Geodesic
On the initial boundary value problem for a nonlinear nonhomogeneous equation of Sobolev type
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 10, pp. 1855-1865 
A. I. Aristov. On the initial boundary value problem for a nonlinear nonhomogeneous equation of Sobolev type. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 10, pp. 1855-1865. http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_10_a7/
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     title = {On the initial boundary value problem for a nonlinear nonhomogeneous equation of {Sobolev} type},
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Voir la notice de l'article provenant de la source Math-Net.Ru

The initial boundary value problem for a nonlinear nonhomogeneous equation of Sobolev type used for modeling nonstationary processes in semiconductors is examined. It is proved that this problem is uniquely solvable at least locally in time. Sufficient conditions for the problem to be solvable globally in time are found, as well as sufficient conditions for the local (but not global) solvability. In the case of only local solvability, upper and lower estimates for the time when a solution exists are determined in the form of either explicit or quadrature formulas.

[1] Sobolev S. L., “Ob odnoi novoi zadache matematicheskoi fiziki”, Izv. AN SSSR. Ser. matem., 1954, no. 18

[2] Sveshnikov A. G., Alshin A. B., Korpusov M. O., Nelineinyi funktsionalnyi analiz i ego prilozheniya k uravneniyam v chastnykh proizvodnykh, Nauchnyi mir, M., 2008

[3] Sveshnikov A. G., Alshin A. B., Korpusov M. O., Pletner Yu. D., Lineinye i nelineinye uravneniya sobolevskogo tipa, Fizmatlit, M., 2007

[4] Korpusov M. O., Razrushenie v neklassicheskikh nelokalnykh uravneniyakh, URSS, M., 2010

[5] Aristov A. I., “Issledovanie kachestvennykh svoistv reshenii odnogo nelineinogo sobolevskogo uravneniya”, Sb. statei molodykh uchenykh fakulteta VMK MGU 2010 g., Maks-Press, M., 2010

[6] Aristov A. I., “Otsenki vremeni suschestvovaniya reshenii nachalno-kraevoi zadachi dlya odnogo nelineinogo sobolevskogo uravneniya”, Tikhonovskie chteniya, Materialy konferentsii (25–29 oktyabrya 2010 g.), Maks-Press, M., 2010

[7] Zaitsev V. F., Polyanin A. D., Spravochnik po nelineinym differentsialnym uravneniyam, Fizmatlit, M., 1993 | MR

[8] Bekkenbakh E., Bellman R., Neravenstva, URSS, M., 2007

[9] Yanpolskii A. R., Giperbolicheskie funktsii, Fizmatgiz, M., 1960

[10] Fikhtengolts G. M., Kurs differentsialnogo i integralnogo ischisleniya, v. 2, Fizmatlit, M., 2003