Vestnik of Samara University. Natural Science SeriesVestnik of Samara University. Natural Science Series2541-75252712-8954Samara National Research University919610.18287/2541-7525-2020-26-4-25-35Research ArticleA NONLOCAL PROBLEM FOR A HYPERBOLIC EQUATION WITH A DOMINANT MIXED DERIVATIVEGilevA. V.<p>postgraduate student of the Department of Differential Equations and Control Theory</p>toshqaaa@gmail.comhttps://orcid.org/0000-0001-6747-5826Samara National Research University0712202026425351708202117082021Copyright © 2021, Gilev A.V.2021<p>In this article, we consider the Goursat problem with nonlocal integral conditions for a hyperbolic equation with a dominant mixed derivative. Research methods of solvability of classical boundary value problems for partial differential equations cannot be applied without serious modifications. The choice of a research method of solvability of a nonlocal problem depends on the form of the integral condition. In the process of developing methods that are effective for nonlocal problems, integral conditions of various types were identified [1]. The solvability of the nonlocal Goursat problem with integral conditions of the first kind for a general equation with dominant mixed derivative of the second order was investigated in [2]. In our problem, the integral conditions are nonlocal conditions of the second kind, therefore, to investigate the solvability of the problem, we propose another method, which consists in reducing the stated nonlocal problem to the classical Goursat problem, but for a loaded equation. In this article, we obtain conditions that guarantee the existence of a unique solution of the problem. The main instrument of the proof is the a priori estimates obtained in the paper.</p>неклассическая задачанелокальные условиянагруженное уравнениезадача Гурсаинтегральные условия второго родасуществование и единственность решенияметод последовательных приближенийредукцияnon-classical problemnon-local conditionsloaded equationGoursat problemintegral conditions of the second kindexistence and uniqueness of a solutionmethod of successive approximationsreduction[[1] Pulkina L.S. Problems with nonclassical conditions for hyperbolic equations. Samara: Izdatel’stvo "Samarskii universitet 2012, 194 p. (In Russ.)][[2] Pulkina L.S. The L2 solvability of a nonlocal problem with integral conditions for a hyperbolic equation. Differential Equations, 2000, vol. 36, no. 2. pp. 316–318. DOI: http://doi.org/10.1007/BF02754219. (English; Russian original)][[3] Cannon J.R. The solution of the heat equation subject to the specification of energy. Quarterly of Applied Mathematics, 1963, vol. 21, no. 2, pp. 155-–160. DOI: http://doi.org/10.1090/QAM][[4] Byszewski L. Existance and uniqueness of solutions of nonlocal problems for hyperbolic equation uxt == F(x; t; u; ux). Journal of Applied Mathematics and Stochastic Analysis, 1990, vol. 3, no. 3, pp. 163–168. Available at: https://www.univie.ac.at/EMIS/journals/HOA/JAMSA/Volume3_3/168.pdf.][[5] Ilin V.A., Moiseev E.I. Uniqueness of the solution of a mixed problem for the wave equation with nonlocal boundary conditions. Differential Equations, 2000, vol. 36, no. 5, pp. 728–733. DOI: https://doi.org/10.1007/BF02754231 (English; Russian original)][[6] Gordeziani D.G., Avalishvili G.A. On the constructing of solutions of the nonlocal initial boundary value problems for one-dimensional medium oscillation equations. Matem. Mod., 2000, vol. 12, no. 1, pp. 94–103. Available at: http://mi.mathnet.ru/eng/mm832. (In Russ.)][[7] Bouziani A., Benouar N. Probleme mixte avec conditions integrales pour une classe d’equations hyperboliques. Bull. Belg. Math. Soc., 1996. no. 3, pp. 137–145. DOI: https://doi.org/10.36045/BBMS][[8] Nakhushev A.M. Loaded equations and their application. Moscow: Nauka, 2012, 232 p. Available at: https://obuchalka.org/20210213129285/nagrujennie-uravneniya-i-ih-primenenie-nahushev-a-m-2012.html; https://www.elibrary.ru/item.asp?id=20886619. (In Russ.)][[9] Kozhanov A.I., Pulkina L.S. On the solvability of boundary value problems with a nonlocal boundary condition of integral form for multidimensional hyperbolic equations. Differential Equations, 2006, vol. 42, no. 9. pp. 1233—1246. DOI: https://doi.org/10.1134/S0012266106090023. (English; Russian original)][[10] Nahusheva Z.A. A nonlocal problem for partial differential equations. Differential Equations, 1986, vol. 22, no. 1, pp. 171–174. Available at: http://mi.mathnet.ru/eng/de5770. (In Russ.)][[11] Assanova A.T. Nonlocal problem with integral conditions for a system of hyperbolic equations in characteristic rectangle. Russian Mathematics, 2017, vol. 61, no. 5, pp. 7–20. DOI: https://doi.org/10.3103/S1066369X17050024. (English; Russian original)][[12] Mikhlin S.G. Lectures on linear integral equations. Moscow: Fizmatgiz, 1959, 232 p. Available at: https://1lib.education/book/571663/1c6922?id=571663&secret=1c6922. (In Russ.)]