
}
TY  - JOUR
AU  - M. O. Korpusov
AU  - S. G. Mikhailenko
TI  - Instantaneous blow-up of classical solutions to the Cauchy problem for the Khokhlov–Zabolotskaya equation
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2017
SP  - 1170
EP  - 1175
VL  - 57
IS  - 7
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_7_a7/
LA  - ru
ID  - ZVMMF_2017_57_7_a7
ER  - 
%0 Journal Article
%A M. O. Korpusov
%A S. G. Mikhailenko
%T Instantaneous blow-up of classical solutions to the Cauchy problem for the Khokhlov–Zabolotskaya equation
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2017
%P 1170-1175
%V 57
%N 7
%U http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_7_a7/
%G ru
%F ZVMMF_2017_57_7_a7

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

The Cauchy problem for a second-order nonlinear equation with mixed derivatives is considered. It is proved that its classical local-in-time solution does not exist. The blow-up of the solution is proved by applying S. I. Pohozaev and E. L. Mitidieri's nonlinear capacity method.

[1] Gurbatov S. N., Rudenko O. V., Saichev A. I., Volny i struktury v nelineinykh sredakh bez dispersii, Fizmatlit, M., 2008

[2] Gibbons J., “The Zabolotskaya–Khokhlov equation and the inverse scattering problem of classical mechanics”, Dynamical problems in soliton systems (Kyoto, 1984), Springer Ser. Synergetics, 30, Springer, Berlin, 1985, 36–41 | DOI | MR

[3] Clarkson P. A., Hood S., “Nonclassical symmetry reductions and exact solutions of the Zabolotskaya–Khokhlov equation”, European J. Appl. Math., 3:4 (1992), 381–414 | DOI | MR | Zbl

[4] Güngör F., Özemir C., “Lie symmetries of a generalized Kuznetsov–Zabolotskaya–Khokhlov equation”, J. Math. Anal. Appl., 423:1 (2015), 623–638 | DOI | MR | Zbl

[5] Mitidieri E., Pokhozhaev C. B., “Apriornye otsenki i otsutstvie reshenii nelineinykh uravnenii i neravenstv v chastnykh proizvodnykh”, Tr. MIAN, 234, 2001, 3–383