Geodesic
A~problem with a~nonlocal with respect to time condition for multidimensional hyperbolic equations
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2016), pp. 41-52.

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

We study the boundary-value problem for hyperbolic equation with nonlocal with respect to time-variable condition in integral form. We obtain sufficient conditions for the unique solvability of the nonlocal problem. The proof is based on possibility to reduce a nonlocal condition of the first kind to the second kind one. This allows to reduce the nonlocal problem to an operator equation. We show that unique solvability of the operator equation implies the existence of a unique solution to the posed problem.
Keywords: hyperbolic equation, nonlocal problem, integral conditions, generalized solution. 
@article{IVM_2016_10_a5,
     author = {L. S. Pul'kina and A. E. Savenkova},
     title = {A~problem with a~nonlocal with respect to time condition for multidimensional hyperbolic equations},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {41--52},
     publisher = {mathdoc},
     number = {10},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2016_10_a5/}
}
TY  - JOUR
AU  - L. S. Pul'kina
AU  - A. E. Savenkova
TI  - A~problem with a~nonlocal with respect to time condition for multidimensional hyperbolic equations
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2016
SP  - 41
EP  - 52
IS  - 10
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2016_10_a5/
LA  - ru
ID  - IVM_2016_10_a5
ER  - 
%0 Journal Article
%A L. S. Pul'kina
%A A. E. Savenkova
%T A~problem with a~nonlocal with respect to time condition for multidimensional hyperbolic equations
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2016
%P 41-52
%N 10
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2016_10_a5/
%G ru
%F IVM_2016_10_a5
L. S. Pul'kina; A. E. Savenkova. A~problem with a~nonlocal with respect to time condition for multidimensional hyperbolic equations. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2016), pp. 41-52. http://geodesic.mathdoc.fr/item/IVM_2016_10_a5/

[1] Cannon J. R., “The solution of the heat equation subject to the specification of energy”, Quart. Appl. Math., 21 (1963), 155–160 | MR

[2] Gordeziani D. G., Avalishvili G. A., “Resheniya nelokalnykh zadach dlya odnomernykh kolebanii sredy”, Matem. modelir., 12:1 (2000), 94–103 | MR | Zbl

[3] Kozhanov A. I.,Pulkina L. S., “O razreshimosti kraevykh zadach s nelokalnym granichnym usloviem integralnogo vida dlya mnogomernykh giperbolicheskikh uravnenii”, Differents. uravneniya, 42:9 (2006), 1166–1179 | MR | Zbl

[4] Pulkina L. S., “Nelokalnaya zadacha dlya giperbolicheskogo uravneniya s integralnymi usloviyami 1 roda s yadrami, zavisyaschimi ot vremeni”, Izv. vuzov. Matem., 2012, no. 10, 32–44 | MR | Zbl

[5] Pulkina L. S., Zadachi s neklassicheskimi usloviyami dlya giperbolicheskikh uravnenii, “Samarskii universitet”, Samara, 2012

[6] Kuz A. M., Ptashnik B. I., “Zadacha z integralnimi umovami dlya rivnyannya Kleina–Gordona u klassi funktsii, maizhe periodichnikh za prostorovimi zminnimi”, Prikl. problemi mekh. i mat., 2010, no. 8, 41–53

[7] Abdrakhmanov A. M., Kozhanov A. I., “Zadacha s nelokalnym granichnym usloviem dlya odnogo klassa uravnenii nechetnogo poryadka”, Izv. vuzov. Matem., 2007, no. 5, 3–12 | MR | Zbl

[8] Lukina G. A., “Kraevye zadachi s integralnymi granichnymi usloviyami po vremeni dlya uravnenii tretego poryadka”, Matem. zametki YaGU, 17:2 (2010), 75–97 | Zbl

[9] Prilepko A. I., Kostin A. B., “O nekotorykh obratnykh zadachakh dlya parabolicheskikh uravnenii s finalnym i integralnym pereopredeleniem”, Matem. sb., 183:4 (1992), 49–68 | MR | Zbl

[10] Cannon J. R., Lin Y., “An inverse problem of finding a parameter in a semi-linear heat equation”, J. Math. Anal. Appl., 145:2 (1990), 470–484 | DOI | MR | Zbl

[11] Kamynin V. L., “Obratnaya zadacha opredeleniya mladshego koeffitsienta v parabolicheskom uravnenii pri uslovii integralnogo nablyudeniya”, Matem. zametki, 94:2 (2013), 207–217 | DOI | Zbl

[12] Denisov A. M., Vvedenie v teoriyu obratnykh zadach, Izd-vo Moskovsk. un-ta, M., 1994

[13] Ionkin N. I., “Reshenie odnoi kraevoi zadachi teorii teploprovodnosti s neklassicheskim kraevym usloviem”, Differents. uravneniya, 13:2 (1977), 294–304 | MR | Zbl

[14] Ladyzhenskaya O. A., Kraevye zadachi matematicheskoi fiziki, Nauka, M., 1973 | MR